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Aarhus School of Business
M. Sc. of Finance and International Business
Department of Finance
Fuglesangs Allé 4 DK-8210 Aarhus V
Master Thesis
Capital Asset Pricing Model and Arbitrage Pricing
Theory An application of market equilibrium models to the Polish market
Authors:
Agnieszka Sawa
Slawomir Sklinda
Written under supervision of
Paula Peare, Associate Professor
Department of Finance
Aarhus, 2003
2
INTRODUCTION ..................................................................................................................... 4 CHAPTER I MARKET EQUILIBRIUM MODELS -THEORY AND ASSUMPTIONS .......................... 6
1.1 APPLICATION OF MARKET EQUILIBRIUM MODELS............................................................ 6 1.2 ARBITRAGE ..................................................................................................................... 7
1.2.1 Arbitrage mechanism and market equilibrium ................................................. 7 1.2.2 Limits of Arbitrage ................................................................................................ 9
1.3 CAPITAL ASSET PRICING MODEL (CAPM) .................................................................. 11 1.3.1 Standard version ................................................................................................ 11 1.3.2 Zero beta version of the CAPM model ............................................................ 12 1.3.3 Assumptions of the standard Capital Assets Pricing Model ........................ 14
1.3.3.1 Market efficiency .......................................................................................................14 1.3.3.2 Decisions based on the mean-variance criteria ...................................................15 1.3.3.3 Homogenous beliefs.................................................................................................16
1. 4 ARBITRAGE PRICING THEORY ..................................................................................... 17 1.4.1 No arbitrage opportunities................................................................................. 18 1.4.2 Factor Model ....................................................................................................... 21 1.4.3 Firm- specific risk ............................................................................................... 22 1.4.4 APT relation ........................................................................................................ 22 1.4.5 Methodological concerns .................................................................................. 23
CHAPTER II CAPITAL MARKET EQUILIBRIUM MODELS EMPIRICAL TESTS ............................ 26
2.1 CAPM EMPIRICAL EVIDENCE ....................................................................................... 26 2.1.1 Early CAPM tests ............................................................................................... 26
2.1.1.1 Lintner test (1968).....................................................................................................27 2.1.1.2 Black, Jensen and Scholes test (1972) .................................................................29 2.1.1.3 Fama and MacBeth test (1973) ..............................................................................30
2.1.2 Rolls critique (1977) .......................................................................................... 31 2.1.3 Later tests of the CAPM model ........................................................................ 33
2.1.3.1 Banz test (1981)........................................................................................................33 2.1.3.2 Fama and French test (1992)..................................................................................34 2.1.3.4 Kozickis and Shens test (2002) ............................................................................37
2.2 EMPIRICAL STUDIES ON APT ....................................................................................... 39 2.2.1 Investigation on variables influencing returns ................................................ 39 2.2.2 Approaches to APT model estimation............................................................. 42
2.2.2.1 Statistical estimation of betas and factors .............................................................42 2.2.2.2 Portfolio method of factor estimation .....................................................................43 2.2.2.3 Betas arbitrary choice..............................................................................................43
2.3 APT CONTRA CAPM.................................................................................................... 45 2.4 EMPIRICAL EVIDENCES IN POLAND ............................................................................... 48
2.4.1 Tests of market efficiency ................................................................................. 49 2.4.2 Multifactor models on Warsaw Stock Exchange ........................................... 50
CHAPTER III DATA DESCRIPTION .......................................................................................................... 53
3.1 DATA CHOICE ............................................................................................................... 53 3.1.1 Choice of the proxy for the market portfolio ................................................... 53 3.1.2 Length of estimation period .............................................................................. 55 3.1.3 Observation frequency ...................................................................................... 56
3.2 CHARACTERISTICS OF DATA USED FOR CAPM AND APT TESTS................................ 58 3.2.1 Characteristics of data used for CAPM test ................................................... 59
3.2.1.1 Returns on Shares....................................................................................................59 3.2.1.2 Warsaw Market Index (WIG) ...................................................................................61 3.2.1.3 Risk Free Rate ..........................................................................................................62
3.2.2 Variables used for APT test .............................................................................. 62 3.2.2.1 S&P 500 .....................................................................................................................63 3.2.2.2 Polish Zloty (PLN) Exchange Rate.........................................................................63 3.2.2.3 International Price of Gold .......................................................................................65
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CHAPTER IV EMPIRICAL TEST OF CAPM .............................................................................................67
4.1 CALCULATION PROCEDURE ..........................................................................................67 4.1.1 CAPM test methodology....................................................................................67 4.1.2 Portfolio grouping ...............................................................................................72 4.1.3 Risk free rate variability .....................................................................................73
4.2 EMPIRICAL TEST OF THE CAPM ...................................................................................74 4.2.1 Time-series regression ......................................................................................75 4.2.2 Cross-sectional regression ...............................................................................78
CHAPTER V APT ESTIMATION AND TESTS.........................................................................................87
5.1 METHODOLOGY.............................................................................................................87 5.1.1 Estimation procedure .........................................................................................87 5.1.2 Methods of testing and Estimation ..............................................................88 5.1.3 Factor Analysis overview ..................................................................................91
5.1.3.1 Factor Analysis formal model ..................................................................................92 5.2 FACTORS ESTIMATION- EMPIRICAL RESULTS................................................................94
5.2.1 Variables analyzed.............................................................................................94 5.2.1.2 Suboptimization.........................................................................................................94 5.2.1.2 Number of cases .......................................................................................................95 5.2.1.3 Sampling adequacy ..................................................................................................95
5.2.2 Number of factors ...............................................................................................98 5.2.2.1 Kaiser rule ..................................................................................................................99 5.2.2.2 Cattell rule ................................................................................................................100 5.2.2.3 Variance criterion ....................................................................................................100
5.2.3 Factoring methods ...........................................................................................101 5.2.3.1 Maximum Likelihood Factoring .............................................................................101 5.2.3.2 PCA versus PFA .....................................................................................................102 5.2.3.3 PCA results ..............................................................................................................103
5.3 TIME-SERIES REGRESSION .........................................................................................106 5.4 CROSS-SECTIONAL REGRESSION ...............................................................................114
CHAPTER VI POSSIBLE REASONS FOR CAPM AND APT FAILURE............................................125
6.1 BETA INSTABILITY........................................................................................................126 6.2 INAPPROPRIATE PORTFOLIO GROUPING APT CASE ...............................................128 6.3 MARKET INEFFICIENCY AND LIQUIDITY........................................................................129 6.4 VALUE WEIGHTED INDEX AND CAPITAL DOMINANCE OF A FEW COMPANIES...............131 6.5 LOW SIGNIFICANCE OF THE MARKET AS A SOURCE OF CAPITAL.................................132 6.6 SHORTCOMINGS OF APT FACTOR ANALYSIS .............................................................132 6.7 DIVERSIFICATION OF THE FIRM-SPECIFIC RISK ...........................................................134 6.8 SMALL NUMBER OF VARIABLES ...................................................................................135 6.9 SHORT ESTIMATION PERIOD .......................................................................................136
REFERENCES: ...................................................................................................................137 APPENDIXES ......................................................................................................................144
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Introduction
Polish capital market is very young. It was set up in 1812 however just
before the II. World War it was liquidated for over fifty years. Warsaw
Stock Exchange was reactivated in 1991 and since then it has been
developing constantly. Studies carried out by Szyszka (2003) revealed
that the efficiency of that market is improving as well. If Polish capital
market was efficient enough, the market equilibrium models could be
assumed to work on it.
Market equilibrium models that are Capital Asset Pricing model and
model created on the basis of Arbitrage Pricing Theory have
applications in many fields of finance. They could support in the
decision making process of Polish corporate managers and investors.
Due to implementation of APT or CAPM models, decisions concerning
the choice of a portfolio that meets certain investor criteria could be
made. Moreover, investors using these models would be able to
identify overpriced and underpriced assets. Furthermore, asset pricing
theories could be applied in budgeting process, as they help with cost
of equity estimation.
Thus, the aim of this paper is to test the standard versions of the
market equilibrium models on Warsaw Stock Exchange and therefore
answer the question if standard CAPM or APT could be useful for
Polish investors and corporate managers. Despite of the fact that such
studies were conducted on many foreign markets, this research is the
first that empirically analyses both market equilibrium models on Polish
capital market.
However, the study conducted in this thesis faced a few important
problems. The study limitations are associated mainly with
characteristics of the WSE that is still developing. The fact Polish
capital market is only twelve years old resulted in a relatively small
5
number of firms analyzed and short estimation period. Those are
respectively 100 companies examined within three-year period from
2000 to 2003.
The research objective determined the structure of this paper.
Therefore, chapter one concerns theoretical issues associated with
market equilibrium models. It describes shortly their application and
assumptions required to create these models. Furthermore, it
discusses these issues with relation to Polish business environment.
The second chapter analyzes studies concerning CAPM and APT
empirical tests that might suggest the methodology of developing these
models. Chapter three discusses the choice of data employed in the
models. Two next sections discuss testing techniques that are
traditionally used when estimating these models, chose the method
most appropriate to Polish capital market and then describe tests
results. Finally, the last chapter focuses on problems faced while
developing and testing both models. Moreover, it suggests
improvements that could be made in order to create models delivering
more reliable results.
Based on the methodology applied in this paper both market
equilibrium models do not describe expected rates of returns on the
WSE. Neither market beta nor other macroeconomic factors examined
satisfactory explain the returns. According to obtained results, standard
models should not be applied by Polish investors and corporate
managers in their decision making process.
However, similar research can be carried out in a few years. It is likely
that Polish capital market will be more efficient because of the
increasing integration of the capital markets. Furthermore, it is believed
that market economy in Poland will be much more liberalized and
therefore market efficiency will be bigger. Due to the fact that the
estimation period would be longer, the estimated results would be more
reliable.
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CHAPTER I
Market Equilibrium Models -Theory and assumptions
This chapter presents market equilibrium models, their application and
assumptions discussing them with relation to Warsaw Stock Exchange.
1.1 Application of market equilibrium models
In spite of the fact that CAPM and APT models tested in this study
were implemented for the first time over thirty years ago, they are still
applied in finance.
First of all, they are used in a modern portfolio theory that is an attempt
to understand the market as a whole. According to this theory
investments are described statistically, in terms of their expected return
rate and their expected volatilities. Market equilibrium models help to
identify acceptable level of risk tolerance, and then find a portfolio with
the maximum expected return at that level of risk.
Second, application of CAPM and APT is a tool on the identification of
unique opportunities, when shares are either over- or undervalued. The
capital market equilibrium models enable estimate the expected value
of the equity in terms of returns.
Finally, these models are of crucial importance in capital budgeting
processes. Applying time value techniques involves estimating a
discounting rate. A very common approach is weighted average cost of
7
capital, which is composed of the cost of equity and the cost of debt.
The first one can be estimated by using capital market models.
As market equilibrium models are useful in so many areas of finance,
they could support Polish investors and corporate managers in their
decision processes as well.
1.2 Arbitrage
Both market equilibrium models assume that markets arbitrageurs are
able to ensure market equilibrium and therefore to prevent mispricing.
There is, however, the question that needs to be answered, namely
whether the market equilibrium is a fact or only a theory.
1.2.1 Arbitrage mechanism and market equilibrium
The theory of arbitrage provides the answer. It says that the
mechanism of arbitrage prevents any deviations from the equilibrium,
as the actions undertaken by investors will immediately increase the
price of undervalued assets and decrease the price of the overvalued
ones. The mechanism of arbitrage was discussed for example by Elton
and Gruber (1998), Francis (2000) and Grinblatt and Titman (1998). An
example of the mechanism is presented below.
Assume that Company 1 has overvalued stocks (Asset 1) and stocks
of Company 2 are undervalued (Asset2). Hence the market is not in
equilibrium and the assets must be placed beyond the Security Market
Line. This situation is graphically presented on Diagram 1.1:
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Diagram No 1.1: Pricing arbitrage undervalued asset (green) and overvalued (red)
Source: M. Grinblatt i and S. Titman: Financial Markets and Corporate Strategy, Irvin McGraw-Hill Companies, 1998: 120
If the above situation occurs, following the arbitrage opportunity the
abnormal profit could be earned without bearing risk.
If assets are overvalued, an investor can obtain any point on the
Security Market Line. It can be done through either purchasing /selling
of individual shares or creating (from other accessible assets) a
portfolio, which can be placed in any point on the SML. In such case,
selling short the asset/portfolio placed on the SML (with the same beta
coefficient as Asset 1) and buying long Asset 1, an investor can earn
profits on the higher expected return. The value of the gain is marked
by the red line and equals R1- R1. This investor will receive the
abnormal return till the price of the Asset 1 reaches its true level R1.
This mechanism is analogical for undervalued assets. In this situation
an investor using the arbitrage mechanism can receive the abnormal
return as well. Now one would sell short Asset 2 buying long portfolio
on the SML with the same beta coefficient as Asset 2. Such investment
will generate abnormal returns without any additional risk. The premium
is marked green and equals R2- RF.
E(R)
β
SML
1
RM . Asset 1
. Asset 2
R1
R1
R2
RF
9
However, proponents of behavioural finance noticed that in a real
business world prices can diverge from equilibrium.
1.2.2 Limits of Arbitrage
First of all, there are risks and costs associated with arbitrage that
might limit it. Furthermore, according to behavioural finance the
deviations from the fundamental value of assets can be caused by
traders that are not fully rational and it might be difficult for rational
traders to undo the capital dislocations made by less rational ones.
According to the theory, a rational investor, who is taking advantage of
the arbitrage profit opportunity ensures market equilibrium. Following
the behavioral finance theory, not all the arbitrage opportunities can
generate profits (Barberis and Thaler, 2002). The investment strategies
may sometimes be costly and risky. Barberis and Thaler (2002) believe
that traders perusing arbitrage strategy usually face fundamental and
noise-trader risk. Furthermore, they need to pay the implementation
costs.
Fundamental risk might not be fully hedged as securities are not
perfect substitutes and the strategy of selling one asset and purchasing
another can incur the fundamental risk.
The noise-trader risk is a risk that irrational investors can make the
mispriced assets even more deviate from equilibrium. This risk may
force arbitrageurs to liquidate their positions too early as there is a
separation of brains and capital (Shleifer and Vishny, 2001). It is an
agency problem, since portfolios managers do not operate their own
money and return maximization instead of ensuring pricing equilibrium
is of crucial importance for them. However, according to De Long et al.
10
(1990) arbitrageurs might make the price diverge from equilibrium, as
well. If the market is dominated by positive feedback traders,
overpriced assets might be purchased by investors making arbitrage
profit, as they would expect that higher price will be pushed up even
higher.
Moreover, exploiting arbitrage is not costless. For example transaction
costs that overwhelm potential arbitrage profit, would deter traders from
exploiting arbitrage opportunity. Furthermore, it might be costly to learn
about mispricing. It was believed that returns predictability is a sign of
incorrectly set prices. However, Summers (1986) and Shiller (1984)
presented that the demand of the irrational traders might be so strong
that the returns can be unpredictable.
Nevertheless, behavioral finance blames for mispricing not only
investors but corporate managers as well who are responsible for
assets issue. Theoretically, if assets are overpriced managers would
decide to issue more assets in order to sell them at attractive prices.
Then the oversupply should push prices back to equilibrium. However,
the issue incurs costs and managers cannot be sure that investors
overestimate their shares. Therefore, they might not decide to issue
equity.
Behavioral finance picks up the weaknesses of market equilibrium
models focusing mostly on investor psychology and beliefs. To date,
there are no behavioral models that might be applied instead of CAPM
or APT in all of their applications. The reason for that might be
psychological factors that cannot be quantified easily. It is possible that
models combining both approaches will be developed in the future.
11
1.3 Capital Asset Pricing Model (CAPM)
CAPM model was implemented almost 40 years ago by Sharpe (1964),
Lintner (1965) and Mossin (1966) independently from each other. This
model was the first and, as its popularity proves, successful attempt of
defining the risk of cash flows from an investment tantamount to the
expected rate of return. The CAPM model is the simplest version of the
capital market equilibrium models, and is also called one factor capital
market equilibrium model. Zero beta model1, which is one of many
derivative of a standard version will be presented further in this section.
1.3.1 Standard version
The CAPM model describes the relationship between risk and
expected return. Expected return of a security or a portfolio is defined
by the systematic risk affecting the company and equals the sum of the
rate on a risk-free asset and risk premium. The capital market
equilibrium takes the following form:
Ri = rf + ( rm - rf)β i ,
Where Ri is the expected return on the equity (of a single company or
portfolio), rf is a risk free rate, rm defines the expected return on the
market portfolio and β measures the sensitivity (risk) of expected return
on equity Ri to the return on the market portfolio rm. Formal derivation o
the model does not make many problems and is available in the
literature ( for example: Elton and Gruber 1998, Francis 2000, Berndt
1996, Haugen 1996).
Application of the CAPM model requires defining a risk free rate,
expected market risk premium ( rm - rf) and calculation of the beta
coefficient. It is computed based on the historical values as the slope
coefficient in the regression of returns on the equity against market risk 1 well known as Black CAPM.
12
premium. The Capital Asset Pricing Model is the most popular among
American companies when estimating the cost of equity in the capital
budgeting process. A recent study conducted by Graham and Harvey
(2001) confirms the popularity of CAPM. According to their results over
73 percent of US firms apply the model while estimating the cost of
equity. However, there was no such study conducted on the Polish
market, hence no comments on the models popularity can be done.
1.3.2 Zero beta version of the CAPM model
The standard version of CAPM model may be considered unrealistic,
as it is rather impossible for a single investor to borrow or lend without
any limitation at the risk free rate. Black (1972) releases this
assumption presenting a version of the CAPM model where all assets
are risky. The equilibrium can be achieved by substitution of any of the
zero beta portfolios, which are placed on the continuous section RFC,
instead of RF from the CAPM equation. Since RFZ is unrealistic, the
minimum variance portfolio with beta coefficient that equals zero is
marked with a sign Z and is placed on the crossing point of the curve K
and the line RFC and the corresponding expected return rate equals
Rz= RF. The zero beta coefficients mean a lack of correlation with a
market portfolio. A graphical explanation of the situation is presented in
Diagram No. 1.2
13
E(R)
σ
RZ = RF
M .
. Z C.S
K
Diagram No1.2 Zero beta portfolio (Z) on the efficient portfolio curve
Source: E. J. Elton and M.J. Gruber: Nowoczesna teoria portfelowa i analiza
papierów wartościowych, WIG-Press, Warszawa, 1998, p. 378.
Hence, the security market line can be presented as follows:
)( ZMiZi RRRR −+= β
The portfolio Z has certainly lower expected rate of return than the
market portfolio, since its return is represented by the crossing point of
the tangent to the curve K and the vertical axis. As the expected market
return is a tangent point with the curve K, it must be placed above Rz.
The portfolio Z will not be an efficient portfolio as well, because it is
placed below the lowest variance portfolio, so it is possible to find a
portfolio of the same variance but higher expected return. Hence,
portfolio Z, although placed on the lowest variance curve is not efficient
(formal derivation in Elton and Gruber 1998).
Assuming that point S represents the lowest variance portfolio, all
investors will hold portfolios placed on the efficient portfolios curve
(SMK). Although portfolio Z is not efficient it is used in analysis
because of its zero correlation with market portfolio. Since a
combination of minimum variance portfolios gives a minimum variance
portfolio, the aim of the analysis is to cerate such a combination that
would be placated on the curve SMK. Investors that chose returns
between RS and RM, will have a combination of zero beta portfolio and
14
market portfolio. On the other hand, investors that own a portfolio on
the right side from the market portfolio M, will have a portfolio
composed by short sale of Z portfolio and long purchase of market
portfolio.
Since the portfolio Z is inefficient, the overall sum of short and long
positions must equalize. That is because in equilibrium all investors will
posses only market portfolio.
1.3.3 Assumptions of the standard Capital Assets Pricing
Model
The reality of financial events is so complex that in order to build any
model describing it in a plain way simplifying assumptions need to be
adopted. These assumptions are to eliminate factors, which marginally
affect the modelled event. The CAPM assumptions can be grouped in
three general conditions (Bailey 2001):
A. Markets are efficient.
B. All investors make their decisions on the basis of the
mean-variance criteria.
C. Investors have homogenous beliefs.
1.3.3.1 Market efficiency
This assumption is the least real and is composed of a number of
interrelated factors:
• No transaction costs, which means that there are no costs
associated with sale or purchase of an asset. This assumption is
justified by the fact that transaction costs might change the return
15
rate from an instrument depending on the investors position at the
beginning of a considered period. Taking these costs into account
significantly increase the complexity of the model. However,
recognising their relatively low value low significance can be
assumed.
• No institutional barriers in assets trade, which is directly related to
the unlimited short sales possibility. This assumption in practice is
rarely fulfilled, since investors are usually not able to sale short any
amount of assets.
• Unlimited short sale and long buys at the risk free rate. As in the
previous assumption the first part of this assumption is rather
impossible, however its second part may be satisfied. Investors may
lend their money at the risk free or even higher rate. Black (1972)
relaxed this assumption and built a well known and accepted
Market model.
• Assets are infinitely divisible, which means that investors can hold
any amount of the asset. This assumption was created because
predictions of the model would be inaccurate, as the indivisible
assets would require a great amount of initial wealth of an investor.
• All assets can be bought or sold at the observed market price.
• There is a market for all kind of assets, even human capital.
• Individual investors decisions about their position in any assets will
not affect their price(Hsu et al. 1974). This assumption implicates
that the market is perfectly competitive with no mono- or oligopoly.
The price is a result of all actions, not a particular investor.
• Taxes are neutral, hence all investors pay the same tax from all
forms of income: dividends, interests and capital gains.
1.3.3.2 Decisions based on the mean-variance criteria
Should an investor be able to make his/her decision based only on
these two variables, the returns must have normal distribution. Risk
16
that accompanies the investment can be defined by the distribution of
possible returns. As this distribution is assumed to be normal, it can be
described by two measures: mean and variance (or its square root
called standard deviation, which is much more popular in finance).
Since all normal distributions are virtually the same, they differ from
each other only in their means and variances.
While all investors prefer higher returns to lower returns, ceteris
paribus, it is true that they do prefer lower risk. It is described by the
standard deviation of returns. This leads to the conclusion that if
investment/portfolio/share risk is high, investors would accept it only if
they would be rewarded by a high expected return. Consequently, if the
expected return is low it would be accepted only if the related risk is low
as well. Commonly applied mean-variance analysis is therefore a trade
off analysis between the accepted risk and the required rate of return.
Even if an investment bears no risk, investors would still expect
nonzero return as an incentive to delay the current consumption. Short-
term government bonds may be considered an example of this kind of
investment as their default risk is virtually zero.
It is additionally assumed that all investors make their decisions only
one period ahead and all of them define this period in the same
manner. This assumption fringe in its classification upon the next
group, introducing homogenous beliefs.
1.3.3.3 Homogenous beliefs
This condition states that all investors have the same, homogenous
believes about the primary data, which are necessary to make portfolio
decisions. These are mainly three groups of data characteristics:
expected returns, variance (or standard deviation) of the returns and
17
the matrix including correlation coefficients between the returns on all
pairs of shares.
The expectations are a result of all available information and therefore
the discussed assumption refers directly to efficient market theory,
which presumes that each investor has an access to the same
information.
In practice this assumption is unlikely to be fulfilled, since information is
not distributed among all of the investors to the same extend. Big
financial institutions have much better access to the information than
individuals. On the other hand individuals state too small percentage of
market players to consider them statistically insignificant. However,
banks may have more information about the companies they service
than other investors, as banks remain in a close business contact with
their customers.
The problem of lack of information among the small investors is usually
solved by observation of the investment decisions made by bigger and
better informed institutions. Although the latter have time advantage, it
must be discounted by the price of gaining information.
1. 4 Arbitrage Pricing Theory
The classic APT was implemented by Ross in 1976 as an alternative to
Capital Asset Pricing Model. It considers more than one factor
influencing assets returns.
The theory deduces that firm-specific risk is fairly unimportant to
investors holding well-diversified portfolios and it might be pretended
that firm-specific- risk is not present. Thus, the risk of securities can be
described by factor beta coefficients only. In equilibrium, where
18
arbitrage opportunity does not exist, assets returns will satisfy an
equation relating expected returns of securities to their factor betas.
This risk-expected return relation was called APT and can be written
formally:
=
++=K
jikikFi RR
1
ελγ
The derivation of APT requires only three general conditions to be met:
1. No arbitrage opportunities.
2. Returns can be described by a factor model.
3. There is large number of securities, so that it is possible to form
portfolios that diversify the firm specific risk of individual stocks.
1.4.1 No arbitrage opportunities
The Arbitrage Pricing Theory is based on the Law of One Price. The
rule says that all goods of the same risk should be sold at the same
price thus market can reach equilibrium preventing arbitrage. There are
further requirements similar to assumptions concerning CAPM that
relate to market efficiency, investors homogenous beliefs and their
mean- variance investment criteria. These assumptions ensure that
there is no arbitrage opportunity and market is in equilibrium what is
crucial for APT. According to Jajuga and Jajuga (1999) there are eight
such requirements for APT significance:
No transactional costs.
Assets divisibility
No taxes on incomes generated by capital market
Unlimited short sale and long buys
Investors can borrow at the risk free rate
No barriers on assets sale or purchase
19
Decisions based on the mean-variance criteria
Individual investors decisions about their position in any assets
will not affect their price
Above assumptions are partly met by Warsaw Stock Exchange. First of
all, there are transactional costs in Polish market but if the transaction
size is large, costs are relatively small and they can be neglected
(Rubaszek, 2002).
Furthermore, assets divisibility condition can be assumed as well. In a
real business world, the smallest unit that can be traded on a real
market is one share that cannot be divided into parts and then traded.
Nevertheless, it can be supposed that market participants invest in the
purchase of one expensive share. Assuming this situation shares could
be seen as divisible.
Moreover, the no taxes condition could be supposed. Taxes on capital
gains are going to be introduced in 2004 or 2005 but there is no explicit
regulation of this issue right now. Nowadays, only dividends and
interests on bank deposits are taxed.
There is a visible impact of important institutional investors on stock
prices of the companies with the greatest capitalisation in Poland.
However, firms characterized by smaller capitalisation are very
sensitive to speculative actions of individuals or a group of non-
institutional investors. As an example, it might be said that the group
manipulated the price of Efekts stocks making it few times greater.
This incident took place ten years ago, but it is a proof that Polish
market does not meet the assumption that individual investor is not
able to influence stock prices (Rubaszek, 2002).
Moreover, APT allows short sale of securities. Polish law has regulated
this issue on 21st December 1999. Warsaw Stock Exchange opened a
20
special internet platform in order to collect investors orders and
encourage market participants to short sale or purchase of securities.
However, there are only five brokerage houses dealing with short sale.
These transactions accounted for 0.01 or 1.18 percent of all
transactions on WSE. According to Maciejewski and Mejszutowicz
(2003) the reason for this situation is that the whole system is too
complicated. Furthermore, borrowers charge high fees what deter
lenders. Therefore, short sale of securities is not very popular among
Polish investors. This fact might imply that arbitrage can be limited and
thus making prices diverge from equilibrium.
Furthermore, in real business world investors that borrow funds need to
pay premium to the borrower as a price of the loan. Thus, the
assumption of borrowing at risk free rate is not fulfilled in reality.
The next condition requires securities to be traded without any barriers.
It is believed that this requirement is met on Polish market as it is
ensured by law (Dz. U. of 2002 year. No 49, position. 447).
The last issue refers to investors. They are assumed to allocate their
funds taking into account only expected returns and securities risks.
That is true that the majority of investors while making financial
decisions focuses mostly on these two issues (Rubaszek, 2002).
However, it cannot be assumed that all investors behave in the same
manner.
Having in mind the assumptions presented above, the APT model can
be implemented.
Nevertheless, these eight assumptions are usually not met in a real
business world. APT proponents believe that the basic advantage of
the theory is the fact that not all of the assumptions need to be met
(Haugen, 1996).
21
1.4.2 Factor Model
APT begins with the assumption on the return generating process. If
individuals believe that the random returns on the set of assets are
explained by K-factor linear model:
=
++=K
kikikii IaR
1εγ
where:
i=1,, n
iR is random return on asset i
ia is the expected return on the asset i
ikγ factors coefficients
iε are the mean zero asset specific disturbances assumed to
uncorrelate with the K
≈δ and with each other
Then, the security is differently sensitive to each kI factor. However, all
of kI factors have the same value for all securities. Moreover, each kI
variable have impact on more than one security. Returns of all
securities depend on kI that are changing constantly and ikγ
coefficients that are specific for each security. Terms iε are assumed to
reflect the random information that is unrelated to other assets. Too
strong dependence on iε would suggest that there are more than k
common factors. Furthermore, n should be much bigger than number
of factors k.
According to Roll and Ross (1980) the formula reflects the nature of
assets in different states of nature.
22
1.4.3 Firm- specific risk
The assumption of diversified firm-specific risk is of utmost importance
for APT, as it allows for relating returns to factor betas. Roll and Ross
assumed that the number of assets analysed is approximate to infinity
and the portfolios are perfectly diversified. Moreover, all variances of
residuals have weights equal squared amount invested in asset, as
residuals are uncorrelated. Thus, for the perfectly diversified portfolios
the residuals risk would be close to zero.
If iε were excluded from the model, the formula would say that each
asset i has returns that are an exact linear combination of the returns
on riskless asset and the returns on k other factors. Thus, the riskless
return and each of the k-factors can be described as linear combination
of k+1 others returns. Any other assets return, since it is a lineal
combination of the factors, must be also a combination of the first k+1
returns. Therefore, the portfolios of the first k+1 assets can be a perfect
substitute for all of the assets in the market. Such substitute should be
priced equally. Thus, the APT suggest that only limited number of risk
components exists. Therefore, if there are only a few systematic risk
components, economic aggregates (for example GDP, inflation rate,
interest rates etc.) could be expected to be such factors.
1.4.4 APT relation In order to track the return on the portfolio with no firm-specific risk a
tracking portfolio with weights of =
−K
jik
1
1 γ on the risk free security, 1iγ
on factor portfolio 1 and 2iγ on factor portfolio 2,., finally ikγ on factor
portfolio k can be constructed. Therefore, the expected return of the
portfolio that tracks the investment would be:
23
=
+K
jkikFR
1
λγ
where 1λ kλ are risk premiums of factor portfolios.
If the investment and this tracking portfolio have the same expected
return, there is no arbitrage opportunity.
Thus, the APT equation for all investments with no- firm specific risk
can be formulated as follows:
=
+=K
jkikFi RR
1λγ
This relation should hold in the absence of arbitrage opportunities. On
the left-hand side an expected return on investment is presented and
on the right-hand side there is the expected return of a tracking
portfolio depending on the same factor coefficients.
If there is only one significant factor in APT model then the asset
pricing equation can be presented as a straight line. Two-factor models
graphical presentation will be a plane as there are three points
necessary to describe a plane. These points will be two coefficients
and expected return. More than two factor model presents a
hyperplane.
1.4.5 Methodological concerns
The theory proponents believe that there are two most important
advantages of APT. The first one is the liberal character of its
assumptions compared to CAPM assumptions. The second benefit is
24
that the theory significance can be verified statistically. The second
issue is discussed by the theory opponents. There were economists
trying to assess if the APT model is testable. The first studies on APT
statistical tests show that a researcher carrying out a factor analysis
may face methodological difficulties.
APT opponents usually believe that the assumption of diversified firm-
specific risk is weak APT opponents such as Schanken (1982) and
Dhrymes et all (1985), researched how the theory works when limited
number of assets is assumed or when the economy analysed is of the
limited size.
Shanken (1982) criticized the idea of APT testability. The return-factors
linearity assumption was pinpointed as the mistake in the theory
formulation. They concluded that employing the infinite number of
assets is not enough to neglect the firm-specific risk. Furthermore, APT
models are prone to manipulation, as neither the factors generating
returns nor their number were specified in the theory.
One of the most important problems concerning the arbitrage pricing
theory testability is the number of assets in the analysed portfolios.
Dhrymes et al. (1984, 1985) stated that for the number of assets
ranging from 15-60 the number of significant factors increases from 3
to 60. Therefore, the number of assets analysed in groups is of utmost
importance in the model estimation. Furthermore, Dhrymes et al show
that the number of factors generated by factor analysis depends on the
number of observations throughout the time and the numbers of
analysed macroeconomic variables.
However, Haugen (1996) believes that the most significant factors will
be estimated even on small samples. Weaker and therefore less
important factors can surely disappear, unless the sample analysed is
large enough. The less visible factors are not valuable for empirical
25
researchers, thus APT proponents believe that the sample size does
not matter.
Furthermore, APT in the contrary to CAPM gives explicit predictions
about the portfolios efficiency. Haugen (1996) gives the following
example. It was assumed that there are n factors and n portfolios and
that each of them is a substitute for one of the factors. According to
Grinblatt and Titman (1998) these portfolios will be efficient portfolios
only if they were created according to APT. Thus, the empirical
verification of the theory is easier than CAPM empirical tests. However,
the problem of APT testability is still unsolved.
26
CHAPTER II
Capital market equilibrium models empirical tests
This chapter focuses on CAPM and APT empirical tests. Presented
studies might be useful in testing these models on Warsaw Stock
Exchange as they present different methodologies of estimation and
tests. Furthermore, the empirical basis of researches includes these
that were carried out on Polish market. These papers might suggest,
whether these models can be implemented on the WSE.
2.1 CAPM empirical evidence
In this section the results of the most significant CAPM tests will be
chronologically presented and discussed. The literature can be divided
into two parts: early CAPM tests, conducted in 70s and later tests
(after the Rolls critique).
2.1.1 Early CAPM tests
In the early CAPM tests the technique of two-stage regression analysis
was commonly applied. The first phase of this analysis was the time-
series regression, which was to estimate the beta coefficients of each
analysed company. In the second phase the cross-sectional regression
was run, while the average rate of return was a dependent variable and
a corresponding beta coefficient became an independent variable. The
27
aim of the regression was to estimate the Security Market Line, which
would allow to state if its theoretical values were consistent with the
empirical findings. The most significant empirical studies on this topic
were conducted by: Lintner, quoted by Douglas (1968), Black, Jensen
and Scholes (1972), Blume and Friend (1973) as well as Fama and
MacBeth (1973).
2.1.1.1 Lintner test (1968)
Based on the sample of 301 randomly chosen companies Lintner
estimated beta coefficients regressing yearly returns on shares against
yearly returns on market index. Years 1954-1963 were the estimation
period for the equation:
titMiiti eRbR ,,, ++= α
where bi is the beta coefficient for the company i. The second phase
was the cross-sectional regression:
ieiii SabaaR η+++= 2321
where 2eiS is a variance of the residual ei. The obtained results stay in
contradiction to the CAPM theory because of three reasons.
First of all, the coefficient a1 should approximately equal the value of
the risk free rate, but it appeared to be higher of any value possibly
taken by RF in the examined period.
Second, the coefficient a2, which defines the price of the accepted risk,
although statistically significant, had much lower value than expected.
28
Finally, assuming that the CAPM is true a3 as a coefficient of additional
independent variable should be statistically insignificant. Therefore,
Lintners results are not coherent with the theory, since a3 is positive
and statistically significant.
A response to the above analysis was a study conducted by Miller and
Scholes (1972), who concentrated their efforts on a critique of the
methodology applied by Lintner. There were three planes of the
critique.
First, the notation of tested equation was incorrect, since it did not
include the models reliance on the risk free rate in the proper manner.
If the original model takes the form of:
( )tFtMitFti RRRR ,,,, −+= β
then
( ) tMitFiti RRR ,,, 1 ββ +−=
The situation gets more complicated if RF is not constant over time.
Second, heteroskedasticity, which is often present in the financial time-
series, is interpreted as an inconstant variance of the returns over the
estimation period. Although Miller and Scholes found the
heteroskedasticity, they decided that it is not a cause of the CAPM
rejection.
The last, and as it appeared the most important reason for the CAPM
failure was a beta estimation error in the time-series regression. The
estimated beta coefficient, biased with the estimation error, becomes
an independent variable, which must lead to the false estimation of the
parameter describing the variable.
29
2.1.1.2 Black, Jensen and Scholes test (1972)
Black, Jensen and Scholes (BJS) overcame the problem of beta
coefficients estimation error, which was a cause of Lintners study
failure. The methodology of this study will be discussed in details later
on, since based on a similar methodology the test of the CAPM model
on the Polish market will be conducted. BJS used only the shares,
which were listed on the NYSE within the period 1926-1965. Based on
the data from the subperiod 1926-1930 beta coefficients of the
individual shares were estimated. These parameters were computed
against the unweighted market index, composed of all shares listed on
NYSE. The next stage was to sort the companies according to their
beta value and dividing them into ten portfolios, so that the first portfolio
contains the decile of the companies with the highest beta coefficient
and the last portfolio was created by the decile of companies with the
lowest values of this coefficient.
Next, for each of the portfolios, series of twelve monthly returns
realised in the next year 1931, were calculated. This process was
repeated shifting the sequence one year ahead, which means that
based on the period 1927-1931 beta coefficients were estimated.
Based on the estimated beta coefficients companies were sorted in
portfolios for which twelve monthly returns were computed. Having the
portfolios monthly returns calculated, BJS could estimate the portfolios
beta coefficients using the same index, which was used to beta
coefficients estimation of individual companies. The final version of the
tested model took the following form:
( )tFtMPtFtP RRRR ,,,, −+= β
30
where RP is a return on the portfolio P. Using the estimated beta
parameters of each portfolio, in the cross-sectional regression the
Security Market Line was estimated:
PP aaR β10 +=
where a0 is a risk free rate, if the one exists. The value of this
coefficient was 0.0519, which is 6.225 percent yearly. Because it is
statistically more than the average interest rate of the government
bonds in the studied period, the results support Black CAPM version.
Black allows long buying of the government bonds at the risk free rate
but forbids their short sailing. The obtained market risk premium was
0.01081, which is 12.972 percent yearly.
The results strongly support the zero beta version of CAPM model. The
estimated SML does not reveal any signs of curvilinearity and the
determination coefficient for the cross-sectional regression equals
almost unity.
2.1.1.3 Fama and MacBeth test (1973)
Applying a similar procedure to BJS Fama and MacBeth (FM) formed
20 portfolios, for which the beta coefficients were estimated in the first
phase. The difference comes from the fact that the beta coefficients
computed against data from the period t were a basis to form the
predictions of the rates of return in the period t+1. Unlike BJS, Fama
and MacBeth in the second phase repeated the regression separately
for each month in the period 1935-1968. Due to the fact that this
technique was employed, FM could analyse the changes in the
parameter values over time. The estimated cross-sectional equation
was as follows:
31
tpPetPtPtttP SR ,,,32
,2,1,0, ηγβγβγγ ++−+=
Having the monthly regressions estimated, the average values of
estimated parameters were calculated in order to test the hypothesis
regarding all four coefficients. The results can be summarised in four
points.
First, the average value of the intercept γ0,t should be equal to (for the
standard version of the CAPM) or greater than (for the Black CAPM)
the risk free rate. Second, the average value of γ1,t coefficient should be
positive. Third, the average value of γ2,t coefficient determines if there
are any signs of curvilinearity. According to the theory this coefficient
should be statistically insignificant and eliminating this variable should
not decrease the value of the determination coefficient. Finally, the
residual variance should not be statistically significant while estimating
the average returns on the portfolio. It is because investors can
eliminate this factor through diversification of their portfolios. Hence,
the average value of the coefficient γ3,t should not be statistically
different from zero.
The results obtained in the FM studies are consistent with CAPM
theory. Similarly to the BJS research, the outcomes support the Blacks
version of the CAPM model. However, one aspect of the study should
be criticised, namely FM did not use a weighted index, hence it can not
be treated as the reliable proxy of the market portfolio. This argument
is presented by Roll (1977), who criticised the early tests of the CAPM
model.
2.1.2 Rolls critique (1977)
Roll criticised tests of the CAPM model of that time arguing that they
are mathematical tautologies. The presented prove confirms that even
32
if the index applied is the market portfolio, but some other on the
efficient portfolio, then there always will be a linear relationship found
between the expected return on a share and its beta coefficient
estimated with the respect to the efficient portfolio. Furthermore, the
indices used are not market portfolios. According to the definition of the
market portfolio, it should consist of all assets available to the investor.
Market indices do not include bonds, real estates, gold and many other
investment opportunities. Hence, the so far conducted tests may only
verify the hypothesis of the particular index efficiency, but can not be
considered a CAPM model tests. The conclusion is that testing the
CAPM model is not possible because of the purely theoretical idea of
the market portfolio. Since the market portfolio grouping all risky assets
is non-existent, it is impossible to calculate its return and therefore
CAPM can not be a testable theory.
However, Stambaugh (1982) proved that the CAPM model test is not
sensitive to the enrichment of the proxy for the market portfolio in
additional investment opportunities. He built a few versions of the
CAPM model starting with NYSE index as a proxy for the market
portfolio, next extending it by the government and corporate bonds
market, then adding the real estate market and finally including even
durable goods market2. Applying Lagrange multiplier tests to verify the
hypothesis, Stambaugh could not reject the Black CAPM version,
concluding that the Rolls critique was too strong. Increasing the
composition of the proxy for the market portfolio did not influence
Stambaughs results.
2 Such as vehicle market.
33
2.1.3 Later tests of the CAPM model
A brand new series of empirical attacks on the CAPM model consisted
in identifying variables other than beta coefficient that could explain the
average expected returns on shares.
2.1.3.1 Banz test (1981)
One of the first studies of this type was conducted by Banz (1981), who
decided to test if the firm size can explain this part of variance in the
returns, which is not explained by the beta coefficient. It was found that
in the period 1936-1975 the average returns on the companies with low
capitalisation were statistically higher than the average returns on the
big companies, after adjusting for the risk in both groups. This
relationship is commonly known as a size effect.
The procedure of portfolios building applied in Banz (1981) test is
similar to the BJS. All portfolios consist of companies listed on NYSE
and the cross-sectional regression defines the relationship between the
average returns, beta coefficient and relative size of the portfolios.
Since the coefficient of the relative size variable is statistically
significant even at the low levels of significance, Banz (1981)
concludes that the CAPM model is not fully specified hence fails. The
negative value of the coefficient should be interpreted as follows: the
shares of the companies with higher capitalisation are characterised by
lower, on average, rates of return than shares of the small companies.
To support the results Banz conducted one more test. Two portfolios
were created, each consisting of 20 shares. The first portfolio included
shares of the companies with low capitalisation, whereas the second
one included big firms. Both were constructed in that their betas were
equal. Based on the same period as in the previous study Banz found
34
that the first portfolio indicates monthly on average 1.48 percent higher
return than the second one and the difference is statistically significant.
This outcome is consistent with previously obtained.
The subsequent studies supported doubts about model
misspecification. Basu (1983) found that the ratios defining the firm size
and E/P are interrelated, hence E/P should explain the expected
returns as well. Additionally, Bhandari (1988) proves that the financial
leverage ratio is positively correlated with expected rate of return.
On the other hand, the same year as the Banzs study was released,
Christie and Hertzel (1981) published a paper, in which they indicated
that the companies decreasing their capitalisation became more risky
and since beta coefficient was measured based on the historical data, it
could not capture an increase in risk over the estimation period, hence
the beta was lower. Reiganum (1981) and Roll (1981) indicated that the
beta coefficient of small companies would be lower as it was an effect
of thin trading.
2.1.3.2 Fama and French test (1992)
A sample, on which Fama and French (FF) conducted their test of the
CAPM model, was constituted of companies listed on NYSE, AMEX
and NASDAQU in the period July 1963 December 1990. FF created
100 portfolios first sorting the shares in ten portfolios with respect to
their capitalisation and then, within each group, shares were sorted
with regard to their beta coefficient value. Based on the cross-sectional
regression analysis of the equation:
PPPPR ηψγβγγ +++= 210
35
where ψP is an independent variable defining the firm size, they came
to the conclusion that γ2 coefficient is negative (-0.17) and statistically
significant (t statistics = -3.41). The beta coefficient, however, is
statistically insignificant and this conclusion will not change even after
exclusion of the size variable ψP.
FF include one more factor in their analysis, book-to-market equity ratio
(BV/P) and conclude that this variable explain much better the variance
of average returns than the size variable. Shares characterised by the
high value of the BV/P ratio generate higher returns on average. Even
though this relationship does not necessarily have to be true in the
short-term, it is held in the long-term. Unexpected might be the fact that
FF applying the same methodology as FM (1973) came to completely
inconsistent conclusions. This inconsistency is assigned to different
estimation period. FF repeated the test for the period applied in FM
studies and reached coherent results.
Studies that are a response to the FF critique refer mostly to the data
used. Three years after FF publication Korthari, Shanken, and Sloan
(KSS) (1995) wrote a paper, in which they prove that the results
obtained by FF depend mainly on the interpretation of the statistical
tests. KSS conclude that beta coefficient from the estimated form of
equation has a very high standard error, which does not allow to reject
statistically a high range of the risk premiums. For example the
estimated coefficient γ1 with a value of 0.24 percent, has a standard
deviation of 0.23 percent, which means that values of γ1 may range
from zero to 0.5.
Amihud et al. (1992) share the view about the statistical noise,
concluding that when applying more sophisticated estimation
techniques the value of γ1 coefficient would be positive and statistically
significant. The same results were obtained by Black (1993), who
suggested that the size effect might have been related only to the
36
estimation period applied by FF. He proved that for the period 1981-
1990 the size variable did not affect the average rate of return and was
statistically insignificant.
Even if the size effect exists, there is a remaining question if its
significance is high enough, because of the relatively low value of the
small companies. Jagannathan and Wang (JW) (1993) state that in
each of the groups tested by FF, 40 percent of the biggest companies
were more than 90 percent of the market value of all companies listed
on the NYSE and AMEX. In this case, the CAPM model holds its
empirical validity. JW criticise that the market indices are used as a
proxy for the market portfolio. They indicate that in the U.S.A. only one
third of non-governmental assets is held by the industrial sector and
only 30 percent of this amount is financed by the capital markets.
Furthermore, the intangible assets like human capital can not be
reflected by market indices. Finally, they conclude that beta coefficient
of an individual company is not a constant value over time and the
reality can be much better described by the CAPM model that allow the
coefficient to vary over time.
The KSS critique refers to the second variable as well proving that the
companies with high BV/P ratio at the beginning of the estimation
period had much lower chances to survive, hence the lower chances of
being included in Compustat, which was used in the survey. On the
other hand, the companies that managed to survive together with
companies added to the survey in the later period indicated on average
higher returns. Taking into consideration the above reasoning Breen
and Korajczyk (1993) verify this hypothesis using the same software
and data as FF. Their conclusion is that the BV/P variable should be
definitely less significant.
37
2.1.3.4 Kozickis and Shens test (2002)
The authors of this test question the contemporary methodology
applied while testing the CAPM model. Kozicki and Shen (KS) consider
the hypotheses stated incorrectly by FM were the basis for many
further studies. They argue that insufficient evidence to reject the null
hypothesis was considered sufficient to reject the model. KS state that
this manner of testing leads to false rejection of the model in at least
half of the studies. They suggest to test the CAPM based on the
statistical test in which the theory is true under the null hypothesis.
Applying this alternative statistical test, the model can not be rejected
based on the data used by FF.
The inverted formulation of the null hypothesis, in which the beta
coefficient equals zero is following:
0:0 =iH β
It causes that the test is a subject to the error type I and II, which
means that rejects the model when it is true and accepts when CAPM
is false.
There are four most popular reasons for the type I error to occur:
• Low value of expected premium for the market risk.
• High variance of the errors in expected premium for the market
risk3 - 2)var( mση =
• High variance of the error in the CAPM model.
• Small number of surveyed periods.
A high frequency of the error type I occurrence reflects the problem of
limited access to information, which causes a lack of sufficient
3 ttMtM RER η+= )( ,,
38
evidence to reject the null hypothesis leading the researcher to the
conclusion that the CAPM model is true.
The error type II arises when the statistical test rejects the null
hypothesis and results in accepting CAPM model when in fact it is not
true. It is because rejecting the null:
0: ,10 =tEH γ
from the cross-sectional regression:
tpPetPtPtttP SR ,,,32
,2,1,0, ηγβγβγγ ++−+=
in favour of the hypothesis:
0: ,11 >tEH γ
model CAPM is considered true, although ( )tFtMt RREE ,,,1 −≠γ , which
means, that in the reality CAPM is a false theory. Hence, the statistical
rejection of the null hypothesis in this case is not unequivocal evidence
that the alternative hypothesis is true.
Finally, KS suggest an alternative test, in which the null hypothesis that
the CAPM model is true takes the following form:
0)(: ,,1,00 =−+ tmtt rEHA γγ
And the formula for the t-test:
Trr
ttm
mA
ACAPM /)(
)( 10
σγγγ −+
=≡
39
where T is a number of observations. Applying this test KS conclude
that the CAPM model is true.
Most of the latest tests of the CAPM model refer to the empirical
verification of the sophisticated derivatives of the standard model
(Jagannathan et al. 1995; Campbell R. Harvey's research papers
http://www.duke.edu/~charvey/research.htm), which is far beyond the
scope of this thesis. The above presented analysis of a standard
CAPM model does not allow to unequivocally state if the theory truly
reflects the financial reality. Even when rejecting the model using a
proxy for the market, because of a highly theoretical definition of a
market portfolio, there is no possibility of testing the theory.
Nevertheless, the CAPM gained the acceptance of the financial
practitioners such as: portfolio managers, investment advisors or
financial analysts. The popularity of the model results from its simplicity
resulting from linear relationship based on only one factor (Javed
2000).
2.2 Empirical Studies on APT
As the most universal group of tests, the early tests of the multiple
factor generating returns are briefly discussed. These studies
motivated researchers to consider other than only beta and risk free
rate factors that might explain assets returns. The empirical
investigation of the APT theory statistical significance should be
considered together with its method of creation. The method of
estimation of factors and coefficients influences the tests results.
2.2.1 Investigation on variables influencing returns The possibility that expected returns are generated by multiple factors
was recognized over twenty years ago. The early empirical studies
40
were carried out by Brennan who concluded that the return generating
process has to be described by at least two factors. Furthermore,
Rosenberg and Marathe searching for the presence of components
that have an impact on assets returns shown that there are many
factors influencing market portfolio. Moreover, studies of Langetieg,
Lee and Vinso, Mayers evidence that there is more than only one factor
in the returns generating model (Roll and Ross, 1980).
There were other researchers investigating variables that were likely to
influence asset return. Chen, Roll and Ross in their research published
in 1986 suggested four-factor model and then tested it. (Francis, 2000).
Measures of unexpected changes in the following variables were
suggested in their paper:
1. Inflation- as it has impact on discount rate and future cashflows
for investors
2. Interest rates term structure- the differences between short
and long term bonds influencing the value of future liabilities
compared to liabilities due within shorter period of time.
3. Risk premium- the difference in low- and high-grade corporate
bonds approximates the reaction of market to risk.
4. Industrial production- the differences in industrial production
have an impact on investment opportunities and the real value of
cashflows.
This study revealed that there is a significant relationship between
these macroeconomic variables and factors estimated statistically in
the previous Ross and Roll analysis. Nevertheless, there is no
evidence that the suggested selection of variables that influence assets
valuation is perfectly correct.
Their research was continued by Burmeister and McElroy. They
implemented a model that generates returns using five factors (Elton
and Gruber 1998)
41
1. The risk of not paying liabilities 2. Premium rewarding long term investments 3. Deflation 4. Change in expected level of sale
The fifth variable represents all unobservable variables that are
estimated on the basis of residuals of diversified portfolios regressed
against four others.
The first variable was measured by the difference in long term rates of
governmental bonds and long term corporate bonds interest rate plus
five percent. The second was approximated by the difference between
long term rates of governmental bonds and one month treasury bonds.
The third one was described by the difference between the expected
inflation rate at the beginning of each month and the real monthly
inflation rate
The first four factors explained 25 percent of the returns variability and
all five described from 30 to 50 percent depending on particular
portfolios. Thus that is the next study confirming that the return
generating process accounts for more than one factor including
unobservable variables. Their later studies introduced three
unobservable factors instead of only one. According to Elton and
Gruber (1998) this survey is one of the strongest empirical evidences
showing that multifactor models are useful in explaining returns.
The variable set that was tested in these studies was selected arbitrary;
hence it should not be treated as the only one that explains returns
correctly. Nonetheless, the analyses can give an idea which variables
may influence returns and therefore should not be neglected by future
researchers.
42
The mentioned studies tried to define factors that influence returns on
the basis of economic theory. Researchers defined factors that are
likely to explain assets returns. Despite of the fact that these studies
did not create one selection of macroeconomic factors generating
returns, their results shown that there is more that one factor significant
in the return generating process.
2.2.2 Approaches to APT model estimation This section analyses empirical studies concerning APT in relation to
their methods of estimation. The following researches were carried out
using two different approaches to model implementation and testing.
There are three general approaches to the estimation of coefficients
applied in empirical studies. The first technique allows for simultaneous
statistical estimation of factors and beta coefficients. The second one
defines factors or coefficients a priori.
2.2.2.1 Statistical estimation of betas and factors
There are empirical studies that use this method. Factor analysis is
usually applied in order to estimate factors and their coefficients
simultaneously. Further details of this method are presented in details
in empirical part of this study. This section summarizes findings of
example tests carried out on the basis of this method.
One of the first APT tests applying this technique was carried out by
Roll and Ross (1980). They surveyed 1260 stocks quoted on the New
York Stock Exchange grouped alphabetically into 42 groups.
The maximum likelihood analysis was performed in order to estimate
factors and factor loadings (coefficients). Then, the factor loadings
estimates were applied to explain the cross-sectional variation of
43
individual estimated expected returns. Finally, the estimates from the
cross-sectional model were used in order to calculate the significance
of the risk premia associated with each factor.
The APT reflects the reality if there are one or more significant risk
coefficients that significantly differ from zero. The results of this test
show that there are no more than four significant risk premia
coefficients.
Cho, Elton and Gruber carried out a similar analysis as Roll and Ross
did but they examined a newer data set. More than four significant
factors were found (Elton Gruber, 1998).
2.2.2.2 Portfolio method of factor estimation
The next group of arbitrary methods that can be used in APT models
estimation assumes that factors having an impact on returns are
reflected in portfolios. These portfolios are defined on the basis of
investors expectations and macroeconomic factors that are likely to
influence returns. This method was applied by Fama and French in
1993 and discussed in details in section 2.1.3.2 of this paper.
Their approach is arbitrary as the portfolio grouping is subjective, but it
confirms that returns can be explained by more than single-factor
model.
2.2.2.3 Betas arbitrary choice Sharp (Elton and Gruber, 1998) carried out a research defining beta
coefficients arbitrary; hence the APT model could be implemented quite
easily. Having betas defined, the model could be created by regressing
returns against betas.
44
The sample consisted of monthly 2197 stocks returns analyzed over
1931-1979. Sharp assumed that equilibrium returns are influenced by
arbitrary selected variables such as:
1. Beta of assets measured with relation to S&P index
2. Dividend payout
3. Firm’s size measured by capitalization
4. Beta estimated against long term governmental bonds
5. Historical alpha parameter (the intercept of regression function
estimated on abnormal returns of securities and of market index
S&P)
Furthermore, the industry specific variables were analyzed as well.
Taking into account the economics theory, beta is expected to
influence return positively as low-grade assets should generate higher
returns in equilibrium. The impact of dividend payout is expected to be
positive as well, as increasing dividend is likely to be interpreted as a
signal of increasing expected future cashflows. The big firm size as
approximation of company liquidity would influence the return
negatively. If the beta measured against bonds is significant, the
abnormals would be sensitive to interest rates and exchange rates. The
alpha significance implies that autocorrelation of residuals in CAPM
occurred; hence there is more than one factor that influences returns.
The empirical results confirmed that the outcomes, expected on the
basis of the economics and all variables, were significant. The
determination coefficient of the model explaining returns related only to
market beta amounted to 0.037. When the model includes beta,
dividend payout, firm size, beta on bonds and alpha intercept the value
of that coefficient increased to 0.079. Moreover, model implemented
on the basis of all analyzed variables explained over ten percent of the
return analyzed.
45
Creation of Sharp model included arbitrary chosen variables but it
showed that there was more than one significant variable explaining
equilibrium returns. Furthermore, the research showed that it was
possible to find arbitrary significant factors reflecting economic
relationships.
2.3 APT contra CAPM
A lot of empirical studies investigate the question, if APT is a better
model than CAPM. They usually confirm that APT overperforms CAPM.
Reinganum (1981) carried out an empirical research that scrutinized
whether the arbitrage pricing model can account for the differences in
average returns between small and large firms traded on American
Stock Exchanges. Reinganum assumed that APT would be better in
explaining returns than CAPM as it explains the CAPM anomalies.
Thus, the firm size effect was analyzed. The research was carried out
within the estimation period of fourteen years that is since 1964 to
1978. Furthermore, the sample was being changed for each year. The
number of firms examined ranges from 1457 in 1963 to over 2500 in
the mid 70s.
To estimate factor loadings, factor analysis was introduced. The test
results indicate that APT fails statistical verification. Portfolios
constructed of small firms earn on average 20 percent more than large
firms portfolios.
However, the research results should be taken with caution as the
Reinganum analysis tested a few hypotheses simultaneously. Thus
there is no certainty which of them have failed. For example the
process generating returns might not have been linear, the firm-specific
variance might not be diversified or the arbitrage opportunity might
46
subsist on the market. Therefore, the reason of APT failure was not
discovered.
Chen (1983) analysed daily returns during the 1963-1978 period. First,
the APT model was implemented and tested. Betas were computed
with market proxies such as: the S&P 500 index, the value weighted
stock index of the equally weighted stocks. The cross-sectional
regression of average stock returns was applied according to both
market equilibrium models.
The significance level of the first risk factor was the highest, and it was
concluded that the first factor is related to market portfolio.
Furthermore, the hypothesis of constant expected return across assets
was rejected. Therefore, the model explains expected returns across
assets.
The next test allowed for comparison of APT with CAPM as models
explaining the expected returns. Chen used the following regression:
iCAPMiAPTi ewwi rrr +−+=−
,
^
,
^
)1(
The past return on security i was related to weighted average of returns
calculated according to both market models. Estimated returns were
always bigger than 0.9. Therefore it can be concluded that APT
performed better than CAPM.
The last test carried out by Chan was based on residuals generated by
both models. Residual estimates equations should follow white noise
for well-estimated models. The CAPM model did not follow a random
walk. The APT explained a part of residuals generated by CAPM but
APT residuals were not explained by CAPM. Therefore, it might be
concluded that CAPM faced a misspecification problem.
47
One of the latest APT tests was carried out by Haugen (2000). He
investigated which model CAPM or APT predicts returns better.
Returns over the same time period 1980 - 1999 and over the same
stock population (the largest 3,500 companies in the United States)
were analyzed.
CAPM was estimated by regression of each stock return against the
S&P 500 return and then recalculating betas each month. Then stocks
were ranked by betas and divided into deciles. Tests results concluded
that the payoff and risk were negatively related on the stock market.
Then APT was implemented. It included the following macroeconomic
factors:
1. The monthly return on Treasury bills
2. The difference in the monthly return on long- and short- term
Treasury bills
3. The difference in the monthly return on the Treasury bonds
and low-grade corporate bonds of the same maturity
4. The monthly change in the consumer price index
5. The monthly change in industrial production
6. The beginning-of-month dividend-to-price ratio for the S&P 500
The model was constructed by regression of stock returns against
these six factors. The APT appeared to predict return better than
CAPM although the negative risk premia for some of the examined
stocks occurred.
Connnor and Korajczyk carried out the APT test applying the Principal
Components Analysis to find the additional returns on small companies
stocks in January. They concluded that, based on weighted index,
developed APT explains the returns to such firms better than CAPM
(Elton and Gruber 1998).
48
However, Gultekin and Gultekin (1987) came to a completely different
conclusion concerning APT and the January effect. Their study
analyses the impact of the return stock seasonality on the empirical
tests of APT. The results show that the APT model can explain risk-
return relation in January only independently on the size of the group
tested. If the January returns are excluded from the sample, there is no
significant relationship between expected stock returns and the risk
measures predicted by APT.
There are empirical studies confirming that arbitrage pricing theory is a
better model for returns predictons than the capital asset pricing one.
This finding is not surprising as there are so many research papers
confirming that one-factor model fails, and that the return of an asset is
generated by the multiple factors process. However, some empirical
studies reject APT as well. Arbitrage pricing theory is not ideal in
explaining asset returns. Nevertheless, it usually overperforms CAPM
in empirical verification.
2.4 Empirical evidences in Poland
The basis of empirical studies concerning market equilibrium models in
Poland is extremely poor right now. The most important reason for that
situation is the fact that Warsaw Stock Exchange is still in early stages
of its development. Furthermore, it is a question whether this market is
efficient enough to implement and test CAPM or APT.
49
2.4.1 Tests of market efficiency
Nowadays, first studies on market efficiency on WSE are being carried
out. The issue of market efficiency was investigated by Szyszka (2003).
Some of his findings are relevant to research carried out in this paper.
The efficiency of WSE was investigated in two subperiods first to
October 1994 and the second from October 1994 to October 1999 as in
the first subperiod stocks were quoted less than five times a week.
First, Szyszka concluded that in the first years of WSE even the weak
efficiency was not present. This finding was based on test of correlation
of daily returns and then series test. However, due to data availability
only 14 companies were examined. There was a significant positive
correlation between returns and historical returns from one to five
sessions. Nonparametric tests confirmed these results. The second
subperiod brought different conclusions. Instead of 14, 29 companies
stocks were examined. 16 firms out of 29 had returns significantly
correlated with historical returns. Furthermore, the correlation for
annual subperiods generated insignificant coefficients.
Significant correlation coefficients between returns within time series
imply that stock prices do not follow random walk. Coefficients were
high in early stages of WSE development when stocks were not quoted
daily. The correlation of returns for Universal mounted to 0.44 these
days. This example is usually presented as a proof of market
inefficiency these days.
After 1999 the correlation coefficients were lower and not all of them
were significant. Szyszka study indicates that the development of the
Warsaw Stock Exchange goes along with better efficiency. There was
not sufficient empirical evidence that allowed the rejection of the weak
form of efficiency.
50
2.4.2 Multifactor models on Warsaw Stock Exchange
Czekaj et al. (2001) analyzing WSE discovered that multifactor model
could have been implemented in order to explain returns. They
examined a sample of 44 stocks in 1995 and 119 stocks in 2000.
Weekly rates of return were analyzed.
Stocks were divided into decile portfolios that were recalculated and
updated each quarter. Portfolios were grouped according to four
characteristics. The first one was based on portfolios created on the
basis of company capitalization, the second on Price to Book Value
ratio (P/BV), third one on the Price to Earnings ratio (P/E) and the last
grouping reflected betas estimated against index WIG. Market premium
on portfolios was estimated and its significance was verified
statistically.
There was a monotonic relation between average rate of return and
average capitalization of companies with yearly premium. The highest
decile portfolios had premium of over six percent with comparison to
WIG and for low decile companies that premium accounted for 11.8
percent. Therefore the choice of portfolio of stocks characterized by
higher capitalization might create value. Thus choosing lower
capitalization portfolio, losses can be expected. Therefore, it was
suggested that beta and capitalization are significant risk measures
and they might be introduced as factors in market equilibrium models.
Tests based on P/E portfolios discovered that premium on low and
high-decile portfolios is negative and lower for high decile portfolios.
Thus, investments in low decile portfolios should generate gains.
According to these results P/E ratio explains rates of return on
portfolios and therefore it can be employed into factor model.
Portfolios grouped according to their P/BV ratio revealed that there is a
U-shaped relationship between returns generated on decile portfolios.
51
It can be concluded that company capitalization and Price to Earnings
ratio are better estimates of expected returns than beta calculated
according to CAPM. Therefore, it might be expected that multifactor
model for WSE would work better than CAPM. However, the
methodology of beta estimation was not presented in details in this
study and it might suffer from methodological mistakes. Thus, it is hard
to say if these two factors overperformed beta when explaining rates of
return.
The first attempt of applying Arbitrage Pricing Theory to Polish market
was made by Rubaszek (2002). He surveyed 73 monthly observations
from 1994- 2000 period. The APT model was created using factor
analysis with maximum likelihood factoring (MLF) method. The
research was carried out on five portfolios grouped according to
company capitalization. Variables concerning investment environment
and the value of companies listed were introduced:
1. Warsaw Stock Exchange Index (WIG)
2. World markets indices (DAX Xetra, NASDAQ, DJ, Standard&
Poor 500)
3. CRB Spot Rate (Commodity Research Bureau Index)
4. Term structure of interest rates
5. Prices (Consumer Price Index, Production Price Index)
6. Aggregated money supply
7. Exchange rates (Polish Zloty against German Mark and US
Dollar)
8. Industrial Production
9. Unemployment Rate
On the basis of these variables (26 together) two significant factors
were created. The first one was correlated with the return on WIG.
Therefore, it can represent the economic environment in Poland. The
second one relates strongly to WIG as well to NASDAQ, PPI and
52
money supply. Moreover, Chow test of factor coefficients stability
revealed that these coefficients were unstable during the estimation
period.
The findings should be taken with caution, as the study was conducted
on the sample of only 38 companies. Besides, the examined period of
six years included quotations from the time of Russian crises that
strongly destabilized Warsaw Stock Exchange. Furthermore, more
explicit results could be obtained if weekly returns were used instead of
monthly.
It is possible that there are academic papers concerning CAPM or APT.
However, they were not published and could not be referred to in this
paper (as Msc and PhD dissertations are available only for PhD
students).
All the discussed studies analyzed very small samples from 14
(Szyszka, 2003) to 119 (Czekaj et al., 2001) securities. That was due to
availability of data as it was usually not possible to find more
companies traded at the same time. This limitation concerned also
estimation period that could not be longer. Findings based on such
samples are very weak.
It is impossible to compare results presented in the literature, as they
all concern different issues and employ completely different
methodologies. However, they may be the basis for future studies on
market equilibrium models.
Szyszkas (2003) findings concerning market efficiency that is getting
better constantly encourage researchers to create and to test market
models. Furthermore, Czekaj et al. (2001) postulates that multifactor
models might perform better than model based only on market beta.
Positive results on arbitrage pricing theory model obtained by
Rubaszek (2002) should rather be taken with caution.
53
CHAPTER III
Data Description
In this chapter a description of the data used in both empirical tests of
CAPM and APT on the Polish market will be presented. The core
discussion will be preceded by the debate about the length of
estimation period, data frequency and the choice of the proxy for the
market portfolio. Data analysed was found in Datastream and Reuter,
basis and cover the period from November 1998 up to the end of 2002.
3.1 Data Choice
Before tests of empirical validity of both models will be conducted, it is
crucial to consider a few issues, which are not specified by the theory.
Decisions, which are to be taken by the researcher, regard the choice
of:
1. A proxy of the market portfolio
2. Length of the estimation period
3. The frequency of the returns
3.1.1 Choice of the proxy for the market portfolio
In practice, capital market index is the most commonly used, however,
there are no such indices that can truly reflect the market portfolio.
Market indices are usually related to the returns on shares or returns
on the debt instruments. In USA the most popular index is S&P 500,
54
which takes into account only 500 of thousands of shares quoted on
the US markets. The problem becomes more complex when one
includes the fact that with a globalisation and free capital flows most
investors have a global access to investment opportunities. Domodaran
(1999) on the example of Disney company showed that the value of
estimated beta depends heavily on the choice of index against which
returns were regressed: Table 3.1 Beta estimated for different indices
* estimated on the monthly data from 1st of. January 1993 to 31st of December 1997 Source: http://pages.stern.nyu.edu/~adamodar/pdfiles/eqnotes/discrate2.pdf
The index, which includes the highest number of shares and is value
weighted average, which means that it takes into account the
companies capitalisation, should provide better estimates. This is the
reason for the S&P 500 popularity, which although does not include as
many companies as NYSE or Wilshire 5000, makes up on its
competitiveness because it is weighted. Furthermore, inclusion of the
unweighted index in this study might be inappropriate for three
reasons.
First, the application of the unweighted index is in contradiction to the
classical CAPM theory, that empirical verification is the aim of this
study. Following the formal derivation of the mode in equilibrium, the
share of each asset in the portfolio will be proportional to its share in
the market portfolio:
i
N
iiM xXRR
=
=1
Where:
Index Estimated Beta*
Dow 30 0.99 S&P 500 1.13 NYSE Composite 1.14 Wilshire 5000 1.05 MS Capital Index 1.06
55
Second, the correlation between unweighted index and returns on
shares would be miscalculated as Polish capital market is dominated
by a few companies. This problem is partially solved by regulation,
which set a maximum share of a single company or industry in the
indices.
Finally, all indices quoted on the GPW (Warsaw Stock Exchange) are
value weighted indices. This suggests that practitioners apply weighted
index as a proper tool in the capital market analysis.
Furthermore, for some small, segmented capital markets local index
should be applied, because it describes better the financial reality.
3.1.2 Length of estimation period
According to Kozicki and Shen (2002) most of financial institutions use
two- up to five-year period for beta estimation purpose. Damodaran
(1999) on the example of Dinsey shows, that beta is time-varying
coefficient and the choice of the estimation period influences
considerably its value:
Table 3.2 Beta estimated for different estimation periods
* estimated on the monthly data from 1st January 1993 to 31st December 1997 using S&P 500 index as a market proxy Source: http://pages.stern.nyu.edu/~adamodar/pdfiles/eqnotes/discrate2.pdf There is no consensus between academics regarding the length of
estimation period. Bartholdy and Peare (BP) provide evidences that are
in favour of five- year period, when estimating expected returns based
Estimation period Estimated Beta* 3 years 1.04 5 years 1.13 7 years 1.09 10 years 1.18
assetsallofvalueMarketassettheofvalueMarketX i
i =
56
on CAPM. However, Daves et al. (2000) strongly suggest that an
estimation period of three years captures most of the maximum
reduction in the standard error of estimated beta from a one-year
estimation period to an eight-year estimation period.
Taking decision about an estimation period a kind of trade off should be
consider. The longer the estimation period, the more observations are
collected for the regression, which increases the quality of estimates,
as standard error of beta estimates is smaller. On the other hand, the
longer the estimation period, the more time-varying characteristics of
the firm might be subjected to changes. These could be for example a
change of the industry, in which the company was primary active,
diversification or change of the financial leverage ratio. Thus, the beta
estimated over longer estimation periods is more likely to be biased.
Therefore, for blue chip companies longer estimation period should be
better. However, for the companies that recently were restructured,
were a target of an acquisition, merged with other firm, changed the
industry or financial leverage, shorter period should be more accurate.
3.1.3 Observation frequency
The last decision that can affect the value of estimated beta is the
frequency of the data used. The most often frequencies are: daily,
weekly, monthly, quarterly or yearly. The example of Disney shows the
difference of estimated beta with relation to the frequency of the
observations:
57
Table 1.3 Beta estimated for different indices frequencies of the observations
* estimated on the monthly data from 1st January 1993 to 31st December 1997 using S&P 500 index as a market proxy Source: http://pages.stern.nyu.edu/~adamodar/pdfiles/eqnotes/discrate2.pdf
BP (2002) found the opposite tendency. Applying Standard and Poors
index the average beta for daily returns of 0.77 increased up to 1.10
when monthly returns were used. BP argue that the best estimates are
achieved by applying data with monthly frequency. This result is once
again in contradiction to Daves et al. (2000) that conclude the financial
manager should always select daily returns because daily returns result
in the smallest standard error of beta or greatest precision of beta
estimates.
Applying greater frequency increases the number of the observations in
the regression, but at the cost of the quality of data, especially if shares
analysed are not traded at the frequency applied. This problem exists
not only because of the lack of daily quotations, but might be a problem
even with shares traded on a daily basis. This is because the moments
of sale/purchase are not synchronised with index movements. For
example if on a particular day shares of a single company are for the
last time traded at 14.30,, the index value might be still changing till
16.00. This will influence the correlation between the share and the
index.
Frequency of the observations Estimated Beta* Daily 1.33 Weekly 1.38 Monthly 1.33 Quarterly 0.44 Yearly 0.77
58
3.2 Characteristics of data used for CAPM and APT tests
Since relatively short history of capital market in Poland and its rather
small significance in the first years of existence4, data used for the test
is from the period after the Russian crisis (August 1998). This is
because the Russian crisis significantly influenced Polish economy and
capital market. It would be impossible to build a representative sample
containing companies traded on the WSE before and after the crisis.
There are three most important reasons that can be summarised as
follows.
First, the short history of the capital market in Poland (12 years) causes
that there were not many companies quoted before the crisis. Till the
end of 1995 there were less than 30 companies listed and till the end of
1997 (half a year before the crisis) there were around 80 companies
listed.
Furthermore, many companies went bankrupt. Hence the sample
would include only a part of companies listed on the stock exchange
before the crisis.
Moreover, the crisis affected almost all companies causing either a
changed the primary industry or diversification. With no doubt the crisis
altered the correlation with the market index.
Although there are techniques that could deal with the crisis problem
(like introducing a binary dummy variable for the period after the crisis
had begun), because of the above mentioned reasons, this would
improve the quality of the study only slightly.
4 Even now the role of the capital market as a source of capital is rather small.
59
Therefore the first observations are from November 1998, as since
then market has recovered gaining stability, and end up with the year
2002.
As the estimation period is relatively short, to ensure necessary
number of degrees of freedom, observations can be either of weekly or
of a daily frequency. As previously described, one can give reasons in
favour of shorter or longer frequency. However, for the purpose of this
study the shorter possible frequency is chosen and therefore weekly
data are going to be used. The reason for this decision is to avoid noise
in data analyzed, which occurs when longer frequency applied.
3.2.1 Characteristics of data used for CAPM test
There are three types of primary variables used in the CAPM test:
1. Returns on Shares
2. Warsaw Market Index (WIG)
3. Risk Free Rate
3.2.1.1 Returns on Shares
From all 209 (http://www.gpw.com.pl/xml/spolki/listaspolek_baza.xml)
of firms quoted on the Warsaw Stock Exchange only these, quoted on
a daily basis are considered. There were three criteria for exclusion of
the companies in the sample:
1) The companys equity has not been quoted on the WSE
since November 1998. The aim of the selection process
is to create a sample that would be constant over time.
2) The company went bankrupt over the estimation period.
The same reasoning as in the previous point.
60
3) The companys equity had gaps in quotations over the
estimation period. Including shares quoted infrequently
would cause estimation bias.
4) The companys equity is quoted in foreign currency.
Including shares denominated in foreign currency would
impose the necessity of simultaneous adjustments to the
variable exchange rate. Excluding these kinds of shares
should not influence the result, as there are only two
companies on the market, which denominate their assets
in foreign currency.
A reason mentioned in point one was the most important for
exclusion from the sample. Over 70 firms currently quoted on the
WSE had to be excluded because of not being listed since
November 1998.
After precise selection process only returns of 100 companies are
examined in the study. The chosen shares are rather liquid since
during the period of two years particular shares had a few, single or
double observation gaps in a row. Therefore, the number of gaps in
quotations, which is a proxy for liquidity, is less then five percent of
all observations for the single company, which allows for conclusion
that shares in the sample can be assumed to be liquid. Thanks to
this selection process the sample is constant over the estimation
period.
The variable was constructed in the following way:
1
1
PrPrPr
−
−−=
t
ttt ice
iceiceR
61
3.2.1.2 Warsaw Market Index (WIG)
WIG is chosen as a proxy for the market portfolio. When accounted,
price changes of all companies quoted are considered. As discussed in
chapter 1.1.1 WIG is an index weighted with the market value of the
single firm, hence the impact of the particular company on its value is
directly proportional to the company capitalisation. Such a construction
of the index results in the fact that the change in its value reflects
changes in the total value of shares included in the portfolios of all
investors. From the beginning of the second quarter of the Next year
1994 the impact of one company on the index was limited to 10 percent
of the value of the all shares portfolio. changes, from the beginning of
the second quarter of the 1995, restricted the share of single industries
in the index to 30 percent and introduced the rule that the basis for
defining impact of the single company on the WIG is the number of the
shares introduced to active trade not just the registered number of
shares. The basis date for WIG is 16 April 1991 the date of the first
session of Warsaw Stock Exchange and its basis value was 1000. WIG
is accounted at the end of each session and published with an
accuracy of 0.1 basis point. The formula for the index calculation is
(www.gpw.com.pl):
1000*)(*)0(
)()(tKM
tMtWIG =
where:
M(t) - capitalisation of index portfolio on session t
M(0) - capitalisation of index portfolio on the base date (16 April
91)
K(t) - adjustment coefficient for session t
62
Thus, the variable was calculated according to the formula:
1
1
−
−−=
t
ttt WIG
WIGWIGdWIG
3.2.1.3 Risk Free Rate
A common proxy for the risk free rate are government bonds. In this
study five-year government bond is chosen. Although shorter, one-
month bonds are recommended as the best proxy for the risk free rate,
since in the practice the probability of their default is zero, it appeared
impossible to obtain their quotations on the secondary market. The
oldest, available and accessible five-year government bond was issued
on 2nd February of 2000 and is marked with a sign PS0205. To
calculate the variable the following formula was applied:
1
1
−
−−=
t
ttt RF
RFRFdRF
3.2.2 Variables used for APT test
Arbitrage Pricing Theory test is implemented and tested on the
following set of variables:
1. Stocks returns
2. WIG
3. Risk Free Rate
4. S&P 500
5. Polish Zloty (PLN) Exchange Rate (against USD).
6. International Price of Gold
63
Therefore the APT model in comparison to CAPM will include three
additional variables.
3.2.2.1 S&P 500
The impact of international markets on the value of assets in Poland
was considered. The American index was found the most influential, as
it has a strong impact on the world economy. The economic results of
events such as Great Depression and Terrorist Attack of September 11
were immediately transferred overseas and expectations of the daily
investors moods were reflected on the WSE. The S&P 500 Indexs
weekly quotations were chosen as a proxy of general condition of
American economy. It is calculated using a base-weighted aggregate
technique.
This method implies that the level of the index reflects the total market
value of all 500 component stocks relative to a particular base period
(www.cftech.com/BrainBank/FINANCE/SandPIndexCalc.html - 40k).
As the APT will be based on the stock returns thus variables in
absolute values should not be employed. Thus, the following variable
was constructed:
1
1
&&&
&−
−−=
t
ttt PS
PSPSPdS
3.2.2.2 Polish Zloty (PLN) Exchange Rate
Many Polish companies are indebted in foreign currencies
(http://www.nbp.pl/statystyka/czasowe/zadluz.html).
64
Thus, if the Polish Zloty exchange rate against USD depreciates, the
country overall indebtedness will increase. On the other hand, the PLN
depreciation is reflected in improvements of competitiveness of Polish
export goods, as lower relative price makes them more attractive for
foreign buyers. In this situation, Polish companies stock prices would
be likely to go up.
The following variable is analyzed:
1
1
−
−−=
t
ttt PLN
PLNPLNdPLN
Where tPLN is the Polish Zloty exchange rate denominated in US
Dollars at time t.
Moreover, due to the fact that exchange rate was employed, a variable
delivering information on price level, was introduced. Macroeconomics
theory states that exchange rate is a price of domestic currency for
foreign investors and that the inflation rate is its domestic price.
According to Purchasing Power Parity (PPP) theory there is a
relationship between the level of prices of domestic and foreign market.
PPP is an equilibrium condition in the market of tradable goods and
forms a basic building block for several models of the exchange rate
based on economic fundamentals. In essence, PPP states that it
should be possible to buy the same collection of goods and services in
any economy for the same amount of home currency.
There are two different interpretations of PPP: absolute and relative.
Empirical evidences usually support the relative version of PPP. In this
interpretation, the changes in exchange rates are related to changes in
the relative prices. Empirical findings confirm that exchange rate may
diverge from PPP, but will tend to return to PPP over time. Proponents
65
of the theory argue that PPP is a long run determinant of the exchange
rate, but it does not hold in a short one. Manzur research showed that
PPP could not be rejected over a long run. Furthermore, it is revealed
that PPP holds well for countries with high inflation relative to trading
partners. The inflation rate in Poland was high in comparison to its
trading partners (McKenzie, 2002).
Assuming that PPP holds in Poland, the variable presenting relative
exchange rate of the Polish Zloty delivers information on a long-term
price level as well.
3.2.2.3 International Price of Gold
The international gold price is introduced to the potential variable set
used for APT tests.
Carruths et al study of 1998 analysed the possibility that movements in
the real price of gold reflect uncertainty in financial and other traded
commodity markets. The research explored UK industrial and
commercial companies (ICC). The investigation of this issue indicates
that price of gold can enhance the explanation of investment spending
by the ICC sector. The size of the relationship between investment and
gold price movements is small but significant. Carruths et al (1998)
results obtained for UK market suggest important and significant effects
for both real profits and the real gold price in both the short and the
long-run.
Therefore, this research assumes that the gold price is an indirect
proxy for aggregated investment uncertainty. As a measure of
uncertainty, this variable has the advantage that it has a global
dimension and might therefore be considered as exogenous. Since
gold is usually regarded as a low-risk hedge, movements in its price
ought to reveal important information about market sentiment vis à vis
66
other asset returns. Furthermore, gold is highly inelastic in supply. Its
price movements are expected to reflect changes in demand. Thus, in
contrast to other price series, gold price movements are expected to
reflect more closely the demand-driven substitution for other assets
(Carruth et al, 1998).
The following variable is employed in the primary data set:
1
1
−
−−=
t
ttt GOLD
GOLDGOLDdGOLD
where tdGOLD is a price of gold measured in USD at week t.
67
CHAPTER IV
Empirical Test of CAPM
In this chapter test of the CAPM model on the Polish market will be
carried out. The methodology applied is based on the Black, Jensen
and Scholes (1972) with some adjustments described below. The test
is performed on the Microsoft econometric software: Eviews 4.1.
Ordinary Least Squares was applied as the estimation method.
4.1 Calculation procedure
In this section methodology of the CAPM test will be presented.
Further, reasons for grouping shares in portfolios will be discussed and
finally time-varying risk free rate and its implications will be commented
on.
4.1.1 CAPM test methodology The methodology of the test that is conducted in the next section is
primary based on a technique developed by Black, Jensen and
Scholes (1972). BJS technique is applied as it deals with the estimation
bias (discussed in the point 3.1.1) to some extent. This method is a
two-stage-regression test, which allows for testing the model not only
for a single company, but for the whole market as well. The two-stage-
regression BJS methodology (discussed in chapter 2.1.1.2) is still a
mainframe for the researchers. Despite the fact that there are many
adjustments to the BJS technique in this paper, they were justified by
68
the availability of data on the Polish capital market. There are many
differences resulting from this fact.
First, the estimation period for beta coefficient calculation is much
shorter then in BJS studies. In order to provide the same number of
observations their frequency was increased from monthly to weekly.
Hence, betas of the companies in the first sub-period will be estimated
using weekly returns over thirteen and a half months to assure the
number of observations at least equal the number used by BJS. The
first estimation sub-period starts on 12th of November 1998 and lasts to
30th of December 1999. This gives exactly 60 observations. BJS used
as well 60 observations to estimate the value of an individual company
betas, because they applied five year period with monthly frequency
(5x12).
Second, betas, unlike in the BJS studies, are estimated not on the
basis of CAPM, but the market model (zero-beta model). The
differences between the implied techniques do not affect significantly
the results of the test, as an individual companies betas are required
only to form the portfolios. Therefore, the exact values of CAPM betas
are not crucial at this stage, as they are only needed to rank the
companies with growing systematic risk. In this context the same
results would be obtained with CAPM model. Unlike in BJS studies,
CAPM model was not used to rank the shares. It is because there is no
publicly available information about risk free rate before year 2000. Due
to the application of the market model, the number of observations in
the CAPM testing phase could be significantly increased, as the
observations used after portfolios are formed derive from the year 2000
onwards, which date is determined by the availability of the government
bonds quotations.
Finally, because the history of the capital market in Poland is relatively
short, the test is conducted over the five-year period, which is
significantly shorter than in the BJS study (35 years).
69
Except for the mentioned differences in the estimation period, the
testing procedure is the same as in the BJS study and will be described
further in this chapter.
All shares used in the study are grouped increasingly with respect to
their beta coefficients and then divided into ten groups that form ten
portfolios. Therefore, the first portfolio is composed of ten shares
characterised by the lowest correlation coefficients with the market, the
second one is formed of companies which were in the second decile of
sorted shares. This is continued till the tenth portfolio, which includes
last ten shares with the highest value of beta coefficients.
The procedure of beta estimation, shares sorting and finally
classification of companies to the portfolios is repeated twice more.
This enables to increase the elasticity of the model, as the companies,
particularly in countries like Poland, fluctuate in their relation to the
market index. Unlike in BJS study, yearly updating of beta values
should significantly raise the quality of obtained results. In each of the
following two sub-periods, the estimation interval is lengthened up to
one and a half year, which gives 78 observations per sub-period.
On the basis of the estimated betas for individual shares in the sub-
period November 1998 December 1999, ten portfolios composed of
ten shares each will be formed. In the year 2000 for each portfolio
weekly rates of returns tpR , will be calculated, which will provide 52
observations. Next, in the sub-period from 8th June 1999 to 28th
December 2000, once again betas of all 100 previously chosen shares
are estimated. Using these values ten new portfolios will be created in
the same way as the previous portfolios and for the year 2001 their
weekly returns are to be worked out. This calculations will give
additional 52 observations. For the last time, this procedure is repeated
for the sub-period from 13th June 2000 27th December 2001, for
70
which the betas are estimated and the portfolios for the year 2002 are
formed. All in, 156 observations for each portfolio will be obtained.
These portfolios will be used for further analysis that is CAPM
regression in time series. The regression will be run on the weekly
market risk premiums (weekly returns on the WIG index decreased by
the weekly returns on the government bonds risk free rate) against
the weekly portfolio risk premiums (weekly returns on each portfolio
reduced by the weekly returns on the government bonds risk free
rate) within the estimation period January 2000 December 2002. The
following estimated equation will be applied:
ttFtMpptFtp RRRR εβα +−+=− )( ,,,,
Therefore, ten betas for each portfolio pβ are calculated, as well as the
average rate of returns of every single portfolio pR for the researched
period of three years. These results are used in the cross-sectional
regression:
pppR ηβγγ ++= 10
If model CAPM is true for the Polish market, the estimated coefficient
0γ should equal the risk free rate for standard version of CAPM or the
lowest borrowing rate in case of Black CAPM. The coefficient 1γ
defines the price of the market risk, hence its value should be
significantly positive.
It might seem that the sample of 10 observations used when estimating
above parameters is far too small and the number of thresholds might
indicate that the t-tests are not powerful. However, BJS conducted their
tests on the sample of the same size and FM (1973) used the sample
of 20 portfolios. This part of their study has never been criticised in later
papers.
71
Furthermore, cross-sectional analysis will be broadened by the two
additional factors taken into consideration by FM. The first one is the
residual variance pSe of the time-series regression, in which the weekly
returns on the portfolios are dependent variables. Residual variance is
a proxy for the non-systematic risk. The second additional variable is
squared value of portfolio beta p2β . This variable is added to check if
there is any non-linearity in the relationship between portfolio returns
premiums and market risk premium. The non-linearity is assumed to be
parabolic, as the most often expected form. Both supplementary
independent variables are added to the model separately, so there are
three types of final cross-sectional equations to be tested:
1. pppR ηβγγ ++= 10
2. ppppR ηβγβγγ +++= 2310
3. pppp SeR ηγβγγ +++= 310
Since variables p2β and pSe do not indicate that there is any
correlation, there is no need to include them in one model
simultaneously. The creation of separate models for any new variable
will improve the quality of obtained results, as for the same number of
degrees of freedom there are fewer parameters to be estimated.
The method of estimation used for all regressions is Ordinary Least
Squares. Both tested models (CAPM and APT) are linear and
according to Gauss-Markov theorem OLS estimators are best linear
unbiased estimators (Wooldridge 2000) and. There are, however, many
other techniques the most common of which is General Method of
Moments and models with autoregressive conditional
heteroskedasticity.
72
The methodology presented above is the best one that could be
applied on the Polish market. Despite relatively short estimation period,
the necessary number of degrees of freedom is provided for the tests
to assure the high power of the conducted tests. The market index is
weighted, lack of this idex characteristic, was the main virtue of the BJS
studies. Furthermore, unlike in BJS studies, there are two additional
factors used, which might give the first insight into any possibly
missing information.
4.1.2 Portfolio grouping
The main problem associated with empirical test of CAPM model is the
bias of estimated coefficients in the second stage of tests - sectional
regression. For thos reason shares are grouped in portfolios and the
test is not conducted on the variables represented by single
companies. The bias is always present when two-stage testing
procedure is implemented. It is a result of applying beta estimated in
the time-series regression, as an independent variable in cross-
sectional regression. Since the parameter always will be estimated with
an error, it is obvious that independent variable in the second, cross-
section regression will be burdened with an error. As a consequence,
the parameter in the cross-section regression determining the price of
the market risk will be estimated with a bias. This problem can be
presented more formally (Larsson 2002):
Time-series regression titMiiti RbR ,,, εα ++=
Cross-sectional regression PPiR ηβγγ ++= 10
Where bi is the estimator of beta and it takes the form of:
iiib υβ +=
73
Where iυ is estimation error with zero mean and zero covariance with
iη and iβ . This estimated coefficient becomes an independent variable
in the second equation. Hence, the estimated parameter 1γ takes the
following form:
21 )(),cov(
i
ii
bbR
σγ = ,
)(),cov(
21
1ii
iiii
νβσνβηβγγ
+++
= ,
22
2
11ii
i
υβ
β
σσσ
γγ+
= .
The value of the fraction is lower, hence 1γ is a biased downward. On
the other hand the estimator 0γ will be biased upward, hence higher
than the true parameter of the population.
Using an estimated beta instead of its true value in the cross-sectional
regression, bias downward the estimated parameter 1γ . Although
grouping shares in portfolios does not eliminate completely the
problem, it does limit its significance. This bias is present in both stages
of the CAPM tests. In the subsequent years the same problem is faced
by FM and FF (Pasquariello 1999).
4.1.3 Risk free rate variability
In the conducted test, unlike in the standard CAPM model, the risk free
rate does not have a constant value. This approach makes the model
much more real, as one of the feature of the Polish economy is the
time-varying interest rate, even in short term. This variability is a result
74
of a continuous and gradual decrease in the rate of inflation from 252.2
percent in 1990 down to projected 0.6 percent in the year 20035. The
other factor influencing the level of the risk free rate is the monetary
policy. Its guidelines are to cut down the interest rate for the last few
years in order to stimulate the economic growth. As the risk free rate
becomes a new independent variable, the model changes its form
from:
tFtMpFtp RRRR εβ +−+= )( ,,
to
ttFtMpptFtp RRRR εβα +−+=− )( ,,,,
where the constant RF becomes a variable RF,t, which when moved on
the other side of the equation, creates a model where the dependent
variable is the risk premium for the portfolio/shares tFtp RR ,, − and the
independent variable premium for the market risk tFtM RR ,, − . As
mentioned before, this technique is adapted only in the time-series
regression of the portfolios.
4.2 Empirical test of the CAPM
In this section a detailed test of CAPM model is presented. The first
phase of the test is time-series regression. Estimated beta parameters
of all ten portfolios are the result of the first phase. These parameters
become independent variables in the second stage, which is cross-
sectional regression of averaged returns against betas. The outcome of
this regression will allow for assessment if the model is statistically
significant on the Polish market.
5 http://bossa.pl/rynki/inflacja.html
75
4.2.1 Time-series regression
On the basis of estimated betas for 100 companies ten portfolios were
formed for that the average weekly returns were calculated6 and then
betas of each portfolio were computed7. Results summary is presented
in the table below:
Table No 4.1: Beta and their t-statistics values for all portfolios Portfolio Beta t-statistics 1 0.304299 4.123798 2 0.326055 4.015498 3 0.473718 7.621019 4 0.278166 3.386901 5 0.3174 4.461146 6 0.230514 3.972888 7 0.569277 9.358641 8 0.574836 6.589265 9 0.599535 9.005915 10 1.074353 12.02110 Source: Own calculations
The results shown in Table 4.1 indicate that all betas are statistically
significant and different from zero. High values of the t-statistics allow
for making this conclusion even at the significance level of α=0.01. On
the other hand, from the further analysis of the results, unlike expected,
values of the betas do not increase with portfolio number. In spite of the
applied technique of yearly portfolio sorting, betas of the shares
forming the portfolios must indicate high variability and instability.
Since in a simple linear regression t-statistics of the estimated
parameter describes the quality of the model, it can be concluded that it
is high in all of the cases. However, doubts are raised when analysing
the coefficients of determination of each regression presented in table
4.2:
6 Appendix No 1 7 results of all time-series regressions of average weekly returns of the portfolios against weekly returns on the WIG index are presented in appendix No 2
76
Table No 4.2: The coefficients of determination R2 of the regression
ttFtMpptFtp RRRR εβα +−+=− )( ,,,,
Portfolio R2
1 0.101827 2 0.097061 3 0.279123 4 0.071041 5 0.117137 6 0.095207 7 0.368645 8 0.306541 9 0.350949 10 0.609768
Source: Own calculations
Low values of R2 for the portfolios 1, 2 and 4 - 6 suggest that there
might be some other variables describing the premium tFtp RR ,, −
better, than premium for the market risk tFtM RR ,, − . Results for the
portfolios 5 and 7-10 are satisfactory, as the financial econometric
practice allows for recognising the CAPM model as a good one,
because it makes possible to explain the arbitrary determined 30
percent of the financial reality. The 10th portfolio has a particularly good
outcome with the coefficient of determination of above 60 percent. The
possible reasons of such a good result are discussed on the basis of
the tests presented below.
Before the results of the time-series regression are used in the next
stage of the study, the tests on the residuals of estimated models
should be run in order to determine the quality of estimated
parameters. Tests on the normal distribution, heteroskedasticity and
autocorrelation are conducted with the level of significance α=0.058.
Table 4.3 contains the summary of the results:
8 Detailed results of all tests in Appendix No 3
77
Table No 4.3: Summary of results of the tests on the normal distribution, heteroskedasticity and autocorrelation of the residuals form regression
ttFtMpptFtp RRRR εβα +−+=− )( ,,,,
Portfolio Normal Distribution Autocorrelation Heteroskedasticity
1 Non-existing Non-existing Non-existing 2 Non-existing Non-existing Non-existing 3 Non-existing Non-existing Non-existing 4 Non-existing Existing AR(2) Non-existing 5 Non-existing Existing AR(1) Non-existing 6 Non-existing Non-existing Non-existing 7 Non-existing Non-existing Non-existing 8 Non-existing Non-existing Existing - 2
,, )( tFtM RR −9 Non-existing Non-existing Non-existing 10 Non-existing Existing - AR(1) Existing - 2
,, )( tFtM RR −Source: Own calculations
The non-existence of normal distribution leads to the serious
consequences, because the value of t-statistics can not be considered
true. This conclusion results from the fact that t-test is based on the
normal distribution. However, if there are around 25-30 observations for
each estimated parameter, Central Limit Theorem can be implied and
conclude that the distribution of residuals is aiming to have normal
distribution. As time-series regressions are run on 152 observations
and there are two parameters estimated pα and pβ , the above theory
can be applied. At this point it should be added that most of the
financial data does not have features of normal distribution and daily
returns even if the number of observations tend to infinity, will never be
approximated to normal distribution (Fama 1976).
The existence of autocorrelation is associated with serious
consequences for the quality of the estimated model. Correlation of the
residuals leads to the higher values of t-statistics, as the variance of the
estimated parameters should be higher. As a result, the estimated
parameters will not be efficient. Autocorrelation is a cause of the higher
coefficient of determination value as well, an example of which might
78
be R2 for the portfolio number ten. Test for autocorrelation was
conducted with four lags because four weekly returns create
approximately one month period. In seven out of ten tested models
autocorrelation is not a problem and although it exists in portfolios four,
five and ten, it is very low. To solve the problem of autocorrelation
additional terms AR (1) or AR (2) are added to the estimated equations.
The test for the heteroskedasticity of residuals is the last one. Its
presence does not allow for trusting the t-statistics, because time-
varying variance can lead to contrary conclusions on the significance of
the estimated parameters. Heteroskedasticity in the above models was
virtually not existing, except for the models for portfolios eighth and ten.
These results are not constant with expectations, as the most of
financial data time-series are characterised by variable variance. It was
impossible to remove heteroskedasticity in order to achieve efficient
estimators of the parameters in the models for portfolio eight and ten.
Therefore, the Newey-West technique is applied to OLS estimation.
In all models, beta was a statistically significant variable and due to the
sufficient number of observations, the tests were of a high statistical
power. However, the coefficients of determination are relatively low.
This fact allows to assume that there are some missing independent
variables.
4.2.2 Cross-sectional regression
The subsequent phase of the CAPM test is cross-sectional regression.
It is based on results obtained in the previous stage. Both standard
errors of regression and beta coefficients in the table four include
changes caused by applying techniques dealing with autocorrelation
and heteroskedasticity9. This data will be used to estimate the final
9 Appendix No 4
79
model and test the hypothesis, which will provide arguments rejecting
or supporting the CAPM theory.
Table No 4.4: Data used in cross-sectional regression
Portfolio Average pR Standard error of regression Beta 1 -0.00263 0.02904 0.304299 2 -0.00531 0.031955 0.326055 3 0.001207 0.024462 0.473718 4 -0.00033 0.031476 0.319304 5 0.000165 0.026856 0.271354 6 0.002446 0,022834 0.230514 7 -0.00596 0.023939 0.569277 8 -0.00215 0.027782 0.574836 9 -0.00094 0.026199 0.599535 10 -0.00405 0.026474 1.092972
Source: Own calculations
The standard version of the CAPM model is the first to be tested:
pppR ηβγγ ++= 10
where according to the classical tests the value of the parameter 0γ
should equal the average weekly risk free rate or higher in case of
Black CAPM and 1γ should be statistically bigger then zero. The results
are presented in Table 4.5: Table No 4.5: Results of the cross-sectional regression: pppR ηβγγ ++= 10
Dependent Variable: SREDNIA Method: Least Squares Date: 04/28/03 Time: 11:27 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C 0.000429 0.001851 0.231727 0.8226 BETA -0.004588 0.003461 -1.325568 0.2216 R-squared 0.180087 Mean dependent var -0.001756 Adjusted R-squared 0.077598 S.D. dependent var 0.002775 S.E. of regression 0.002665 Akaike info criterion -8.840017 Sum squared resid 5.68E-05 Schwarz criterion -8.779500 Log likelihood 46.20008 F-statistic 1.757130 Durbin-Watson stat 2.321095 Prob(F-statistic) 0.221584
Source: Own calculations
80
Ho: 0γ =0 the parameter 0γ is statistically insignificant.
H1: 0γ ≠0 the parameter 0γ is statistically significant.
At the level of significance α=0.05 the P-value of the t-statistics equals
0.8226 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence the parameter 0γ is statistically insignificant.
Ho: 1γ =0 variable pβ does not affect pR
H1: 1γ ≠0 variable pβ does affect pR
At the level of significance α=0.05 the P-value of the t-statistics equals
0.2216 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence variable pβ does not affect average portfolio returns.
The results of classical tests indicate that the CAPM theory does not
hold on the Polish market. The conclusion does not change even if
testing procedure developed by Kozicki and Shen (2002) is applied:
0)(: 100 =−+ mrEHA γγ , the CAPM model holds
0)(: 101 ≠−+ mrEHA γγ , the CAPM model does not hold
20.4651560.00094/
-0.00157)(004588.0000429.0/)(
)( 10 =−−=
−+=≡
Trr
ttm
mA
ACAPM σ
γγγ
At the level of significance α=0.05 the t-statistics equals 20.465 and
leads to a conclusion that there is sufficient evidence to reject Ho in
favour of 1HA . The CAPM model does not hold.
81
Graphical results of the regression presented in diagram 4.1 and 4.2
support the statistics: Diagram No 4.1: Graphical results of the regression of the average returns on portfolios against their betas as independent variable.
Source: Own calculations
Diagram No 4.2: Actual, residual and fitted graph from the model
pppR ηβγγ ++= 10
-.006
-.004
-.002
.000
.002
.004
-.008
-.006
-.004
-.002
.000
.002
.004
1 2 3 4 5 6 7 8 9 10
Residual Actual Fitted
Source: Own calculations
In order to be assured of the correctness of obtained results tests of the
residual values are conducted:
-.008
-.006
-.004
-.002
.000
.002
.004
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
BETA
SR
ED
NIA
S R E D N IA vs . B E TA
82
1. Distribution of the residuals graphically and statistically is
presented in Diagram 4.3: Diagram No 4.3: Histogram of residual values from the model: pppR ηβγγ ++= 10
Source: Own calculations
Ho: pη have normal distribution
H1: pη have not normal distribution
At the level of significance α=0.05 the P-value of the χ2 statistics
equals 0.670771 and leads to a conclusion that there is no sufficient
evidence to reject Ho, hence the residuals pη have normal distribution.
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
-0 .004 -0 .002 0.000 0.002 0.004
S eries : R es idualsS am ple 1 10O bservations 10
M ean -5 .64E -19M edian 0 .000505M axim um 0 .003063M inim um -0 .004246S td . D ev. 0 .002516S kewness -0 .553095K urtos is 2 .167464
J arque-B era 0 .798655P robab ility 0 .670771
83
2. Heteroskedasticity The outcome of the test for the
heteroskedasticity is presented in Table 4.6 Table No 4.6: Results of the test on the heteroskedasticity of the residuals from the model pppR ηβγγ ++= 10
White Heteroskedasticity Test: F-statistic 0.372913 Probability 0.701627 Obs*R-squared 0.962875 Probability 0.617895 Test Equation:Dependent Variable: RESID^2 Method: Least Squares Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C 4.62E-06 1.18E-05 0.390724 0.7076 BETA 1.04E-05 4.35E-05 0.239174 0.8178 BETA^2 -1.37E-05 3.33E-05 -0.410860 0.6935 R-squared 0.096288 Mean dependent var 5.70E-06 Adjusted R-squared -0.161916 S.D. dependent var 6.49E-06 S.E. of regression 7.00E-06 Akaike info criterion -20.65941 Sum squared resid 3.43E-10 Schwarz criterion -20.56863 Log likelihood 106.2970 F-statistic 0.372913 Durbin-Watson stat 1.737492 Prob(F-statistic) 0.701627
Source: Own calculations
Ho: the variance of pη is constant
H1: the variance of pη is not constant
At the level of significance α=0.05 the P-value of the χ2 statistics
equals 0.617895 and leads to a conclusion that there is no sufficient
evidence to reject Ho, hence the variance of residuals pη is constant.
84
3. Autocorrelation Outcome of the test for the autocorrelation of
the residuals is presented in Table 4.7:
Table No 4.7: Results of the test on the autocorrelation of residuals in model
pppR ηβγγ ++= 10
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.256624 Probability 0.628012 Obs*R-squared 0.353641 Probability 0.552059 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 04/28/03 Time: 11:32 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C -3.76E-05 0.001945 -0.019332 0.9851 BETA 5.79E-05 0.003636 0.015925 0.9877 RESID(-1) -0.188663 0.372425 -0.506581 0.6280 R-squared 0.035364 Mean dependent var 3.79E-19 Adjusted R-squared -0.240246 S.D. dependent var 0.002513 S.E. of regression 0.002799 Akaike info criterion -8.676021 Sum squared resid 5.48E-05 Schwarz criterion -8.585246 Log likelihood 46.38011 F-statistic 0.128312 Durbin-Watson stat 2.034878 Prob(F-statistic) 0.881601
Source: Own calculations
Ho: errors are independent
H1: errors are autocorrelated
At the level of significance α=0,05 the P-value of the Lagrange
Multiplier with one lag statistics equals 0.552 and leads to a conclusion
that there is no sufficient evidence to reject Ho, hence the errors are
independent As the regression is built on the cross-section data, the
achieved results are consistent with expectations.
The hypothesis concerning the statistical significance of the variables
p2β and pSe will be tested by adopting the testing procedure for
omitted variables.
85
Table 4.8 contains the results:
Table No 4.8: Results of the test for omitted variable p2β
Omitted Variables: BETA^2
F-statistic 0.222555 Probability 0.651451 Log likelihood ratio 0.312986 Probability 0.575853 Test Equation: Dependent Variable: SREDNIA Method: Least Squares Date: 04/28/03 Time: 11:36 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C 0.002419 0.004647 0.520605 0.6187 BETA -0.012390 0.016934 -0.731650 0.4881 BETA^2 0.006030 0.012781 0.471758 0.6515 R-squared 0.205351 Mean dependent var -0.001756 Adjusted R-squared -0.021691 S.D. dependent var 0.002775 S.E. of regression 0.002805 Akaike info criterion -8.671315 Sum squared resid 5.51E-05 Schwarz criterion -8.580540 Log likelihood 46.35658 F-statistic 0.904463 Durbin-Watson stat 2.362831 Prob(F-statistic) 0.447314
Source: Own calculations
H0: omitted variable p2β is statistically insignificant
H1: omitted variable p2β is statistically significant
At the level of significance α=0.05 the P-value of the F-statistics equals
0.651451and leads to the conclusion that there is no sufficient
evidence to reject Ho, hence the omitted variable p2β is statistically
insignificant. The hypothesis of the nonlinear relationship between
average returns and betas is therefore rejected.
Applying the same procedure, the hypothesis of the statistical
significance of the second added variable - residual variance is rejected
as well, which is presented in the Table 4.9:
86
Table No 4.9 Results of the test for omitted variable pSe
Omitted Variables: SE
F-statistic 1.940882 Probability 0.206213 Log likelihood ratio 2.447241 Probability 0.117732 Test Equation: Dependent Variable: SREDNIA Method: Least Squares Date: 04/28/03 Time: 11:37 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C 0.011490 0.008130 1.413221 0.2005 BETA -0.005493 0.003338 -1.645715 0.1438 SE -0.392221 0.281534 -1.393155 0.2062 R-squared 0.358073 Mean dependent var -0.001756 Adjusted R-squared 0.174665 S.D. dependent var 0.002775 S.E. of regression 0.002521 Akaike info criterion -8.884741 Sum squared resid 4.45E-05 Schwarz criterion -8.793965 Log likelihood 47.42370 F-statistic 1.952334 Durbin-Watson stat 2.036590 Prob(F-statistic) 0.211933
Source: Own calculations
H0: omitted variable pSe is statistically insignificant
H1: omitted variable pSe is statistically significant
At the level of significance α=0.05 the P-value of the F-statistics equals
0.206213 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence the omitted variable pSe is statistically insignificant.
Thus, the residual variance has no impact on the value of the expected
return of the portfolio.
Results of the CAPM test conducted on the Polish market suggest that
the model does not hold in the Polish economy. All presented variables
appeared to be statistically insignificant. The overall conclusion is that
there is no statistically significant relation between expected average
returns and beta as a market risk measure of the individual portfolios.
87
CHAPTER V APT Estimation and Tests
Due to the fact that APT theory defines only asset valuation structure
without defining economic or company characteristics that influence
rates of returns, APT test presented in this chapter is only one of many
versions that can be developed for the Polish market.
5.1 Methodology Methodology that is going to be applied in the empirical research is
presented briefly in this section. As there are different methods of factor
estimation, the choice of factor analysis in comparison to portfolio and
macroeconomic factors theories is discussed.
5.1.1 Estimation procedure In order to implement and test the APT model (assuming that firm
specific risk is diversified) the equation presented in the second chapter
needs to be estimated. That is:
=
+=K
jkikFi RR
1
λγ
The ikγ can be found with the use of the following formula:
=
++=K
kikikii IaR
1εγ
88
The last equation shows that ikγ can be calculated either after or
simultaneously with factors kI estimation. Therefore, the first step of
APT model estimation is to specify factors that influence returns
assuming that the theoretical assumptions are met.
After calculation of factors their coefficients should be estimated for
each individual security using regression. Instead of returns on
securities they are to be regressed against portfolios. Assuming that
the Law of Large Numbers holds, this would allow for diversification of
the firm specific risk.
The last step will be the estimation of the risk premiums for all factors.
It would be done applying the cross-sectional regression between the
time-series of the returns on portfolios. Finally, the estimated model is
going to be statistically verified with Chow and Lagrange Multiplier
tests.
The next sections of this chapter present the estimation procedure in
details introducing the empirical results that were obtained.
5.1.2 Methods of testing and Estimation
As mentioned in chapter two there are three general approaches to
factor estimation:
1. macroeconomic factors
2. portfolios based on company characteristics
3. statistical choice of factor proxies such as factor analysis
The first technique defines factors on the basis of macroeconomic
theory. This approach chooses arbitrary several economic variables as
89
proxies of factors explaining asset returns. The most important
advantage of this approach is that it names factors. This characteristic
is quite useful for corporate managers, as it presents how their
company returns relate directly to individual macroeconomic factors.
However, this method is strongly disadvantaged. It is also difficult to
identify the unexpected changes in variables that are factor proxies
(Grinblatt and Titman, 1998). Furthermore, this technique might result
in neglecting of some potentially important non-quantifiable factors.
Taking into account that Polish economy might be strongly influenced
by political uncertainty, factors defined arbitrary and limited only to a
few variables might not catch the political risk at all.
The portfolio method of factor estimation includes intuitional choice of
variables as well. According to this technique, factors are estimated on
the basis of firm characteristics. This method assumes that risk
premium ( kλ ) associated with the firm characteristic (for example firm
size, dividends or earnings ratio etc.) represents compensation for that
specific type of factor risk and therefore for portfolios constructed of
assets described by the characteristics (Elton and Gruber 1998). Elton
and Gruber (1998) suggest that this approach will give better factor
proxies than other methods if covariances change over time.
Furthermore, this method might overperform macroeconomic factor
method because these factors are able to catch unpredicted
macroeconomic changes, as they are based on stock returns that are
unpredictable as well. The only disadvantage of this method is that if
there is no link between return premiums and factor sensitivities, the
approach picks out mispriced portfolios (Grinblatt and Titman, 1998).
This drawback might significantly influence the asset returns on the
Polish market. The empirical basis of empirical studies on Polish
market is extremely poor as discussed in Chapter II. Thus, it might be
very difficult to look intuitively for firm characteristics that are likely to
generate factors in Poland.
90
Finally, factors might be estimated statistically using for example factor
analysis. If covariances between stocks do not change over time, this
method generates exactly desired factors. However, this technique
does not name factors and therefore is less useful for corporate
managers. Even though, when testing the theory on a whole capital
market, factors interpretation is not so important, as the technique will
show whether returns are generated by some factors and therefore
answer the question if APT holds in Poland. Furthermore, unless there
are hidden factors that are created on the basis of a few
macroeconomic variables, the methodology will estimate factors based
on individual macroeconomic variables giving similar model as
macroeconomic factors technique. Moreover, macroeconomic variables
are likely to be correlated with each other. If macroeconomic variables
are likely to autocorrelate, factor analysis could create statistically
significant model. Political risk that is of crucial importance in Poland
might be for instance the hidden variable. Therefore, factor analysis
might be a good technique of the factors estimation.
The portfolio method might be a good alternative but it seems to be too
intuitive. If there were more studies on that topic, they might give an
idea of company characteristics that should be included in the model.
It might be inappropriate to examine the same factors as those, which
were analysed when implementing APT in the United States. For
example dividend payout ratio is not likely to be a significant factor in
Poland. Polish investors do not pay attention to the fact that company
pays dividends or not. The number of companies paying dividends is
substantially decreasing as it is presented in Table 5.1.
91
Table 5.1 Companies paying dividends in Poland
Year Number of companies paying dividends
1999 61
2000 49
2001 38
2002 14
Source: own calculations
The reason for that situation might be that dividends are taxied at 15
percent when other capital market gains are tax free. Furthermore,
Polish companies pay the dividends rather accidentally that is usually
less frequent than once a year. Therefore, it is hard to state if they have
a dividend policy at all.
Moreover, there were no studies analysing the impact of earnings etc.
on stock prices. Thus it would be difficult to assume that this company
characteristic may influence the stock prices or not. As there were no
empirical researches that might give an idea on which characteristics
can be factor proxies, the portfolio method will not be applied.
However, after better understanding of processes and relationships on
the WSE, this method could be implemented.
Taking into account advantages and disadvantages of the three
techniques discussed above, factor analysis is going to be applied as
the most appropriate for Polish capital market.
5.1.3 Factor Analysis overview
Factor Analysis was presented for this time in Spearmans article of
1904 (Rubaszek, 2002). He carried out a survey on unobservable
factors influencing tests results of the high school students.
92
This statistical technique is used to uncover the latent structure
(dimensions) of a set of variables. It reduces attribute space from a
larger number of variables to a smaller number of factors and as such
is a "non-dependent" procedure. Moreover, factor analysis generates
observed raw indicator variables and the factors or latent variables
which explain the variance in these variables as good as possible.
The kI are called here factors and ikγ factor loadings (for example
Rószkiewicz, 1998).
5.1.3.1 Factor Analysis formal model
This section is based on papers of Rószkiewicz (1998), Rubaszek
(2002) and Electronic Textbook StatSoft (2003).
Assuming that there are N observable variables in the sample, the
observation matrix is as follows:
NTNTTT
N
N
xxx
xxxxxx
X
×
=
...............
...
...
21
22212
12111
where itx is a value of i-th variable observed at time t = 1,2,,T.
Assume that vector TtiX )( (T means here transposed matrix) is
distributed in N- dimensions, where ( )Ω;µN .
Factor analysis assumes that all variables are a function of common
factors and idiosyncratic factor.
There are three requirements that need to be met in factor analysis.
( ) iki IIIfX ε+= ,...,, 21
93
1. Uncorrelated thus orthogonal factors:
( ) 0,cov =lj II
where lj ≠ .
2. Uncorrelated unique variance:
( ) 0,cov =ji εε
where ji ≠
3. Factors are not correlated with the unique variance:
( ) 0,cov =ijI ε .
The most popular approach that was employed in this paper, assumes
that there is a lineal relationship between variables and factors.
Therefore, the factor analysis model can be formulated as follows:
ikikiiit IIIX εγγγ ++++= ....2211
If a model meets the requirements presented above, the following
equation is true:
=
Ψ+=Ψ++++=k
jijikiii
1
2222
21
2 ... γγγγσ
jkikjiij γγγγσ ++= ...11
where:
94
2iσ - the variance of variable iX
ijσ - the covariance of variables iX and jX
=
k
jij
1
2γ - common variance
Ψ - unique variance
The greater the impact of common variance in comparison to unique
variance is, the better the factor analysis.
5.2 Factors estimation- empirical results The empirical findings and detailed calculation procedure are
presented in this section.
5.2.1 Variables analyzed
In order to perform factor analysis certain requirements should be met
concerning the number of variables and cases examined and so called
sampling adequacy.
5.2.1.2 Suboptimization
The rule that "The more variables, the better factor analysis" may not
be appropriate, if there is a chance of suboptimal factor solutions
("bloated factors"). Too many too similar items will camouflage true
basic factors, leading to suboptimal solutions. To avoid
suboptimization, it should be started with a small set of the soundest
items that represent the range of the factors. Data employed in this
research are expected to convey the information content that is not too
analogous. Therefore, it is assumed that the data selection, applied in
this study, will eliminate the feasibility of suboptimization.
95
5.2.1.2 Number of cases
The selection of variables is based not only on their economic
meaning. There is a required number of cases that need to be
examined in factor analysis. However, methodologists differ in this
issue (Garson):
1. Rule of 10. There should be at least 10 cases for each item in
the instrument that is being used.
2. Rule of 100: The number of subjects should be the larger of 5
times the number of variables, or 100. Even more subjects are
needed when communalities are low and/or few variables load
on each factor.
3. Rule of 150: At least 150 - 300 cases, more toward the 150 end
when there are a few highly correlated variables.
4. Significance rule. There should be 51 more cases than the
number of variables, to support chi-square testing
152 cases for each of five variables were examined in this study. This
is in accordance to almost all rules that were suggested by
methodologists. Therefore, the sample employed is assumed to be
large enough to deliver interpretable factors.
5.2.1.3 Sampling adequacy
There are statistical requirements related to variables used in factor
analysis (Rószkiewicz, 2002). Sampling adequacy predicts if data is
likely to factor well on the basis of correlation and partial correlation. In
order to measure sampling adequacy the Kaiser-Meyer-Olkin (KMO)
statistics was employed. KMO can be applied, to assess which
variables need to be excluded from the model because they are too
multicollinear.
96
There is a KMO statistic for each individual variable as well. KMO
overall statistic is a sum of individual statistics. The value of KMO
varies from 0 to 1.0. There is a rule that the KMO overall should
amount to 0.5 or bigger to proceed with factor analysis. If it does not,
the indicator variables with the lowest individual KMO statistic values
should be excluded from the sample, until KMO overall rises above 0.5.
In order to compute KMO overall the following formula is used:
≠ ≠≠ ≠
≠ ≠
+=
ij jiij
ij jiij
ij jiij
ar
rKMO 22
2
where ijr is an element of the correlation matrix R and ija is a partial
correlation coefficient between variable i and j estimated when others
variables do not influence the results of the correlation.
The numerator is the sum of squared correlations of all variables in the
analysis (except the 1.0 that implies self-correlations of variables). The
denominator is the same sum plus the sum of squared partial
correlations of each variable with each variable. According to the theory
the partial correlation should not be very large if separate factors are
anticipated to emerge from factor analysis.
The variable set that meets the KMO test was computed using SPSS
for Windows software. First the anti-image matrices were computed
(see table one Appendix No 5). They contain the negative partial
covariances and correlations that can give an indication of correlations
that are not due to the common factors. The diagonal elements on the
97
Anti-image correlation matrix are the KMO individual statistics for each
variable.
The overall KMO amounts to 0.4909951087217. Therefore, factor
analysis should not be conducted on this sample. The KMO statistics of
individual variable are presented below (Diagram 5.1.)
Diagram: 5.1.KMO statistics five variables
Individual KMO Statistics
0,44
0,46
0,48
0,5
0,52
0,54
WIG risk free GOLD exchangerate
S&P
Variables
KM
O e
stim
ates
Source: own calculations
To improve the overall KMO the return on S&P500 was excluded from
the sample as the variable with the lowest KMO value. Anti-image
matrices are presented in appendix No 5 (table 2).
After exclusion of this variable the estimation of the statistic increased
to the level of 0.5411158754155. Thus, the sample is assessed to be
good enough to perform factor analysis. The final sample employed in
the study consists of the following variables: WIG, RF, GOLD and EX.
98
5.2.2 Number of factors
There are three best known strategies of the statistical selection of the
right number of factors such as variance explained, Kaiser rule of
eigenvalues and Cattell criterion.
First strategy uses the first n factors that explain 80 percent (or some
other arbitrary percentage) of the variance. This rule of thumb is a
middle ground between the two below ones (Rószkiewicz, 1998).
The second one uses only the factors whose eigenvalues10 are at or
above the mean eigenvalue (the Kaiser rule). This is the strictest rule of
the three and may cause using too few factors.
Finally, the third one applies a scree plot. It is a plot in which the x axis
represents the factors arranged in a way that they would be
descending eigenvalue and the y axis is the value of the eigenvalues.
This plot will demonstrate a sharp decrease leveling off to a flat tail as
each consecutive component's eigenvalue explains less and less of the
variances. The Cattell rule is to choose all factors prior to where the
plot levels off. Nevertheless, this rule is very arbitrary. Picking the
"elbow" can be prejudiced because of the fact that the curve has
multiple elbows or it is a smooth curve. Therefore, it is feasible that
the researcher may be tempted to set the cut-off at the number of
factors according to the more desirable outcomes. Furthermore, the
criterion tends to result in more factors than the Kaiser criterion.
In practice, an additional important aspect is the extent to which a
solution is interpretable. Therefore, generally several solutions with
more or less factors are examined. Then the best one that makes the
model most logical is analyzed.
10 Eigenvalues are a variances extracted by the factors ( Rószkiewicz, 1998).
99
5.2.2.1 Kaiser rule
First, Kaiser criterion was applied. Estimated eigenvalues are
presented in Table 5.1:
Table 5.1: Eigenvalues for different number of factors
Factor Eigenvalue 1 1.288 2 1.034 3 0.882 4 0.796
Source: own calculations
The eigenvalues of two factors were bigger than average. Thus, two
factors were extracted. However, the components of these factors
imply that the first factor is correlated strongly with the risk free rate.
The second factor is correlated strongly with WIG. These findings are
presented in Table 5.2:
Table 5.2: Factor Loadings matrix for two factors extracted
Variable F1 F2 WIG -0.156 0.872 RF 0.720 0.112 GOLD 0.654 -0.282 EX 0.563 0.425
Source: own calculations
Therefore, APT created in this way would be very similar to CAPM.
Since Capital Asset Pricing Model was tested in previous chapter, there
is no point in creating a similar model.
100
5.2.2.2 Cattell rule
The scree plot method (Diagram 5.2 ) brings similar results. However,
this criterion allows for arbitrary choice of two or three factors as the
Cattell rule indicates. The choice of two factors would create model
very similar to CAPM, thus three factors would be selected.
Diagram 5.2 Scree Plot
Plot
S c r e e P lo t
C o m p o n e n t N u m b e r
4321
Eig
env
alu
e
1 , 4
1 , 3
1 , 2
1 , 1
1 , 0
, 9
, 8
, 7
Source: own calculations
If three factors are extracted, the curve is still descending and not flat.
Therefore, the extraction of three factors is in accordance with Cattell
rule.
5.2.2.3 Variance criterion
The criterion of variance explained by the individual factors indicates
that three factors explain over 80 percent of the cumulative variance.
101
Table 5.3 presents the percentage of variance explained by each factor
and the cumulative percentage of variance explained. For example it
states that two factors explain over 50 percent of the variance and
three explain 80.1 percent.
Table 5.3. Eigenvalues and the total variance three factors extracted
Factor % of variance explained Cumulative % 1 32.20 32.20 2 25.84 58.05 3 22.04 80.10 4 19.89 100.00 Source: own calculations
5.2.3 Factoring methods
There are different methods of extracting factors from a dataset. The
most popular are Maximum Likelihood, Principal Factor Analysis and
Principal Component Analysis.
5.2.3.1 Maximum Likelihood Factoring The maximum likelihood estimators used to be applied in studies that
tested APT. This methodology was employed for example in studies
carried out by Roll and Ross (1980) and Rubaszek (2002).
However, the maximum likelihood method should not be applied in this
paper. If it is employed, the model would fail. The ratio of variables to
estimated factors is of the value that makes the model insignificant.
Bartlett Statistic is usually used to check if the estimated factors are
good enough in explaining the variance-covariance matrix. The
Bartletts test uses the 2χ statistics with the number of the degrees of
freedom:
v = ½ [(N-K)(N-K)-N-K]
102
The number of thresholds would be negative in this research, if the
maximum likelihood methodology applied. As there are four analyzed
variables and three common factors were decided to be extracted. If N
= 4 and K = 3, there is a negative number of degrees of freedom that is
v = -3.
Due to the negative number of degrees of freedom, the results of factor
analysis should be interpreted with caution. Therefore, it was decided
to proceed with a different factoring method.
For one or two factors the model would be significant but it would be
similar to CAPM as the two first factors depend strongly on market
index and risk free rate.
5.2.3.2 PCA versus PFA The Principal Components Analysis was introduced by Hoteling
(Rószkiewicz, 1998). It assumes that k-dimensional variable may be
transformed into p-dimensional one where p is not bigger than k. Due
to that transformation a new selection of variables is created. The
Principal Component Analysis reflects both, common and unique
variance of the variables, and may be seen as a variance-focused
approach that seeks to reproduce both, the total variable variance with
all components and the correlations. This approach seeks such a linear
combination of variables that the maximum variance is extracted from
the variables. The variance is then removed and a second linear
combination, which explains the maximum proportion of the remaining
variance, is looked for.
Principal Factor Analysis also called principal axis factoring, PAF, alias
common factor analysis, Principal Factor Analysis is a form of factor
analysis which seeks the smallest number of factors. It can account for
103
the common variance (correlation) of a set of variables, whereas the
more common Principal Components Analysis in its full form seeks the
set of factors which can account for all the common and unique
(specific plus error) variance in a set of variables.
PCA establishes the factors that can account for the total (unique and
common) variance in a selection of variables. It is an appropriate
approach for creating a typology of variables or reducing attribute
space. PCA is appropriate for most social science research purposes
and is the most often used form of factor analysis.
PFA determines the least number of factors that can account for the
common variance in a set of variables. This is suitable for determining
the dimensionality of a set of variables, such as a set of items in a
scale, explicitly to test whether one factor can account for the bulk of
the common variance in the set. Thus, PCA can also be used to test
dimensionality. PFA has the drawback that it can produce negative
eigenvalues, which are meaningless.
The principal components analysis, as usually applied in social
sciences and commonly known was assessed as a suitable one for this
study.
5.2.3.3 PCA results
The correlations between variables and the three created factors are
presented in table 5.4. These correlations are also called factor
loadings.
104
Table 3.4: Factors loadings- three factors extracted
Variable F1 F2 F3
RF 0.720 0.112 0.172
GOLD 0.654 -0.282 0.477
WIG -0.156 0.872 0.449
EX 0.563 0.425 -0.650
Source: Own calculations
It seems that the first factor is generally more correlated with variables
than the second and the third one. This could be expected because the
factors are extracted sequentially and will account for less and less
variance overall. Moreover, there is a strong correlation between the
first factor and the risk free rate, the second factor is correlated strongly
with the variable based on WIG and the last factor with the exchange
rate.
The performed analysis enabled to reduce the number of variables
from initially five to finally three factors (components). Reduction in the
number of variables is quite useful when explaining reality. If there are
more than two variables, a "space" is defined, as two variables define a
plane. Therefore, if the number of variables is reduced to three factors,
a three- dimensional scatterplot (Diagram 5.3) can be plotted.
105
Diagram 5.3: Component Plot
gold
risk free
Component 2
1,01,0
-,5
0,0
wig
,5,5
,5 exchange rate
1,0
Component 3Component 10,00,0
-,5-,5
Source: own calculations
The diagram presents three new factors that were based on variables
that are correlated with them.
Moreover, to calculate the value of factors that would be applied in
further APT testing, factor scores coefficients were estimated. Factor
scores coefficient matrix (Table 5.5.) presents the coefficients by which
variables are multiplied to obtain factor scores.
Table No 5.5: Factor Scores Coefficients
Variables F1 F2 F3
WIG -0.121 0.844 0.510
RF 0.559 0.108 0.195
GOLD 0.508 -0.273 0.541
EX 0.437 0.410 -0.737
Source: own calculations.
106
Factor scores are also called component scores. There are different
alternative methods for calculating the factor scores such as
regression, Bartlett, or Anderson-Rubin. The regression method was
employed as the most popular one. Finally, three factors were
estimated as the final solution of factor analysis.
In order to check the models quality the Bartletts test is employed. It
indicates that these results should be treated with caution, as the
significance level is very high that is nine percent. Thus, estimated
factors deliver reliable information only if such a liberal significance
level is assumed.
5.3 Time-series regression
On the basis of estimated betas for 100 companies ten portfolios were
formed the same way as for the CAPM test were formed. For each of
ten portfolios the average weekly returns were calculated11. Then, the
coefficients and the t-statistics for all three factors for each portfolio
were computed12, summary of which is presented in the Table 5.6:
Table No 5.6: Factors coefficients and their t-statistics values for all portfolios
tttttp FFFR εγγγα ++++= ,33,22,110,
Portfolio 1γ t-stat 2γ t-stat
3γ t-stat 1 -0.025368 -0.153133 0.385412 3.356710 -0.123005 -0.806169 2 -0.267580 -1.503911 0.304808 2.471697 -0.078523 -0.479159 3 0.018577 0.134705 0.569930 5.962446 -0.047475 -0.373755 4 0.025362 0.139012 0.343685 2.717870 -0.114360 -0.680549 5 -0.286785 -1.840305 0.363908 3.369192 -0.184273 -1.283841 6 -0.03681 -0.287014 0.344764 3.878438 -0.224909 -1.903958 7 -0.313953 -2.355602 0.601654 6.513072 -0.056357 -0.459098 8 -0.580418 -3.660831 0.556596 5.065011 -0.013661 -0.093549 9 -0.446433 -3.033983 0.641715 6.292173 -0.096088 -0.708995 10 -0.649553 -4.078258 1.017440 9.216588 0.317883 2.166932 Source: Own calculations
11 Appendix No 1 12 results of all time-series regressions of average weekly returns of the portfolios against weekly returns on the WIG index are presented in appendix No 6
107
The results from Table 5.6 indicate that only second factor was
statistically significant at α=0.05. The two other factors F1 and F3 in
most cases were not significantly different from zero at the same level
of significance. F1 was statistically insignificant for the first six portfolios
and the same conclusion must be made about F3 for portfolios 1-9.
Very low p-values of F-statistics indicate that although the two out of
three variables explain the average weekly returns on portfolios, all
models are considered to be of a high quality. At the level of
significance α=0.05 there is statistically sufficient evidence to reject the
null hypothesis of joint insignificance of the estimated parameters. The
results are summarised in the Table 5.7:
Table No 5.7: P-value for the F-test on the joint significance of the model
tttttp FFFR εγγγα ++++= ,33,22,110,
Portfolio P-value of f-test
1 0.003036 2 0.018271 3 0.000000 4 0.026342 5 0.002079 6 0.001842 7 0.000000 8 0.000000 9 0.000000 10 0.000000
Source: Own calculations
As in CAPM model tested in this paper, doubts are raised when
analysing the coefficients of determination of each regression, which
are presented in Table 5.8:
108
Table No 5.8: The coefficients of determination R2 of regression
tttttp FFFR εγγγα ++++= ,33,22,110, Portfolio R2
1 0.089397 2 0.065345 3 0.277215 4 0.060287 5 0.094334 6 0.095900 7 0.327815 8 0.277328 9 0.314343 10 0.597561 Source: Own calculations
The portfolios which achieved low values of R2 in the test of CAPM
achieve them here too. It suggests that the variables are strongly
dominated by the same factor as in CAPM test. Indeed, it might be
true, because two out of four factors are repeated in both studies:
Polish market index WIG and proxy for the risk free rate. The
suspicions are confirmed by the correlation matrix between the three
factors and variables representing risk free rate and market index,
displayed in the Table 5.9:
Table No 5.9: The Correlation matrix between factors, risk free rate, WIG index F1 F2 F3 RF WIG
1.000000 0.004317 -0.129932 0.531905 -0.297325
F2 0.004317 1.000000 0.643287 0.092447 0.948004 F3 -0.129932 0.643287 1.000000 0.075375 0.712005 RF 0.531905 0.092447 0.075375 1.000000 -0.012807 WIG -0.297325 0.948004 0.712005 -0.012807 1.000000 Source: Own calculations
There is a high correlation between F2 and WIG that is likely to be the
reason for its statistical significance and similar values of R2. High
correlation between F2 and F3 is the most probable reason for its
insignificance.
109
The next step in the statistical verification is a test for redundant
variables. The test is for the joint insignificance of F1 and F3. It is
designed in the manner that any of the significant variables would be
detected. The F-test was used and P-values are presented in Table
5.1013:
Ho: 1γ = 3γ =0 variables F1 and F3 do not affect tpR ,
H1: Either or both 1γ and 3γ equal 0, either or both F1 and
F3 affect tpR ,
The tests are conducted at the level of significance α=0.05
Table No 5.10: The P-value of F-test on the redundant variables F1 and F3 from regression tttttp FFFR εγγγα ++++= ,33,22,110, Portfolio P-value of F-test
1 0.723008 2 0.317730 3 0.913652 4 0.767099 5 0.117426 6 0.166690 7 0.065548 8 0.001426 9 0.011318 10 0.000009 Source: Own calculations
The null hypothesis can not be rejected for most of the models, namely
for portfolios 1 - 7. Further tests show that in the case of portfolios 8 -
10 the F1 is significant, whereas F3 remains insignificant for portfolios 8
- 9. Before factors F1 and F3 will be rejected, tests on residuals are
conducted. Tests on the normal distribution, heteroskedasticity and
autocorrelation are conducted with the level of significance α=0.05. The
results are displayed in Table 5.1114:
13 Detailed results in Appendix No 7 14 Detailed results in appendix No 8
110
Table No 5.11: Summary of results of the tests on the normal distribution, heteroskedasticity and autocorrelation of the residuals form regression
tttttp FFFR εγγγα ++++= ,33,22,110,
Portfolio Normal Distribution Autocorrelation Heteroskedasticity 1 Non-existing Non-existing Non-existing 2 Non-existing Existing AR(2) Non-existing 3 Non-existing Non-existing Existing (F2^2) 4 Non-existing Existing AR(2) Non-existing 5 Non-existing Existing AR(1) Non-existing 6 Non-existing Non-existing Non-existing 7 Non-existing Non-existing Non-existing 8 Non-existing Non-existing Existing 9 Non-existing Non-existing Non-existing 10 Non-existing Existing AR(1) Non-existing
Source: Own calculations
All consequences related to the lack of normal distribution,
heteroskedasticity and autocorrelation were described in the previous
chapter. Distribution is assumed to tend to be normal, problems arising
from heteroskedasticity are reduced by applying Newey-West
technique and finally autocorrelation is solved by adding to the
estimated equations terms AR(1) or AR(2), if needed.
In order to solve autocorrelation problem term AR(1) was added to
equations 5 and 10. Similarly, AR(2) was inserted to models for
portfolios 2 and 4. Application of Newey-West technique to the models
3 and 8 assures that the estimates are efficient15. The changes made
to the models are effective enough to reconsider the results:
Table No 5.12: factors coefficients and their p-values for t-statistics after including changes: tttttp FFFR εγγγα ++++= ,33,22,110,
Portfolio 1γ p-value 2γ p-value
3γ p-value AR-terms p-value
1 -0,025368 0.8785 0.385412 0.0010 -0,123005 0.4214 No changes 2 -0.441216 0.0214 0.443532 0.0002 -0.240978 0.1577 -0.292197 0.0005 3 0.018577 0.9096 0.569930 0.0000 -0.047475 0.7234 Newey-West technique 4 -0.004101 0.9832 0.370791 0.0023 -0.076247 0.6635 -0.245680 0.0028 5 -0.183777 0.2171 0.309504 0.0024 -0.144080 0.2931 -0.255977 0.0017 6 -0.03681 0.7745 0.344764 0.0002 -0.224909 0.0589 7 -0.313953 0.0198 0.601654 0.0000 -0.056357 0.6468
No changes No changes
8 -0.580418 0.0025 0.556596 0.0001 -0.013661 0.9459 Newey-West technique 9 -0.446433 0.0029 0.641715 0.0000 -0.096088 0.4794 No changes 10 -0.624755 0.0000 0.996932 0.0000 0.405940 0.0038 -0.282885 0.0005
Source: Own calculations
15 The final version of all corrected three-factor models in appendix No 9
111
According to the results the variable F3 is redundant, since it remains
significant only for one portfolio. These results are coherent with
previously obtained. It is still inconclusive factor F1 should be included
in the next stage, as it is significant for a half of the models. The
decision will be based on the Schwarz Criterion. All ten regressions are
re-estimated for one independent factor F2 and alternatively for two
independent factors F1 and F2.
Models are re-estimated in order to choose between one or two factors
for further analysis and to find the best estimates for the next stage of
the testing procedure. Hence, if only one factor is examined to find its
best value any bias that may occur while estimating should be reduced.
This bias is definitely caused by redundant variables. Before the
coefficients and their P-values for both versions are presented16,
summary of residual tests in both models are depicted in Tables 5.13
and 5.1417:
Table No 5.13: Summary of results of the tests on the normal distribution, heteroskedasticity and autocorrelation of the residuals form regression
tttp FR εγα ++= ,220,
Portfolio Normal Distribution Autocorrelation Heteroskedasticity 1 Non-existing Non-existing Non-existing 2 Non-existing Non-existing Non-existing 3 Non-existing Non-existing Existing (F2^2) 4 Non-existing Existing AR(2) Non-existing 5 Non-existing Existing AR(1) Non-existing 6 Non-existing Non-existing Non-existing 7 Non-existing Non-existing Non-existing 8 Non-existing Non-existing Existing 9 Non-existing Non-existing Non-existing 10 Non-existing Existing AR(1) Non-existing
Source: Own calculations
16 Results of the time-series regressions of two factor models and one-factor models in Appendix No 10 and 11 respectively 17 Detailed results of residuals tests in Appendix No 12 for two-factor model and No 13 for one-factor model
112
Table No 5.14: Summary of results of the tests on the normal distribution, heteroskedasticity and autocorrelation of the residuals form regression
ttttp FFR εγγα +++= ,22,110,
Portfolio Normal Distribution Autocorrelation Heteroskedasticity 1 Non-existing Non-existing Non-existing 2 Non-existing Existing AR(2) Non-existing 3 Non-existing Non-existing Existing (F2^2) 4 Non-existing Existing AR(2) Non-existing 5 Non-existing Existing AR(1) Non-existing 6 Non-existing Non-existing Non-existing 7 Non-existing Non-existing Non-existing 8 Non-existing Non-existing Existing 9 Non-existing Non-existing Non-existing 10 Non-existing Existing AR(1) Non-existing
Source: Own calculations
Analysing the above tables, it can be concluded that both kinds of
models face the same problems with residuals for the same portfolios.
Coefficients estimated after application of the procedure that improves
the models quality, are presented in Table 5.15 and 5.1618:
Table No 5.15: 2γ coefficient and its P-value for t-statistics
tttp FR εγα ++= ,220, Portfolio 2γ P-value 1 0,325300 0.0003 2 0,265853 0.0053 3 0,546794 0.0000 4 0,336301 0.0002 5 0,235964 0.0018 6 0,234874 0.0007 7 0,573414 0.0000 8 0,548586 0.0000 9 0,593772 0.0000 10 1,171754 0.0000 Source: Own calculations
18 Results of all regressions for one- and two-factor models after adjusting for residuals are in the Appendixes No 14 and 15 respectively
113
Table No 5.16: 1γ and 2γ coefficients and their P-value for t-statistics
ttttp FFR εγγα +++= ,22,110, Portfolio 1γ P-value
2γ P-value
1 -0,002219 0.9892 0,325305 0.0003 2 -0,349877 0.0571 0,327616 0.0002 3 0,027512 0.8583 0,546731 0.0000 4 0,020165 0.9136 0,336209 0.0002 5 -0,158216 0.2791 0,239777 0.0016 6 0,005516 0.9655 0,234861 0.0007 7 -0,303347 0.0219 0,574115 0.0000 8 -0,577847 0.0031 0,549921 0.0000 9 -0,428349 0.0036 0,594761 0.0000 10 -0,690301 0.0000 1,186652 0.0000 Source: Own calculations
Comparing results from Tables 5.6 and 5.12 the values of the
coefficient 2γ are the smallest for the one-factor model, whereas their t-
statistics are the highest for the two-factor model. Hence, it still remains
inconclusive if variable F1 should be included in further studies. In order
to make decision which of the two models: tttp FR εγα ++= ,220, or
ttttp FFR εγγα +++= ,22,110, , is better Schwarz Criterion (SBC) is
applied. The comparison is based on the models after adjusting for
residuals. The lower value of the SBC indicates better model: Table No 5.17: Comparison of the one-factor model tttp FR εγα ++= ,220, with
two-factor model ttttp FFR εγγα +++= ,22,110, . Comparison method: Schwarz Criterion
Portfolio Schwarz Criterion value for one factor model
Schwarz Criterion value for two factors model
1 -4.197212 -4.164162 2 -4.043250 -4.039216 3 -4.567012 -4.534237 4 -4.027585 -3.994260 5 -4.379106 -4.353943 6 -4.689233 -4.656194 7 -4.599729 -4.602078 8 -4.200778 -4.256219 9 -4.378062 -4.402179 10 -4.167651 -4.266619
Source: Own calculations
114
For models 7 - 10 SBC is lower in two-factor model, which can be
attributed to the statistical significance of the variable F1. Although for
portfolios 1 6 one-factor model is slightly better, once again the
results cannot definitely indicate superior model.
In conclusion, F3 was not statistically significant in nine out of ten
models. Furthermore, despite some problems with residuals, after their
correction the conclusions cannot be changed. As there is no final
conclusion achieved concerning F1, the cross-sectional tests are run
with and without F1. The only significant variable even at the level of
significance α=0.01 is F2, which shows high correlation with the weekly
return on the WIG index. In both one- and two-factor models, the
coefficient of determination remains close to or below the values
obtained for three-factor models, which was expected. It may suggest
that some variables are missing and therefore model is likely to be not
fully specified.
5.4 Cross-sectional regression
The subsequent phase of the APT test is the same as for CAPM
model, namely cross-sectional regression based on the data obtained
in the previous stage. Observations from the Table 3.18 will be used to
estimate the final model and test the hypothesis, which will provide
arguments rejecting or supporting this version of APT model. As in the
previous chapter the conclusion about the significance of the factor F1
was not achieved, there are two sets of data used in cross-sectional
study.
115
Table No 5.18: Data used in cross-sectional regression
Portfolio Average pR 2γ from one-factor model
1γ from two-factor model
2γ from two-factor model
1 -0,00263 0,325300 -0,002219 0,325305 2 -0,00531 0,265853 -0,349877 0,327616 3 0,001207 0,546794 0,027512 0,546731 4 -0,00033 0,336301 0,020165 0,336209 5 0,000165 0,235964 -0,158216 0,239777 6 0,002446 0,234874 0,005516 0,234861 7 -0,00596 0,573414 -0,303347 0,574115 8 -0,00215 0,548586 -0,577847 0,549921 9 -0,00094 0,593772 -0,428349 0,594761 10 -0,00405 1,171754 -0,690301 1,186652
Source: Own calculations
For the one-factor model the estimated equation takes the form of:
pppR ηγδδ ++= ,210
and for the two-factor model:
ppppR ηγϕγϕϕ +++= ,22,110
where according to the theory the coefficients should be statistically
bigger then zero.
The results for the one-factor model are presented in Table 5.19:
Table No 5.19: Results of the cross-sectional regression: pppR ηγδδ ++= ,210
Dependent Variable: AVERAGE Method: Least Squares Date: 04/27/03 Time: 17:34 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C -5.81E-05 0.001791 -0.032416 0.9749 Γ 2 -0.003513 0.003241 -1.083982 0.3100 R-squared 0.128067 Mean dependent var -0.001756 Adjusted R-squared 0.019075 S.D. dependent var 0.002775 S.E. of regression 0.002749 Akaike info criterion -8.778503 Sum squared resid 6.04E-05 Schwarz criterion -8.717986 Log likelihood 45.89251 F-statistic 1.175017 Durbin-Watson stat 2.380740 Prob(F-statistic) 0.309957
Source: Own calculations
116
Ho: 0δ =0 the parameter 0δ is statistically insignificant
H1: 0δ ≠0 the parameter 0δ is statistically significant
At the level of significance α=0.05 the P-value of the t-statistics equals
0.9749 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence the parameter 0δ is statistically insignificant.
Ho: 1δ =0 variable 2γ does not affect pR
H1: 1δ ≠0 variable 2γ does affect pR
At the level of significance α=0.05 the P-value of the t-statistics equals
0.31 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence the variable 2γ does not affect average portfolio
returns. As in the simple linear regression t-statistics of the estimated
parameter reflects the overall quality of the model, its value of
1.083982 indicates its insignificance.
The graphical explanation of the regression in diagrams 5.4 and 5.5
supports the statistical results:
Diagram No 5.4: Graphical result of the regression of the average returns on portfolios against 2γ variable.
-.008
-.006
-.004
-.002
.000
.002
.004
0.2 0.4 0.6 0.8 1.0 1.2
V2
AV
ER
AG
E
AVERAGE vs. V2
Source: Own calculations
117
Diagram No 5.5: Actual, residual and fitted graph from the model
pppR ηγδδ ++= ,210
-.006
-.004
-.002
.000
.002
.004
-.008
-.006
-.004
-.002
.000
.002
.004
1 2 3 4 5 6 7 8 9 10
Residual Actual Fitted
Source: Own calculations
In order to be assured of the correctness of obtained results tests of the
residual values are conducted:
4. Distribution of the residuals graphically and statistically is
presented in Diagram 5.6: Diagram No 5.6: Histogram of residual values from the model:
pppR ηγδδ ++= ,210
0
1
2
3
4
5
-0.0050 -0.0025 0.0000 0.0025 0.0050
Series: ResidualsSample 1 10Observations 10
Mean 2.40E-19Median 0.000517Maximum 0.003329Minimum -0.004316Std. Dev. 0.002592Skewness -0.465285Kurtosis 2.214968
Jarque-Bera 0.617598Probability 0.734328
Source: Own calculations
118
Ho: pη have normal distribution
H1: pη have not normal distribution
At the level of significance α=0.05 the P-value of the χ2-statistics
equals 0.7343 and leads to a conclusion that there is no sufficient
evidence to reject Ho, hence the residuals pη have normal distribution.
5. Heteroskedasticity - the outcome of the test for the
heteroskedasticity is presented in Table 5.20
Table No 5.20: Results of the test on the heteroskedasticity of the residuals from the model pppR ηγδδ ++= ,210
White Heteroskedasticity Test: F-statistic 0.361530 Probability 0.708892 Obs*R-squared 0.936235 Probability 0.626180 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 04/27/03 Time: 17:50 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C 7.89E-06 1.09E-05 0.721636 0.4939 Γ 2 -6.74E-07 3.88E-05 -0.017365 0.9866 Γ 2^2 -4.98E-06 2.78E-05 -0.178848 0.8631 R-squared 0.093623 Mean dependent var 6.04E-06 Adjusted R-squared -0.165341 S.D. dependent var 7.02E-06 S.E. of regression 7.58E-06 Akaike info criterion -20.49847 Sum squared resid 4.02E-10 Schwarz criterion -20.40769 Log likelihood 105.4923 F-statistic 0.361530 Durbin-Watson stat 2.010468 Prob(F-statistic) 0.708892
Source: Own calculations
Ho: the variance of pη is constant
H1: the variance of pη is not constant
At the level of significance α=0.05 the P-value of the χ2-statistics
equals 0.626180 and leads to a conclusion that there is no sufficient
evidence to reject Ho, hence the variance of residuals pη is constant.
119
6. Autocorrelation Outcome of the test on the autocorrelation of
the residuals is displayed in Table 5.21.
Table No 5.21: Results of the test on the autocorrelation of residuals in model
pppR ηγδδ ++= ,210
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.314934 Probability 0.592171 Obs*R-squared 0.430535 Probability 0.511726 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 04/27/03 Time: 17:57 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C 6.37E-06 0.001873 0.003403 0.9974 γ 2 -1.84E-05 0.003389 -0.005419 0.9958 RESID(-1) -0.207531 0.369805 -0.561190 0.5922 R-squared 0.043054 Mean dependent var 2.40E-19 Adjusted R-squared -0.230360 S.D. dependent var 0.002592 S.E. of regression 0.002875 Akaike info criterion -8.622511 Sum squared resid 5.78E-05 Schwarz criterion -8.531735 Log likelihood 46.11255 F-statistic 0.157467 Durbin-Watson stat 2.062926 Prob(F-statistic) 0.857249
Source: Own calculations
Ho: errors are independent
H1: errors are autocorrelated
At the level of significance α=0.05 the P-value of the Lagrange
Multiplier with one lag statistics equals 0.511726 and leads to a
conclusion that there is no sufficient evidence to reject Ho, hence errors
are independent. As the regression is built on the cross-section data
the achieved results are consistent with expectations.
The overall conclusion is that the one factor model does not explain
much and is statistically inappropriate. T-statistic for 2γ coefficient
indicates that this variable does not explain the variance of average
returns on portfolios. Although the sample is relatively small, the tests
are of high statistical power as no problems with residuals are
uncovered.
120
The two-factor model was tested as an alternative. As previously, its
coefficients from time-series regression are now regressed as
independent variables against average returns on portfolios. The
results are presented in Table 5.22.
Table No 5.22: Results of the cross-sectional regression:
ppppR ηγϕγϕϕ +++= ,22,110 Dependent Variable: AVERAGE Method: Least Squares Date: 04/27/03 Time: 18:01 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C -0.000428 0.001751 -0.244250 0.8140 γ1 0.006457 0.004706 1.372128 0.2124 γ 2 0.000525 0.004420 0.118878 0.9087 R-squared 0.334514 Mean dependent var -0.001756 Adjusted R-squared 0.144375 S.D. dependent var 0.002775 S.E. of regression 0.002567 Akaike info criterion -8.848697 Sum squared resid 4.61E-05 Schwarz criterion -8.757921 Log likelihood 47.24349 F-statistic 1.759311 Durbin-Watson stat 2.164912 Prob(F-statistic) 0.240429
Source: Own calculations
To test the significance of the estimated parameters, t-statistics is
applied:
Ho: 0ϕ =0 the parameter 0ϕ is statistically insignificant
H1: 0ϕ ≠0 the parameter 0ϕ is statistically significant
At the level of significance α=0.05 the P-value of the t-statistics equals
0.8140 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence the parameter 0ϕ is statistically insignificant.
Ho: 1ϕ =0 variable 1γ does not affect pR
H1: 1ϕ ≠0 variable 1γ does affect pR
121
At the level of significance α=0.05 the P-value of the t-statistics equals
0.2124 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence variable 1γ does not affect average portfolio returns.
Ho: 2ϕ =0 variable 2γ does not affect pR
H1: 2ϕ ≠0 variable 2γ does affect pR
At the level of significance α=0.05 the P-value of the t-statistics equals
0.9087 and leads to a conclusion that there is no sufficient evidence to
reject Ho, hence variable 2γ does not affect average portfolio returns.
The general quality of the model is poor, as expected:
Ho: 1ϕ = 2ϕ = 0 the model is miss-specified
H1: Either or both of 1ϕ and 2ϕ ≠ 0
At the level of significance α=0.05 the P-value of the F-statistics equals
0.2404 and leads to a conclusion that there is no sufficient evidence to
reject Ho, the model is miss-specified.
Graphical results in Diagram 5.7 support the hypothesis of the model
insignificance:
122
Diagram No 5.7: Actual, residual and fitted graph from the model
-.004
-.002
.000
.002
.004
-.008
-.006
-.004
-.002
.000
.002
.004
1 2 3 4 5 6 7 8 9 10
R es idual A c tual F itted
Source: Own calculations
In order to be assured of the correctness of obtained results tests of the
residual values are conducted:
1. Residuals distribution is presented graphically and statistically in
Diagram 5.8: Diagram No 5.8: Histogram of residual values from the model:
ppppR ηγϕγϕϕ +++= ,22,110
0
1
2
3
4
5
-0.004 -0.002 0.000 0.002
Series: ResidualsSample 1 10Observations 10
Mean 7.86E-20Median 0.000689Maximum 0.002715Minimum -0.003874Std. Dev. 0.002264Skewness -0.564016Kurtosis 1.885116
Jarque-Bera 1.048092Probability 0.592120
Source: Own calculations
Ho: pη have normal distribution
H1: pη have not normal distribution
123
At the level of significance α=0.05 the P-value of the χ2-statistics
equals 0.5921 and leads to a conclusion that there is no sufficient
evidence to reject Ho, the residuals pη have normal distribution.
2. Heteroskedasticity - the outcome of the test for the
heteroskedasticity is presented in Table 5.23 Table No 5.23: Results of the test on the heteroskedasticity of the residuals from the
model ppppR ηγϕγϕϕ +++= ,22,110
White Heteroskedasticity Test: F-statistic 0.937605 Probability 0.511254 Obs*R-squared 4.285988 Probability 0.368683 Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 04/27/03 Time: 18:19 Sample: 1 10 Included observations: 10 Variable Coefficient Std. Error t-Statistic Prob. C 1.63E-06 7.20E-06 0.227156 0.8293 γ 1 -3.61E-05 2.24E-05 -1.613891 0.1675 γ 1^2 -7.01E-05 4.55E-05 -1.539498 0.1843 γ 2 5.41E-06 2.60E-05 0.207639 0.8437 γ 2^2 3.47E-07 1.92E-05 0.018091 0.9863 R-squared 0.428599 Mean dependent var 4.61E-06 Adjusted R-squared -0.028522 S.D. dependent var 4.57E-06 S.E. of regression 4.64E-06 Akaike info criterion -21.41697 Sum squared resid 1.08E-10 Schwarz criterion -21.26568 Log likelihood 112.0849 F-statistic 0.937605 Durbin-Watson stat 1.501012 Prob(F-statistic) 0.511254
Source: Own calculations
Ho: the variance of pη is constant
H1: the variance of pη is not constant
At the level of significance α=0.05 the P-value of the χ2-statistics
equals 0.3686 and leads to a conclusion that there is no sufficient
evidence to reject Ho, hence the variance of residuals pη is constant.
3. Autocorrelation the outcome of the test on the autocorrelation
of the residuals is presented in Table 5.24:
124
Table No 5.24: Results of the test on the autocorrelation of residuals in model
ppppR ηγϕγϕϕ +++= ,22,110
Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.142565 Probability 0.718740 Obs*R-squared 0.232093 Probability 0.629976 Test Equation: Dependent Variable: RESID Method: Least Squares Date: 04/27/03 Time: 18:21 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-Statistic Prob. C -0.000188 0.001935 -0.097376 0.9256 γ 1 0.000519 0.005208 0.099610 0.9239 γ 2 0.000636 0.005010 0.126872 0.9032 RESID(-1) -0.161865 0.428692 -0.377578 0.7187 R-squared 0.023209 Mean dependent var 7.86E-20 Adjusted R-squared -0.465186 S.D. dependent var 0.002264 S.E. of regression 0.002740 Akaike info criterion -8.672180 Sum squared resid 4.51E-05 Schwarz criterion -8.551146 Log likelihood 47.36090 F-statistic 0.047522 Durbin-Watson stat 1.915817 Prob(F-statistic) 0.984958
Source: Own calculations
Ho: errors are independent
H1: errors are autocorrelated
At the level of significance α=0.05 the P-value of the Lagrange
Multiplier with one lag statistics equals 0.62998 and leads to a
conclusion that there is no sufficient evidence to reject Ho, hence errors
are independent. As the regression is built on the cross-section data
the achieved results are consistent with expectations.
None of the APT model versions presented in this paper works on
Polish market. All tested factors appeared not to be statistically
significant. The overall conclusion after testing classical CAPM and
APT models is that classical capital market equilibrium models do not
work on the Warsaw Stock Exchange. Discussion on the possible
reasons for the market equilibrium models failure is carried out in the
next chapter.
125
CHAPTER VI
Possible Reasons for CAPM and APT Failure
The validity of both models: standard version CAPM as well as
proposed version of APT is rejected. The aim of this chapter is to point
out the possible explanations for the failure of both models.
Furthermore, in this chapter improvements that could be applied in
order to deliver more reliable results will be discussed.
There is a number of reasons why the models can not be applied to
Polish market. Each of these reasons will be a subject of detailed
discussion:
1. Instability of companies betas even in short term
2. Inappropriate portfolio grouping on the beta basis
3. Market inefficiency and liquidity
o Individual investor impact on the asset price
o Small volume low liquidity
o Autocorrelation
4. Weighted index capital dominance of a few companies
5. Generally low significance of the market as a source of capital
6. Shortcomings of APT factor analysis
7. Short estimation period
126
6. 1 Beta instability In the CAPM case, time-series models of the portfolios are stable over
the estimation period. For the arbitrary chosen time that is on 4th July
2002 applying Chow Breakpoint Test only for two out of ten models:
seventh and tenth for which the null hypothesis about the stability of
estimated parameters can not be rejected even at a very high level of
significance19. The results are presented in Table 6.1:
Table No 6.1: Results of the model stability test Chow Breakpoint Test: 7/04/2002 Estimated model: )( ,,,, tFtMpptFtp RRRR −+=− βα - Portfolio P-Value 1 0,543011 2 0,298215 3 0,331163 4 0,507972 5 0,738408 6 0,598632 7 0,000184 8 0,371525 9 0,240888 10 0,022043 Source: Own calculations
Problems, however, emerge much earlier, in the first stage and were
associated with beta estimation for the individual shares, which
afterwards form the portfolios. Betas estimated for the shares in the
period subsequent to forming portfolios are different from the betas of
the same shares in the period when these shares are prescribed to the
particular portfolios. This is a cause of inconsistency within a group. It
leads to results, which are in contradiction to the expectations, for
example: portfolio, which at the forming phase was in the sixth decile
based on its relation to the market risk, after estimation appeared to
have the lowest beta, which would be expected for the portfolio
containing first decile of shares. Although CAPM model does not
impose the requirement of constant beta of an asset, such a high
variability makes it impossible to find a true relation between beta and
average returns. Table 6.2 should clarify the magnitude of the problem: 19 Appendix No 16
127
Table No 6.2: Differences in shares betas composing fourth portfolio before and after it is formed.
Beta estimated for the period from 13. July 2000 to 27. December 2001
Beta estimated for the period from 3. January 2002 to 26. December 2002
OCEAN SA 0.29048631 0.713788012 STALPRODKUT 0.304070251 0.461836357 BUDOPOL-WROCLAW 0.305519608 -0.403168579
WISTIL 0.306052531 0.033646917 OBORNIKI 0.314800398 0.046852433 HANDLOWY 0.320958834 0.494068836 FORTE 0.329261567 -0.042074626 EKODROB 0.330939013 0.43268908 DEBICA 0.335007843 0.478581477 FORTIS BANK POLSKA 0.337494666 0.188541371
Source: Own calculations
Instability of stock betas in long term is a natural order resulting from
either business or economic cycle. This time-varying characteristic of
beta can result from the thee following reasons:
• Investments in new projects, mergers and acquisitions as well
as industry diversification.
• Change in financial leverage, not only by increasing or
decreasing debt, but by paying out dividends and buying back
shares.
• Company growth, reflecting the change in the structure of
operational costs influence the beta value.
However, high variability of beta coefficient in short term implies that
the relation between beta of the portfolio and beta of the previous
period estimated for the particular shares will be sustained.
128
6. 2 Inappropriate portfolio grouping APT case
The other source of problems might be the basis caused by wrong
portfolios grouping. As in the case of CAPM test there is not much
choice, as most of the empirical studies use portfolios formed on the
basis of increasing correlation with market risk, there is no reason to do
so for APT model. This is because this model does not assume the
necessity of close relation to the one particular factor, which is to be
market index. Therefore, the time-series regressions are conducted
once again for portfolios formed by alphabetically sorted shares. As the
logic suggests sorting procedure is applied only once. Forming
portfolios on the alphabetical basis assures that shares included in
portfolios are chosen randomly. The results are presented in Table
6.320:
Table No 6.3: factors coefficients and P-values of t-statistics for all portfolios
tttttp FFFR εγγγα ++++= ,33,22,110, , portfolios formed alphabetically.
Portfolio 1γ P-value 2γ P-value
3γ P-value
1 -0.020109 0.0920 -0.027823 0.8838 0.178185 0.1610 2 0.078409 0.6773 0.015397 0.9061 0.175062 0.3137 3 0.014168 0.9212 0.098606 0.3211 0.060672 0.6455 4 0.078000 0.7208 -0.089273 0.5552 0.204282 0.3102 5 0.123644 0.3771 0.075355 0.4372 0.162521 0.2081 6 -0.102236 0.4999 0.080018 0.4463 0.200052 0.1529 7 -0.268465 0.1579 -0.067171 0.6092 0.150678 0.3885 8 -0.105581 0.5530 -0.065707 0.5942 0.156367 0.3406 9 -0.061129 0.7568 0.046841 0.7321 0.308565 0.0912 10 0.135191 0.5177 -0.028143 0.8459 0.217079 0.2601 Source: Own calculations
Analysis of Table 6.3 suggests that none of the factors is statistically
significant for all portfolios. High power of the t-test is ensured by non-
existence of the errors disturbances21:
20 detailed estimation output in Appendix No 17 21 Detailed results of the residuals tests in Appendix No 18
129
Table No 6.4: Summary of results of the tests on the normal distribution, heteroskedasticity and autocorrelation of the residuals form regression
tttttp FFFR εγγγα ++++= ,33,22,110, , portfolios formed alphabetically.
Portfolio Normal Distribution Autocorrelation Heteroskedasticity 1 Non-existing Non-existing Non-existing 2 Non-existing Non-existing Non-existing 3 Existing Non-existing Existing 4 Non-existing Non-existing Non-existing 5 Existing Non-existing Non-existing 6 Non-existing Non-existing Existing 7 Non-existing Non-existing Non-existing 8 Non-existing Non-existing Non-existing 9 Existing Non-existing Non-existing 10 Non-existing Non-existing Non-existing
Source: Own calculations
The conclusion cannot be changed after correction of the
heteroskedasticity in models three and six22.
The results support the hypothesis that none of the factors presented in
APT model is statistically significant. There is no relation between
these factors and average returns.
6. 3 Market inefficiency and liquidity
Market efficiency is one of the assumptions underlying capital market
equilibrium models. There is a number of conditions that must be
satisfied for the market to be efficient, some of which are discussed in
the first chapter of this paper. Market is considered efficient, if the
prices rationally reflect all available information. However, Polish capital
market has features indicating its inefficiency, which might be a cause
for the models failure. Three sources of inefficiency are detected and
shortly discussed.
Efficient market is liquid, which means that all investors can buy and
sell any amount of shares any time. Therefore, investors funds for
22 Estimation output for the models after correction in Appendix No 19
130
trading purposes should not be locked in, because an asset is illiquid.
Difficulty in trading on a stock exchange causes mispricing, as the
value of the share is determined by other factors then available
information (Damodaran 2001). The liquidities of the stocks traded on
the Warsaw Stock Exchange are relatively low. High-capitalisation
companies are traded most often whereas frequent gaps in quotation of
many small companies indicate the low liquidity of their shares.
Low liquidity of the Polish capital market is supported by the low
volume of the trade, which is on average merely 150mln23 USD daily
and in the last months around 150mln PLN (≈ 39.5mln USD)24. This is
very little when comparing with average daily trading on NYSE 41,3
billion USD25. On a randomly chosen day that is 31st March 2002 the
volume of trade on the WSE reached approximately 27mln USD, of
which close to 7 mln USD is accounted for Bank Pekao S.A. and
almost 7 mln USD is a turnover for TP S.A a Polish Telecommunication
company stocks. Both companies are included in WIG 20, twenty
biggest firms listed on the Warsaw Stock Exchange.
Small volume of trade may result in other source of market inefficiency,
namely the impact of a particular investor on a stock price. Under
efficient market assumption none of individual trader is able to affect
the price. Although since the beginning of the second quarter 1994
impact of one company on the index was limited to 10 percent of the
value of the general shares portfolio. For such a small market turnover
as on the Warsaw Stock Exchange liquidation of one big investors
positions can disturb the equilibrium significantly.
23 http://www.igte.com.pl/Biuletyny/Biuletyn_002_2002/8_4.htm 24 average exchange rate on 05.05.2003, https://www.bm.bphpbk.pl/pieniadz/kalkulator/ 25 Ibidem
131
6. 4 Value weighted index and capital dominance of a few companies
Unlike in JBS studies, the index used, WIG, is a weighted index, hence
its value depends on the market capitalisation of any individual
company listed on the Warsaw Stock Exchange. A capital dominance
of a few biggest companies is a feature of the Polish market, which is
presented in Table 6.5:
Table No 6.5: Capital share in market portfolio of ten biggest companies
State on 5. May 2003 Company Capital share in a market portfolio in %
TPSA 10,71 PKNORLEN 10,51 PEKAO 9,28 BPHPBK 6,76 KGHM 6,09 PROKOM 4,23 SWIECIE 3,25 AGORA 2,97 STOMIL 2,85 BZWBK 2,85 ∑ 59,5 Source: http://www.gpw.com.pl/xml/dane/indeksy/wig.xml
A capital dominance of a few biggest companies might be a source of
additional problems, as the relationship built in CAPM model and
partially in the proposed version of APT model becomes close to the
relation between a return on an individual stock and a few dominating
companies. This is in contradiction to the theory, which implies the
relationship with well diversified portfolio. This effect increases on its
significance, if the overall trade volume is relatively small. The problem
of low volume of trade was discussed in previous section.
Furthermore, assuming that the weighted average beta equals one and
betas of the dominating companies are bigger than one, majority of
listed companies on the stock exchange, will have betas lower than
one. Such situation occurred in this study. Big and stable companies
132
have relatively high betas with comparison to small, risky firms, betas
of which are relatively low.
6. 5 Low significance of the market as a source of capital
The importance of the Warsaw Stock Exchange as a source of capital
is still relatively small. Its current capitalisation of the companies
shares is roughly 28 bln USD. Polish capital market is very small when
comparing with NYSE with its capitalization of 14 900 mld USD. This
comparison stresses rather marginal importance of WSE. For example
the number of listed shares on NYSE is 258626 and it is only one of a
few capital markets in USA.
Furthermore, Warsaw Stock Exchange does not fulfil a definition of a
market portfolio. Stocks of publicly listed companies in Poland reflect
only a small part of investment opportunities. For example the debt
market is much bigger with its 38 bln USD27. Furthermore, Polish
investors do not have to invest only in Poland as they have global
access to investment opportunities in Western countries. On the other
hand, as most of the investments in Poland are made by foreign
capital, it might be concluded that Warsaw Stock Exchange serves only
for a diversification for international investors. In such situation
investment horizon becomes much wider than the region of Poland.
6. 6 Shortcomings of APT factor analysis
APT significance should be assessed simultaneously with the
estimation methodology used to create this model. Factor analysis is a
method that allows for identification of factors simultaneously with the
26 http://www.igte.com.pl/Biuletyny/Biuletyn_002_2002/8_4.htm 27 state at the end of 2002, http://www.rk.pl/fua/fundacja/dzialalnosc/vii.asp
133
estimation of coefficients influencing the returns on individual stocks
(Elton and Gruber, 1998). The fact that the estimated model did not
support the arbitrage pricing theory, can result from biases of the
estimation technique applied.
Factor analysis has disadvantages that may influence APT tests. First
of all, ikγ are biased thus kλ are significant only asymptotically.
Furthermore, these parameters can be rescaled arbitrary (for example
multiplied by two). Third, the researcher cannot be sure if factors were
obtained in the right rank. Therefore, the analysis of different samples
may bring different results. That may lead to the situation that first
factor in first study may be a second one in the next research.
Nevertheless, the most important disadvantage of factor analysis is the
complexity of mathematical calculations that need to be carried out.
This difficulty made researchers introduce simplifications. Therefore,
the study usually employs a limited sample of securities. Portfolio
aggregation is likely to generate information losses, as the results will
describe only a small sample of returns leaving the most risky assets
beyond the model. Chen (1981) presented a technique that enables
APT testing with a large number of securities. However, this
methodology was criticized by Dryhmes, Friend and Gultekin (Elton,
Gruber, 1998). They concluded that the number of significant factors
depends on the sample size. Thus, the more securities in portfolio, the
bigger the number of significant results. Their research revealed that
there are three significant factors for groups of 15 securities and seven
factors for groups of 60 securities. The portfolios analyzed in this paper
consisted of ten securities, thus the results obtained may not be
reliable. According to Dryhmes et al the portfolios created are expected
to neglect the covariance between securities included in different
groups. Furthermore, factors for one portfolio are likely to differ from
factors estimated for another. In order to improve the APT model a
bigger number of securities should be grouped.
134
6.7 Diversification of the firm-specific risk
Arbitrage pricing model tested in this research assumed that firm-
specific risk is diversified, as portfolios of stocks were tested, instead of
individual shares.
However, portfolios consisted of only ten shares and therefore this
assumption might not be met in all groups of stocks.
There are big companies in Poland, whose business activities,
influence other firms. Any scandals associated with these companies
might have a significant impact on Polish market. Therefore, it would be
difficult to diversify firm-specific risk of such a company.
For example PKN Orlen, Polish gas producer and retailer, has a great
impact on a whole industry because of its monopolistic position.
However, this company is well known for corporate scandals. Its former
president Mr. Modrzejewski was arrested and dismissed because of
revealing secret information. As a result of that it was possible to buy
shares of one of the companies at competitive price. The information
about the arresting of that executive caused a drop in stock prices of
seven percent. Furthermore, Orlen incumbent president Mr. Wróbel is
accused of collusion while bargaining. Moreover, his advisors are
accused of manipulation with stock prices (Indulski and Koczot, 2003).
Due to that scandals companys investors are exposed to additional
uncertainty. However, the problem is that this firm has a very strong
impact on other companies and incidents of that kind could influence
test results as well. Moreover, corporate scandals take place in other
companies as well. For example Drosed (Mielczarek, 1999) stocks
prices were manipulated as well.
It is questionable if the risk of such mispricing can be fully diversified
when grouping stocks into portfolios.
135
6.8 Small number of variables
Moreover, the factor analysis that was carried out may not explain the
reality well enough because there were only five variables introduced.
More variables especially these delivering different information are
expected to improve the model. There were no variables on income (for
example GDP), no variables linking Polish economy directly to
European markets or no information on industrial production. The
application of such data is likely to improve created APT model.
Furthermore, it would be advantageous to employ variables that could
describe political risk in Poland. This risk is reflected to a certain extent
in the exchange rate as political uncertainty results usually in the PLN
depreciation. That is because foreign investors take their capital back,
when they find political situation in Poland uncertain. There are no
empirical studies confirming this hypothesis but there is some evidence
that suggests this opinion.
For example rotation on crucial positions in Polish government,
especially those directly linked to Ministry of Finance was deterring
foreign investors from investments on the WSE.
Such situation occurred when the government of the incumbent Prime
Minister Leszek Miller or former Minister of Finance Grzegorz Kołodko
was expected to resign (http://waluty.onet.pl/731083.wiadomości.html,
http://waluty.onet.pl/732360.wiadomości.html).
However, political risk can also refer to unexpected changes of
legislation that may influence investors or certain industries. Assuming
that markets are efficient these law modifications should be discounted
by investors and included in stock prices. Nevertheless, the legislation
process is sometimes not as clear as it should be.
136
For example capital gains on stock prices will be taxied in 2004.
Despite the fact that the new tax rate will be charged next year,
investors are not sure what the tax rate will be. Internet chats with
Minister of Finance representatives were their source of information.
Having no access to more reliable information and therefore being not
able to estimate their future net profits, they might try to make gains as
soon as possible. Such strategy would result in making investors less
rational (Brycki, 2003).
The inclusion of more variables could describe better the political risk in
Poland and therefore the constructed model would perform better.
However, implementing such variables might cause additional
problems as it would be difficult to quantify such measures. Factor
analysis creating unobservable variables could include political risk
factors to some extent but introducing for example binary variables
describing changes in law or in the government could improve the
model even more.
6.9 Short estimation period
The last possible reason for the failures of the examined models might
be short estimation period and a small number of shares included in
the study. Because of the problems resulting from the low liquidity data
of a monthly frequency might be better as it would give more averaged
returns. On the other hand, short estimation period does not allow for
using monthly rates of return, as the number of observations would be
far too small and as a consequence the insufficient number of degrees
of freedom would be obtained. The trade off problem, between the
number of observations (degrees of freedom) and the size of the
sample of companies included, is caused by the short history of
Warsaw Stock Exchange. Most of the studies were conducted on the
markets with long history and therefore without such a limitation.
137
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recently accessed on 21.04.2003
144
APPENDIXES
Appendix No 1
Data used in time series regression
Data risk free Index Portfolio
1 Portfolio
2 Portfolio
3 Portfolio
4 Portfolio
5 Portfolio
6 Portfolio
7 Portfolio
8 Portfolio
9 Portfolio
10 00-02-03 0,00205 0,03710 0,02705 0,01145 0,05663 0,03858 0,07741 0,00245 0,15550 0,07398 0,05104 0,09903 00-02-10 0,02727 0,03245 0,03306 -0,01353 0,09737 0,00142 -0,02278 -0,03027 0,00312 -0,04835 -0,02507 0,00450 00-02-17 0,01462 0,00885 0,02632 0,04853 0,03648 -0,02809 -0,01709 0,00235 0,01016 0,01537 0,02911 0,04829 00-02-24 -0,01614 0,07890 -0,04848 -0,05462 0,06219 -0,05864 -0,04085 -0,07287 0,01338 -0,07148 -0,00136 0,00525 00-03-02 0,00117 -0,02833 -0,02107 0,01265 0,04269 0,00132 0,04437 -0,01320 -0,00691 -0,01431 0,05919 -0,03765 00-03-09 0,00410 -0,00425 0,04999 0,07290 0,07189 0,04552 0,00607 0,08715 0,00966 0,02898 0,11437 0,00581 00-03-16 -0,02564 -0,00279 0,02951 0,01146 0,00444 -0,03236 0,03422 -0,01866 -0,00221 -0,02647 0,02253 0,00800 00-03-23 -0,00263 0,00509 -0,02468 -0,01010 -0,00779 0,04188 -0,00140 0,07125 0,01165 0,06061 0,00791 0,03332 00-03-30 -0,02219 -0,00437 0,00703 0,01642 -0,00187 -0,03068 -0,01064 -0,01047 -0,01465 0,02157 -0,00746 -0,01924 00-04-06 0,00515 -0,05941 -0,01729 -0,04605 -0,04286 -0,00756 -0,02167 -0,00559 -0,04959 0,01147 -0,04191 -0,03546 00-04-13 -0,00940 -0,00977 -0,00035 0,01427 0,01242 0,02397 0,01415 0,02409 0,00375 -0,03294 -0,00253 -0,00705 00-04-20 -0,01256 -0,06738 -0,01788 -0,03772 -0,06705 0,00138 -0,01999 0,00248 -0,04815 -0,04995 -0,02798 -0,05414 00-04-27 0,01422 0,01104 -0,00261 -0,01497 0,00762 0,02582 -0,02413 0,00488 -0,01262 0,00512 -0,01091 0,01135 00-05-04 -0,01415 0,01890 0,00504 0,01341 0,00018 -0,02683 0,00847 0,00400 0,01433 0,00246 0,04317 0,00454 00-05-11 -0,00437 -0,01804 -0,03084 -0,02060 0,01393 -0,01721 0,00209 -0,01393 0,01145 0,00354 -0,02231 0,01383 00-05-18 -0,01128 0,03443 -0,00765 0,04419 -0,01211 0,02643 0,03935 -0,00648 0,02930 0,01463 0,02587 0,03482 00-05-25 0,01584 -0,06428 -0,02622 -0,04240 0,04340 -0,01946 -0,06597 -0,04398 -0,05709 -0,03226 -0,04143 -0,06971 00-06-01 0,02084 0,05024 0,03022 -0,01633 0,03217 0,02674 0,01963 0,03667 0,06165 0,02846 0,03185 0,07653 00-06-08 -0,00269 0,01455 -0,01499 -0,00480 -0,00116 -0,00238 -0,03418 -0,01424 -0,01032 -0,01940 -0,00879 -0,02247 00-06-15 -0,00735 -0,00926 0,04532 0,05349 0,01687 0,02381 0,01672 0,00456 0,01151 0,01128 -0,00269 0,00336 00-06-22 0,01235 -0,00615 -0,00352 -0,02623 -0,01754 -0,02519 0,00886 -0,01278 0,00481 0,00577 -0,01876 0,00916 00-06-29 -0,01341 0,01520 0,00311 0,00958 -0,00502 -0,00344 0,02260 -0,01329 0,00454 -0,01968 -0,00900 -0,00625 00-07-06 -0,00062 -0,01590 0,01939 -0,01016 -0,00126 0,02668 0,02329 0,03826 0,00362 0,03176 0,02169 0,00729 00-07-13 -0,02165 -0,00051 0,01843 -0,00270 0,00611 0,02469 -0,00009 0,02379 0,01282 0,00348 0,02657 0,00877 00-07-20 0,00860 0,02084 -0,00285 0,00804 -0,01203 0,02608 0,00571 -0,00747 -0,00258 0,03192 -0,01844 0,00923 00-07-27 0,00777 -0,02826 0,00608 -0,05699 -0,01406 -0,00856 0,18051 -0,00733 -0,01731 0,00053 0,02280 -0,03093 00-08-03 0,00199 -0,02597 -0,02021 -0,05946 -0,00541 0,00726 -0,07056 0,02047 -0,05260 0,01513 -0,04095 -0,03037 00-08-10 0,00422 0,01053 0,01463 0,03121 0,03081 0,01730 0,06909 0,03244 0,02344 0,04371 0,01664 0,01734 00-08-17 -0,01051 -0,03219 -0,01107 0,00539 -0,01593 -0,02270 -0,02232 -0,00870 -0,04546 -0,01081 -0,00996 -0,01696 00-08-24 -0,01562 -0,01063 -0,02357 -0,01896 -0,00064 0,00380 -0,03258 -0,00215 0,01563 -0,02186 -0,00732 0,00444 00-08-31 -0,00888 0,00867 -0,00344 0,00840 0,01266 0,00941 -0,00212 -0,00855 -0,00131 -0,01671 0,02106 0,01207 00-09-07 -0,01665 -0,00789 -0,00803 -0,02643 -0,01329 -0,00603 -0,00688 -0,02079 0,00167 -0,01674 -0,00851 -0,01236 00-09-14 0,00859 0,00043 0,00599 -0,03939 -0,02464 0,01577 -0,02890 -0,00217 -0,02821 -0,02011 -0,00204 0,00499 00-09-21 0,00633 -0,02349 -0,00505 -0,06308 -0,02483 0,03067 0,04186 -0,00323 0,00466 -0,04101 -0,04564 -0,04094 00-09-28 0,01219 -0,06032 -0,00982 -0,06454 -0,06491 -0,07691 -0,07561 -0,01897 -0,03337 -0,05265 -0,07212 -0,04312 00-10-05 0,00380 -0,02129 -0,03890 0,12518 -0,01448 -0,00143 0,03274 -0,02382 0,01073 -0,00146 0,00505 -0,00517 00-10-12 0,00000 -0,06694 -0,02641 0,01868 -0,08507 -0,01037 -0,03479 -0,04582 -0,04401 -0,07829 -0,04747 -0,07068 00-10-19 0,01010 0,03385 -0,00050 -0,08101 0,04462 -0,02410 -0,01940 0,01769 -0,02413 0,00678 -0,03474 0,03891 00-10-26 0,00750 0,01856 0,03456 0,04375 0,01346 0,06601 0,00292 -0,00458 0,03881 -0,00787 0,04109 -0,02355 00-11-02 0,00868 -0,00904 0,02743 0,04120 0,05031 0,02720 0,00073 0,00318 0,00974 0,04272 0,05560 0,03802 00-11-09 0,01476 0,04386 -0,01342 -0,00891 0,04370 -0,03665 0,01067 0,02994 -0,00364 0,00257 -0,02653 -0,00247 00-11-16 0,02061 -0,02771 -0,02378 -0,00981 -0,01390 -0,03144 -0,00606 -0,00211 -0,03983 -0,04258 -0,00782 -0,00581 00-11-23 0,01010 0,00786 -0,01383 0,00435 0,02232 -0,00556 -0,00863 0,01790 -0,02005 -0,01280 0,01168 -0,00281 00-11-30 0,01634 -0,00373 -0,00811 0,00022 0,02196 0,00165 0,01924 -0,01758 -0,01426 -0,00593 0,00202 0,02587 00-12-07 -0,00856 0,02590 -0,00349 -0,01150 -0,02001 -0,02552 -0,04699 -0,02745 0,01236 0,01487 -0,01719 0,02773 00-12-14 -0,00817 0,05752 -0,01389 -0,03121 0,05508 -0,02934 0,00019 0,00974 0,01945 -0,00479 0,01739 0,04659
145
Data risk free Index Portfolio
1 Portfolio
2 Portfolio
3 Portfolio
4 Portfolio
5 Portfolio
6 Portfolio
7 Portfolio
8 Portfolio
9 Portfolio
10 00-12-21 0,01588 -0,01293 -0,01319 -0,00187 -0,01259 0,01804 0,01267 0,02154 0,02723 -0,01357 -0,01915 -0,0124500-12-28 -0,00984 0,02645 -0,00346 0,07132 0,03852 0,02104 0,04054 0,00804 0,01847 0,03528 0,03444 0,02102 01-01-04 0,01275 -0,00183 0,08131 0,00494 0,02174 0,00640 0,01097 0,01717 -0,02469 0,05824 0,01447 -0,0333101-01-11 0,00104 -0,05440 -0,03024 0,00147 -0,03195 -0,01034 -0,01365 -0,01518 -0,03421 -0,01359 -0,04926 -0,0673501-01-18 0,00311 0,01174 -0,04480 0,01478 -0,02637 -0,02742 -0,02187 0,01222 -0,01403 0,01854 -0,03990 0,01605 01-01-25 -0,01553 0,01502 0,00583 -0,01737 0,01549 0,01312 0,00971 -0,01771 -0,00198 0,02666 0,01729 0,00757 01-02-01 0,00409 0,01635 0,02441 0,01994 -0,00958 0,02479 0,01017 -0,00117 0,00815 -0,03341 0,01868 0,00397 01-02-08 -0,00465 -0,04545 -0,06873 -0,03291 -0,03452 -0,04346 -0,01229 -0,04322 -0,03928 -0,01734 -0,03317 -0,0812201-02-15 0,00701 0,00123 -0,01421 0,01043 -0,01357 -0,01863 -0,00145 -0,00812 -0,01146 -0,00987 -0,02127 0,01249 01-02-22 -0,00638 -0,07068 -0,01151 -0,01681 -0,03323 -0,03047 -0,06559 -0,00715 -0,04303 -0,04855 -0,01261 -0,0830101-03-01 0,00117 -0,04291 -0,03871 -0,00514 -0,00975 -0,03005 -0,02163 0,00108 -0,03355 -0,01842 -0,01446 -0,0556701-03-08 -0,00233 0,01985 -0,03512 -0,04447 -0,05693 -0,01925 -0,00854 -0,01110 -0,00200 -0,01105 0,00903 -0,0340001-03-15 0,00000 -0,05572 -0,06957 -0,05318 -0,09424 -0,00080 -0,03959 -0,07721 -0,05913 -0,06323 -0,02828 -0,0917101-03-22 0,00819 0,00813 0,07078 0,05681 0,04142 0,02060 0,02767 0,00511 0,00968 0,04366 0,05430 0,01130 01-03-29 0,00116 0,00422 0,00424 -0,01863 0,01997 0,00314 0,00547 0,01555 0,00295 0,01774 0,04576 0,01080 01-04-05 -0,01159 -0,01520 0,00213 0,02759 -0,00561 0,01451 0,00094 -0,02078 0,00003 0,00551 -0,00301 -0,0128101-04-12 0,00586 0,05441 -0,01978 -0,00536 0,01450 0,00878 0,03982 0,00874 0,04070 0,04792 0,02472 0,07768 01-04-19 0,00000 0,02785 -0,01970 0,01544 0,01291 -0,00045 0,01677 0,01359 -0,00222 -0,00388 0,00612 0,03072 01-04-26 -0,00932 -0,00233 -0,03487 -0,01730 0,07071 0,00509 0,00381 0,01945 -0,00033 0,01092 0,01007 -0,0007201-05-03 -0,00471 -0,02149 -0,03390 -0,00282 0,00296 -0,00043 -0,00402 0,00359 0,00142 -0,03124 0,00182 -0,0128901-05-10 0,00059 -0,01707 -0,03612 -0,04471 -0,03090 -0,01847 -0,02411 -0,00292 -0,01877 -0,00758 -0,01951 -0,0306201-05-17 0,01063 -0,00101 -0,04086 0,00575 0,03302 -0,00893 -0,01162 0,00416 -0,02199 0,00064 0,01334 0,00033 01-05-24 -0,00584 0,04956 -0,01392 -0,01447 0,00839 -0,00306 0,00456 0,02751 0,01539 0,00367 0,02276 0,04310 01-05-31 -0,00176 -0,00451 -0,00027 -0,07263 -0,00407 -0,00432 -0,03969 -0,01383 0,00140 -0,00926 -0,01161 -0,0184301-06-07 0,00471 -0,04176 -0,03587 0,00588 0,02435 -0,02211 0,00238 -0,01306 -0,02202 -0,00971 0,00877 -0,0649301-06-14 0,00352 -0,02888 -0,03333 0,00078 -0,02432 -0,03534 -0,03220 -0,00378 -0,06547 -0,04287 -0,03479 -0,0340701-06-21 -0,00759 -0,02467 -0,02295 -0,02927 -0,00127 -0,01683 -0,01424 -0,03022 -0,04066 -0,00336 -0,00685 -0,0231101-06-28 -0,01236 -0,01258 0,00326 -0,02864 0,00891 0,01088 0,01282 -0,02706 0,00668 -0,01338 -0,00718 -0,0123001-07-05 0,01669 -0,04657 -0,04006 -0,01295 -0,04286 -0,01945 -0,01959 -0,00104 -0,05446 -0,06923 -0,00753 -0,0693701-07-12 0,00586 0,00458 -0,02233 -0,02458 -0,00080 -0,02823 -0,00102 -0,01197 -0,00052 0,08248 -0,01587 0,05333 01-07-19 0,00350 0,00189 -0,02417 0,00670 -0,02152 -0,00755 -0,01994 -0,01631 -0,02746 -0,05347 -0,00272 -0,0615701-07-26 0,00000 -0,02769 0,00333 -0,01504 -0,02630 -0,04086 -0,01559 -0,02401 -0,02479 -0,02849 -0,03847 -0,0243501-08-02 -0,01394 0,01136 0,00883 0,06406 0,04509 0,01497 0,01146 -0,00643 0,02949 0,04560 -0,00922 0,04280 01-08-09 0,01178 -0,05030 0,02255 -0,00183 -0,04277 -0,01094 -0,02502 -0,01851 -0,00392 -0,02636 -0,01056 -0,0446401-08-16 0,00698 -0,02988 -0,03497 0,00611 0,00190 -0,00167 -0,00931 -0,04325 -0,02852 -0,01265 -0,02712 -0,0115101-08-23 0,00578 0,02435 0,00058 -0,01952 0,00462 0,02165 0,00881 -0,00577 0,03385 0,02174 -0,01673 0,01171 01-08-30 0,00920 0,01832 0,04310 -0,01847 -0,00282 0,03422 0,01003 0,00904 0,02685 0,01748 -0,00922 0,03884 01-09-06 0,01082 0,03800 -0,00392 -0,02251 0,02726 0,01394 -0,00747 -0,00754 0,00736 0,00465 0,04502 0,02863 01-09-13 0,00394 -0,04455 -0,01071 -0,07282 -0,03046 -0,03592 -0,02266 -0,00087 -0,05332 -0,03202 -0,02507 -0,0617101-09-20 0,01403 -0,04040 -0,01124 0,04980 0,00266 -0,00651 0,04154 -0,01265 -0,02299 -0,03188 -0,04519 -0,0641201-09-27 0,01162 -0,02484 -0,00817 -0,01444 -0,00696 -0,00053 0,00959 -0,06518 -0,02585 0,01235 -0,00010 -0,0362101-10-04 0,00274 -0,00570 -0,02222 -0,02455 -0,01398 0,00135 0,00390 0,01177 0,01490 -0,03991 -0,00243 -0,0161401-10-11 0,01037 0,09440 0,10149 0,00124 0,01981 0,02315 0,00559 -0,01145 0,03473 0,06083 0,02646 0,15063 01-10-18 0,01836 0,01320 -0,03187 0,01486 0,00844 0,04858 0,01261 0,03903 -0,00956 0,02500 0,01396 0,04647 01-10-25 -0,00212 0,03431 -0,04269 0,01738 0,01620 0,00329 -0,00914 -0,03341 0,03949 0,05187 0,02212 0,02358 01-11-01 0,01700 0,02174 0,00053 0,00092 0,02643 0,00829 0,02339 0,03832 0,00686 0,03132 0,00467 0,01510 01-11-08 0,00104 0,00630 0,02826 -0,00124 -0,01905 -0,01089 0,02732 -0,00656 0,00719 0,01031 0,01072 0,00015 01-11-15 -0,00626 0,05305 0,02483 0,05910 0,02926 0,01533 0,05329 0,00754 0,02644 0,02091 -0,01433 0,07348 01-11-22 0,01261 -0,01885 0,08495 -0,01740 -0,00526 -0,01569 -0,02742 0,02532 -0,00555 -0,00884 -0,01525 -0,0586201-11-29 -0,00674 -0,01174 -0,02282 -0,03655 0,01422 0,00280 -0,04044 -0,01541 0,01062 -0,03025 -0,05208 0,00208 01-12-06 0,00888 0,01659 0,01905 0,00469 -0,01058 0,00416 0,01315 -0,01721 0,03147 -0,01076 0,03792 0,00125 01-12-13 0,00000 -0,02647 -0,05801 0,00369 -0,02481 -0,04245 0,01422 -0,01697 -0,03957 -0,03221 -0,03486 -0,1025001-12-20 0,00414 -0,01023 0,01134 -0,02430 0,02160 0,02670 0,00099 0,03027 0,00154 -0,04042 0,01140 0,02642 01-12-27 0,00928 -0,00441 0,01096 0,03213 -0,00385 0,00654 0,01676 -0,02575 0,01042 -0,01554 -0,04257 0,00960 02-01-03 0,00613 0,06273 0,00869 0,02802 0,00000 0,05375 0,01993 0,03188 0,05851 0,03209 0,07460 0,05465 02-01-10 0,00203 0,06230 0,00220 0,00410 -0,01286 -0,01173 0,00501 0,01681 0,00402 0,04359 0,05699 0,11039
146
Data risk free Index Portfolio
1 Portfolio
2 Portfolio
3 Portfolio
4 Portfolio
5 Portfolio
6 Portfolio
7 Portfolio
8 Portfolio
9 Portfolio
10 02-01-17 0,00395 0,02554 -0,02632 0,00760 -0,01836 0,02728 -0,01718 0,00926 0,00484 -0,03361 0,01323 0,02960 02-01-24 -0,00959 0,01441 0,04305 -0,00541 -0,02081 0,01940 0,02116 0,01677 -0,03234 -0,00046 0,01035 0,00360 02-01-31 -0,00958 -0,00637 0,00095 0,01083 -0,03013 0,00924 0,01808 0,00162 0,00393 0,00090 -0,00701 -0,08403 02-02-07 0,00309 -0,04636 -0,01579 -0,00980 -0,01638 0,00142 -0,01956 -0,04647 -0,00129 0,01791 -0,03527 0,06212 02-02-14 0,00205 0,00381 -0,00619 -0,04738 -0,00438 -0,00291 -0,00307 -0,01648 0,00411 -0,01449 0,00523 -0,00475 02-02-21 0,00102 -0,01981 0,01975 0,00882 0,00423 -0,01301 -0,01194 -0,01545 -0,00797 -0,01677 -0,01011 -0,02664 02-02-28 0,00409 0,01372 0,02478 0,04916 0,00145 -0,00113 -0,00032 -0,00001 -0,00006 -0,00368 -0,00257 -0,00781 02-03-07 0,00102 0,01100 -0,00194 -0,00251 -0,00463 -0,01982 0,00013 0,00446 -0,00179 0,02440 -0,00364 0,02409 02-03-14 -0,00203 -0,01312 -0,02449 -0,05144 0,00681 0,00762 -0,01893 -0,00528 -0,02676 -0,01490 -0,01934 -0,03120 02-03-21 0,00000 -0,01751 -0,04020 -0,01404 -0,00869 -0,03007 0,00371 -0,01995 -0,05778 0,00169 -0,03772 -0,03929 02-03-28 0,00000 0,00329 -0,00168 -0,01086 -0,00504 0,00919 0,00863 -0,04162 0,02473 -0,00206 -0,00578 -0,00441 02-04-04 0,00306 -0,01747 -0,02254 -0,03989 0,00961 -0,00360 -0,00051 0,03326 -0,04413 -0,07574 -0,02604 -0,03090 02-04-11 0,00203 0,02915 -0,02076 0,00251 0,03923 -0,04527 0,00458 0,01283 -0,01174 0,07283 -0,01158 -0,00194 02-04-18 -0,00152 -0,00918 0,05454 0,00891 -0,01549 0,01977 0,02571 -0,00912 -0,01412 -0,01075 -0,05262 -0,03682 02-04-25 0,00000 -0,00222 0,02385 -0,00852 -0,00829 0,01959 -0,00502 -0,01398 -0,00815 0,00526 0,02853 -0,04240 02-05-02 0,00254 -0,00236 -0,01333 0,00219 0,01715 -0,01803 0,00996 0,01547 0,01345 0,02023 0,01420 0,03300 02-05-09 -0,00051 -0,00333 -0,02921 -0,01652 -0,02686 -0,01932 -0,04721 -0,01918 -0,01827 -0,01337 -0,04377 -0,00840 02-05-16 0,00608 0,03613 -0,02367 -0,01931 0,04249 0,05355 0,02678 0,00261 -0,00004 0,09625 0,07036 0,08730 02-05-23 0,00000 0,00958 0,04633 0,01118 -0,00231 -0,01111 -0,02561 -0,01446 0,00322 0,01471 0,00746 0,02754 02-05-30 0,00957 0,00587 0,01071 -0,00063 -0,01059 0,00503 0,02793 0,00221 -0,02057 0,02289 0,02087 0,00605 02-06-06 0,00000 -0,00148 -0,01352 -0,01471 -0,00126 0,00943 -0,03023 -0,01856 0,00131 0,02867 0,00285 -0,02485 02-06-13 0,00649 -0,01444 -0,03569 0,02336 -0,00937 -0,02115 -0,00087 -0,00992 -0,00828 -0,00416 -0,03339 -0,01941 02-06-20 0,00248 -0,03928 0,05162 -0,03549 0,00419 -0,03354 -0,00858 0,00274 -0,02187 -0,02415 -0,05400 -0,03404 02-06-27 -0,00346 -0,02780 -0,04022 -0,03008 0,02838 -0,04391 -0,01216 -0,02116 -0,00155 -0,01704 -0,04978 -0,02310 02-07-04 0,00050 -0,04232 0,01819 -0,00192 -0,01025 -0,03036 -0,00225 -0,00492 -0,02886 -0,02097 -0,03995 -0,05478 02-07-11 0,00000 -0,00609 -0,01413 -0,02032 -0,01133 -0,02187 0,01554 -0,00641 0,01514 -0,03585 0,00268 -0,02147 02-07-18 0,00000 -0,01026 0,02872 -0,05065 -0,00943 -0,01441 -0,00672 -0,01549 -0,03320 -0,01661 -0,00391 -0,03247 02-07-25 0,00794 -0,07541 -0,02670 -0,05608 -0,03557 -0,01592 -0,00507 -0,02363 -0,04332 -0,06771 -0,10683 -0,12595 02-08-01 -0,00098 0,04056 -0,01573 0,00393 -0,00710 -0,00311 0,02313 -0,02270 0,01563 -0,00878 0,01524 0,05336 02-08-08 0,00197 -0,02815 0,00688 0,00128 -0,00569 0,00413 0,00551 0,06355 0,00849 -0,01585 0,00800 -0,03069 02-08-15 0,00197 0,01464 -0,01855 -0,02836 -0,00014 0,23537 -0,02500 -0,01941 -0,01349 0,02152 0,01983 0,01226 02-08-22 0,00294 0,03245 -0,01253 -0,00194 -0,00694 -0,05957 -0,00698 0,01266 -0,01166 0,03117 0,04115 -0,00478 02-08-29 0,00440 -0,00438 0,00660 -0,01453 0,01690 -0,09465 0,01756 -0,00328 -0,03849 0,00267 0,05481 -0,02469 02-09-05 0,00097 -0,03769 0,01950 -0,03345 0,00088 0,03832 -0,02683 0,00103 -0,02277 -0,01583 -0,04882 -0,13317 02-09-12 0,00389 0,03193 0,05254 -0,01395 -0,00149 0,01307 -0,01446 -0,00443 -0,06709 0,03933 0,03091 0,08439 02-09-19 0,00145 -0,01067 -0,06309 -0,01759 -0,01355 0,00649 -0,00054 -0,02609 0,00871 -0,01404 0,00726 0,00012 02-09-26 -0,00097 -0,00003 0,00230 -0,02767 0,00040 -0,00530 0,01045 -0,01131 -0,02991 0,01445 -0,00626 -0,02367 02-10-03 0,00194 -0,01343 -0,01317 -0,00822 -0,00665 -0,06881 -0,01093 -0,03862 -0,03751 -0,04391 -0,01680 -0,06281 02-10-10 0,00338 0,00238 -0,04571 -0,01357 -0,03220 0,00244 0,00276 0,01025 -0,01628 -0,00857 0,02823 0,04827 02-10-17 0,00530 0,04138 0,09096 0,03946 0,00397 -0,01601 -0,00380 -0,01609 -0,03484 0,04331 0,01300 0,04408 02-10-24 0,00384 0,02257 -0,03232 0,02304 -0,00624 0,01737 -0,00822 0,01203 -0,00802 0,00226 0,01261 -0,00328 02-10-31 0,00764 0,02649 0,03308 0,00857 0,04906 0,03517 0,01259 0,02539 0,00611 0,00550 0,03305 0,04929 02-11-07 0,00047 -0,02323 0,01384 0,04493 -0,00725 0,06091 -0,00734 0,00057 0,01712 0,00074 0,00389 -0,02039 02-11-14 0,00237 0,01309 0,01900 -0,00394 0,01331 0,01607 0,00058 0,01533 -0,02299 0,01126 -0,00156 0,01744 02-11-21 -0,00284 0,03645 0,01554 0,04095 0,00851 -0,03117 0,01867 -0,00822 -0,02969 -0,00241 0,01317 0,03151 02-11-28 0,00190 0,02628 -0,00093 -0,01353 0,02897 0,00595 0,02034 0,00437 0,01945 0,02221 0,01170 0,01362 02-12-05 -0,00378 0,00510 0,00666 -0,01513 0,02174 0,01734 0,01295 0,01640 -0,03165 -0,03621 0,02429 0,01609 02-12-12 -0,00237 -0,02536 0,03029 -0,04215 0,00771 0,04282 0,01332 -0,03061 -0,03411 0,01150 0,00944 -0,02890 02-12-19 0,00714 -0,02723 -0,00611 -0,08219 0,01146 -0,00519 -0,01550 -0,01589 0,01558 -0,09521 -0,02507 -0,05488 02-12-26 -0,00284 0,01121 -0,00490 0,01883 -0,00649 0,01755 0,02487 -0,01643 -0,00088 -0,00070 0,01328 0,01535
147
Appendix No 2 Results of the time-series regression ttFtMpptFtp RRRR εβα +−+=− )( ,,,, Portfolio 1
Dependent Variable: P1-RF Method: Least Squares Date: 03/27/03 Time: 16:55 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003279 0.002367 -1.385033 0.1681
INDEX-RF 0.304299 0.073791 4.123798 0.0001 R-squared 0.101827 Mean dependent var -0.004246 Adjusted R-squared 0.095839 S.D. dependent var 0.030540 S.E. of regression 0.029040 Akaike info criterion -4.227221 Sum squared resid 0.126497 Schwarz criterion -4.187433 Log likelihood 323.2688 F-statistic 17.00571 Durbin-Watson stat 2.171409 Prob(F-statistic) 0.000062
Portfolio 2 Dependent Variable: P2-RF Method: Least Squares Date: 03/27/03 Time: 16:56 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005885 0.002605 -2.259168 0.0253
INDEX-RF 0.326055 0.081199 4.015498 0.0001 R-squared 0.097061 Mean dependent var -0.006922 Adjusted R-squared 0.091042 S.D. dependent var 0.033517 S.E. of regression 0.031955 Akaike info criterion -4.035885 Sum squared resid 0.153172 Schwarz criterion -3.996097 Log likelihood 308.7272 F-statistic 16.12422 Durbin-Watson stat 2.120784 Prob(F-statistic) 0.000093
Portfolio 3 Dependent Variable: P3-RF Method: Least Squares Date: 03/27/03 Time: 16:56 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.001101 0.001994 0.551929 0.5818
INDEX-RF 0.473718 0.062159 7.621019 0.0000 R-squared 0.279123 Mean dependent var -0.000406 Adjusted R-squared 0.274317 S.D. dependent var 0.028716 S.E. of regression 0.024462 Akaike info criterion -4.570293 Sum squared resid 0.089761 Schwarz criterion -4.530506 Log likelihood 349.3423 F-statistic 58.07993 Durbin-Watson stat 1.731651 Prob(F-statistic) 0.000000
148
Portfolio 4 Dependent Variable: P4-RF Method: Least Squares Date: 03/27/03 Time: 16:57 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001054 0.002635 -0.400190 0.6896
INDEX-RF 0.278166 0.082130 3.386901 0.0009 R-squared 0.071041 Mean dependent var -0.001939 Adjusted R-squared 0.064848 S.D. dependent var 0.033424 S.E. of regression 0.032322 Akaike info criterion -4.013088 Sum squared resid 0.156703 Schwarz criterion -3.973300 Log likelihood 306.9947 F-statistic 11.47110 Durbin-Watson stat 2.087148 Prob(F-statistic) 0.000903
Portfolio 5 Dependent Variable: P5-RF Method: Least Squares Date: 03/27/03 Time: 16:57 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000439 0.002282 -0.192162 0.8479
INDEX-RF 0.317400 0.071148 4.461146 0.0000 R-squared 0.117137 Mean dependent var -0.001448 Adjusted R-squared 0.111251 S.D. dependent var 0.029700 S.E. of regression 0.028000 Akaike info criterion -4.300183 Sum squared resid 0.117597 Schwarz criterion -4.260396 Log likelihood 328.8139 F-statistic 19.90183 Durbin-Watson stat 2.421461 Prob(F-statistic) 0.000016
Portfolio 6 Dependent Variable: P6-RF Method: Least Squares Date: 03/27/03 Time: 16:58 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.004437 0.001861 -2.384044 0.0184
INDEX-RF 0.230514 0.058022 3.972888 0.0001 R-squared 0.095207 Mean dependent var -0.005170 Adjusted R-squared 0.089175 S.D. dependent var 0.023926 S.E. of regression 0.022834 Akaike info criterion -4.708060 Sum squared resid 0.078209 Schwarz criterion -4.668272 Log likelihood 359.8126 F-statistic 15.78384 Durbin-Watson stat 1.997001 Prob(F-statistic) 0.000110
Portfolio 7 Dependent Variable: P7-RF Method: Least Squares Date: 03/27/03 Time: 16:59 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005762 0.001951 -2.952685 0.0037
INDEX-RF 0.569277 0.060829 9.358641 0.0000 R-squared 0.368645 Mean dependent var -0.007572 Adjusted R-squared 0.364436 S.D. dependent var 0.030028 S.E. of regression 0.023939 Akaike info criterion -4.613562 Sum squared resid 0.085960 Schwarz criterion -4.573774 Log likelihood 352.6307 F-statistic 87.58416 Durbin-Watson stat 1.931419 Prob(F-statistic) 0.000000
149
Portfolio 8 Dependent Variable: P8-RF Method: Least Squares Date: 03/29/03 Time: 09:19 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.001938 0.002322 -0.834632 0.4053
INDEX-RF 0.574836 0.087238 6.589265 0.0000 R-squared 0.306541 Mean dependent var -0.003766 Adjusted R-squared 0.301918 S.D. dependent var 0.033251 S.E. of regression 0.027782 Akaike info criterion -4.315822 Sum squared resid 0.115772 Schwarz criterion -4.276034 Log likelihood 330.0025 F-statistic 66.30699 Durbin-Watson stat 2.203549 Prob(F-statistic) 0.000000
Portfolio 9 Dependent Variable: P9-RF Method: Least Squares Date: 03/27/03 Time: 16:59 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000650 0.002136 -0.304417 0.7612
INDEX-RF 0.599535 0.066571 9.005915 0.0000 R-squared 0.350949 Mean dependent var -0.002557 Adjusted R-squared 0.346622 S.D. dependent var 0.032411 S.E. of regression 0.026199 Akaike info criterion -4.433151 Sum squared resid 0.102955 Schwarz criterion -4.393363 Log likelihood 338.9195 F-statistic 81.10651 Durbin-Watson stat 1.984412 Prob(F-statistic) 0.000000
Portfolio 10 Dependent Variable: P10-RF Method: Least Squares Date: 03/29/03 Time: 09:23 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.002250 0.001736 -1.296261 0.1969
INDEX-RF 1.074353 0.089372 12.02110 0.0000 R-squared 0.609768 Mean dependent var -0.005667 Adjusted R-squared 0.607166 S.D. dependent var 0.044062 S.E. of regression 0.027617 Akaike info criterion -4.327719 Sum squared resid 0.114403 Schwarz criterion -4.287932 Log likelihood 330.9067 F-statistic 234.3862 Durbin-Watson stat 2.454092 Prob(F-statistic) 0.000000
150
Appendix No 3
Results of the residual tests on the model
ttFtMpptFtp RRRR εβα +−+=− )( ,,,,
1. Autocorrelation
Ho: errors are independent
H1: errors are autocorrelated
Level of significance α=0.05
LaGrange Multiplier with four lags test statistics
Portfolio 1 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.128388 Probability 0.347986 Obs*R-squared 5.692802 Probability 0.337267
Portfolio 2 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.775369 Probability 0.121467 Obs*R-squared 8.768572 Probability 0.118659
Portfolio 3 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.996668 Probability 0.422018 Obs*R-squared 5.050346 Probability 0.409767
Portfolio 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.903403 Probability 0.023887 Obs*R-squared 11.19998 Probability 0.024406
Portfolio 5 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.027994 Probability 0.019604 Obs*R-squared 11.64378 Probability 0.020206
Portfolio 6 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.366132 Probability 0.832440 Obs*R-squared 1.509573 Probability 0.824944
151
Portfolio 7 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.491548 Probability 0.741941 Obs*R-squared 2.019794 Probability 0.732118
Portfolio 8 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.159478 Probability 0.076447 Obs*R-squared 8.490560 Probability 0.075174
Portfolio 9 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.279893 Probability 0.890621 Obs*R-squared 1.156712 Probability 0.885173
Portfolio 10 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.724382 Probability 0.031695 Obs*R-squared 10.55736 Probability 0.032016
2. Heteroskedasticity
Ho: the variance of tε is constant
H1: the variance of tε is not constant
Level of significance α=0.05
χ2 test statistics
Portfolio 1 White Heteroskedasticity Test: F-statistic 1.460309 Probability 0.235468 Obs*R-squared 2.922143 Probability 0.231988
Portfolio 2 White Heteroskedasticity Test: F-statistic 0.401893 Probability 0.669776 Obs*R-squared 0.815569 Probability 0.665122
152
Portfolio 3 White Heteroskedasticity Test: F-statistic 3.064364 Probability 0.049637 Obs*R-squared 6.005121 Probability 0.059660
Portfolio 4 White Heteroskedasticity Test: F-statistic 0.287009 Probability 0.750919 Obs*R-squared 0.583329 Probability 0.747019
Portfolio 5 White Heteroskedasticity Test: F-statistic 0.892155 Probability 0.411949 Obs*R-squared 1.798696 Probability 0.406835
Portfolio 6 White Heteroskedasticity Test: F-statistic 2.079087 Probability 0.128657 Obs*R-squared 4.126730 Probability 0.127026
Portfolio 7 White Heteroskedasticity Test: F-statistic 1.359293 Probability 0.260008 Obs*R-squared 2.723628 Probability 0.256196
Portfolio 8 White Heteroskedasticity Test: F-statistic 6.645796 Probability 0.001719 Obs*R-squared 12.44872 Probability 0.001981
Portfolio 9 White Heteroskedasticity Test: F-statistic 0.207854 Probability 0.812561 Obs*R-squared 0.422898 Probability 0.809411
Portfolio 10 White Heteroskedasticity Test: F-statistic 3.107630 Probability 0.047617 Obs*R-squared 6.086512 Probability 0.047679
153
3. Distribution of the residuals graphically and statistically
Ho: tε have normal distribution
H1: tε have not normal distribution
Level of significance α=0.05
χ2 test statistics
Portfolio 1
0
4
8
12
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -2.16E-18Median -0.002911Maximum 0.085191Minimum -0.057977Std. Dev. 0.028944Skewness 0.552016Kurtosis 3.211695
Jarque-Bera 8.003443Probability 0.018284
Portfolio 2
0
5
10
15
20
25
30
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 3.65E-19Median -0.002456Maximum 0.135440Minimum -0.092975Std. Dev. 0.031849Skewness 0.529423Kurtosis 4.812456
Jarque-Bera 27.90562Probability 0.000001
154
Portfolio 3
0
4
8
12
16
20
24
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 9.02E-19Median -0.002410Maximum 0.075618Minimum -0.068943Std. Dev. 0.024381Skewness 0.343984Kurtosis 3.885111
Jarque-Bera 7.959247Probability 0.018693
Portfolio 4
0
5
10
15
20
25
30
-0.1 0.0 0.1 0.2
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.32E-18Median -0.001229Maximum 0.230936Minimum -0.095557Std. Dev. 0.032214Skewness 2.117807Kurtosis 19.37437
Jarque-Bera 1811.717Probability 0.000000
Portfolio 5
0
4
8
12
16
20
24
28
32
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -6.85E-19Median -0.000562Maximum 0.184610Minimum -0.064348Std. Dev. 0.027907Skewness 1.814073Kurtosis 14.55761
Jarque-Bera 929.3646Probability 0.000000
155
Portfolio 6
0
4
8
12
16
20
24
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 6.39E-19Median -0.001802Maximum 0.089412Minimum -0.074198Std. Dev. 0.022758Skewness 0.352129Kurtosis 5.806068
Jarque-Bera 53.00997Probability 0.000000
Portfolio 7
0
4
8
12
16
20
24
28
32
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 6.66E-18Median 0.001287Maximum 0.139257Minimum -0.081185Std. Dev. 0.023859Skewness 0.907173Kurtosis 9.771453
Jarque-Bera 311.2480Probability 0.000000
Portfolio 8
0
4
8
12
16
20
-0.10 -0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 5.93E-19Median -0.000280Maximum 0.079294Minimum -0.108030Std. Dev. 0.027689Skewness -0.299125Kurtosis 4.767592
Jarque-Bera 22.05448Probability 0.000016
156
Portfolio 9
0
4
8
12
16
20
24
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.85E-18Median 0.000627Maximum 0.115926Minimum -0.064148Std. Dev. 0.026112Skewness 0.505025Kurtosis 5.124037
Jarque-Bera 35.03432Probability 0.000000
Portfolio 10
0
4
8
12
16
20
24
28
32
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -5.71E-20Median 0.000969Maximum 0.114410Minimum -0.090362Std. Dev. 0.027525Skewness 0.009939Kurtosis 5.397909
Jarque-Bera 36.41897Probability 0.000000
157
Appendix No 4
Changes in the estimation output after including errors correction techniques
for portfolios tttp FR εγα ++= ,220,
Portfolio 4
Dependent Variable: P4-RF Method: Least Squares Date: 04/28/03 Time: 10:59 Sample(adjusted): 2/17/2000 12/26/2002 Included observations: 150 after adjusting endpoints Convergence achieved after 7 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.000944 0.002070 -0.455971 0.6491
INDEX-RF 0.319304 0.076968 4.148546 0.0001 AR(2) -0.251764 0.079526 -3.165791 0.0019
R-squared 0.126001 Mean dependent var -0.002036 Adjusted R-squared 0.114109 S.D. dependent var 0.033442 S.E. of regression 0.031476 Akaike info criterion -4.059388 Sum squared resid 0.145638 Schwarz criterion -3.999176 Log likelihood 307.4541 F-statistic 10.59617 Durbin-Watson stat 2.090360 Prob(F-statistic) 0.000050
Portfolio 5 Dependent Variable: P5-RF Method: Least Squares Date: 04/28/03 Time: 11:03 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 5 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.000971 0.001778 -0.546332 0.5857
INDEX-RF 0.271354 0.067812 4.001535 0.0001 AR(1) -0.239612 0.079470 -3.015115 0.0030
R-squared 0.161199 Mean dependent var -0.001957 Adjusted R-squared 0.149864 S.D. dependent var 0.029127 S.E. of regression 0.026856 Akaike info criterion -4.376976 Sum squared resid 0.106745 Schwarz criterion -4.317030 Log likelihood 333.4617 F-statistic 14.22116 Durbin-Watson stat 1.876127 Prob(F-statistic) 0.000002
Portfolio 8
Dependent Variable: P8-RF Method: Least Squares Date: 04/28/03 Time: 11:00 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.001938 0.002322 -0.834632 0.4053
INDEX-RF 0.574836 0.087238 6.589265 0.0000 R-squared 0.306541 Mean dependent var -0.003766 Adjusted R-squared 0.301918 S.D. dependent var 0.033251 S.E. of regression 0.027782 Akaike info criterion -4.315822 Sum squared resid 0.115772 Schwarz criterion -4.276034 Log likelihood 330.0025 F-statistic 66.30699 Durbin-Watson stat 2.203549 Prob(F-statistic) 0.000000
158
Portfolio 10 Dependent Variable: P10-RF Method: Least Squares Date: 04/28/03 Time: 11:01 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 6 iterations Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.002521 0.001657 -1.521716 0.1302
INDEX-RF 1.092972 0.081680 13.38112 0.0000 AR(1) -0.247250 0.072311 -3.419260 0.0008
R-squared 0.632883 Mean dependent var -0.006347 Adjusted R-squared 0.627922 S.D. dependent var 0.043402 S.E. of regression 0.026474 Akaike info criterion -4.405610 Sum squared resid 0.103732 Schwarz criterion -4.345664 Log likelihood 335.6236 F-statistic 127.5707 Durbin-Watson stat 1.998442 Prob(F-statistic) 0.000000
159
Appendix No 5
Table 1: Anti-image matrix
Anti-image
,895 -4,082E- 7,403E- 5,821E- -,278-4,082E- ,925 -,162 -,148 ,1387,403E- -,162 ,958 -5,283E- -2,517E-5,821E- -,148 -5,283E- ,960 -,101
-,278 ,138 -2,517E- -,101 ,874,493 -4,485E- 7,995E- 6,279E- -,315
-4,485E- ,482 -,172 -,157 ,1537,995E- -,172 ,533 -5,508E- -2,752E-6,279E- -,157 -5,508E- ,499 -,110
-,315 ,153 -2,752E- -,110 ,478
VariablesWIG risk freeGOLDexchange S&PWIG risk freeGOLDexchange S&P
AntiAntiAntiAnti----CovarianceCovarianceCovarianceCovariance
AntiAntiAntiAnti----CorrelationCorrelationCorrelationCorrelation
WIGWIGWIGWIG rrrrisk freeisk freeisk freeisk free GOLDGOLDGOLDGOLD exchangexchangexchangexchang
raterateraterate S&PS&PS&PS&P
Source: own calculations
Table 2: Anti-image matrix
Anti-image
.994 3.485E- 7,333E- -2,949E-3.485E- .947 -,162 -,1377.333E- -.162 ,959 -5,645E-
-2.949E- -.137 -5,645E- ,972,489 a 3,592E- 7,514E- -3,001E-
3,592E- ,536 a -,170 -,1437,514E- -,170 ,543 a -5,848E-
-3,001E- -,143 -5,848E- ,560 a
WIGWIGWIGWIG risk freerisk freerisk freerisk freeGOLDGOLDGOLDGOLDexchange exchangeexchangeexchange WIGWIGWIGWIG risk freerisk freerisk freerisk freeGOLDGOLDGOLDGOLDexchange exchange exchange exchange
AntiAntiAntiAnti----CovarianceCovarianceCovarianceCovariance
AntiAntiAntiAnti----CorrelationCorrelationCorrelationCorrelation
WIGWIGWIGWIGriskriskriskriskfreefreefreefree GOLDGOLDGOLDGOLD
exchangexchangexchangexchangraterateraterate
.
Source:own calculations
160
Appendix No 6 Results of the time-series regression tttttp FFFR εγγγα ++++= ,33,22,110, Portfolio 1 Portfolio 2
Dependent Variable: P2 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.004612 0.002556 -1.803965 0.0733 F1 -0.267580 0.177923 -1.503911 0.1347 F2 0.304808 0.123319 2.471697 0.0146 F3 -0.078523 0.163876 -0.479159 0.6325
R-squared 0.065345 Mean dependent var -0.005308 Adjusted R-squared 0.046399 S.D. dependent var 0.031928 S.E. of regression 0.031178 Akaike info criterion -4.072216 Sum squared resid 0.143870 Schwarz criterion -3.992640 Log likelihood 313.4884 F-statistic 3.449067 Durbin-Watson stat 2.129426 Prob(F-statistic) 0.018271
Portfolio 3
Dependent Variable: P3 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.001453 0.001982 0.733273 0.4646 F1 0.018577 0.137911 0.134705 0.8930 F2 0.569930 0.095587 5.962446 0.0000 F3 -0.047475 0.127023 -0.373755 0.7091
R-squared 0.277215 Mean dependent var 0.001207 Adjusted R-squared 0.262564 S.D. dependent var 0.028142 S.E. of regression 0.024167 Akaike info criterion -4.581704 Sum squared resid 0.086438 Schwarz criterion -4.502128 Log likelihood 352.2095 F-statistic 18.92116 Durbin-Watson stat 1.831629 Prob(F-statistic) 0.000000
Dependent Variable: P1 Method: Least Squares Date: 04/25/03 Time: 12:31 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002405 0.002380 -1.010240 0.3140 F1 -0.025368 0.165658 -0.153133 0.8785 F2 0.385412 0.114818 3.356710 0.0010 F3 -0.123005 0.152579 -0.806169 0.4214
R-squared 0.089397 Mean dependent var -0.002633 Adjusted R-squared 0.070939 S.D. dependent var 0.030117 S.E. of regression 0.029029 Akaike info criterion -4.215067 Sum squared resid 0.124719 Schwarz criterion -4.135492 Log likelihood 324.3451 F-statistic 4.843217 Durbin-Watson stat 2.233962 Prob(F-statistic) 0.003036
161
Portfolio 4
Dependent Variable: P4 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000222 0.002621 -0.084754 0.9326 F1 0.025362 0.182445 0.139012 0.8896 F2 0.343685 0.126454 2.717870 0.0074 F3 -0.114360 0.168041 -0.680549 0.4972
R-squared 0.060287 Mean dependent var -0.000326 Adjusted R-squared 0.041238 S.D. dependent var 0.032651 S.E. of regression 0.031971 Akaike info criterion -4.022017 Sum squared resid 0.151277 Schwarz criterion -3.942441 Log likelihood 309.6733 F-statistic 3.164939 Durbin-Watson stat 2.119335 Prob(F-statistic) 0.026342
Portfolio 5
Dependent Variable: P5 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.000913 0.002239 0.407760 0.6840 F1 -0.286785 0.155836 -1.840305 0.0677 F2 0.363908 0.108010 3.369192 0.0010 F3 -0.184273 0.143532 -1.283841 0.2012
R-squared 0.094334 Mean dependent var 0.000165 Adjusted R-squared 0.075976 S.D. dependent var 0.028408 S.E. of regression 0.027308 Akaike info criterion -4.337312 Sum squared resid 0.110367 Schwarz criterion -4.257737 Log likelihood 333.6357 F-statistic 5.138547 Durbin-Watson stat 2.439310 Prob(F-statistic) 0.002079
Portfolio 6
Dependent Variable: P6 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003344 0.001843 -1.814769 0.0716 F1 -0.036810 0.128252 -0.287014 0.7745 F2 0.344764 0.088892 3.878438 0.0002 F3 -0.224909 0.118127 -1.903958 0.0589
R-squared 0.095900 Mean dependent var -0.003557 Adjusted R-squared 0.077574 S.D. dependent var 0.023400 S.E. of regression 0.022474 Akaike info criterion -4.726916 Sum squared resid 0.074755 Schwarz criterion -4.647340 Log likelihood 363.2456 F-statistic 5.232910 Durbin-Watson stat 2.230707 Prob(F-statistic) 0.001842
162
Portfolio 7
Dependent Variable: P7 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005010 0.001915 -2.616154 0.0098 F1 -0.313953 0.133279 -2.355602 0.0198 F2 0.601654 0.092376 6.513072 0.0000 F3 -0.056357 0.122757 -0.459098 0.6468
R-squared 0.327815 Mean dependent var -0.005959 Adjusted R-squared 0.314190 S.D. dependent var 0.028202 S.E. of regression 0.023355 Akaike info criterion -4.650025 Sum squared resid 0.080729 Schwarz criterion -4.570449 Log likelihood 357.4019 F-statistic 24.05921 Durbin-Watson stat 1.875303 Prob(F-statistic) 0.000000
Portfolio 8
Dependent Variable: P8 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000667 0.002278 -0.292996 0.7699 F1 -0.580418 0.158548 -3.660831 0.0003 F2 0.556596 0.109890 5.065011 0.0000 F3 -0.013661 0.146031 -0.093549 0.9256
R-squared 0.277328 Mean dependent var -0.002153 Adjusted R-squared 0.262680 S.D. dependent var 0.032356 S.E. of regression 0.027783 Akaike info criterion -4.302802 Sum squared resid 0.114243 Schwarz criterion -4.223226 Log likelihood 331.0129 F-statistic 18.93188 Durbin-Watson stat 2.229662 Prob(F-statistic) 0.000000
Portfolio 9
Dependent Variable: P9 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.000293 0.002114 0.138684 0.8899 F1 -0.446433 0.147144 -3.033983 0.0029 F2 0.641715 0.101986 6.292173 0.0000 F3 -0.096088 0.135527 -0.708995 0.4794
R-squared 0.314343 Mean dependent var -0.000943 Adjusted R-squared 0.300445 S.D. dependent var 0.030829 S.E. of regression 0.025785 Akaike info criterion -4.452093 Sum squared resid 0.098399 Schwarz criterion -4.372517 Log likelihood 342.3591 F-statistic 22.61717 Durbin-Watson stat 2.029890 Prob(F-statistic) 0.000000
163
Portfolio 10
Dependent Variable: P10 Method: Least Squares Date: 04/25/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002129 0.002288 -0.930508 0.3536 F1 -0.649553 0.159272 -4.078258 0.0001 F2 1.017440 0.110392 9.216588 0.0000 F3 0.317883 0.146697 2.166932 0.0318
R-squared 0.597561 Mean dependent var -0.004054 Adjusted R-squared 0.589404 S.D. dependent var 0.043557 S.E. of regression 0.027910 Akaike info criterion -4.293689 Sum squared resid 0.115289 Schwarz criterion -4.214113 Log likelihood 330.3203 F-statistic 73.25259 Durbin-Watson stat 2.509384 Prob(F-statistic) 0.000000
164
Appendix No 7
Results of the test on the redundant variables F1 and F3 in the model: tttttp FFFR εγγγα ++++= ,33,22,110,
Ho: 1γ = 3γ =0, both variables are redundant H1: Either or both 1γ and 3γ are not equal to 0 Level of significance α=0.05 F test statistics
Portfolio 1
Redundant Variables: F1 F3 F-statistic 0.325047 Probability 0.723008 Log likelihood ratio 0.666202 Probability 0.716698
Test Equation: Dependent Variable: P1 Method: Least Squares Date: 04/25/03 Time: 14:19 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002466 0.002344 -1.051815 0.2946 F2 0.325300 0.086923 3.742406 0.0003
R-squared 0.085397 Mean dependent var -0.002633 Adjusted R-squared 0.079300 S.D. dependent var 0.030117 S.E. of regression 0.028898 Akaike info criterion -4.237000 Sum squared resid 0.125266 Schwarz criterion -4.197212 Log likelihood 324.0120 F-statistic 14.00560 Durbin-Watson stat 2.226116 Prob(F-statistic) 0.000259
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant. Portfolio 2
Redundant Variables: F1 F3 F-statistic 1.155480 Probability 0.317730 Log likelihood ratio 2.355080 Probability 0.308036
Test Equation: Dependent Variable: P2 Method: Least Squares Date: 04/25/03 Time: 14:25 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005172 0.002532 -2.042522 0.0429 F2 0.265853 0.093878 2.831892 0.0053
R-squared 0.050751 Mean dependent var -0.005308 Adjusted R-squared 0.044422 S.D. dependent var 0.031928 S.E. of regression 0.031211 Akaike info criterion -4.083038 Sum squared resid 0.146117 Schwarz criterion -4.043250 Log likelihood 312.3109 F-statistic 8.019615 Durbin-Watson stat 2.137451 Prob(F-statistic) 0.005263
165
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant. Portfolio 3
Redundant Variables: F1 F3 F-statistic 0.090361 Probability 0.913652 Log likelihood ratio 0.185492 Probability 0.911425
Test Equation: Dependent Variable: P3 Method: Least Squares Date: 04/25/03 Time: 14:25 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.001488 0.001949 0.763667 0.4463 F2 0.546794 0.072249 7.568186 0.0000
R-squared 0.276332 Mean dependent var 0.001207 Adjusted R-squared 0.271508 S.D. dependent var 0.028142 S.E. of regression 0.024020 Akaike info criterion -4.606799 Sum squared resid 0.086543 Schwarz criterion -4.567012 Log likelihood 352.1168 F-statistic 57.27744 Durbin-Watson stat 1.812175 Prob(F-statistic) 0.000000
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant. Portfolio 4
Redundant Variables: F1 F3 F-statistic 0.265614 Probability 0.767099 Log likelihood ratio 0.544609 Probability 0.761622
Test Equation: Dependent Variable: P4 Method: Least Squares Date: 04/25/03 Time: 14:26 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000178 0.002581 -0.068874 0.9452 F2 0.287911 0.095693 3.008692 0.0031
R-squared 0.056914 Mean dependent var -0.000326 Adjusted R-squared 0.050626 S.D. dependent var 0.032651 S.E. of regression 0.031814 Akaike info criterion -4.044750 Sum squared resid 0.151820 Schwarz criterion -4.004962 Log likelihood 309.4010 F-statistic 9.052227 Durbin-Watson stat 2.135425 Prob(F-statistic) 0.003078
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant.
166
Portfolio 5
Redundant Variables: F1 F3 F-statistic 2.173248 Probability 0.117426 Log likelihood ratio 4.399675 Probability 0.110821
Test Equation: Dependent Variable: P5 Method: Least Squares Date: 04/25/03 Time: 14:26 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.000306 0.002233 0.136930 0.8913 F2 0.273280 0.082779 3.301311 0.0012
R-squared 0.067736 Mean dependent var 0.000165 Adjusted R-squared 0.061521 S.D. dependent var 0.028408 S.E. of regression 0.027521 Akaike info criterion -4.334683 Sum squared resid 0.113609 Schwarz criterion -4.294895 Log likelihood 331.4359 F-statistic 10.89865 Durbin-Watson stat 2.492303 Prob(F-statistic) 0.001203
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant. Portfolio 6
Redundant Variables: F1 F3 F-statistic 1.813482 Probability 0.166690 Log likelihood ratio 3.680080 Probability 0.158811
Test Equation: Dependent Variable: P6 Method: Least Squares Date: 04/25/03 Time: 14:26 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003436 0.001833 -1.874631 0.0628 F2 0.234874 0.067966 3.455754 0.0007
R-squared 0.073744 Mean dependent var -0.003557 Adjusted R-squared 0.067569 S.D. dependent var 0.023400 S.E. of regression 0.022596 Akaike info criterion -4.729021 Sum squared resid 0.076587 Schwarz criterion -4.689233 Log likelihood 361.4056 F-statistic 11.94223 Durbin-Watson stat 2.242984 Prob(F-statistic) 0.000714
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant.
167
Portfolio 7
Redundant Variables: F1 F3 F-statistic 2.775760 Probability 0.065548 Log likelihood ratio 5.597229 Probability 0.060894
Test Equation: Dependent Variable: P7 Method: Least Squares Date: 04/25/03 Time: 14:27 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005664 0.001917 -2.954767 0.0036 F2 0.573414 0.071077 8.067534 0.0000
R-squared 0.302602 Mean dependent var -0.005959 Adjusted R-squared 0.297952 S.D. dependent var 0.028202 S.E. of regression 0.023630 Akaike info criterion -4.639517 Sum squared resid 0.083758 Schwarz criterion -4.599729 Log likelihood 354.6033 F-statistic 65.08511 Durbin-Watson stat 1.952596 Prob(F-statistic) 0.000000
Conclusion: There is no sufficient evidence to reject Ho, both variables are redundant. Portfolio 8
Redundant Variables: F1 F3 F-statistic 6.851716 Probability 0.001426 Log likelihood ratio 13.45986 Probability 0.001195
Test Equation: Dependent Variable: P8 Method: Least Squares Date: 04/25/03 Time: 14:27 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001871 0.002340 -0.799487 0.4253 F2 0.548586 0.086768 6.322451 0.0000
R-squared 0.210416 Mean dependent var -0.002153 Adjusted R-squared 0.205152 S.D. dependent var 0.032356 S.E. of regression 0.028847 Akaike info criterion -4.240566 Sum squared resid 0.124821 Schwarz criterion -4.200778 Log likelihood 324.2830 F-statistic 39.97338 Durbin-Watson stat 2.298967 Prob(F-statistic) 0.000000
Conclusion: There is sufficient evidence to reject Ho, either or both variables are significant.
168
Portfolio 9
Redundant Variables: F1 F3 F-statistic 4.619808 Probability 0.011318 Log likelihood ratio 9.204904 Probability 0.010027
Test Equation: Dependent Variable: P9 Method: Least Squares Date: 04/25/03 Time: 14:27 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000639 0.002142 -0.298163 0.7660 F2 0.593772 0.079408 7.477518 0.0000
R-squared 0.271538 Mean dependent var -0.000943 Adjusted R-squared 0.266682 S.D. dependent var 0.030829 S.E. of regression 0.026400 Akaike info criterion -4.417850 Sum squared resid 0.104542 Schwarz criterion -4.378062 Log likelihood 337.7566 F-statistic 55.91328 Durbin-Watson stat 2.031500 Prob(F-statistic) 0.000000
Conclusion: There is sufficient evidence to reject Ho, either or both variables are significant. Portfolio 10
Redundant Variables: F1 F3 F-statistic 12.57348 Probability 0.000009 Log likelihood ratio 23.85313 Probability 0.000007
Test Equation: Dependent Variable: P10 Method: Least Squares Date: 04/25/03 Time: 14:24 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003453 0.002433 -1.419261 0.1579 F2 1.171136 0.090196 12.98439 0.0000
R-squared 0.529182 Mean dependent var -0.004054 Adjusted R-squared 0.526043 S.D. dependent var 0.043557 S.E. of regression 0.029986 Akaike info criterion -4.163076 Sum squared resid 0.134878 Schwarz criterion -4.123288 Log likelihood 318.3938 F-statistic 168.5945 Durbin-Watson stat 2.391936 Prob(F-statistic) 0.000000
Conclusion: There is sufficient evidence to reject Ho, either or both variables are significant.
169
Appendix No 8
Results of the residual tests on the model
tttttp FFFR εγγγα ++++= ,33,22,110,
4. Autocorrelation
Ho: errors are independent
H1: errors are autocorrelated
Level of significance α=0.05
LaGrange Multiplier with four lags test statistics
Portfolio 1 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.230697 Probability 0.300481 Obs*R-squared 5.024507 Probability 0.284792
Portfolio 2 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.857992 Probability 0.025718 Obs*R-squared 11.17955 Probability 0.024619
Portfolio 3 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.213874 Probability 0.307529 Obs*R-squared 4.958067 Probability 0.291627
Portfolio 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.158425 Probability 0.015971 Obs*R-squared 12.25996 Probability 0.015519
Portfolio 5 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.908010 Probability 0.023762 Obs*R-squared 11.36058 Probability 0.022797
Portfolio 6 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.703692 Probability 0.590647 Obs*R-squared 2.914182 Probability 0.572288
170
Portfolio 7 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.529102 Probability 0.714520 Obs*R-squared 2.201630 Probability 0.698731
Portfolio 8 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.789412 Probability 0.134124 Obs*R-squared 7.197535 Probability 0.125810
Portfolio 9 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.399189 Probability 0.808982 Obs*R-squared 1.666979 Probability 0.796707
Portfolio 10 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.350398 Probability 0.011764 Obs*R-squared 12.94169 Probability 0.011564
5. Heteroskedasticity
Ho: the variance of tε is constant
H1: the variance of tε is not constant
Level of significance α=0.05
χ2 test statistics
Portfolio 1 White Heteroskedasticity Test: F-statistic 1.318419 Probability 0.252544 Obs*R-squared 7.863411 Probability 0.248282
Portfolio 2 White Heteroskedasticity Test: F-statistic 0.531442 Probability 0.783704 Obs*R-squared 3.270661 Probability 0.774189
171
Portfolio 3 White Heteroskedasticity Test: F-statistic 2.381517 Probability 0.031787 Obs*R-squared 13.63523 Probability 0.033987
Portfolio 4 White Heteroskedasticity Test: F-statistic 0.420098 Probability 0.864734 Obs*R-squared 2.597127 Probability 0.857443
Portfolio 5 White Heteroskedasticity Test: F-statistic 0.192120 Probability 0.978629 Obs*R-squared 1.198838 Probability 0.976942
Portfolio 6 White Heteroskedasticity Test: F-statistic 0.959063 Probability 0.455233 Obs*R-squared 5.801926 Probability 0.445740
Portfolio 7 White Heteroskedasticity Test: F-statistic 0.599668 Probability 0.730247 Obs*R-squared 3.680383 Probability 0.719836
Portfolio 8 White Heteroskedasticity Test: F-statistic 2.511336 Probability 0.024239 Obs*R-squared 14.30853 Probability 0.026373
Portfolio 9 White Heteroskedasticity Test: F-statistic 0.731242 Probability 0.625184 Obs*R-squared 4.464184 Probability 0.614122
Portfolio 10 White Heteroskedasticity Test: F-statistic 1.076716 Probability 0.379176 Obs*R-squared 6.483314 Probability 0.371278
172
6. Distribution of the residuals graphically and statistically
Ho: tε have normal distribution
H1: tε have not normal distribution
Level of significance α=0.05
χ2 test statistics
Portfolio 1
0
2
4
6
8
10
12
14
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.75E-18Median -0.002448Maximum 0.091168Minimum -0.067661Std. Dev. 0.028739Skewness 0.645623Kurtosis 3.882293
Jarque-Bera 15.48980Probability 0.000433
Portfolio 2
0
4
8
12
16
20
24
28
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 2.92E-18Median -0.001358Maximum 0.134339Minimum -0.084265Std. Dev. 0.030867Skewness 0.543737Kurtosis 5.062900
Jarque-Bera 34.44166Probability 0.000000
173
Portfolio 3
0
4
8
12
16
20
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.95E-18Median -0.002441Maximum 0.073214Minimum -0.066815Std. Dev. 0.023926Skewness 0.450151Kurtosis 3.958209
Jarque-Bera 10.94848Probability 0.004193
Portfolio 4
0
4
8
12
16
20
24
28
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.51E-18Median 0.000489Maximum 0.229060Minimum -0.095902Std. Dev. 0.031652Skewness 2.152003Kurtosis 20.14395
Jarque-Bera 1978.783Probability 0.000000
Portfolio 5
0
5
10
15
20
25
30
35
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -9.59E-19Median -0.000110Maximum 0.186205Minimum -0.065530Std. Dev. 0.027035Skewness 2.031336Kurtosis 16.91937
Jarque-Bera 1331.609Probability 0.000000
174
Portfolio 6
0
4
8
12
16
20
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.60E-19Median 3.85E-05Maximum 0.088406Minimum -0.085643Std. Dev. 0.022250Skewness 0.351991Kurtosis 5.973343
Jarque-Bera 59.13030Probability 0.000000
Portfolio 7
0
4
8
12
16
20
24
28
32
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 7.85E-19Median -0.000272Maximum 0.142107Minimum -0.077373Std. Dev. 0.023122Skewness 1.162156Kurtosis 11.51972
Jarque-Bera 493.9244Probability 0.000000
Portfolio 8
0
4
8
12
16
20
24
28
32
-0.10 -0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -5.71E-20Median 0.001506Maximum 0.080765Minimum -0.118237Std. Dev. 0.027506Skewness -0.201049Kurtosis 5.180095
Jarque-Bera 31.12514Probability 0.000000
175
Portfolio 9
0
4
8
12
16
20
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.34E-18Median -0.001641Maximum 0.115874Minimum -0.066109Std. Dev. 0.025527Skewness 0.629736Kurtosis 5.381907
Jarque-Bera 45.97843Probability 0.000000
Portfolio 10
0
5
10
15
20
25
30
35
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.39E-18Median 0.000806Maximum 0.107117Minimum -0.090970Std. Dev. 0.027632Skewness -0.085925Kurtosis 5.025290
Jarque-Bera 26.16510Probability 0.000002
176
Appendix No 9
Changes in the estimation output after including errors correction techniques for portfolios tttttp FFFR εγγγα ++++= ,33,22,110,
Portfolio 2 Dependent Variable: P2 Method: Least Squares Date: 04/27/03 Time: 11:58 Sample(adjusted): 2/17/2000 12/26/2002 Included observations: 150 after adjusting endpoints Convergence achieved after 7 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.004227 0.001947 -2.170725 0.0316 F1 -0.441216 0.189685 -2.326050 0.0214 F2 0.443532 0.116142 3.818865 0.0002 F3 -0.240978 0.169701 -1.420015 0.1577
AR(2) -0.292197 0.082286 -3.551009 0.0005 R-squared 0.130171 Mean dependent var -0.005365 Adjusted R-squared 0.106176 S.D. dependent var 0.032105 S.E. of regression 0.030353 Akaike info criterion -4.119087 Sum squared resid 0.133588 Schwarz criterion -4.018733 Log likelihood 313.9315 F-statistic 5.424870 Durbin-Watson stat 2.134371 Prob(F-statistic) 0.000424
Portfolio 3 Dependent Variable: P3 Method: Least Squares Date: 04/27/03 Time: 12:07 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C 0.001453 0.002277 0.638240 0.5243 F1 0.018577 0.163333 0.113738 0.9096 F2 0.569930 0.111925 5.092062 0.0000 F3 -0.047475 0.133874 -0.354626 0.7234
R-squared 0.277215 Mean dependent var 0.001207 Adjusted R-squared 0.262564 S.D. dependent var 0.028142 S.E. of regression 0.024167 Akaike info criterion -4.581704 Sum squared resid 0.086438 Schwarz criterion -4.502128 Log likelihood 352.2095 F-statistic 18.92116 Durbin-Watson stat 1.831629 Prob(F-statistic) 0.000000
Portfolio 4 Dependent Variable: P4 Method: Least Squares Date: 04/27/03 Time: 12:06 Sample(adjusted): 2/17/2000 12/26/2002 Included observations: 150 after adjusting endpoints Convergence achieved after 8 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.000234 0.002075 -0.112664 0.9105 F1 -0.004101 0.194645 -0.021067 0.9832 F2 0.370791 0.119380 3.105966 0.0023 F3 -0.076247 0.174904 -0.435935 0.6635
AR(2) -0.245680 0.080786 -3.041131 0.0028 R-squared 0.113468 Mean dependent var -0.000597 Adjusted R-squared 0.089012 S.D. dependent var 0.032713 S.E. of regression 0.031223 Akaike info criterion -4.062527 Sum squared resid 0.141361 Schwarz criterion -3.962173 Log likelihood 309.6896 F-statistic 4.639671 Durbin-Watson stat 2.138995 Prob(F-statistic) 0.001489
177
Portfolio 5 Dependent Variable: P5 Method: Least Squares Date: 04/27/03 Time: 11:56 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 5 iterations
Variable Coefficient Std. Error t-Statistic Prob. C 0.000284 0.001715 0.165429 0.8688 F1 -0.183777 0.148259 -1.239563 0.2171 F2 0.309504 0.099968 3.096037 0.0024 F3 -0.144080 0.136542 -1.055208 0.2931
AR(1) -0.255977 0.080109 -3.195379 0.0017 R-squared 0.147579 Mean dependent var -0.000346 Adjusted R-squared 0.124225 S.D. dependent var 0.027792 S.E. of regression 0.026008 Akaike info criterion -4.428255 Sum squared resid 0.098759 Schwarz criterion -4.328345 Log likelihood 339.3333 F-statistic 6.319203 Durbin-Watson stat 1.958521 Prob(F-statistic) 0.000102 Inverted AR Roots -.26
Portfolio 8 Dependent Variable: P8 Method: Least Squares Date: 04/27/03 Time: 12:07 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.000667 0.002201 -0.303270 0.7621 F1 -0.580418 0.188491 -3.079286 0.0025 F2 0.556596 0.138224 4.026762 0.0001 F3 -0.013661 0.200942 -0.067985 0.9459
R-squared 0.277328 Mean dependent var -0.002153 Adjusted R-squared 0.262680 S.D. dependent var 0.032356 S.E. of regression 0.027783 Akaike info criterion -4.302802 Sum squared resid 0.114243 Schwarz criterion -4.223226 Log likelihood 331.0129 F-statistic 18.93188 Durbin-Watson stat 2.229662 Prob(F-statistic) 0.000000
Portfolio 10 Dependent Variable: P10 Method: Least Squares Date: 04/27/03 Time: 11:57 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 7 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.002491 0.001714 -1.453504 0.1482F1 -0.624755 0.148029 -4.220500 0.0000F2 0.996932 0.101094 9.861402 0.0000F3 0.405940 0.138028 2.940997 0.0038
AR(1) -0.282885 0.079962 -3.537753 0.0005R-squared 0.627337 Mean dependent var -0.004737Adjusted R-squared 0.617127 S.D. dependent var 0.042878S.E. of regression 0.026532 Akaike info criterion -4.388414Sum squared resid 0.102773 Schwarz criterion -4.288504Log likelihood 336.3253 F-statistic 61.44379Durbin-Watson stat 1.971545 Prob(F-statistic) 0.000000Inverted AR Roots -.28
178
Appendix No 10 Results of the time-series regression ttttp FFR εγγα +++= ,22,110, Portfolio 1
Dependent Variable: P1 Method: Least Squares Date: 04/27/03 Time: 14:43 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002461 0.002376 -1.035735 0.3020 F1 -0.002219 0.162959 -0.013617 0.9892 F2 0.325305 0.087215 3.729936 0.0003
R-squared 0.085398 Mean dependent var -0.002633 Adjusted R-squared 0.073122 S.D. dependent var 0.030117 S.E. of regression 0.028995 Akaike info criterion -4.223844 Sum squared resid 0.125266 Schwarz criterion -4.164162 Log likelihood 324.0121 F-statistic 6.956216 Durbin-Watson stat 2.225976 Prob(F-statistic) 0.001294
Portfolio 2
Dependent Variable: P2 Method: Least Squares Date: 04/27/03 Time: 14:42 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.004648 0.002549 -1.823641 0.0702 F1 -0.252803 0.174776 -1.446439 0.1502 F2 0.266438 0.093539 2.848401 0.0050
R-squared 0.063895 Mean dependent var -0.005308 Adjusted R-squared 0.051330 S.D. dependent var 0.031928 S.E. of regression 0.031098 Akaike info criterion -4.083824 Sum squared resid 0.144093 Schwarz criterion -4.024142 Log likelihood 313.3706 F-statistic 5.085096 Durbin-Watson stat 2.124975 Prob(F-statistic) 0.007306
Portfolio 3
Dependent Variable: P3 Method: Least Squares Date: 04/27/03 Time: 14:43 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.001431 0.001975 0.724638 0.4698 F1 0.027512 0.135430 0.203143 0.8393 F2 0.546731 0.072482 7.543014 0.0000
R-squared 0.276533 Mean dependent var 0.001207 Adjusted R-squared 0.266822 S.D. dependent var 0.028142 S.E. of regression 0.024097 Akaike info criterion -4.593919 Sum squared resid 0.086519 Schwarz criterion -4.534237 Log likelihood 352.1378 F-statistic 28.47631 Durbin-Watson stat 1.818690 Prob(F-statistic) 0.000000
179
Portfolio 4
Dependent Variable: P4 Method: Least Squares Date: 04/27/03 Time: 14:44 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000275 0.002616 -0.105102 0.9164 F1 0.046884 0.179360 0.261396 0.7941 F2 0.287802 0.095992 2.998177 0.0032
R-squared 0.057346 Mean dependent var -0.000326 Adjusted R-squared 0.044693 S.D. dependent var 0.032651 S.E. of regression 0.031913 Akaike info criterion -4.032050 Sum squared resid 0.151750 Schwarz criterion -3.972369 Log likelihood 309.4358 F-statistic 4.532165 Durbin-Watson stat 2.134060 Prob(F-statistic) 0.012282
Portfolio 5
Dependent Variable: P5 Method: Least Squares Date: 04/27/03 Time: 14:44 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.000828 0.002243 0.369177 0.7125 F1 -0.252106 0.153810 -1.639073 0.1033 F2 0.273863 0.082319 3.326863 0.0011
R-squared 0.084248 Mean dependent var 0.000165 Adjusted R-squared 0.071956 S.D. dependent var 0.028408 S.E. of regression 0.027367 Akaike info criterion -4.339395 Sum squared resid 0.111597 Schwarz criterion -4.279713 Log likelihood 332.7940 F-statistic 6.853878 Durbin-Watson stat 2.461815 Prob(F-statistic) 0.001421
Portfolio 6
Dependent Variable: P6 Method: Least Squares Date: 04/27/03 Time: 14:44 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003448 0.001858 -1.855581 0.0655 F1 0.005516 0.127419 0.043291 0.9655 F2 0.234861 0.068194 3.444018 0.0007
R-squared 0.073755 Mean dependent var -0.003557 Adjusted R-squared 0.061323 S.D. dependent var 0.023400 S.E. of regression 0.022672 Akaike info criterion -4.715876 Sum squared resid 0.076586 Schwarz criterion -4.656194 Log likelihood 361.4065 F-statistic 5.932321 Durbin-Watson stat 2.240875 Prob(F-statistic) 0.003319
180
Portfolio 7
Dependent Variable: P7 Method: Least Squares Date: 04/27/03 Time: 14:44 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005036 0.001909 -2.637866 0.0092 F1 -0.303347 0.130914 -2.317151 0.0219 F2 0.574115 0.070064 8.194112 0.0000
R-squared 0.326858 Mean dependent var -0.005959 Adjusted R-squared 0.317823 S.D. dependent var 0.028202 S.E. of regression 0.023293 Akaike info criterion -4.661760 Sum squared resid 0.080844 Schwarz criterion -4.602078 Log likelihood 357.2937 F-statistic 36.17505 Durbin-Watson stat 1.889881 Prob(F-statistic) 0.000000
Portfolio 8
Dependent Variable: P8 Method: Least Squares Date: 04/27/03 Time: 14:45 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000674 0.002269 -0.296879 0.7670 F1 -0.577847 0.155628 -3.713004 0.0003 F2 0.549921 0.083291 6.602379 0.0000
R-squared 0.277286 Mean dependent var -0.002153 Adjusted R-squared 0.267585 S.D. dependent var 0.032356 S.E. of regression 0.027691 Akaike info criterion -4.315901 Sum squared resid 0.114249 Schwarz criterion -4.256219 Log likelihood 331.0084 F-statistic 28.58361 Durbin-Watson stat 2.230032 Prob(F-statistic) 0.000000
Portfolio 9
Dependent Variable: P9 Method: Least Squares Date: 04/27/03 Time: 14:45 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.000249 0.002110 0.117980 0.9062 F1 -0.428349 0.144675 -2.960777 0.0036 F2 0.594761 0.077429 7.681358 0.0000
R-squared 0.312015 Mean dependent var -0.000943 Adjusted R-squared 0.302780 S.D. dependent var 0.030829 S.E. of regression 0.025742 Akaike info criterion -4.461860 Sum squared resid 0.098734 Schwarz criterion -4.402179 Log likelihood 342.1014 F-statistic 33.78718 Durbin-Watson stat 2.039310 Prob(F-statistic) 0.000000
181
Portfolio 10
Dependent Variable: P10 Method: Least Squares Date: 04/27/03 Time: 14:45 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001983 0.002316 -0.856292 0.3932 F1 -0.709376 0.158795 -4.467258 0.0000 F2 1.172775 0.084986 13.79960 0.0000
R-squared 0.584793 Mean dependent var -0.004054 Adjusted R-squared 0.579220 S.D. dependent var 0.043557 S.E. of regression 0.028254 Akaike info criterion -4.275612 Sum squared resid 0.118946 Schwarz criterion -4.215931 Log likelihood 327.9465 F-statistic 104.9286 Durbin-Watson stat 2.431763 Prob(F-statistic) 0.000000
182
Appendix No 11 Results of the time-series regression tttp FR εγα ++= ,220, Portfolio 1
Dependent Variable: P1 Method: Least Squares Date: 04/27/03 Time: 12:30 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002466 0.002344 -1.051815 0.2946 F2 0.325300 0.086923 3.742406 0.0003
R-squared 0.085397 Mean dependent var -0.002633 Adjusted R-squared 0.079300 S.D. dependent var 0.030117 S.E. of regression 0.028898 Akaike info criterion -4.237000 Sum squared resid 0.125266 Schwarz criterion -4.197212 Log likelihood 324.0120 F-statistic 14.00560 Durbin-Watson stat 2.226116 Prob(F-statistic) 0.000259
Portfolio 2 Dependent Variable: P2 Method: Least Squares Date: 04/27/03 Time: 12:31 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005172 0.002532 -2.042522 0.0429 F2 0.265853 0.093878 2.831892 0.0053
R-squared 0.050751 Mean dependent var -0.005308 Adjusted R-squared 0.044422 S.D. dependent var 0.031928 S.E. of regression 0.031211 Akaike info criterion -4.083038 Sum squared resid 0.146117 Schwarz criterion -4.043250 Log likelihood 312.3109 F-statistic 8.019615 Durbin-Watson stat 2.137451 Prob(F-statistic) 0.005263
Portfolio 3
Dependent Variable: P3 Method: Least Squares Date: 04/27/03 Time: 12:31 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.001488 0.001949 0.763667 0.4463 F2 0.546794 0.072249 7.568186 0.0000
R-squared 0.276332 Mean dependent var 0.001207 Adjusted R-squared 0.271508 S.D. dependent var 0.028142 S.E. of regression 0.024020 Akaike info criterion -4.606799 Sum squared resid 0.086543 Schwarz criterion -4.567012 Log likelihood 352.1168 F-statistic 57.27744 Durbin-Watson stat 1.812175 Prob(F-statistic) 0.000000
183
Portfolio 4 Dependent Variable: P4 Method: Least Squares Date: 04/27/03 Time: 12:31 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000178 0.002581 -0.068874 0.9452 F2 0.287911 0.095693 3.008692 0.0031
R-squared 0.056914 Mean dependent var -0.000326 Adjusted R-squared 0.050626 S.D. dependent var 0.032651 S.E. of regression 0.031814 Akaike info criterion -4.044750 Sum squared resid 0.151820 Schwarz criterion -4.004962 Log likelihood 309.4010 F-statistic 9.052227 Durbin-Watson stat 2.135425 Prob(F-statistic) 0.003078
Portfolio 5 Dependent Variable: P5 Method: Least Squares Date: 04/27/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C 0.000306 0.002233 0.136930 0.8913 F2 0.273280 0.082779 3.301311 0.0012
R-squared 0.067736 Mean dependent var 0.000165 Adjusted R-squared 0.061521 S.D. dependent var 0.028408 S.E. of regression 0.027521 Akaike info criterion -4.334683 Sum squared resid 0.113609 Schwarz criterion -4.294895 Log likelihood 331.4359 F-statistic 10.89865 Durbin-Watson stat 2.492303 Prob(F-statistic) 0.001203
Portfolio 6 Dependent Variable: P6 Method: Least Squares Date: 04/27/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003436 0.001833 -1.874631 0.0628 F2 0.234874 0.067966 3.455754 0.0007
R-squared 0.073744 Mean dependent var -0.003557 Adjusted R-squared 0.067569 S.D. dependent var 0.023400 S.E. of regression 0.022596 Akaike info criterion -4.729021 Sum squared resid 0.076587 Schwarz criterion -4.689233 Log likelihood 361.4056 F-statistic 11.94223 Durbin-Watson stat 2.242984 Prob(F-statistic) 0.000714
184
Portfolio 7 Dependent Variable: P7 Method: Least Squares Date: 04/27/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.005664 0.001917 -2.954767 0.0036 F2 0.573414 0.071077 8.067534 0.0000
R-squared 0.302602 Mean dependent var -0.005959 Adjusted R-squared 0.297952 S.D. dependent var 0.028202 S.E. of regression 0.023630 Akaike info criterion -4.639517 Sum squared resid 0.083758 Schwarz criterion -4.599729 Log likelihood 354.6033 F-statistic 65.08511 Durbin-Watson stat 1.952596 Prob(F-statistic) 0.000000
Portfolio 8
Dependent Variable: P8 Method: Least Squares Date: 04/27/03 Time: 12:32 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001871 0.002340 -0.799487 0.4253 F2 0.548586 0.086768 6.322451 0.0000
R-squared 0.210416 Mean dependent var -0.002153 Adjusted R-squared 0.205152 S.D. dependent var 0.032356 S.E. of regression 0.028847 Akaike info criterion -4.240566 Sum squared resid 0.124821 Schwarz criterion -4.200778 Log likelihood 324.2830 F-statistic 39.97338 Durbin-Watson stat 2.298967 Prob(F-statistic) 0.000000
Portfolio 9
Dependent Variable: P9 Method: Least Squares Date: 04/27/03 Time: 12:33 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000639 0.002142 -0.298163 0.7660 F2 0.593772 0.079408 7.477518 0.0000
R-squared 0.271538 Mean dependent var -0.000943 Adjusted R-squared 0.266682 S.D. dependent var 0.030829 S.E. of regression 0.026400 Akaike info criterion -4.417850 Sum squared resid 0.104542 Schwarz criterion -4.378062 Log likelihood 337.7566 F-statistic 55.91328 Durbin-Watson stat 2.031500 Prob(F-statistic) 0.000000
Portfolio 10
Dependent Variable: P10 Method: Least Squares Date: 04/27/03 Time: 12:33 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003453 0.002433 -1.419261 0.1579 F2 1.171136 0.090196 12.98439 0.0000
R-squared 0.529182 Mean dependent var -0.004054 Adjusted R-squared 0.526043 S.D. dependent var 0.043557 S.E. of regression 0.029986 Akaike info criterion -4.163076 Sum squared resid 0.134878 Schwarz criterion -4.123288 Log likelihood 318.3938 F-statistic 168.5945 Durbin-Watson stat 2.391936 Prob(F-statistic) 0.000000
185
Appendix No 12 Results of the residual tests on the model ttttp FFR εγγα +++= ,22,110,
7. Autocorrelation
Ho: errors are independent
H1: errors are autocorrelated
Level of significance α=0.05
LaGrange Multiplier with four lags test statistics Portfolio 1 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.221666 Probability 0.304213 Obs*R-squared 4.955562 Probability 0.291888 Portfolio 2 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.583442 Probability 0.039588 Obs*R-squared 10.11198 Probability 0.038583 Portfolio 3 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.216437 Probability 0.306412 Obs*R-squared 4.935041 Probability 0.294028 Portfolio 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.204688 Probability 0.014819 Obs*R-squared 12.34613 Probability 0.014955 Portfolio 5 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.136997 Probability 0.016505 Obs*R-squared 12.10612 Probability 0.016579 Portfolio 6 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.948356 Probability 0.437979 Obs*R-squared 3.875175 Probability 0.423163 Portfolio 7 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.550034 Probability 0.699286 Obs*R-squared 2.271877 Probability 0.685894
186
Portfolio 8 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.827581 Probability 0.126635Obs*R-squared 7.295431 Probability 0.121076 Portfolio 9 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.325481 Probability 0.860524Obs*R-squared 1.352629 Probability 0.852385 Portfolio 10 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.464720 Probability 0.047666Obs*R-squared 9.676872 Probability 0.046237
8. Heteroskedasticity
Ho: the variance of tε is constant
H1: the variance of tε is not constant
Level of significance α=0.05
χ2 test statistics
Portfolio 1 White Heteroskedasticity Test: F-statistic 1.564152 Probability 0.186912Obs*R-squared 6.205307 Probability 0.184331
Portfolio 2 White Heteroskedasticity Test: F-statistic 0.401352 Probability 0.807440Obs*R-squared 1.642082 Probability 0.801210
Portfolio 3 White Heteroskedasticity Test: F-statistic 3.266385 Probability 0.013397Obs*R-squared 12.40718 Probability 0.014567
187
Portfolio 4 White Heteroskedasticity Test: F-statistic 0.535495 Probability 0.709858 Obs*R-squared 2.183026 Probability 0.702138
Portfolio 5 White Heteroskedasticity Test: F-statistic 0.229889 Probability 0.921252 Obs*R-squared 0.944923 Probability 0.918029
Portfolio 6 White Heteroskedasticity Test: F-statistic 1.138313 Probability 0.340834 Obs*R-squared 4.566673 Probability 0.334715
Portfolio 7 White Heteroskedasticity Test: F-statistic 0.759964 Probability 0.552948 Obs*R-squared 3.079570 Probability 0.544599
Portfolio 8 White Heteroskedasticity Test: F-statistic 3.392238 Probability 0.010957 Obs*R-squared 12.84483 Probability 0.012059
Portfolio 9 White Heteroskedasticity Test: F-statistic 0.329949 Probability 0.857494 Obs*R-squared 1.352542 Probability 0.852400
Portfolio 10 White Heteroskedasticity Test: F-statistic 0.794805 Probability 0.530340 Obs*R-squared 3.217765 Probability 0.522066
188
9. Distribution of the residuals graphically and statistically
Ho: tε have normal distribution
H1: tε have not normal distribution
Level of significance α=0.05
χ2 test statistics
Portfolio 1
0
4
8
12
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 9.42E-19Median -0.003479Maximum 0.092160Minimum -0.066990Std. Dev. 0.028802Skewness 0.679189Kurtosis 3.881913
Jarque-Bera 16.61210Probability 0.000247
Portfolio 2
0
4
8
12
16
20
24
28
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.42E-18Median -0.001756Maximum 0.134557Minimum -0.084296Std. Dev. 0.030891Skewness 0.553401Kurtosis 5.055777
Jarque-Bera 34.52446Probability 0.000000
189
Portfolio 3
0
4
8
12
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -6.85E-19Median -0.002047Maximum 0.072850Minimum -0.066660Std. Dev. 0.023937Skewness 0.434671Kurtosis 3.947705
Jarque-Bera 10.47470Probability 0.005314
Portfolio 4
0
5
10
15
20
25
30
35
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.20E-18Median 0.000921Maximum 0.229186Minimum -0.096381Std. Dev. 0.031701Skewness 2.156819Kurtosis 20.06758
Jarque-Bera 1962.761Probability 0.000000
Portfolio 5
0
5
10
15
20
25
30
35
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.83E-19Median 0.000244Maximum 0.187494Minimum -0.064967Std. Dev. 0.027185Skewness 2.047667Kurtosis 16.99754
Jarque-Bera 1347.119Probability 0.000000
190
Portfolio 6
0
5
10
15
20
25
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.96E-18Median -0.000370Maximum 0.090224Minimum -0.084416Std. Dev. 0.022521Skewness 0.321071Kurtosis 6.028512
Jarque-Bera 60.70014Probability 0.000000
Portfolio 7
0
4
8
12
16
20
24
28
32
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.01E-18Median -0.000426Maximum 0.141857Minimum -0.077827Std. Dev. 0.023139Skewness 1.149837Kurtosis 11.43901
Jarque-Bera 484.5342Probability 0.000000
Portfolio 8
0
5
10
15
20
25
30
-0.10 -0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 2.29E-18Median 0.001273Maximum 0.080020Minimum -0.118162Std. Dev. 0.027507Skewness -0.208482Kurtosis 5.158508
Jarque-Bera 30.60910Probability 0.000000
191
Portfolio 9
0
4
8
12
16
20
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.03E-19Median -0.000690Maximum 0.116650Minimum -0.064966Std. Dev. 0.025571Skewness 0.646268Kurtosis 5.425238
Jarque-Bera 47.83205Probability 0.000000
Portfolio 10
0
5
10
15
20
25
30
35
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -7.65E-19Median -0.000572Maximum 0.119703Minimum -0.090422Std. Dev. 0.028066Skewness 0.083934Kurtosis 5.681180
Jarque-Bera 45.70708Probability 0.000000
192
Appendix No 13 Results of the residual tests on the model tttp FR εγα ++= ,220,
10. Autocorrelation
Ho: errors are independent
H1: errors are autocorrelated
Level of significance α=0.05
LaGrange Multiplier with four lags test statistics
Portfolio 1 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.227993 Probability 0.301535 Obs*R-squared 4.947387 Probability 0.292739
Portfolio 2 Breusch-Godfrey Serial Correlation LM Test: F-statistic 2.130591 Probability 0.079915 Obs*R-squared 8.383248 Probability 0.078506
Portfolio 3 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.206500 Probability 0.310594 Obs*R-squared 4.863565 Probability 0.301585
Portfolio 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.296958 Probability 0.012776 Obs*R-squared 12.59236 Probability 0.013449
Portfolio 5 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.567267 Probability 0.008294 Obs*R-squared 13.53286 Probability 0.008945
Portfolio 6 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.951795 Probability 0.436029 Obs*R-squared 3.862908 Probability 0.424877
193
Portfolio 7 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.131437 Probability 0.344048 Obs*R-squared 4.570075 Probability 0.334319
Portfolio 8 Breusch-Godfrey Serial Correlation LM Test: F-statistic 3.117081 Probability 0.017016 Obs*R-squared 11.95940 Probability 0.017656
Portfolio 9 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.177065 Probability 0.949920 Obs*R-squared 0.733805 Probability 0.947096
Portfolio 10 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.944017 Probability 0.106215 Obs*R-squared 7.686257 Probability 0.103771
11. Heteroskedasticity
Ho: the variance of tε is constant
H1: the variance of tε is not constant
Level of significance α=0.05
χ2 test statistics
Portfolio 1 White Heteroskedasticity Test: F-statistic 2.184002 Probability 0.116181 Obs*R-squared 4.329043 Probability 0.114805
Portfolio 2 White Heteroskedasticity Test: F-statistic 0.101959 Probability 0.903130 Obs*R-squared 0.207738 Probability 0.901343
194
Portfolio 3 White Heteroskedasticity Test: F-statistic 5.250751 Probability 0.006258 Obs*R-squared 10.00761 Probability 0.006712
Portfolio 4 White Heteroskedasticity Test: F-statistic 0.704061 Probability 0.496210 Obs*R-squared 1.423026 Probability 0.490901
Portfolio 5 White Heteroskedasticity Test: F-statistic 0.483996 Probability 0.617281 Obs*R-squared 0.981108 Probability 0.612287
Portfolio 6 White Heteroskedasticity Test: F-statistic 2.205713 Probability 0.113757 Obs*R-squared 4.370839 Probability 0.112431
Portfolio 7 White Heteroskedasticity Test: F-statistic 1.263923 Probability 0.285555 Obs*R-squared 2.535723 Probability 0.281433
Portfolio 8 White Heteroskedasticity Test: F-statistic 3.975636 Probability 0.020792 Obs*R-squared 7.700437 Probability 0.021275
Portfolio 9 White Heteroskedasticity Test: F-statistic 0.505932 Probability 0.603976 Obs*R-squared 1.025274 Probability 0.598914
Portfolio 10 White Heteroskedasticity Test: F-statistic 1.613802 Probability 0.202590 Obs*R-squared 3.222778 Probability 0.199610
195
12. Distribution of the residuals graphically and statistically
Ho: tε have normal distribution
H1: tε have not normal distribution
Level of significance α=0.05
χ2 test statistics
Portfolio 1
0
4
8
12
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.12E-18Median -0.003514Maximum 0.092148Minimum -0.066939Std. Dev. 0.028802Skewness 0.679665Kurtosis 3.882553
Jarque-Bera 16.63563Probability 0.000244
Portfolio 2
0
4
8
12
16
20
24
28
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.83E-18Median -0.001119Maximum 0.137575Minimum -0.083235Std. Dev. 0.031107Skewness 0.584540Kurtosis 5.155906
Jarque-Bera 38.09296Probability 0.000000
196
Portfolio 3
0
4
8
12
16
20
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 2.18E-18Median -0.002054Maximum 0.073120Minimum -0.066883Std. Dev. 0.023940Skewness 0.440341Kurtosis 3.980442
Jarque-Bera 11.00015Probability 0.004086
Portfolio 4
0
4
8
12
16
20
24
28
32
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 9.36E-19Median 0.001140Maximum 0.229598Minimum -0.095765Std. Dev. 0.031708Skewness 2.170166Kurtosis 20.17607
Jarque-Bera 1987.753Probability 0.000000
Portfolio 5
0
5
10
15
20
25
30
35
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -3.65E-19Median 9.36E-05Maximum 0.186807Minimum -0.065502Std. Dev. 0.027429Skewness 1.976847Kurtosis 16.28152
Jarque-Bera 1216.193Probability 0.000000
197
Portfolio 6
0
4
8
12
16
20
24
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.10E-18Median -0.000362Maximum 0.090256Minimum -0.084542Std. Dev. 0.022521Skewness 0.321672Kurtosis 6.037644
Jarque-Bera 61.06076Probability 0.000000
Portfolio 7
0
5
10
15
20
25
30
35
-0.05 0.00 0.05 0.10 0.15
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.23E-18Median -0.000430Maximum 0.144663Minimum -0.076491Std. Dev. 0.023552Skewness 1.142220Kurtosis 11.34212
Jarque-Bera 473.7944Probability 0.000000
Portfolio 8
0
5
10
15
20
25
30
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 4.99E-19Median 0.000322Maximum 0.097024Minimum -0.104901Std. Dev. 0.028751Skewness -0.144773Kurtosis 5.041746
Jarque-Bera 26.93290Probability 0.000001
198
Portfolio 9
0
4
8
12
16
20
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -6.16E-19Median -4.55E-05Maximum 0.114180Minimum -0.063706Std. Dev. 0.026312Skewness 0.546633Kurtosis 5.047371
Jarque-Bera 34.11740Probability 0.000000
Portfolio 10
0
4
8
12
16
20
-0.10 -0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -7.08E-19Median 0.002026Maximum 0.098345Minimum -0.099663Std. Dev. 0.029887Skewness -0.099234Kurtosis 4.513454
Jarque-Bera 14.75624Probability 0.000625
199
Appendix No 14
Changes in the estimation output after including errors correction techniques
for portfolios tttttp FFFR εγγγα ++++= ,33,22,110,
Portfolio 2
Dependent Variable: P2 Method: Least Squares Date: 04/27/03 Time: 15:03 Sample(adjusted): 2/17/2000 12/26/2002 Included observations: 150 after adjusting endpoints Convergence achieved after 6 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.004344 0.002002 -2.169308 0.0317F1 -0.349877 0.182413 -1.918043 0.0571F2 0.327616 0.086949 3.767915 0.0002
AR(2) -0.258986 0.082456 -3.140891 0.0020R-squared 0.118860 Mean dependent var -0.005365Adjusted R-squared 0.100754 S.D. dependent var 0.032105S.E. of regression 0.030445 Akaike info criterion -4.119500Sum squared resid 0.135325 Schwarz criterion -4.039216Log likelihood 312.9625 F-statistic 6.564804Durbin-Watson stat 2.110067 Prob(F-statistic) 0.000341
Portfolio 3
Dependent Variable: P3 Method: Least Squares Date: 04/27/03 Time: 15:03 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C 0.001431 0.002297 0.623114 0.5342 F1 0.027512 0.153856 0.178815 0.8583 F2 0.546731 0.104342 5.239776 0.0000
R-squared 0.276533 Mean dependent var 0.001207 Adjusted R-squared 0.266822 S.D. dependent var 0.028142 S.E. of regression 0.024097 Akaike info criterion -4.593919 Sum squared resid 0.086519 Schwarz criterion -4.534237 Log likelihood 352.1378 F-statistic 28.47631 Durbin-Watson stat 1.818690 Prob(F-statistic) 0.000000
Portfolio 4
Dependent Variable: P4 Method: Least Squares Date: 04/27/03 Time: 15:03 Sample(adjusted): 2/17/2000 12/26/2002 Included observations: 150 after adjusting endpoints Convergence achieved after 8 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.000258 0.002067 -0.124845 0.9008F1 0.020165 0.185548 0.108677 0.9136F2 0.336209 0.089027 3.776471 0.0002
AR(2) -0.246599 0.080203 -3.074680 0.0025R-squared 0.112300 Mean dependent var -0.000597Adjusted R-squared 0.094059 S.D. dependent var 0.032713S.E. of regression 0.031137 Akaike info criterion -4.074544Sum squared resid 0.141548 Schwarz criterion -3.994260Log likelihood 309.5908 F-statistic 6.156647Durbin-Watson stat 2.153262 Prob(F-statistic) 0.000571
200
Portfolio 5 Dependent Variable: P5 Method: Least Squares Date: 04/27/03 Time: 15:03 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 5 iterations
Variable Coefficient Std. Error t-Statistic Prob. C 0.000226 0.001705 0.132386 0.8949 F1 -0.158216 0.145652 -1.086259 0.2791 F2 0.239777 0.074553 3.216188 0.0016
AR(1) -0.263201 0.079107 -3.327172 0.0011 R-squared 0.141051 Mean dependent var -0.000346 Adjusted R-squared 0.123521 S.D. dependent var 0.027792 S.E. of regression 0.026019 Akaike info criterion -4.433871 Sum squared resid 0.099515 Schwarz criterion -4.353943 Log likelihood 338.7573 F-statistic 8.046443 Durbin-Watson stat 1.938473 Prob(F-statistic) 0.000053 Inverted AR Roots -.26
Portfolio 8
Dependent Variable: P8 Method: Least Squares Date: 04/27/03 Time: 15:03 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.000674 0.002215 -0.304214 0.7614 F1 -0.577847 0.192162 -3.007083 0.0031 F2 0.549921 0.101660 5.409397 0.0000
R-squared 0.277286 Mean dependent var -0.002153 Adjusted R-squared 0.267585 S.D. dependent var 0.032356 S.E. of regression 0.027691 Akaike info criterion -4.315901 Sum squared resid 0.114249 Schwarz criterion -4.256219 Log likelihood 331.0084 F-statistic 28.58361 Durbin-Watson stat 2.230032 Prob(F-statistic) 0.000000
Portfolio 10
Dependent Variable: P10 Method: Least Squares Date: 04/27/03 Time: 15:04 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 6 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.002354 0.001822 -1.291448 0.1986 F1 -0.690301 0.151183 -4.566003 0.0000 F2 1.186652 0.080295 14.77870 0.0000
AR(1) -0.234029 0.080869 -2.893934 0.0044 R-squared 0.606223 Mean dependent var -0.004737 Adjusted R-squared 0.598186 S.D. dependent var 0.042878 S.E. of regression 0.027180 Akaike info criterion -4.346547 Sum squared resid 0.108596 Schwarz criterion -4.266619 Log likelihood 332.1643 F-statistic 75.43582 Durbin-Watson stat 2.008499 Prob(F-statistic) 0.000000 Inverted AR Roots -.23
201
Appendix No 15
Changes in the estimation output after including errors correction techniques
for portfolios tttp FR εγα ++= ,220,
Portfolio 3
Dependent Variable: P3 Method: Least Squares Date: 04/27/03 Time: 15:08 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C 0.001488 0.002298 0.647573 0.5183 F2 0.546794 0.104410 5.236996 0.0000
R-squared 0.276332 Mean dependent var 0.001207 Adjusted R-squared 0.271508 S.D. dependent var 0.028142 S.E. of regression 0.024020 Akaike info criterion -4.606799 Sum squared resid 0.086543 Schwarz criterion -4.567012 Log likelihood 352.1168 F-statistic 57.27744 Durbin-Watson stat 1.812175 Prob(F-statistic) 0.000000
Portfolio 4 Dependent Variable: P4 Method: Least Squares Date: 04/27/03 Time: 15:08 Sample(adjusted): 2/17/2000 12/26/2002 Included observations: 150 after adjusting endpoints Convergence achieved after 6 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.000223 0.002032 -0.109607 0.9129 F2 0.336301 0.088683 3.792188 0.0002
AR(2) -0.247827 0.079788 -3.106071 0.0023 R-squared 0.112229 Mean dependent var -0.000597 Adjusted R-squared 0.100151 S.D. dependent var 0.032713 S.E. of regression 0.031032 Akaike info criterion -4.087798 Sum squared resid 0.141559 Schwarz criterion -4.027585 Log likelihood 309.5848 F-statistic 9.291634 Durbin-Watson stat 2.154799 Prob(F-statistic) 0.000159
Portfolio 5
Dependent Variable: P5 Method: Least Squares Date: 04/27/03 Time: 15:09 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 5 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.000111 0.001665 -0.066925 0.9467 F2 0.235964 0.074278 3.176758 0.0018
AR(1) -0.273346 0.077689 -3.518460 0.0006 R-squared 0.134096 Mean dependent var -0.000346 Adjusted R-squared 0.122394 S.D. dependent var 0.027792 S.E. of regression 0.026035 Akaike info criterion -4.439052 Sum squared resid 0.100321 Schwarz criterion -4.379106 Log likelihood 338.1484 F-statistic 11.45977 Durbin-Watson stat 1.912337 Prob(F-statistic) 0.000024 Inverted AR Roots -.27
202
Portfolio 8 Dependent Variable: P8 Method: Least Squares Date: 04/27/03 Time: 15:09 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.001871 0.002264 -0.826437 0.4099 F2 0.548586 0.110929 4.945370 0.0000
R-squared 0.210416 Mean dependent var -0.002153 Adjusted R-squared 0.205152 S.D. dependent var 0.032356 S.E. of regression 0.028847 Akaike info criterion -4.240566 Sum squared resid 0.124821 Schwarz criterion -4.200778 Log likelihood 324.2830 F-statistic 39.97338 Durbin-Watson stat 2.298967 Prob(F-statistic) 0.000000
Portfolio 10 Dependent Variable: P10 Method: Least Squares Date: 04/27/03 Time: 15:10 Sample(adjusted): 2/10/2000 12/26/2002 Included observations: 151 after adjusting endpoints Convergence achieved after 6 iterations
Variable Coefficient Std. Error t-Statistic Prob. C -0.003838 0.001941 -1.977290 0.0499 F2 1.171754 0.085242 13.74625 0.0000
AR(1) -0.214009 0.080226 -2.667587 0.0085 R-squared 0.550570 Mean dependent var -0.004737 Adjusted R-squared 0.544496 S.D. dependent var 0.042878 S.E. of regression 0.028939 Akaike info criterion -4.227597 Sum squared resid 0.123944 Schwarz criterion -4.167651 Log likelihood 322.1836 F-statistic 90.65287 Durbin-Watson stat 1.920530 Prob(F-statistic) 0.000000 Inverted AR Roots -.21
203
Appendix No 16
Results of the Chow Breakpoint Test for the time-series models on 4 July 2002
Ho: Model is stable over the period
H1: Model is different after 4 July 2002
Level of significance α=0.05
F - test statistics
Portfolio 1
Chow Breakpoint Test: 7/04/2002 F-statistic 0.613151 Probability 0.543011 Log likelihood ratio 1.254267 Probability 0.534121
Portfolio 2
Chow Breakpoint Test: 7/04/2002 F-statistic 1.219886 Probability 0.298215 Log likelihood ratio 2.485294 Probability 0.288619
Portfolio 3 Chow Breakpoint Test: 7/04/2002 F-statistic 1.113439 Probability 0.331163 Log likelihood ratio 2.270039 Probability 0.321416
Portfolio 4 Chow Breakpoint Test: 7/04/2002 F-statistic 0.680438 Probability 0.507972 Log likelihood ratio 1.391280 Probability 0.498755
Portfolio 5 Chow Breakpoint Test: 7/04/2002 F-statistic 0.303882 Probability 0.738408 Log likelihood ratio 0.622921 Probability 0.732377
Portfolio 6 Chow Breakpoint Test: 7/04/2002 F-statistic 0.514891 Probability 0.598632 Log likelihood ratio 1.053961 Probability 0.590385
Portfolio 7 Chow Breakpoint Test: 7/04/2002 F-statistic 9.117991 Probability 0.000184 Log likelihood ratio 17.66182 Probability 0.000146
Portfolio 8
204
Chow Breakpoint Test: 7/04/2002 F-statistic 0.996792 Probability 0.371525Log likelihood ratio 2.033807 Probability 0.361713
Portfolio 9 Chow Breakpoint Test: 7/04/2002 F-statistic 1.437201 Probability 0.240888Log likelihood ratio 2.923797 Probability 0.231796
Portfolio 10 Chow Breakpoint Test: 7/04/2002 F-statistic 3.914811 Probability 0.022043Log likelihood ratio 7.835757 Probability 0.019883
205
Appendix No 17 Results of the time-series regression tttttp FFFR εγγγα ++++= ,33,22,110, , portfolios formed alphabetically. Portfolio 1
Dependent Variable: P1 Method: Least Squares Date: 04/29/03 Time: 08:29 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003346 0.001973 -1.695996 0.0920 F1 -0.020109 0.137311 -0.146445 0.8838 F2 -0.027823 0.095171 -0.292343 0.7704 F3 0.178185 0.126471 1.408901 0.1610
R-squared 0.018631 Mean dependent var -0.003405 Adjusted R-squared -0.001262 S.D. dependent var 0.024047 S.E. of regression 0.024062 Akaike info criterion -4.590414 Sum squared resid 0.085688 Schwarz criterion -4.510838 Log likelihood 352.8714 F-statistic 0.936560 Durbin-Watson stat 1.996753 Prob(F-statistic) 0.424702
Portfolio 2
Dependent Variable: P2 Method: Least Squares Date: 04/29/03 Time: 08:29 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002377 0.002702 -0.879871 0.3804 F1 0.078409 0.188027 0.417009 0.6773 F2 0.015397 0.130323 0.118145 0.9061 F3 0.175062 0.173182 1.010851 0.3137
R-squared 0.014133 Mean dependent var -0.002254 Adjusted R-squared -0.005851 S.D. dependent var 0.032853 S.E. of regression 0.032949 Akaike info criterion -3.961742 Sum squared resid 0.160675 Schwarz criterion -3.882166 Log likelihood 305.0924 F-statistic 0.707204 Durbin-Watson stat 2.106504 Prob(F-statistic) 0.549159
Portfolio 3
Dependent Variable: P3 Method: Least Squares Date: 04/29/03 Time: 08:29 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.000834 0.002053 -0.406276 0.6851 F1 0.014168 0.142892 0.099149 0.9212 F2 0.098606 0.099039 0.995626 0.3211 F3 0.060672 0.131611 0.460993 0.6455
R-squared 0.020600 Mean dependent var -0.000866 Adjusted R-squared 0.000747 S.D. dependent var 0.025049 S.E. of regression 0.025040 Akaike info criterion -4.510733 Sum squared resid 0.092795 Schwarz criterion -4.431157 Log likelihood 346.8157 F-statistic 1.037627 Durbin-Watson stat 2.091272 Prob(F-statistic) 0.377813
206
Portfolio 4
Dependent Variable: P4 Method: Least Squares Date: 04/29/03 Time: 08:29 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001317 0.003129 -0.420834 0.6745 F1 0.078000 0.217802 0.358124 0.7208 F2 -0.089273 0.150960 -0.591367 0.5552 F3 0.204282 0.200606 1.018323 0.3102
R-squared 0.007242 Mean dependent var -0.001146 Adjusted R-squared -0.012882 S.D. dependent var 0.037923 S.E. of regression 0.038167 Akaike info criterion -3.667746 Sum squared resid 0.215591 Schwarz criterion -3.588170 Log likelihood 282.7487 F-statistic 0.359860 Durbin-Watson stat 2.125686 Prob(F-statistic) 0.782077
Portfolio 5
Dependent Variable: P5 Method: Least Squares Date: 04/29/03 Time: 08:29 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.002272 0.002005 -1.133215 0.2590 F1 0.123644 0.139565 0.885920 0.3771 F2 0.075355 0.096733 0.778993 0.4372 F3 0.162521 0.128547 1.264301 0.2081
R-squared 0.042044 Mean dependent var -0.002084 Adjusted R-squared 0.022626 S.D. dependent var 0.024738 S.E. of regression 0.024457 Akaike info criterion -4.557851 Sum squared resid 0.088524 Schwarz criterion -4.478275 Log likelihood 350.3967 F-statistic 2.165195 Durbin-Watson stat 1.913312 Prob(F-statistic) 0.094583
Portfolio 6
Dependent Variable: P6 Method: Least Squares Date: 04/29/03 Time: 08:30 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -1.90E-05 0.002172 -0.008743 0.9930 F1 -0.102236 0.151174 -0.676278 0.4999 F2 0.080018 0.104780 0.763681 0.4463 F3 0.200052 0.139239 1.436754 0.1529
R-squared 0.050713 Mean dependent var -0.000308 Adjusted R-squared 0.031470 S.D. dependent var 0.026918 S.E. of regression 0.026491 Akaike info criterion -4.398052 Sum squared resid 0.103863 Schwarz criterion -4.318476 Log likelihood 338.2520 F-statistic 2.635483 Durbin-Watson stat 1.728309 Prob(F-statistic) 0.051973
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Portfolio 7
Dependent Variable: P7 Method: Least Squares Date: 04/29/03 Time: 08:30 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001423 0.002718 -0.523432 0.6015 F1 -0.268465 0.189159 -1.419255 0.1579 F2 -0.067171 0.131107 -0.512335 0.6092 F3 0.150678 0.174225 0.864848 0.3885
R-squared 0.021745 Mean dependent var -0.001971 Adjusted R-squared 0.001916 S.D. dependent var 0.033179 S.E. of regression 0.033147 Akaike info criterion -3.949737 Sum squared resid 0.162616 Schwarz criterion -3.870162 Log likelihood 304.1800 F-statistic 1.096620 Durbin-Watson stat 1.860688 Prob(F-statistic) 0.352576
Portfolio 8
Dependent Variable: P8 Method: Least Squares Date: 04/29/03 Time: 08:30 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.004039 0.002551 -1.583385 0.1155 F1 -0.105581 0.177558 -0.594630 0.5530 F2 -0.065707 0.123066 -0.533919 0.5942 F3 0.156367 0.163540 0.956144 0.3406
R-squared 0.010183 Mean dependent var -0.004252 Adjusted R-squared -0.009881 S.D. dependent var 0.030962 S.E. of regression 0.031114 Akaike info criterion -4.076323 Sum squared resid 0.143280 Schwarz criterion -3.996747 Log likelihood 313.8006 F-statistic 0.507532 Durbin-Watson stat 2.118429 Prob(F-statistic) 0.677681
Portfolio 9
Dependent Variable: P9 Method: Least Squares Date: 04/29/03 Time: 08:30 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.003266 0.002831 -1.153410 0.2506 F1 -0.061129 0.197058 -0.310206 0.7568 F2 0.046841 0.136582 0.342947 0.7321 F3 0.308565 0.181501 1.700077 0.0912
R-squared 0.044406 Mean dependent var -0.003472 Adjusted R-squared 0.025035 S.D. dependent var 0.034972 S.E. of regression 0.034532 Akaike info criterion -3.867916 Sum squared resid 0.176481 Schwarz criterion -3.788341 Log likelihood 297.9616 F-statistic 2.292474 Durbin-Watson stat 2.278450 Prob(F-statistic) 0.080486
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Portfolio 10
Dependent Variable: P10 Method: Least Squares Date: 04/29/03 Time: 08:30 Sample: 2/03/2000 12/26/2002 Included observations: 152
Variable Coefficient Std. Error t-Statistic Prob. C -0.001690 0.002996 -0.564264 0.5734 F1 0.135191 0.208493 0.648420 0.5177 F2 -0.028143 0.144507 -0.194752 0.8459 F3 0.217079 0.192032 1.130428 0.2601
R-squared 0.013298 Mean dependent var -0.001435 Adjusted R-squared -0.006702 S.D. dependent var 0.036414 S.E. of regression 0.036535 Akaike info criterion -3.755107 Sum squared resid 0.197556 Schwarz criterion -3.675532 Log likelihood 289.3882 F-statistic 0.664892 Durbin-Watson stat 1.915904 Prob(F-statistic) 0.574897
209
Appendix No 18 Results of the residual tests on the model
tttttp FFFR εγγγα ++++= ,33,22,110, , portfolios formed alphabetically.
13. Autocorrelation
Ho: errors are independent
H1: errors are autocorrelated
Level of significance α=0.05
LaGrange Multiplier with four lags test statistics
Portfolio 1 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.173384 Probability 0.951740 Obs*R-squared 0.728559 Probability 0.947761
Portfolio 2 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.648359 Probability 0.628929 Obs*R-squared 2.689087 Probability 0.611126
Portfolio 3 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.867569 Probability 0.485091 Obs*R-squared 3.576870 Probability 0.466287
Portfolio 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.148788 Probability 0.336125 Obs*R-squared 4.700443 Probability 0.319437
Portfolio 5 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.335134 Probability 0.853942 Obs*R-squared 1.401959 Probability 0.843854
Portfolio 6 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.917462 Probability 0.110653 Obs*R-squared 7.686542 Probability 0.103759
210
Portfolio 7 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.676909 Probability 0.609034 Obs*R-squared 2.805313 Probability 0.590916
Portfolio 8 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.401874 Probability 0.807058 Obs*R-squared 1.678067 Probability 0.794698
Portfolio 9 Breusch-Godfrey Serial Correlation LM Test: F-statistic 1.635980 Probability 0.168372 Obs*R-squared 6.607214 Probability 0.158159
Portfolio 10 Breusch-Godfrey Serial Correlation LM Test: F-statistic 0.699477 Probability 0.593522 Obs*R-squared 2.897058 Probability 0.575198
14. Heteroskedasticity
Ho: the variance of tε is constant
H1: the variance of tε is not constant
Level of significance α=0.05
χ2 test statistics
Portfolio 1 White Heteroskedasticity Test: F-statistic 0.924375 Probability 0.479367 Obs*R-squared 5.599810 Probability 0.469476
Portfolio 2 White Heteroskedasticity Test: F-statistic 0.425314 Probability 0.861195 Obs*R-squared 2.628811 Probability 0.853782
211
Portfolio 3 White Heteroskedasticity Test: F-statistic 2.268080 Probability 0.040198 Obs*R-squared 13.04148 Probability 0.042382
Portfolio 4 White Heteroskedasticity Test: F-statistic 0.193005 Probability 0.978376 Obs*R-squared 1.204315 Probability 0.976671
Portfolio 5 White Heteroskedasticity Test: F-statistic 0.568716 Probability 0.754715 Obs*R-squared 3.494785 Probability 0.744663
Portfolio 6 White Heteroskedasticity Test: F-statistic 2.228277 Probability 0.043627 Obs*R-squared 12.83193 Probability 0.045784
Portfolio 7 White Heteroskedasticity Test: F-statistic 0.958776 Probability 0.455430 Obs*R-squared 5.800251 Probability 0.445934
Portfolio 8 White Heteroskedasticity Test: F-statistic 0.679879 Probability 0.666114 Obs*R-squared 4.159196 Probability 0.655143
Portfolio 9 White Heteroskedasticity Test: F-statistic 0.908729 Probability 0.490491 Obs*R-squared 5.508460 Probability 0.480435
Portfolio 10 White Heteroskedasticity Test: F-statistic 0.187734 Probability 0.979862 Obs*R-squared 1.171679 Probability 0.978260
212
15. Distribution of the residuals graphically and statistically
Ho: tε have normal distribution
H1: tε have not normal distribution
Level of significance α=0.05
χ2 test statistics
Portfolio 1
0
4
8
12
16
20
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.55E-18Median -0.002405Maximum 0.074629Minimum -0.068142Std. Dev. 0.023822Skewness 0.311155Kurtosis 3.895838
Jarque-Bera 7.535376Probability 0.023105
Portfolio 2
0
4
8
12
16
20
24
28
32
-0.10 -0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 5.14E-18Median -0.001773Maximum 0.077189Minimum -0.132766Std. Dev. 0.032620Skewness -0.429508Kurtosis 4.386454
Jarque-Bera 16.84770Probability 0.000220
213
Portfolio 3
0
4
8
12
16
20
24
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.05E-18Median 0.001603Maximum 0.074577Minimum -0.072962Std. Dev. 0.024790Skewness -0.243588Kurtosis 3.688469
Jarque-Bera 4.505089Probability 0.105131
Portfolio 4
0
5
10
15
20
25
30
0.0 0.1 0.2
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 2.25E-18Median -0.000683Maximum 0.283947Minimum -0.075088Std. Dev. 0.037786Skewness 2.931883Kurtosis 22.76911
Jarque-Bera 2692.943Probability 0.000000
Portfolio 5
0
4
8
12
16
20
-0.050 -0.025 0.000 0.025 0.050 0.075
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -5.71E-19Median 0.001466Maximum 0.074129Minimum -0.062871Std. Dev. 0.024213Skewness 0.051144Kurtosis 3.139100
Jarque-Bera 0.188808Probability 0.909915
214
Portfolio 6
0
4
8
12
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.83E-19Median -0.000987Maximum 0.092501Minimum -0.074278Std. Dev. 0.026227Skewness 0.078651Kurtosis 4.040973
Jarque-Bera 7.019671Probability 0.029902
Portfolio 7
0
5
10
15
20
25
30
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.48E-18Median -0.004427Maximum 0.127720Minimum -0.071181Std. Dev. 0.032817Skewness 0.931137Kurtosis 4.666932
Jarque-Bera 39.56261Probability 0.000000
Portfolio 8
0
2
4
6
8
10
12
14
16
-0.05 0.00 0.05
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean 1.95E-18Median -0.000274Maximum 0.092972Minimum -0.086084Std. Dev. 0.030804Skewness 0.174309Kurtosis 3.909603
Jarque-Bera 6.009781Probability 0.049544
215
Portfolio 9
0
2
4
6
8
10
12
14
16
-0.05 0.00 0.05 0.10
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.92E-18Median -0.001396Maximum 0.101669Minimum -0.080126Std. Dev. 0.034187Skewness 0.307682Kurtosis 3.365505
Jarque-Bera 3.244354Probability 0.197468
Portfolio 10
0
4
8
12
16
20
24
-0.05 0.00 0.05 0.10 0.15 0.20
Series: ResidualsSample 2/03/2000 12/26/2002Observations 152
Mean -1.51E-18Median -0.001168Maximum 0.211403Minimum -0.073034Std. Dev. 0.036171Skewness 1.376739Kurtosis 9.669236
Jarque-Bera 329.7155Probability 0.000000
216
Appendix No 19
Changes in the estimation output after including errors correction techniques for portfolios tttttp FFFR εγγγα ++++= ,33,22,110, , portfolios formed alphabetically.
Portfolio 3 Dependent Variable: P3 Method: Least Squares Date: 04/29/03 Time: 08:47 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -0.000834 0.002118 -0.393796 0.6943 F1 0.014168 0.129529 0.109379 0.9131 F2 0.098606 0.110099 0.895610 0.3719 F3 0.060672 0.119734 0.506720 0.6131
R-squared 0.020600 Mean dependent var -0.000866 Adjusted R-squared 0.000747 S.D. dependent var 0.025049 S.E. of regression 0.025040 Akaike info criterion -4.510733 Sum squared resid 0.092795 Schwarz criterion -4.431157 Log likelihood 346.8157 F-statistic 1.037627 Durbin-Watson stat 2.091272 Prob(F-statistic) 0.377813
Portfolio 6 Dependent Variable: P6 Method: Least Squares Date: 04/29/03 Time: 08:48 Sample: 2/03/2000 12/26/2002 Included observations: 152 Newey-West HAC Standard Errors & Covariance (lag truncation=4)
Variable Coefficient Std. Error t-Statistic Prob. C -1.90E-05 0.002331 -0.008147 0.9935 F1 -0.102236 0.178843 -0.571652 0.5684 F2 0.080018 0.109564 0.730329 0.4663 F3 0.200052 0.125117 1.598916 0.1120
R-squared 0.050713 Mean dependent var -0.000308 Adjusted R-squared 0.031470 S.D. dependent var 0.026918 S.E. of regression 0.026491 Akaike info criterion -4.398052 Sum squared resid 0.103863 Schwarz criterion -4.318476 Log likelihood 338.2520 F-statistic 2.635483 Durbin-Watson stat 1.728309 Prob(F-statistic) 0.051973
