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Feedback Trading and the Behavioral ICAPM:
Multivariate Evidence across International Equity and
Bond Markets
WARREN G. DEAN
ROBERT W. FAFF *
Department of Accounting and Finance
Monash University
* Corresponding author:
Department of Accounting and Finance
PO Box 11E
Monash University
Victoria 3800 Australia.
(Email : [email protected])
1
Feedback Trading and the Behavioral ICAPM:
Multivariate Evidence across International Equity and
Bond Markets
ABSTRACT
In this paper we develop a ‘behavioural’ ICAPM in which the behavioural impetus comes
from the feedback trading implications for the autocorrelation of returns. We apply the model
in a setting of paired equity and bond investments, employing a bivariate diagonal BEKK
framework. Our empirics rely on daily equity and bond index returns across six major
economies, over the period 1 January 1990 to 30 June 2005. We find evidence supporting the
theory that the observed dynamics of serial correlation can be a function of both volatility and
conditional covariance (between equity and bonds). Moreover, our behavioural ICAPM
shows empirical promise as a useful model of asset pricing in markets that display the
feedback trading phenomenon.
2
1. Introduction
The primary goal of this paper is to jointly model the feedback trading implications for return
autocorrelations in equity and bond markets. We achieve this goal by using daily data across
six major international markets: Australia; Canada; Germany; Japan; the UK and the US. The
major innovation on previous research is our development of a ‘behavioural’ version of
Merton’s two-factor intertemporal CAPM (BICAPM) which incorporates the feedback
trading phenomenon, in a bivariate GARCH setting for pairs of equity and bond index
returns.
The serial correlation properties of stock returns have proven to be a challenging area
of financial research, with potentially important implications for trading strategies and market
efficiency. Indeed, plausible, robust explanations of why return autocorrelation is so widely
observed have been elusive. Early analysis of this issue focussed on the presence of
conditional risk premia [Fama (1971)]. However, numerous papers have reported evidence
suggesting that the magnitude of observed autocorrelations is too large to be compatible with
time-varying expected returns [see, for example, Atchison, Butler and Simonds (1987);
Conrad and Kaul (1988, 1989) and Lo and MacKinlay (1988, 1990)].1 A second area of
research which has attempted (but similarly failed) to explain the widespread occurrence of
non-zero return autocorrelation, focuses on non-synchronous trading [see, for example,
Atchison, Butler and Simonds (1987) and Ogden (1997)]. Likewise, market microstructure
biases have also been discounted as viable explanations [see Cohen et al (1986), Mech (1993)
and Ogden (1997)].
Cutler et al. (1991) argue that the serial correlation patterns observed in asset returns
can be accounted for by models with feedback traders who do not base their asset decisions
3
on fundamental values but instead react to recent price changes. There is little doubt that
some investors follow such naïve strategies in which they trade according to trends in stock
prices and base their buying and selling decision on the expectation that these trends will
continue. This type of investment philosophy can manifest either in a positive feedback
trading strategy wherein such investors buy in a rising market and sell in a falling market, or
a negative feedback trading strategy where they do the opposite. Such trading in sufficiently
large volumes will induce autocorrelation of returns, leading to (partial) predictability of
aggregate stock returns.
Sentana and Wadhwani (1992) extend the logic of Cutler et al. (1991) and find
evidence of a linkage between volatility and serial correlation within the US equity market.
Koutmos (1997) broadens the application of the Sentana and Wadhwani (1992) model to
confirm their result across a group of six foreign equity markets (Australia, Belgium,
Germany, Italy, Japan and the UK). Specifically, Koutmos (1997) documents negative first
order autocorrelation in stock returns across all six markets, and also finds that this negative
autocorrelation becomes more negative as volatility rises in four of these markets.
The Sentana and Wadhwani (1992) approach to examining the impact of feedback
traders assumes that investors’ risk premium can be modelled by a conditional single factor
CAPM: effectively creating a form of ‘behavioural’ CAPM (BCAPM). While such a
parsimonious setting has its attractions, many financial theorists have proposed multi-factor
models as alternative measures of investors risk premia, which may better describe the
buying and selling decisions of investors. Accordingly, in the current paper, we are motivated
by one such model – the intertemporal CAPM (ICAPM) of Merton (1973), to further extend
the logic of Cutler et al. (1991) and Sentana and Wadhwani (1992). This version of the
1 Mech (1993) also points out that an economic model of time varying expected returns is unlikely to explain the
documented asymmetric serial cross covariances between large and small firm returns, nor why portfolio
autocorrelation can be used to predict negative portfolio returns.
4
ICAPM allows for any potential linkage connecting the covariance between stocks and bonds
and serial correlation of returns.
A broad literature suggests that Merton’s (1973) ICAPM represents a legitimate and
productive asset pricing framework [see, for example, Fama (1996, 1998) and Fama and
French (2004)]. Moreover, we argue that the ICAPM is ideal for allowing a more
sophisticated version of feedback trading to condition the dynamic structure of returns:
thereby effectively creating a ‘behavioural’ ICAPM.
The ICAPM incorporates expectations of future changes in the investment
opportunity set, captured by at least one ‘state’ variable, which influences the expected risk
premium demanded by investors. Of particular relevance to our study is the fact that Merton
proposed interest rates as a likely candidate for one state variable. Notably, Rubio (1989),
Shanken (1990), Song (1994), Elyasiani and Mansur (1998), Scruggs (1998) and others have,
with varying degrees of success, empirically investigated such a version of the ICAPM.
Further, very supportive evidence for the ICAPM in the Australian equity and bond markets
has been reported by Dean and Faff (2001). Informationally linked markets, such as bond and
equity markets, often react to the same information set, and it is reasonable to suggest that the
movement of both markets will be correlated to some extent and will impact upon investors’
decisions. Therefore, for each country we use the domestic bond market as a second ICAPM-
type factor and, given our feedback trading hypothesis, this second factor will enable us to
capture, or extract, more information with regard to the autocorrelation properties of stock
returns (than would be possible in the single-factor CAPM setting).
To our knowledge, this is the first time that the two-factor model of Merton has been
transformed into a ‘behavioural’ ICAPM. As such, its use in examining the serial correlation
properties of returns is novel, and the successful application of this methodology is our main
contribution to this field of research. Moreover, through our empirical tests we exploit the
5
fact that our BICAPM nests three alternative specifications: (a) the conventional CAPM; (b)
the conventional ICAPM and (c) the BCAPM of Sentana and Wadhwani (1992).
The remainder of this paper is structured as follows: Section 2 provides the theoretical
framework. Section 3 presents the empirical framework. Section 4 presents the data and the
empirical findings, while Section 5 concludes.
2. The ICAPM and Feedback Trading
The Intertemporal CAPM (ICAPM) of Merton (1973) extended the static or single period
CAPM proposed by Sharpe (1964) and Lintner (1965) into a multi-period world. Whereas the
CAPM predicts that the expected return on an asset above the risk free rate is proportional to
the non-diversifiable risk as measured by the covariance of the asset return with a portfolio
composed of all available assets, the ICAPM incorporates the ‘price’ of covariance between
the asset return and other forecasting, or state, variables.
Merton develops the ICAPM by assuming that investors’ demands for assets are
affected by the possibility of uncertain changes in the investors’ opportunity set. In such an
environment, rational investors look beyond maximising wealth in one period – they seek to
maximise utility over the total investment horizon. Merton shows that investors would seek to
hedge against adverse changes in the future investment opportunity set, and would therefore
price assets not only by their systematic risk (covariance with the market return), but also by
their covariance with the changing investment environment as evidenced by a set of state
variables.
This covariance between an asset and the state variable represents how asset returns
(and therefore prices) change in response to changing future investment opportunities. Here
the multi-factor ICAPM provides researchers with another source of changing risk premia,
and one that is both intuitively reasonable and economically plausible. Investors live in a
6
dynamic world that responds immediately to news and events that describe possible changes
to future investment opportunities. This news arrives on a continuous basis and is impounded
immediately into stock prices. The use of a state variable that indicates such future changing
investment opportunity sets, one that impounds market information immediately, provides a
great deal of substance to the original CAPM.
Merton’s model is set in a continuous time economy where both asset prices and state
variables are assumed to follow standard diffusion processes and exist in market equilibrium.
Merton assumes the existence of a risk averse representative agent with a utility of wealth
function )),(),(( ttFtWJ , where )(tW is wealth and )(tF is a variable describing the state of
investment opportunities in the economy. In equilibrium the expected market risk premium is
tMF
W
WF
tM
W
WW
tMtJ
J
J
WJrE ,
2
,,1 ][ σσ
−+
−=− (1)
where ][1 ⋅−tE is the expectation operator conditional on information available at time t-1, rM,t
is the return to representative asset M, and 2
,tMσ and tMF ,σ are the conditional variance and
conditional covariance with a state variable F conditional on information available at time t-
1. The subscripts of J denote partial derivatives.
The first term [ ]WWW JWJ− in equation (1) is the coefficient of relative risk aversion.
Non-satiation and risk aversion implies, respectively, 0>WJ and 0<WWJ , suggesting a
positive relationship between the market risk premium and conditional market variance. The
second term [ ]WWF JJ− in (1) is the market risk premium attaching to the conditional
covariance between the market and a hedging, or state, variable that may be used by investors
to (partially) offset the risk associated with holding the market asset in isolation. The sign of
this relationship is indeterminate as it depends upon investors assumed utility functions and
the state of the conditional covariance. Firstly, we observe that if the marginal utility of
wealth is independent of the assumed state variable )0( =WFJ then (1) reduces to the familiar
7
CAPM of Sharpe (1964) where the expected market risk premium is a function solely of
conditional market variance. However, if 0≠WFJ , the expected market risk premium is also a
function of conditional market covariance with the state variable F. If 0>WFJ and
0, >tMFσ , or if 0<WFJ and 0, <tMFσ , investors will demand a lower risk premium on the
market portfolio because some of the risk associated with holding the static market portfolio
is being diversified away in an intertemporal setting. Conversely, investors will demand a
higher risk premium if 0>WFJ and 0, <tMFσ , or if 0<WFJ and 0, >tMFσ , since the ability
to diversify the market portfolio over time is reduced, increasing the risk of holding the static
market portfolio. The assumption of non-satiation implies that 0>WJ , so that all we can say
about [ ]WWF JJ− is that it’s sign is the opposite of WFJ , where WFJ represents how the risk
profile of the representative investor changes with respect to changes in the state variable.
Of course, the selection of the ‘hedging variable’ is, as Merton points out, a subjective
one, however the bond market is a natural choice as there is overwhelming evidence that both
short term interest rates and long term bonds are priced into the equity market.2 As noted by
Scruggs (1998) long-term bonds are a natural instrument for hedging interest rate risk since
their returns are negatively correlated with changes in interest rates. Additionally, support for
the ICAPM within an Australian context has been reported by Dean and Faff (2001) who find
evidence that the conditional covariance of equity and bond market returns is a significant
risk factor priced into equity market returns. Accordingly, we focus upon this support of the
ICAPM in conditional risk premia in an effort to further explain the observed autocorrelation
in asset prices.
2 In a US context Fama and Schwert (1977), Christie (1982), Chen, Roll and Ross (1986), Shanken (1990),
Glosten et al. (1993) all find that both the level and volatility of short-term interest rates are associated with
significant shifts in the bond and equity markets. In the Australian context, Faff and Howard (1999) report that
the financial and banking sector of the Australian market is sensitive to long-term interest rates.
8
To incorporate feedback trading into an ICAPM setting, we look to the model
developed by Sentana and Wadhwani (1992). Sentana and Wadhwani propose a model in
which two groups of agents trade shares. The first group are ‘information’ traders who value
shares in the context of the CAPM. Specifically, they have a demand function for shares
given by:
t
tt
t
rEI
µ
α−= − )(1 (2)
where It is the fraction of shares they hold; rt is the ex-post return in period t; α is the return at
which the demand for shares by this group is zero (risk-free rate) and µt is the risk premium
needed to induce them to hold shares. In a conventional CAPM setting it is assumed that the
risk premium is a function of volatility (σ2) viz:
)( 2
tt σµµ = (3)
and given the usual risk aversion assumption, µ’ ( ) > 0. In an ICAPM setting we assume that
the risk premium is a linear function of the market volatility (σM2) and the covariance
between the return on the market and state variable, F, (σMF2) viz:
µt = µ (σMt2, σMFt) = µ1(σMt
2) + µ2(σMFt) (4)
and given the usual risk aversion assumption, µ1’ ( ) > 0, while the sign of µ2’( ) is
indeterminate as discussed above. In an ICAPM world, all investors have the same demand
function given by (2) with the risk premium now defined by (4) - then in market equilibrium
(It = 1) we have the ICAPM:
)()()( ,2
2
,1,1 tMFtMtMt rE σµσµα +=−− (5)
In the Sentana and Wadhwani (1992) model, the second group of agents is a group of
traders who naïvely base their decisions on past price information (naïve traders). Feedback
trading can be neatly partitioned into two strategies. On the one hand there is a group of naïve
traders that systematically follow the strategy of buying after price rises and of selling
9
following price falls – they are classed as ‘positive’ feedback traders. A number of possible
explanations for this type of behaviour have been proposed including the presence of
technical analysts’, extrapolative expectations, dynamic trading strategies, the liquidation of
positions held by traders unable to meet margin calls, or the use of stop loss orders by
investors. Evidence of this type of behaviour for both individual investors and institutions
can be found in Bange (2000) and Nofsinger and Sias (1999), respectively. The demand
function for shares by feedback traders can be expressed as:
1−⋅= tt rF γ (6)
where for positive feedback traders the expected sign on γ is positive. Positive feedback
traders reinforce price movements such that prices will continually overshoot the levels
suggested by current publicly available information. This over-reaction phenomenon may be
exacerbated where rational speculators anticipate feedback traders decisions [see DeLong,
Shleifer, Summers and Waldmann (1990)]. As the market corrects for this over-reaction in
the following days trading, prices tend to move in the opposite direction and so negative
autocorrelation results.
On the other hand, naïve traders may choose to engage in ‘negative’ feedback trading
which implies a negative sign on γ in Equation (6). In this case, traders sell following a price
rise to close out their positions and lock in profits. This selling pressure causes price to close
lower than it otherwise would have based on the market information set. In the following
days trading, price will again rise to correct for this profit taking on the part of traders
resulting in positive autocorrelation. Thus, positive autocorrelation results from negative
feedback trading.
In our more general setting, market equilibrium requires that the shares on issue be
held by information traders (It) or by feedback traders (Ft), i.e.:
It + Ft = 1 (7)
10
In our ICAPM setting with both information traders and feedback traders in market
equilibrium, substituting (2), (4) and (6) into (7) produces:
12
2
12
2
11 )]σ(µ)(σγ[µ)σ(µ)(σµα)( −− +−+=− tMFtMtMFtMttt rrE (8)
Compared to the standard ICAPM of (5), the ‘behavioral’ ICAPM (BICAPM) of (8)
has an extra term induced by allowing for the existence of feedback traders. This extra term
in the BICAPM is an autocorrelation term wherein the autocorrelation coefficient
is )]σ(µ)(σγ[µ 2
2
1 MFtMt +− . Notably, it is a function of (a) the dominant type of feedback
trading; (b) the volatility of market returns, and (c) the covariance of market returns with the
state variable. Of further interest is the fact that its sign is now determined by three factors –
(a) the dominant type of feedback trading (i.e. by the sign of γ); (b) the sign of the covariance
term and (c) the sign of µ2’( ).
Referring to the model of Sentana and Wadhwani (1992) [equation (6) in their paper]
1
2
,1
2
,11 )]([)()( −− −=− ttMtMtt rrE σµγσµα (9)
and comparing this ‘behavioral’ CAPM (BCAPM) to our equation (8) we see that basing
investors risk preferences upon the ICAPM incorporates an additional autocorrelation factor
)]([ ,2 tMFσµ that should provide additional explanation of the autocorrelation properties of
returns.
11
3. Empirical Framework
The empirical form of equation (8) in our proposed bivariate GARCH-M setting gives the
following system of equations for the joint process of conditional returns:
tbtbtebbtbbbtebbtbbbtb
tetetebeteeetebeteeete
rr
rr
,1,,3
2
,21,3
2
,21,
,1,,3
2
,21,3
2
,21,
)(
)(
εσγσγγσλσλλ
εσγσγγσλσλλ
++++++=
++++++=
−
− (10)
=
tb
te
t
,
,
ε
εε
),0(~| 1 ttt HN−Ωε
=
2
,,
,
2
,
tbteb
tebte
tHσσ
σσ
where re,t (rb,t) is the equity (bond) market return at time t; σ2
e,t (σ2
b,t) is conditional volatility
in the equity (bond) market return at time t; and σeb,t is conditional covariance between the
equity market and bond market returns at time t.
For the parameterization of the variance-covariance matrix Ht, there is a trade-off
between ease of estimation, model complexity and the ability to reliably compute large
number of parameters subject to restrictions (i.e. the variance must be positive). Such
complexity in multivariate models has seen the common use of the constant correlation
model of Bollerslev (1990) and although the assumption of constant correlation through time
is open to criticism, it has yielded consistent results as reported in the literature.
However, the assumption of constant correlation is highly simplistic and, for our
purposes, too restrictive in examining the impact of changing conditional covariance. Indeed,
in examining the interaction effects in contemporaneous and informationally linked markets
like the bond and equity markets, it is reasonable to expect that the movement of both
markets will be correlated to some extent, i.e. 0)( ,, ≠tetbE εε and that this correlation will not
necessarily be constant.
12
In our particular case, the conditional mean equation contains a large number of
parameters which, combined with a complex variance-covariance matrix, would be difficult
to achieve convergence and/or reliable results. We therefore propose to use a GARCH (1,1)
version of the diagonal BEKK model of Engle and Kroner (1995) which is sufficiently
general to allow conditional variances and covariances of both markets to influence each
other, whilst being relatively parsimonious compared to other ARCH models.3 The diagonal
BEKK is modeled as
GHGAACCH tttt 111 −−−′+′′+′= εε (11)
where C, A and G are diagonal matrices,
=
=
=
22
11
22
11
22
11
0
0,
0
0,
0
0
g
gG
a
aA
c
cC ,
This parameterisation allows for the possibility that conditional covariance can change from
positive to negative or vice versa over the sample period – something that the constant
conditional correlation model would not allow. This is an important aspect since we are
looking for the (possibly changing) impact of conditional covariance in the serial correlation
properties of the returns.
Numerical procedures are used to maximise the log likelihood functions using the
BHHH estimation procedure (Berndt, Hall, Hall and Hausman, 1974). We estimate the model
by maximising the log likelihood function assuming the errors follow a N(0,1) distribution
and standard errors are calculated by inverting the Hessian at the maximum likelihood
parameter estimates.
3 Our choice of the symmetric GARCH (1,1) is in keeping with Koutmos (1997). We experimented with
asymmetric GARCH specifications, but encountered non-trivial convergence problems in our bivariate setting
for some of our data series. As revealed later, the diagnostics for our reported model estimates indicate that the
parsimonious symmetric framework is justified.
13
4. Estimation
4.1 Preliminaries
In order to avoid potential concerns of data mining and to seek broader relevance of our
analysis, we apply equation (10) to returns on broad equity and bond indices of six major
world markets: the US, UK, Canada, Australia, Japan and Germany.
Daily prices representing the total return to equity and the total return to government
bonds, as constructed by Datastream, were obtained from Datastream. The total return bond
index was an ‘all lives’ index representing the average total return across all government
bonds of varying maturities. The period under investigation extends from 1 January 1990 to
30 June 2005 producing a dataset numbering 4041 observations. In addition, for each index in
each country a relevant risk free rate was subtracted to obtain a daily excess return. Table 1
presents a summary of the relevant series used for each country and market as provided by
Datastream.
Table 2 presents preliminary descriptive statistics for the daily returns data. Several
features of interest are conveyed in the table. First, it is evident that over our sample period
the average return to the equity index exceeds the average return to the counterpart bond
index for four of our six countries – notably, Japan and Germany represent the two
exceptions. Second, in all cases the standard deviation of equity market returns is higher than
the associated bond market returns. We would expect this to be the case. Third, also as
expected, for financial time series, all return distributions are mildly skewed and quite
leptokurtotic relative to the normal distribution. Fourth, stationarity of all return series was
confirmed by the Augmented Dickey-Fuller test. Fifth, intertemporal dependencies in both
the daily returns and squared daily returns for equity and bond markets are indicated by the
Ljung-Box statistics (12 lags), on both the raw data and the square of the data series. Also,
the presence of ARCH effects in the residuals was universally confirmed using the LM
14
ARCH test of Engle (1982) – particularly so, for all equity return series. Overall the data
appear consistent with autoregressive conditional heteroskedasticity and volatility clustering
characteristics of high frequency data (Bollerslev, Chou and Kroner, 1992) suggesting that
the GARCH class of models are appropriate.
4.2 Unconditional Serial Correlation in Returns
Before proceeding to estimate the more complex conditional mean equation (10) and to
provide some preliminary evidence of the existence of autocorrelation in bond and equity
market returns, we estimate the following model with a GARCH(1,1) conditional variance:
ttt rr εγλ ++= −111 (12)
The presence of serial autocorrelation of returns is indicated by 1γ being significantly
different from zero. Table 3 presents the results of the estimation and evidence of
autocorrelation exists across all countries for both bond and equity markets. With regard to
the equity markets represented in our sample, the autocorrelation parameter estimates are all
positive and range from 0.0444 (US) to 0.1738 (Canada). Similarly, the bond market
autocorrelation parameter estimates are (nearly) all positive and range from -0.0387
(Australia) to 0.0831 (Canada). According to the model of Sentana and Wadhwani (1992)
these preliminary results suggest that, on average, negative feedback traders generally
dominate these markets over our time period. Of course what this likely hides is the potential
time variation in feedback trading behaviour that may take place, depending on the level of
volatility (‘traditional’ BCAPM) and the extent of equity/bond correlation (our BICAPM).
15
4.3 Sentana and Wadhwani (1992) Behavioral CAPM Estimation Results
In this section we apply the model of Sentana and Wadhwani (1992) across the equity
markets, similar to that reported by Koutmos (1997). However, more notably, our paper is to
the best of our knowledge the first to see if there is evidence supporting their feedback
trading hypothesis across the bond markets. Specifically, using the following model of
Sentana and Wadhwani
ttttt rr εσγγσλλ ++++= −1
2
21
2
21 )( (13)
where 2
tσ is estimated using a GARCH(1,1) process [consistent with Koutmos (1997)],
evidence consistent with feedback trading exists if the parameter, γ2, is significant. Table 4
reports the outcome of estimating this model.
It is quite apparent from the table that there is widespread support for the feedback
trading hypothesis – the estimated coefficient γ2, is significant across all equity and bond
markets with two exceptions: the German equity market and the US bond market. That is,
there is significant evidence of feedback trading for 10 of the 12 markets examined. Of these
cases, we see that four of the five significant equity-based coefficients are negative –
suggesting that, in these equity markets, as volatility increases it is more likely that serial
correlation will become negative. That is, in these four cases (Australia, Canada, Japan and
the UK) equity markets tend to become more influenced by positive feedback traders as
volatility increases. For the lone other significant case, the US, the large positive coefficient
suggests the converse: namely, that its equity market will become more influenced by
negative feedback traders as volatility increases. These findings are qualitatively the same as
those reported by Sentana and Wadhwani (1992) in the US equity market and Koutmos
16
(1997) in the Australian, Japanese and UK equity markets where higher volatility is more
likely to induce returns to exhibit negative serial correlation.4
With regard to the five bond market cases that show a significant estimated γ2
coefficient in Table 4, the most notable feature is that (like their equity market counterparts)
four of the five are negative in sign. Once more this indicates that increases in volatility tend
to induce lower and even negative serial correlation (Australia, Canada, Germany and the
UK), whereas for Japan the opposite is true. Hence, these new findings suggest that in the
bond markets (except for Japan and the US), positive feedback traders tend to become more
influential/prevalent as volatility increases.
We can use the parameter estimates in Table 4 to help us understand from where
some of the autocorrelation observed in Table 3 comes. As an example, let us look at the
Australian equity and bond markets noting that Table 3 reported positive autocorrelation for
the Australian equity market (γ1 = 0.0656) and negative autocorrelation (γ1 = -0.0387) for the
Australian bond market. Table 4 parameter estimates for the Australian equity market are 1γ
= 0.0841 and 2γ = -0.0319, that is, 1γ is positive and in absolute terms almost three times as
large as 2γ . These parameter estimates imply that for values of 2
tσ < 2.6 there will be positive
autocorrelation, while variances larger than 2.6 produce negative autocorrelation.
Interestingly, the GARCH variance series estimated for the Australian equity market
shows an average 2
tσ of 0.798 with a standard deviation of 0.644 – indicating that when
variance is less than approximately three standard deviations above its mean (i.e. a very
common event) then the Australian equity market exhibits positive serial correlation. Only
when variance increases to more than three standard deviations above the mean, does
4 These findings for equity markets are remarkably similar to Koutmos (1997) – he reports negative coefficients
of very similar magnitude to ours, for his earlier/shorter sample period (1986 to 1991). The only contradictory
case is Germany – Koutmos (1997) finds German equities to also have a negative and significant estimated γ2
coefficient.
17
autocorrelation of returns become negative. Clearly, the strong suggestion is that negative
feedback traders are a dominant feature of trading in the Australian equity market. Moreover,
according to the findings reported in Table 4, this conclusion is basically applicable to all
equity markets in our sample with the exception of Germany and the US. In the case of the
US equity market, it appears that positive feedback trading dominates.
Looking similarly at Australian bond market returns as an illustrative example, Table
4 shows 1γ = 0.308 and 2γ = -2.332 from which we can see that 1γ is relatively much smaller
in magnitude than 2γ and clearly showing from where the observed negative autocorrelation
in Table 3 comes. Specifically, for values of 2
tσ < 0.132 there will be positive autocorrelation,
while for values of 2
tσ > 0.132 negative autocorrelation results. Interestingly, the average
level of Australian bond market variance from our GARCH model is 0.101, with a standard
deviation of 0.090. Although this level of variance is (predictably) much lower than the
counterpart equity market value, it suggests that negative autocorrelation in returns will be
relatively common across many volatility scenarios. Indeed, it takes abnormally low levels of
volatility for this bond market to demonstrate economically important positive
autocorrelation. This conclusion is basically applicable to all bond markets except Japan and
the US.
4.4 Specification Tests of the Bivariate Model
Prior to discussing the parameter estimates for equation (10), we examine the specification
tests on the residuals. To have a situation in which the specification is acceptable, we require
that the distribution of standardised residuals ,tiz satisfy 0][ , =tizE , 1][ 2
, =tizE and that
,tiz and 2
,tiz are serially uncorrelated. Table 5 shows a summary of such diagnostic metrics
on the standardised residuals for the diagonal BEKK model and reports the p-values
associated with these tests. The mean and variance tests are all satisfied. Further, Ljung-Box
18
tests (lag 12) for serial correlation of raw and square residuals show little evidence of residual
autocorrelation or heteroskedasticity remaining. Similarly, ARCH LM tests of the residuals
reveal that ARCH effects have dramatically diminished (compared to corresponding analysis
of the raw returns reported in Table 2) and in all cases but two, have totally disappeared.
Collectively, the diagnostics reported in Table 5, strongly suggest that the model is well
specified.
4.5 Behavioral ICAPM Estimation Results
Equation (10) was estimated maximising the log likelihood of the bivariate system using a
diagonal BEKK formulation for the conditional variance/covariance matrix. Due to the large
number of parameters being estimated, only the conditional mean estimates are reported in
Table 6.5
Recalling the theory of feedback trading within an ICAPM framework developed in
Section 2 above, our main parameter of interest is γ3, which, if significant, provides evidence
that conditional covariance could also account for some of the observed autocorrelation of
returns. That is, our main proposition is that the observed dynamic of autocorrelation
associated with financial time series may also be attributable to another factor beyond
conditional variance as reported by Sentana and Wadhwani (1992) and Koutmos (1997). We
note from Table 6 that eight of the twelve series examined report γ3 as being significant at the
5% level (with one other significant at 10%), which provides encouraging support for our
ICAPM / feedback trading model.
5 The conditional variance estimates are suppressed to conserve space.
19
Looking more closely at Table 6, consider the Australian market as an illustrative
case. First, for the Australian equity market we find that equity market conditional variance
and conditional covariance (with the bond market) are priced into the equity market (that is,
λ2 and λ3 are significant) – this is in support of the ICAPM of Merton (1973). We also find
neither conditional variance or covariance is priced into the Australian bond market (that is,
λ2 and λ3 are not significant). This result is similar to that reported in Dean and Faff (2001).
Furthermore, looking at the serial correlation parameters, 1γ , 2γ and 3γ , we find
evidence that conditional covariance does play a role in determining the serial correlation of
returns for both the equity market and the bond market in Australia. With regard to the equity
market, we can see that while the estimated value of 1γ is not statistically significant, 2γ and
γ3 are both significant with estimated values of -0.077 and 0.366, respectively. These results
suggest that during high periods of volatility, there is enough feedback trading to produce
negative first order autocorrelation, but an additional factor is bond market conditional
covariance which can reduce or strengthen the apparent autocorrelation, depending on the
sign of the covariance. Thus conditional covariance as observed here can significantly
influence the first order correlation properties of equity returns. Specifically, when
covariance is negative (positive) then the impact of feedback traders tends to induce negative
(positive) autocorrelation in equity returns.
Next looking at the estimated values for the Australian bond market, we can see that
1γ is not significant, 2γ is negative and significant at the 10% level, and 3γ is negative and
significant at the 5% level. That is, we find evidence that serial correlation in the bond market
is a function of conditional volatility and conditional covariance, similar to that of the equity
market. Similar to its equity market counterpart, at times of sufficiently increased bond
market volatility there is enough feedback trading to produce negative first order correlation
in bond market returns. However, we note that the sign of 3γ is the opposite of that found in
20
the equity market – therefore the impact of the changing dynamics of conditional covariance
will be different for the bond market than for the equity market. Specifically, in this
Australian case, negative (positive) conditional covariance will tend to induce negative
(positive) serial correlation in the equity market. Conversely for the Australian bond market,
negative (positive) conditional covariance will tend to induce positive (negative) serial
correlation.
Five of our six equity markets display the same covariance effect in serial correlation,
as seen in the Australian case just described i.e. a significantly positive role – only Germany
lacks such evidence. Interestingly, in most cases this covariance effect totally dominates the
variance effect, for example, in the US the variance term, 2γ , is starkly insignificant. Notably,
the role of covariance is less prominent in the autocorrelation of bond returns – only three
cases are significant: namely, Australia – which suggests a negative role, while Japan and the
UK indicate a positive role.
4.6 Formal Tests of Nested Models within the BICAPM Framework
As a final piece of analysis that will hopefully allow a tighter conclusion to be drawn
regarding the usefulness and insights delivered by our model, we perform some formal
testing of the asset pricing models nested within our overall behavioural ICAPM (BICAPM)
framework. In essence we have a two x two matrix of pricing models open for scrutiny,
relative to each other, across our sample of six countries. Along one dimension is the
distinction between ‘traditional’ versus behavioural asset pricing models (where the
behavioural models are confined to the feedback trading related models). Along the second
dimension is the distinction between the CAPM versus the ICAPM. Accordingly, the
hypotheses tested within our BICAPM framework of equation (10) are:
21
H10: λ3 = γ2 = γ3 = 0 (i.e. CAPM is the null model)
H20: γ2 = γ3 = 0 (i.e. ICAPM is the null model)
H30: λ3 = γ3 = 0 (i.e. BCAPM is the null model)
Collectively, if all three restrictions above are rejected, then evidence falls in favour of the
BICAPM. We apply these tests to equity markets only.6
The outcome of performing these nested asset pricing tests on each equity market is
reported in Table 7 and several features are evident. First, with regard to the CAPM
restriction (H10) we see that all cases show rejection at the 5% level, with the exception of
Japan which rejects at the 10% level. Second, with regard to the ICAPM restriction (H20)
universal rejection is recorded at the 5% level of significance. Third, the BCAPM restriction
(H30) is rejected at the 5% level for all equity markets with the exception of the US in which
the p-value is 0.055. In sum, it is apparent that the BICAPM is a reasonable model for all six
equity markets – particularly for Australia; Canada; Germany and the UK. For Japanese
equities there is some suggestion that the CAPM may suffice, while for the US the original
BCAPM of Sentana and Wadhwani (1992) would probably be preferred on the principle of
parsimony.
6 Application of these tests to the bond markets is problematic and potentially misleading. One way to see this is
to note that the results in Table 6 show that the bond market in-mean variance term is only significant in one
case (Germany) – but even then the estimated coefficient takes a negative sign, which is difficult to rationalize
from an asset pricing point of view. In contrast, the equity in-mean variance term is positive and significant in
all six cases (see Table 6) – thereby, making sense of the H10 CAPM restriction.
22
5. Summary and Conclusion
This paper further investigates the properties and determinants of the observed
autocorrelation structure of returns as previously reported by Sentana and Wadhwani (1992)
and Koutmos (1997), amongst others. We contribute to the literature by applying the
Intertemporal CAPM of Merton (1973) to the feedback trading model of Sentana and
Wadhwani and derive a behavioural version of the ICAPM (BICAPM). Our BICAPM
presents a model of conditional returns that includes conditional covariance as a determinant
of the serial correlation properties of returns.
Applying this model to the excess daily returns in six major equity and bond markets
(Australia; Canada; Germany; Japan; the UK and the US) we find that the BICAPM is a
useful framework for exploring behavioural influences on asset pricing. This is particularly
so for equity markets. Further, we find evidence to support the inclusion of conditional
covariance as an additional determinant of serial correlation in returns. We find that
covariance with the bond market can be important to feedback traders in the equity market,
and that covariance with the equity market can be important for feedback traders in the bond
market.
23
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26
Table 1: Data Series Obtained from Datastream
Datastream Series Name Datastream Code
Australia
Equity Market AUSTRALIA - DS MARKET TOTMKAU(RI)
Bond Market AU TOTAL ALL LIVES DS GOVT. INDEX AAUGVAL(RI)
Risk-Free Proxy AUSTRALIA DEALER BILL90 DAY ADBR090
Canada
Equity Market CANADA - DS MARKET TOTMKCN(RI)
Bond Market CN TOTAL ALL LIVES DS GOVT. INDEX ACNGVAL(RI)
Risk-Free Proxy CANADA TREASURY BILL 1 MONTH CN13883
Germany
Equity Market GERMANY - DS MARKET TOTMKBD(RI)
Bond Market BD TRACKER ALL LIVES DS GOVT. INDEX TBDGVAL(RI)
Risk-Free Proxy GERMANY EURO - MARK 1 MTH ECWGM1M
Japan
Equity Market JPN – DS MARKET TOTMKJPN(RI)
Bond Market JPN TRACKER ALL LIVES DS GOVT. INDEX TJPGBAL(RI)
Risk-Free Proxy JPN EURO YEN 3 MTH ECJAP3M
UK
Equity Market UK – DS MARKET TOTMKUK(RI)
Bond Market UK TOTAL ALL LIVES DS GOVT. INDEX AUKGVAL(RI)
Risk-Free Proxy UK TREASURY BILL DISCOUNT 3 MTH LDNTB3M
US
Equity Market US - DS MARKET TOTMKUS(RI)
Bond Market US TRACKER ALL LIVES DS GOVT. INDEX TUSGVAL(RI)
Risk-Free Proxy US CD 3 MONTH USCOD3M
27
Table 2: Descriptive Statistics This table reports some basic descriptive statistics using daily excess returns from 1 January 1990 to 30 June
2005 for six major equity and bond market indices obtained from Datastream. Excess daily returns (x100) are
based on the equity Total Return Index and All Lives Government Bond Total Return Index minus the relevant
daily return on a proxy for the risk free rate (see Table 1). ADF is the augmented Dickey-Fuller test for a unit
root; Q(z)12 is the Ljung-Box statistic of order 12 for returns; Q(z)2
12 is the Ljung-Box statistic of order 12 for
squared returns; ARCH (2) is the Engle (1982) test for ARCH up to order 2.
Mean Std Dev Skew Kurt J-Bera ADF Q(z)12 Q(z)
212 ARCH(2)
Australia
Equity Rtns 0.0254 0.7919 -0.257 7.534 3508 60.29* 31.45* 453.5* 157.7*
Bond Rtns 0.0187 0.3175 -0.259 5.908 1470 67.49* 26.71 257.1* 25.1*
Canada
Equity Rtns 0.0270 0.8080 -0.638 10.09 8750 57.1* 64.5* 937.7* 107.6*
Bond Rtns 0.0198 0.3067 -0.372 6.226 1848 58.9* 32.8* 344.1* 45.6*
Germany
Equity Rtns 0.0108 1.1323 -0.410 7.382 3349 59.8* 37.5* 1299* 118.9*
Bond Rtns 0.0156 0.2519 -0.777 7.631 4022 62.2* 7.21 189.5* 8.5*
Japan
Equity Rtns -0.020 1.2269 0.0781 6.915 2587 45.2* 58.2* 485.3* 59.9*
Bond Rtns 0.0132 0.1837 -0.467 7.560 3651 58.7* 80.0* 791.2* 84.1*
UK
Equity Rtns 0.0179 0.9260 -0.148 6.438 2007 61.3* 45.5* 2603* 254.2*
Bond Rtns 0.0169 0.3172 -0.015 6.520 2066 59.3* 34.5* 242.7* 27.8*
US
Equity Rtns 0.0300 1.0048 -0.128 7.169 2940 62.3* 28.5* 1196* 151.6*
Bond Rtns 0.0151 0.3032 -0.283 5.589 1184 60.0* 39.9* 255.7* 24.7*
* indicates statistical significance at the 5% level.
28
Table 3: Estimation of Unconditional AR (1) Model This table reports the results of a univariate AR (1) model [equation (12)] using daily
excess returns from 1 January 1990 to 30 June 2005 for six major equity and bond market
indices obtained from Datastream.
ttt rr εγλ ++= −111 (12)
2
1
2
1
2
, −− ++= ttte βσαεωσ
λ1 γ1 ω α β LogL
Australia
Equity Rtns 0.0368* 0.0656* 0.0128* 0.0691* 0.9120* -4574.89
t-stat 3.1880 4.2220 6.2223 14.05 131.02
Bond Rtns 0.0223* -0.0387* 0.0013* 0.0337* 0.9535* -927.41
t-stat 4.8093 -2.3833 7.0098 11.89 260.52
Canada
Equity Rtns 0.0340* 0.1738* 0.0054* 0.0721* 0.9211* -4197.18
t-stat 3.5416 10.4001 6.8743 17.89 230.35
Bond Rtns 0.0200* 0.0831* 0.0011* 0.0580* 0.9380* -734.37
t-stat 4.6176 5.2030 5.2541 12.75 189.55
Germany
Equity Rtns 0.0306* 0.0652* 0.0256* 0.0882* 0.8910* -5662.53
t-stat 2.0913 3.6206 10.0914 13.46 113.15
Bond Rtns 0.0197* 0.0346* 0.0006* 0.0544* 0.9377* - 125.11
t-stat 5.7504 1.9743 6.8972 15.70 243.26
Japan
Equity Rtns 0.0094 0.1000* 0.0445* 0.1102* 0.8645* -6221.15
t-stat 0.5784 5.9349 8.4075 14.35 91.59
Bond Rtns 0.0113* 0.0713* 0.0001* 0.0711* 0.9317* - 1715.73
t-stat 5.7692 4.3688 6.6050 19.17 346.62
UK
Equity Rtns 0.0338* 0.0488* 0.0105* 0.0786* 0.9089* -4867.47
t-stat 2.9186 2.9521 5.2936 12.33 119.85
Bond Rtns 0.0198* 0.0708* 0.0014* 0.0385* 0.9474* -916.12
t-stat 4.3351 4.1893 6.5672 14.11 271.65
US
Equity Rtns 0.0478* 0.0444* 0.0059* 0.0581* 0.9368* -5190.91
t-stat 3.8405 2.5750 5.7443 13.63 206.23
Bond Rtns 0.0151* 0.0488* 0.0001 0.0356* 0.9642* -608.50
t-stat 3.9435 2.9522 1.9226 12.47 337.18
* indicates statistical significance at the 5% level.
29
Table 4: Estimation of Sentana and Wadhwani Behavioral CAPM This table reports the maximum likelihood estimates of the GARCH (1,1)-M model [equation (13) in the text] from
Sentana and Wadhwani (1992).
ttttt rr εσγγσλλ ++++= −1
2
21
2
21 )( (13)
2
1
2
1
2
−− ++= ttt βσαεωσ
The sample involves daily excess returns from 1 January 1990 to 30 June 2005 for six major equity and bond market
indices obtained from Datastream.
λ1 λ2 γ1 γ2 ω α β LogL
Australia
Equity Rtns 0.0592* 0.0517 0.0841* -0.0319* 0.1125* 0.0684* 0.9130* -4572.6195 t-stat 2.3660 1.0709 2.5319 -2.6344 12.2134 13.9946 131.2544
Bond Rtns 0.1289* -0.8905* 0.3078* -2.3321* 0.0403* 0.0643* 0.9325* -1337.6888 t-stat 13.0866 -39.9337 52.2612 -57.8098 8.8761 21.9977 159.7057
Canada
Equity Rtns 0.0258 0.0205 0.2307* -0.0886* 0.0720* 0.0715* 0.9219* -4186.2802 t-stat 1.7825 0.6935 9.7730 -3.3276 13.4015 18.0354 234.2300
Bond Rtns 0.0508 -0.2637* 0.5277* -1.9804* 0.0244 0.1309* 0.8963* -1070.4417 t-stat 5.6917* -29.3842 14.3948 -11.2018 1.6115 9.7648 84.3441
Germany
Equity Rtns 0.0795 -0.0484 0.1437 -0.0431 0.1998* 0.1241* 0.8496* -6123.0826 t-stat 0.8902 -0.6531 1.3873 -0.7062 4.2679 4.6388 25.8988
Bond Rtns 0.1282* -1.1841* 0.2144* -1.2773* 0.0151 0.1203* 0.9099* -346.6372 t-stat 14.7095 -12.2981 4.5819 -8.6227 1.4506 9.7457 77.8308
Japan
Equity Rtns -0.0460* 0.0512* 0.1309* -0.0213* -0.2129* 0.1106* 0.8633* -6216.9340 t-stat -2.4601 2.1709 4.1975 -2.3276 -16.6584 14.0236 89.6521
Bond Rtns 0.0097 0.1006 0.0701* 0.0257* 0.0093* 0.0723* 0.9304* 1714.1759 t-stat 3.6250* 0.8443 2.9222 3.0459 12.9773 19.0024 339.6716
UK
Equity Rtns 0.0140* 0.0362 0.0733* -0.0286* 0.1026* 0.0787* 0.9086* -4864.1441 t-stat 2.7405 1.3175 3.0615 -3.4419 10.4780 12.2748 118.7780
Bond Rtns 0.0266 -0.0842 0.1395* -0.6730* 0.0370* 0.0366* 0.9495* -911.4146 t-stat 2.2200* -0.6426 3.6042 -2.0297 13.3710 13.7712 286.5580
US
Equity Rtns 0.2839 -0.2773* -0.3772* 0.3552* 0.1869* 0.0327* 0.9330* -5857.9518 t-stat 14.6442* -63.4601 -31.7891 78.7196 116.1699 96.4147 771.7169
Bond Rtns 0.0168 -0.0434 0.0616 -0.1221 0.0097* 0.0317* 0.9677* -602.7475 t-stat 2.4467* -0.4741 1.7458 -0.3466 3.7289 11.6930 361.0699
* indicates statistical significance at the 5% level.
30
Table 5: Specification and Diagnostic Tests in the Behavioral ICAPM
setting This table reports the specification and diagnostic tests of the residuals from the estimation of
the bivariate GARCH-M model [equation (10) in the text], representing the behavioural
ICAPM. The variance-covariance matrix is estimated using the GARCH (1,1) version of the
diagonal BEKK formulation. Conditional errors are assumed to follow the Normal
distribution.
tbtbtebbtbbbtebbtbbbtb
tetetebeteeetebeteeete
rr
rr
,1,,3
2
,21,3
2
,21,
,1,,3
2
,21,3
2
,21,
)(
)(
εσγσγγσλσλλ
εσγσγγσλσλλ
++++++=
++++++=
−
−
),0(~1,
1,
1 HNtb
te
t
=
−
−
− ε
εε
The sample involves daily excess returns from 1 January 1990 to 30 June 2005 for six major
equity and bond market indices obtained from Datastream. Q(z)12 is the Ljung-Box statistic of
order 12 for returns; Q(z)2
12 is the Ljung-Box statistic of order 12 for squared returns; ARCH
(4) is the Engle (1982) test for ARCH up to order 4.
E(z) = 0
p-value
E(z2) = 1
p-value
E(z|z|) = 0
p-value
Q(z)12
p-value
Q2(z)12
p-value
ARCH(4)
p-value
Australia
Equity Rtns 0.5334 0.3655 0.3126 0.369 0.950 0.6749
Bond Rtns 0.7893 0.5598 0.9075 0.415 0.975 0.4551
Canada
Equity Rtns 0.4489 0.1803 0.7745 0.334 0.978 0.5864
Bond Rtns 0.6074 0.1975 0.2456 0.097 0.296 0.0225*
Germany
Equity Rtns 0.6888 0.8871 0.4454 0.230 0.541 0.9901
Bond Rtns 0.4668 0.3369 0.2050 0.257 0.209 0.0530
Japan
Equity Rtns 0.9223 0.2730 0.8595 0.449 0.916 0.9345
Bond Rtns 0.3285 0.1683 0.3353 0.250 0.169 0.8520
UK
Equity Rtns 0.3313 0.4106 0.4712 0.442 0.208 0.0089
Bond Rtns 0.6026 0.4051 0.5480 0.302 0.384 0.7669
US
Equity Rtns 0.4573 0.2756 0.7458 0.027* 0.398 0.1061
Bond Rtns 0.2790 0.3801 0.7433 0.159 0.049* 0.5963
* indicates statistical significance at the 5% level.
31
Table 6: Conditional Mean Estimates of Behavioral ICAPM in a Multivariate
setting This table reports the maximum likelihood estimates for the mean equation of the bivariate GARCH-M
model [equation (10) in the text], representing the behavioural ICAPM. The variance-covariance matrix is
estimated using the GARCH (1,1) version of the diagonal BEKK formulation. Conditional errors are
assumed to follow the Normal distribution and standard errors are calculated from the inverse of computed
Hessian.
tbtbtebbtbbbtebbtbbbtb
tetetebeteeetebeteeete
rr
rr
,1,,3
2
,21,3
2
,21,
,1,,3
2
,21,3
2
,21,
)(
)(
εσγσγγσλσλλ
εσγσγγσλσλλ
++++++=
++++++=
−
−
),0(~1,
1,
1 HNtb
te
t
=
−
−
− ε
εε
The sample involves daily excess returns from 1 January 1990 to 30 June 2005 for six major equity and
bond market indices obtained from Datastream.
λ1 λ2 λ3 γ1 γ2 γ3 LogL
Australia
Equity Rtns 0.0393* 0.0797* -0.5030* 0.0024 -0.0769* 0.3660* -5360.41
p-value 0.0000 0.0000 0.0000 0.9274 0.0003 0.0274
Bond Rtns 0.0320* -0.1727 0.1405 0.0261 -0.6571 -0.4657*
p-value 0.0078 0.1584 0.0562 0.5332 0.0707 0.0213
Canada
Equity Rtns 0.0426* 0.0789* -0.1012 0.1855* -0.0402 0.6760* -4784.17
p-value 0.0000 0.0000 0.4015 0.0000 0.0616 0.0003
Bond Rtns 0.0265* -0.1288 0.0718 0.1211* -0.6010 0.1101
p-value 0.0061 0.2166 0.1991 0.0009 0.0569 0.3881
Germany
Equity Rtns 0.0331* 0.0800* -0.4501* 0.0627* 0.0039 -0.2025 -5248.46
p-value 0.0000 0.0000 0.0001 0.0280 0.6740 0.0542
Bond Rtns 0.0435* -0.7240* 0.0477 0.0991* -0.7000* 0.0315
p-value 0.0000 0.0000 0.0686 0.0005 0.0134 0.7245
Japan
Equity Rtns 0.0081* 0.1021* 0.0688 0.0886* -0.0031 0.3733* -4460.86
p-value 0.0000 0.0000 0.6474 0.0030 0.7741 0.0070
Bond Rtns 0.0085* 0.0308 -0.0552* 0.0686* -0.0157 0.7076*
p-value 0.0020 0.7474 0.0326 0.0024 0.9667 0.0000
UK
Equity Rtns 0.0353* 0.0268* 0.0020 0.0223 0.0039 0.5542* -5428.36
p-value 0.0000 0.0000 0.9727 0.4016 0.8180 0.0000
Bond Rtns 0.0216* -0.0570 0.1229* 0.0530 -0.1348 0.4823*
p-value 0.0353 0.5748 0.0003 0.1056 0.5930 0.0000
US
Equity Rtns 0.0208* 0.0322* 0.1044 0.0307 -0.0004 0.2949* -5468.35
p-value 0.0000 0.0287 0.3501 0.2845 0.9829 0.0225
Bond Rtns 0.0181* -0.0755 0.0077 0.0222 0.2724 -0.0221
p-value 0.0037 0.3121 0.8057 0.4919 0.2862 0.7840
* indicates statistical significance at the 5% level.
32
Table 7: Tests of Asset Pricing Model Restrictions on Equity Market Returns in
the BICAPM Setting This table reports the results of testing various nested asset pricing models, applied to equity markets, in the
context of the bivariate GARCH-M model [equation (10) in the text], representing the behavioural ICAPM.
The variance-covariance matrix is estimated using the GARCH (1,1) version of the diagonal BEKK
formulation. Conditional errors are assumed to follow the Normal distribution.
tbtbtebbtbbbtebbtbbbtb
tetetebeteeetebeteeete
rr
rr
,1,,3
2
,21,3
2
,21,
,1,,3
2
,21,3
2
,21,
)(
)(
εσγσγγσλσλλ
εσγσγγσλσλλ
++++++=
++++++=
−
−
),0(~1,
1,
1 HNtb
te
t
=
−
−
− ε
εε
The sample involves daily excess returns from 1 January 1990 to 30 June 2005 for six major equity and
bond market indices obtained from Datastream. The table reports the χ2
statistic for the specified test, with
the associated p-value below. Null Model
CAPM ICAPM BCAPM
H10: λ3 = γ2 = γ3 = 0 H20: γ2 = γ3 = 0 H30: λ3 = γ3 = 0
Australia 4273.32* 14.56* 1780.33* p-value 0.0000 0.0007 0.0000
Canada 20.00* 19.04* 14.55* p-value 0.0002 0.0001 0.0007
Germany 18.66* 6.09* 17.13* p-value 0.0003 0.0476 0.0002
Japan 7.56 7.31* 7.53* p-value 0.0559 0.0259 0.0231
UK 35.23* 35.21* 28.35* p-value 0.0000 0.0000 0.0000
US 8.54* 7.81* 5.78 p-value 0.0361 0.0202 0.0551
* indicates statistical significance at the 5% level.
