Essays in Petroleum Futures Market, Convenience Yield, and Long Memory

Ataollah Mazaheri

A Thesis submitted in conformity with the requirement of the degree of Doctor of Philosophy Graduate Department of Economics

University of Toronto

OCopyright by A. Mazaheri, 1998

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Essays in Petroleum F'utures Market, Convenience Yield, and Long Memory

By Ataollah Mnzaheri, Ph-D. Thesis Submitted to Graduate Department of Economics,

University of Toronto, 1998

Abstract

This thesis is a collection of three essays which address some empirical applications

of long memory processes with specific interest in financial economics of energy futures

market. The first essay 'Evidence of Long bfemory in the Petroleum Market' studies

evidence of long memory in the energy market using daily and weekly futures data. This

essay concentrates on the question of interdependence between crude oil futures and the

corresponding products. The empirical results provide strong support for long memory in

the energy futures market. The cointegrating relations between crude oil and heating oil

futures as well as crude oil and unleaded gasoline futures exhibit long memory, whereas

the individual series are unit-roo t .

The second essay 'Convenience Yield, Mean Reversion and Long Memory in the

Petroleum Market' analyzes convenience yields in the petroleum market. The focus

of this essay is the behavior of the spot and futures prices over the long run. The implied

convenience yield for petroleum and petroleum products is found to be driven by a non-

stationary and mean reverting long memory process. The theoretical implication of this

finding is established. It is discussed that this might be attributed to the fact that the

market is expecting mean reversion in the spot prices. Furthermore, the volatility process

and its relation with the mean process and the corresponding direction of causality have

been studied in detail.

The third essay 'Long Memory and Conditional Heterosked~tici ty , A Monte Carlo

Investigation', unlike the fist two, looks at the econometrics of the estimators of the long

memory process. It evaluates performance of three methods of estimating the parameter

of fractionally integrated noise: the exact maximum likelihood estimator(MLE) , the

quasi maximurn likelihood estimator(Q MLE) , and the G PH under different realizations

for variance.

ACKNOWLEDGMENTS

I must first thank my parents for their tolerance, support, and encouragement. I also

would like to thank my wife, Elham and my son, Alireza for their constant love and

support.

Special thanks goes to the members of my committee. In particular, I must thank

Professor Thomas McCurdy for his time and effort and invaluable comments. I offer

special thanks to my advisor Professor Peter Pauly for all his guidance.

1 also thank Jim Pesando and Michael Berkowitz for their effort. In addition, I am

thankfd to many of my fellow friends among them Kazem Yavari, Walid Hejazi,

Mike Campdieti, Henry Li, and Jack Parkinson for their constant encouragement.

Contents

I Introduction and Overview

2 Evidence of Long Memory in the Petroleum Market

Introduction

Analytical Framework

2.2.1 Periodogram Regression

2.2.2 Modified R/S Test

Fractionai Cointegration

Data and Empirical Results

2.4.1 Univariate Analysis

2.4.2 Multivariate Analysis

Conclusion

3 Convenience Yield, Long Memory, and Mean Reversion in the

Petroleum Market

Introduction

Theoretical Considerations

Econometrics and Estimation Procedure

Data and Empirical Results

3.4.1 Estimating ARFIMA-GARCH

3 A.2 Impulse Responses

3.4.3 Spread, Volatility and the Direction of Causality

Conclusion

4 Long Memory and Conditional Heteroskedasticity, A Monte Carlo

Investigation 95

4.1 htroduction 95

4.2 Analytical Framework 97

4.3 Simuiation Results 102

4.3.1 GARCH Model 103

4.3.2 EGARCH Model 104

4.4 Conclusion 106

List of Tables

Size of GPH Test for Cointegration(Nu1l Hypothesis of d=l) 42

Size of modified R/S Test for Fractional Cointegration 43

Results of Unitroot Tests on Daily Energy Futtres(1) 43

Results of Unitroot Tests on Daily Energy Fumes(2) 44

Results of Unitroot Tests on Weekly Energy Futures(1) 44

Results of Unitroot Tests on Weekly Energy Fuhlres(2) 45

Descriptive Statistics, One Month Ahead Energy Futures 45

Descriptive Statistics, Three Month Ahead Energy Futures 45

Short Term dependency and Conditional Heteroskedasticity in Energy Futures, One

Month Ahead Series 46

Short Term dependency and Conditional Heteroskedasticity in Energy Futures, Three

Month Ahead Series

Results of Long Memory Tests on Daily Energy Futures(1)

Results of Long Memory Tests on Daily Energy Futures(2)

Results of Long Memory Tests on Weekly Energy Fuhlres(1)

Results of Long Memory Tests on Weekly Energy Futures(1)

Results of OLS Regression Between Energy Futures

Autocorrelation Coefficients, p(t), One month Ahead Daily

Autocorrelation Coefficients, p(t), Three month Ahead Daily

Autocorrelation Coefficients, p(t), One month Ahead Weekly

Autocorrelation Coefficients, p(t), Three month Ahead Weekly

Results of Unitroot Tests on Error Correction Terms, Daily Energy Futures 5 1

Results of Unitroot Tests on Error Correction Terms, Weekly Energy Futures 5 1

Results of Fractional Integration Using GPH and MRR Tests(1) 52

VI

2.23 Results of Fractional Integration Using GPH and MRR Tests(2) 52

Testing Order of Integration for Convenience yields in the Petroleum Market 84

Estimation of ARFIMA Model 84

Estimation of ARFIMA-GARCH(l ,1) Model 85

Estimation of ARFIMA-EGARCH(1, I ) Model 86

Estimation of ARFIMA-GJR(1, I ) Model 87

Results of Causality Tests between the Mean and the Voiatility Process 88

Distribution of ARFIMA(O,d,O) When the Underlying Processes are GARCH(1, I), GPH

Estimation 110

Distribution of ARFIMA(O,d,O) When the Underlying Processes are GARCH(1, I), MLE

Estimation 111

Distribution of ARFIMA(O,d,O) When the Underlying Processes are GARCH(1, I),

QMLE Estimation 112

Distribution of ARFIMA(O,d,O) When the Underlying Processes are EGARCH(1, 1), GPH

Estimation 113

Distribution of ARFIMA(O,d,O) When the Underlying Processes are EGARCH(1,I ),

MLE Estimation 115

Distribution of ARFIMA(O,d,O) When the Underlying Processes are EGARCH( 1.1 ),

QMLE Estimation 117

List of Figures

Correlogram for the Cointegrating Residuals 53

Evolution of the Convenience Yield and the Conditional Heteroskedasticity , Crude Oil

89

Evolution of the Convenience Yield and the Conditional Heteroskedasticity, Unleaded

Gasoline 90

Evolution of the Convenience Yield and the Conditional Heteroskedasticity, Heating Oil

91

Correlogram for the Convenience Yield of Crude oil 91

Correlogram for the Convenience Yield of Unleaded Gasoline 92

Correlogram for the Convenience Yield of Heating oil 92

The Cumulative Impulse Response, ARFIMA(0,0.753, I), Crude Oil 93

The Cumulative Impulse Response, ARFIMA(0,0.753, I), Unleaded Gasoline 93

The Cumulative Impulse Response, ARFIMA(0,0.80 1, I), Heating Oil 94

Efficiency of Methods for Estimation of ARFIMA When the Underlying Processes are

EGARCH, High Volatility T= 1 00 119

Efficiency of Methods for Estimation of ARFIMA When the Underlying Processes are

EGARCH, High Volatility T=500 120

Efficiency of Methods for Estimation of ARFIMA When the Underlying Processes are

EGARCH, Low Volatility T= 1 00 121

Efficiency of Methods for Estimation of ARFIMA When the Underlying Processes are

EGARCH, Low Volatility T=500 122

VIII

Chapter 1

Introduction and Overview

This thesis consists of three essays that discuss some theory and applications of the long

memory process to the energy futures market. The thesis contributes to the d i n g

literature by addressing three main issues: the evidence of long memory in the energy

futures market with specific interest in the interdependence of crude oil and products,

the application of the long memory to the theory of storage and the energy market, and

the properties of some of the existing estimators of the long memory p-cess. Three

essays address these aforementioned issues. The first essay, presented in the second

chapter, focuses on the evidence of long memory in the error correction term for the

relation between crude oil-unleaded gasoline and crude oil-heating oil. The second essay,

presented in the third chapter, discusses the applications of the long memory to the

theory of storage and the corresponding convenience yield. The third essay(Chapter

Four) presents the properties of the long memory estimators when the process has a time

varying variance structure. In this chapter, a short discussion and an overview of each

chapter is presented. We start with a brief description of the long memory process and

then pursue with each chapter.

A long memory process can be thought of as a specid form of commonly used non-

Linear dynamics known as f radal dynamics. Fractal dynamics are characterized by

non-periodic cyclical behaviors and long-term dependency and show persistence in their

first moments. The implications of fractal dynamics are far-reaching: if a process ex-

hibits fractal dynamics, conventional statistical inference concerning unit-root tests and

vector autoregression, as extensively as they are used, may no longer be applicable. F'ur-

thennore, with the presence of a long memory process? behavior of the series will be

partly predictable over the long horizon. For a fkactal process, to the extent that a de-

parture from equilibrium is not permanent, series will eventually exhibit mean reversion

and the autocorrelation and hence the impulse response will converge to zero over infi-

nite horizons. In fact Mandelbrot(l971), in the context of asset pricing models, argues

that in the presence of fsactal dynamics the arrival of new market information can not

be fully arbitraged. The thesis uses ARFIMA(Autoregressive-Ftactionally integrated-

Moving Average) process to analyze the long memory.' For this process. the dependence

between observations decays at a very slow-hyperbolic rate, while the dependence pro-

duced by and ARMA process decays at a geometric rate and those for an ARIICIA

process never dies out. This implies that the impulse response for a long memory process

converges asymptotically to zero. The speed of this convergence depends on the parame-

ter of fkactional integration(d). A kactionally differenced process exhibits tong memory

in the sense that its Wold decomposition and autocorrelation coefficients will all show a

very slow rate of hyperbolic decay.'

The estimation of long-memory process has also received considerable attention in

recent years since they were first introduced by Granger and Joyeux(l980) and Hosk-

ing(1981). A number of procedures have been introduced to estimate the differencing

parameter (d) in the hamework of an ARFIMA process. One can classify these estima-

'As pointed out in Granger and Ding(19961, a number of other processes can also be long memory, including generalized fiactionaily integrated models arising from aggregation, time-changing coefficient models, and possibly nonlinear models. In this dissertation we apply ARFIiWA process mainly due to practical issues. For other long memory processes, the estimation procedures are not derived or have unknown properties. Furthermore, in our empirical analysis, we use an approximate 10 years span of data (daily or weekly). We acknowledge the fact t k t in the absense of data constraint this span may not be adequate.

*In this dissertation we differentiate between unit root and long memory since impulse responses of a unit root process do not show hyperbolic decay, although one may argue that a unit root process exhibits persistence.

tion procedures in two general categories: frequency domain estimations and time domain

estimators. Exact maximum likelihood estimator (M LE) as proposed by Sowell(l992)

and quasi maximum likelihood estimator (QMLE) as suggested by Baillie et aL(1994)

are among time domain estimators, while the GPH estimator as proposed by Geweke

and Porter-Hudak (1983) is among frequency domain estimators. A more formal analysis

of the behavior of the long memory process in the kamework of ARFIlLfA process will

be presented in Chapter Two.

These characteristics of the long memory process and the their estimation procedures

provide an attractive tool for the analysis of financial series. Recent applications include

the analysis of inflation rates(Hass1er and Wolters (1995)), interest rate futures market

( h g and Lo(1993), Fang, Lai and Lai (1994), and Booth and Tse(1995)), pwchas-

ing power parity(Cheung and Lai(1993)), and stock returns (Cheung and Lai(1995)).

Chapter Two and Chapter Three are among some other applications.

Chapter two discusses the evidence of long memory in the petroleum market. The

behavior of energy futures has been studied rather extensively. Schwarz(l994) performed

a univariate analysis on the energy futures, Litzenberger(l995) discussed the evidence of

strong backwardation in the oil futures market, and Quan(1992) explored the price dis

covery of futures prices. Crowder and Hamed(1995) discussed the issue of efficiency in the

futures market and Cho and McDougail(l992) examined the efficient market theorem for

the energy market in the context of the theory of storage. Moosa and Al-Loughani(l995)

analyzed the theory of arbitrage and Moosa(1994) considered time varying risk premia

in the crude oil futures market. Furthermore, the behavior of energy prices over the

business cycle has been discussed by Serbtis(l994a), and the short term dynamics of the

prices have been considered by Antoniou(l992) using G ARCH models. More recently,

Ng and Pirrong(l996) characterize the spot and futures price dynamics of gasoline and

heating oil using a non-linear error correction model.

One aspect which has received rather less attention is the interdependence among

sub-markets (heating oil-crude oil and unleaded gasoline-crude oil) in the energy market.

In a recent article, Serletis (1994b) argues for a common stochastic trend in a system of

three petroleum futures prices(crude oil, heating oil, and unleaded gasoline) using an error

correction model as proposed by Johansen(l988). In Line with Bacon(l99l), Borenstein,

Cameron, and Gilbert (1997) construct the cumulative impulse response of gasoline prices

to shocks to the crude oil price. They argue for an asymmetric-slow response of unleaded

gasoline to shocks to crude oil. In their approximation it takes about three weeks for

gasoline prices to adjust to a change in price of crude oil. This type of analysis relies

heavily on the assumption of a long run relation as implied by cointegration between oil

products and crude oil. This assumption is a natural implication of the fact that crude

oil is a major component of oil products; any shock to the crude oil price should transfer

to the price of oil products quickly enough to ensure cointegration.

This view of the long run relationship between crude oil and products, attributes

patterns of adjustment to some short term disequilibrium. Although this might shed

some light on the behavior of energy market in the short run, it will unnecessarily simpw

its long run behavior. In the absence of strong theoretical reasoning, the assumption of

quick adjustment between crude oil and heating oil and between crude oil and unleaded

gasoline inherited in traditional cointegration analysis seems too restrictive.

The theoretical considerations which may justlfy lagged adjustment of the prices to

new information are two-fold. The macroeconomic literature focuses on the notion of the

"sticky prices" whereas the industrial organization concentrates on the relevant market

process which may lead to lagged adjustment. In general, the combined research suggests

several reasons for the lagged adjustment among them menu cost, long-t e m relationship

between buyers and sellers , search costs, and product ion stickiness. Borenst ein and

Shepard(l996) add to this literature and demonstrates that a £inn with market power

will have a different price adjustment path than a competitive b. In addition, they

theoretically show that if inventory is used to minimize costly changes to production

schedules when there are input cost shocks, the refined gasoline price does not adjust

quickly to the shocks in the crude oil price to reflect both production and inventory

adjustment costs. As might be concluded, these theoretical considerations justify a lagged

adjustment irrespective of how long this adjustment may take.

Chapter two relaxes the assumption of cointegration between crude oil and products to

provide a more comprehensive and systematic analysis of the dynamics of energy market.

A fkactional cointegration approach provides the basis of this study. The implied long

memory process in the fkactional cointegration allows for the possibility of dependence

among distant observations and therefore accommodates slow adjustment among the

markets in contrast to the conventional cointegration approach which assumes a more

rapid adjustment path. More formally, by allowing the fkactional differencing parameter

(d) to take any value between zero and one, the fractional cointegration analysis goes

beyond the dichotomy of cointegration and non-cointegration to allow for the process to

exhibit long memory.

The findings of this chapter point to the existence of long-memory in the cointegrat-

ing relations between the crude oil and heating oil, and between crude oil and unleaded

gasoline, although individual series exhibit unit-root. The results point to the fact that

the response to a shock to the system is not as fast as might be inferred from a conven-

tional cointegration analysis. For instance, an innovation in the price of crude oil has no

permanent effect on the price of heating oil or unleaded gasoline but it takes a long time

before it dies out. These findings are important from two main aspects: First, fractional

cointegration implies that patterns of adjustment in the energy market is very slow so

that micro-based interpretations such as sticky production and inventory as proposed by

Borenstein et a1.(1996) may not be sufEcient. Second, these results cast doubt on con-

ventional cointegration approaches such as S d e t is (l994b) and Borenst ein et a1. (lgg?'),

since with the process exhibiting long memory, the error correction mechanism reflecting

fkactional cointegration will have to be included in the vector a~toregression.~

The third chapter examines the behavior of the convenience yield as implied by the

a c i e n t market theorem in the kamework of long memory process. The efficient market

"ee Baillie and Bollerslev(l994) for more detail.

5

theorem and its implications towards the basis, risk premium, and price volatility have

been discussed extensively in the literature. Engel (1995) brings a summary of the

main issues for the exchange rate, and Brewer and Kroner(1995) provide a systematic

analysis of the empirical testing for the efficient market hypothesis in the framework of a

nearbitrage condition implied by the theory of storage. The commodity market, though

to a lesser extent than the exchange rate, has also been the target of much discussions.

This essay tends to take a further step in this direction using recent developments in the

theory of long memory time series. The essay focuses on the largest futures contracts

traded on physical commodities market, namely crude oil and refined petroleum complex

futures*

The literature on the efficient market theorem is vast, while the literature on the

petroleum and petroleum products is considerably more Limited. Among the papers

that exclusively consider the &cient market theorem in the petroleum market one can

mention Ng and Pirrong(l996) who consider the interaction of the daily spot and futures

prices for refined petroleum products in the framework of an error correction model

based on a no-arbitrage condition. Schwan and Szakmary(1994) follow the same path in

a slightly difTerent framework. These papers both assume cointegrated contemporaneous

spot and one month ahead futures prices, based on the results provided by standard unit-

root tests such as Augmented Dickey-FUler test. Crowder and Hamed(1993) perform

a Johansen-based cointegration test on one month ahead futures prices and realized

spot prices for the crude oil. They argue that their analysis supports a simple efficient

market hypothesis but not the nearbitrage equilibrium hypothesis. The difference, in

their framework, relies on the fact that the arbitrage equilibrium hypothesis implies that

expected returns to speculation in the oil futures market should equal the risk-fiee rate

of return, whereas a simple efficiency hypothesis implies that the expected return to

futures speculation in the oil futures market should be zero. Cho and McDougall(l990),

on the other hand, use the realized inventory data to test for the storage theorem and

its implications regarding the spread between the futures price and the contemporaneous

spot rate, using a weekly data-set. Their results confirm the existence of a supply of

storage curve in both the gasoline and heating oil markets. However, the evidence of the

supply of storage curve in the crude oil market is weaker.

Furthermore, Gibson and Schwartz (1989) discuss the time series properties of the

forward convenience yield of crude oil and find strong evidence in favor of a mean-

reverting pattern. Gibson and Schwartz(l990) develop and test a mefactor model for

pricing financial and real assets contingent on the price of oil. The factors are the spot

price of oil and the instantaneous convenience yield. They assume that the spot prices

follow a lognormal-stationary distribution and that the convenience yield has a Omstein-

Uhlenbeck mean-reverting process. These assumptions imply that the evolution of the

spot prices is random walk whereas that of the convenience yield is mean reverting. Their

empirical test confirms both of these assumptions.

This essay contributes to the existing literature by considering analytically how mean

reversion in the convenience yield as implied in the theory of storage might be attributed

to mean reversion in the spot price and by empirically estimating the corresponding Long

memory process for the petroleum market. To this aim the paper uses the ARFIM A

( Aut oregressive- Fkact ionally Integrat ed-Moving Average) process in analyzing the prop

erties of convenience yield. Furthermore, the theory of storage has some strong implica-

tions as for the volatility of the convenience yield and its relation with the mean process.

By extending the ARFIMA process to include GARCH family, we also produce a com-

prehensive study of the predictions and implications of the theory of storage in a unified

approach.

The findings of this essay hint to strong evidence in favor of long memory process

in the convenience yield. The estimated differencing parameter, a measure of degree of

persistence, is found to be between 0.70 and 0.80, indicating a rather strong persistence.

We attribute this non-stationary and mean-reverting convenience yields to expected mean

reversion of the spot prices. F'urthermore, the volatility process and its relation with the

convenience yield is also estimated. It is found that the heating oil and unleaded gasoline

very weli conform with the predictions of the theory of storage that their convenience yield

is more volatile for positive price shocks. While crude oil also confirms this prediction,

this type of asymmetric behavior for the volatility process fails to be sigdicant.

One can also apply this procedure to non-commodity assets such as exchange rate and

financial assets by modifying the no-mbitrage condition, in Line with Brenner and Kroner

(1995). In the case of exchange rate, the possible mean reversion can be attributed to

some other factors such as interest rate differentials. In fact, Baillie and Bollerslev(l994)

found strong persistence in the forward premium for five selected exchange rates. Baillie

and Bollerslev(l994) attributed this persistence to a possible persistence in the variance

of the spot price, which theoretically may constitute a portion of the risk premium. One

may also look for persistence in interest rate differentials as a source for this phenomenon.

Consistent with our analysis and based on the nearbitrage condition, this persistence can

contribute to mean reversion in equilibrium exchange rates. In fact, Cheung(l.993) doc-

uments some evidence in favor of mean reverting spot prices using long-memory analysis

for some selected exchange rates which is consistent with our view that mean-reverting

forward premium implies expected mean-revert ing spot prices.

The difference between our interpretation for the persistence in the forward premium,

and the one provided by Baillie and Bollerslev(l994), can be attributed to the long debate

between the two main rivals in the financial economics namely the CAPM models and

the theory of storage. Baillie and BoUerslev(l994) try to explain the long memory in

the forward premium using CAPM models by explicitly deriving the risk premium and

attributing the persistence in forward premium to a portion of risk premium i-e. the

variance of spot price. However, the same phenomenon can be adequately explained by

the theory of storage, by attributing the long memory in the forward premium to the

long memory in the interest rate differentiald Some evidence for this long memory can

be found in recent literature such as Booth and Tse(1995)

4The no - arbitrage condition implies that the forward premium in the currency market is a function of interest rate differentials. Considering this, the persistence in the forward premium can be attributed to the persistence in the interest differentials.

The third essay(Chapter Four) discusses properties of the long memory estimators,

specifically those used in the first two essays. The performance of these estimators has

been studied extensively in the Literature. However, one aspect which has received very

limited effort is the effect of different realizations of the variance process on the esti-

mation of the differencing parameter. There are two minor exceptions: Cheung(1993)

who studied the effect of ARCH on GPH and MRR, and Hauser et al. (1994) who

investigate the effect of GARCH(1,l) on GPH estimation. Hauser et aL(1994) argue

that long run dependency and heteroskedasticity can be severely misleading if one looks

exclusively at the other. However, their study is confined to GARCH models and the

estimation procedure is restricted to GPH. Cheung(1993) study, on the other hand,

finds no significant distortionary effect from the ARCH factor on the estimated differ-

encing parameter. These studies shed some light on how misspecificcation of the variance

structure may affect estimation of the differencing parameter. However, prevalence of

time-varying variance structure in the applied time series specifically in the financial se-

ries including those studied in the first two essays warrants a further study for which this

essay is devoted to.

This essay extends the existing literature to include different characterizations of

the variance process in addition to comparing the performance of different estimation

procedures. Of specific importance is the inclusion of an asymmetric response variance

structure such as that implied in EGARCH model. FVe extensively analyze the effect

of asymmetry in the variance structure on the estimation of the fkactional differencing

parameter. In fact, for the empirical series studied in applied economics and finance, a

wide range for the parameters of volatility and asymmetry in the variance response is

found. For instance, Cheung and Ng(1992) found a very wide range of asymmetry and

volatility in their study which includes 300 US stock returns. These empirical studies

specifically call for broader analysis of the ARCH effect on the estimation of the differ-

encing parameter than those performed by Cheung(1993) and Hauser et a1.(1994) and

justifies our empirical design.

The main findings of this Monte Carlo exercise are: First, the relative efficiency of

the MLE estimator with respect to Q MLE estimator disappears when the underlying

process exhibits timevarying variance. This might be attributed to the fact that the

Vasiance-covariance required for the M L E estimation is bound to values of the differenc-

ing parameter less than 0.5. Second, it is found that the bias increases with an increase in

the conditional heteroskedasticity and asymmetry. Third, it is found the robust standard

errors for the Q M L E procedure tend to over-estimate(under-estimate) the true standard

errors when the degree of asymmetry is high(1ow). This is more transparent when the

sample size is smaller.

Further research should concentrate on developing ma1ytica.l models which may j u s

tify the empirical finding of this dissertation. For instance, the first essay demonstrates

that adjustment of prices in the petroleum market is time consuming. The theoretical

considerations, as mentioned earlier, just@ this finding however they are silent as to if

this pattern of adjustment is a short term or a long term phenomenon. The second essay

demonstrates that mean reverting convenience yield implies mean reverting anticipated

spot prices. Although the theory of storage predicts mean reverting convenience yield,

it is silent as to if the corresponding stochastic process exhibits long memory. Further

research is required to develop models which can analytically yield long memory process.

Furthermore, in this dissertation we use ARFIMA process whereas some may argue

that in the presence of frequent external shocks other models such as regime switching

or threshold cointegration models may provide more insight. However our focus on

ARFIMA is for two main reasons; First, it is flexible enough to accommodate aIl aspects

we are seeking to analyze. Second, we do not think that structural change, within the

Eramework of our analysis, is serious enough. In fact, we have repeated our analysis for

ditferent sub-periods to make sure that the results do not change qualitatively. M h e r

research may concentrate on comparing our results with those obtained if the alternative

madels are used.

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Chapter 2

Evidence of Long Memory in the

Petroleum Market

2.1 Introduction

The behavior of energy futures has been studied rather extensively. One aspect which

has received rather less attention is the interdependence among submarkets (heating

oil-crude oil and unleaded gasolinecmde oil) in the energy market. In a recent article,

Serletis (1994) argues for a common stochastic trend in a system of three petroleum

futures prices(cmde oil, heating oil, and unleaded gasoline) using an error correction

model as proposed by Johansen(l988). In line with Bacon(l991), Borenstein, Cameron,

and Gilbert (1997) construct the cumulative impulse response of gasoline prices to shocks

to the crude oil price. They argue for an asymmetric-slow response of unleaded gasoline

to shocks to crude oil. In their approximation it takes about three weeks for gasoline

prices to adjust to a change in price of crude oil. This type of analysis relies heavily on

the assumption of a long run relation as implied by cointegration between oil products

and crude oil. This assumption is a natural implication of the fact that crude oil is a

major component of oil products; any shock to the crude oil price &odd transfer to the

price of oil products quickly enough to ensure cointegration.

This view of the long run relationship between crude oil and products, attributes

patterns of adjustment to some short term disequilibrium. Although this might shed

some light on the behavior of energy market in the short run, it will unnecessarily simplify

its long run behavior. As discussed in Chapter 1, the lagged adjustment of the prices

to new information in the petroleum market has been well established theoretically in

the economic literature. However, these theories are silent as to the pattern of this

adjustment. In the absence of strong theoretical reasoning, the assumption of quick

adjustment between crude oil and heating oil and between crude oil and unleaded gasoline

inherited in traditional coint egration analysis seems too restrictive.

This study relaxes the assumption of cointegration between crude oil and products to

provide a more comprehensive and systematic analysis of the dynamics of energy market.

A fractional cointegration approach provides the basis of this study. The implied long

memory process in the kactional cointegration allows for the possibility of dependence

among distant observations and therefore accommodates slow adjustment among the

markets in contrast to the conventional cointegration approach which assumes a more

rapid adjustment path. More formdy, by allowing the fractional differencing parameter

(d) to take any value between zero and one, the fractional cointegration analysis goes

beyond the dichotomy of cointegration and non-cointegration to allow for the process to

exhibit long memory. In fact, with the low power of traditional unit root tests such as the

Augmented Dickey-Ner test against fractionally integrated alternatives1, the applica-

tion of the long memory process seems natural and can potentially provide more insight.

In this paper we intend to contribute to the literature by examining the evidence of long

memory in energy futures with the help of the semi-nonparametric method of Geweke and

Porter-Hudak(l983) (G P H) known as periodogram regression. However, since the G PH

test might have low power against autoregressive alternatives, a more conventional long

memory test, the modified resealed range test (M R R) analyzed extensively by Lo ( 199 I),

' See Diebold and Rudebusch(l990), Cheung and Lai(1993), and Hassler and Wolters(1994) for more detail.

is also performed to confirm the results.

The hdings of the study point to the existence of long-memory in the cointegrating

relations between the crude oil and heating oil, and between crude oil and unleaded gas*

line, although individual series exhibit unit-root, and it rejects the idea of cointegration

in the energy market. The results point to the fact that the response to a shock to the

system is not as fast as might be inferred from a conventional cointegration analysis. For

instance, an innovation in the price of crude oil has no permanent effect on the price of

heating oil or unleaded gasoline but it takes a long time before it dies out. These findings

are important from two main aspects: First, fractional cointegration implies that pat-

terns of adjustment in the energy market is very slow so that micro-based interpretations

such as sticky production and inventory as proposed by Borenstein et a1.(1996) may not

be sufficient. Second, these results cast doubt on conventional cointegration approaches

such as Serletis(l994) and Borenstein et a1. (1997) : since with the process exhibiting long

memory, the error correction mechanism reflecting fractional cointegration will have to

be included in the vector autoregression*.

The paper proceeds as follows; the next section provides the analytical framework,

it discusses the characteristics of long memory process and provides the fundamentals

of periodogram regression and modified re-scaled range analysis. The following sect ion

presents the data and the empirical results in both univariat e and multivariate present a-

tions. A brief conclusion ends the paper.

Analytical Framework

This paper uses fractionally differenced processes to analyze the long mn behavior of

energy futures. A long memory process [Granger and Joye~~( l980 ) and Hosking(1981)]

depends on an estimable fkactional differencing parameter represented by (d). A processes

is defined to have long memory if its autocorrelation coefficients at long lags are not neg-

*See Baillie and Bollerslev(l994) for more detail.

17

ligible. Clearly, an ARMA process does not show persistence in its autocorrelation, and

the autocorrelation coeffcients of an ARIMA show much more persistence. More for-

mally, the dependence between observations produced by the ARMA process decays at

a geometric rate, while the dependence produced by ARFIMA(Autoregressive Fkaction-

ally Integrated Moving Average) at a much slower hyperbolic rate. Hence, the long-range

dependence between observations is eventually determined by the fkactionally differenc-

ing parameter. A kactionally differenced process exhibits long memory in the sense

that its Wold decomposition and autoco~elation co&cients will all show a very slow

rate of hyperbolic decay. A more formal analysis of this feature will follow. A general

ARFIMA(p, d, q) process can be represented by:

Where d denotes the hactional differencing parameter, @ (L) = 1 - alL - ... - k L m ,

and @ (L) = 1 + b1 L + . . . +bn Ln are polynomials in the lag operator L with roots outside

the unit circle, and Q is a white noise random variable. For -0.5 < d < 0.5 the process

is covariance stationary while with d < 1 the process exhibits mean reversion with an

infinite variance. If 0 < d < 0.5, the process eventually shows strong positive dependency

between distant observations, whereas if -0.5 c d < 0, the process eventually exhibits

negative dependence between distant observations. The fractional differencing operator

(1 - ~ 5 ) ~ can be expanded as a binomial series:

By rewriting dj as a gamma function, the fractional differencing operator can be

defined as r(j - d ) P /r( j + 1)Q-d) where r(.) represents a gamma function. j=O

This ARFIMA@, dl q ) representation includes the ARMA(p, 0, q) process as a special

case where d = 0, ARIMA@, 1 , q ) where d = 1 and covers a whole range of long memory

processes. In this way the ARFIMA@, d , q) process can be defined as an intermediate

representation between I(1) and I (0) processes, in which the long memory property of a

process depends crucially on the value of d. Hence, the existence of long memory can be

tested conveniently by examining the est h a t ed differencing parameter d.

In the absence of short term dependence, i.e. p = q = 0, one can define the autocor-

relation function of a fractionally differenced process (2.1) as:

where d < 0.5, and c is a constant. Clearly for d # 0, the autocorrelation p of xt decays

hyperbolically. This feature implies that for positive values of the differencing parameter

the autocovariances of the process are not absolutely summable. Furthermore, it insures

that the impulse response converges to zero in infinite horizon. For a simple A R M A

process, in contrastt it is well known that for large values of r the autocorrelations decay

approximately geometrically as:

where k is a constant with an absolute value less than one. This apparent difference

between the autocorrelations is the primary reason for the ARFIMA process to exhibit

strong dependency between distant observations and hence a logarithmically decaying

impulse responses whereas ARMA process does not exhibit strong dependency.

2.2.1 Periodogram Regression

In estimating the differencing parameter d, this paper follows a semi-nonparametric a p

proach first introduced by Geweke Porter-Hudak(l983). It is semi-nonparametric in the

sense that the estimated differencing parameter does not require an explicit parameteri-

zation of the underlying /IRMA@, q) process. Define the spectrum of xt by:

Where e is the stationary linear ARMA@, q) process represented by a-'(L) @ ( L ) E ~

and f,(X) is its spectral density. Taking logarithm of the spectral density evaluated at

harmonic fkequencies Aj = 2nj/Tl (j = 1, ..., T - 1) one can write:

Following Geweke Porter-Hudak(GPH), for low harmonic fkequencies A, close to zero,

the last term in (4) is negligible, so that adding the periodogram to the two sides of the

equation will lead to the following periodogram regression:

Here n represents the number of ordinates and I, represents the periodogram which

can be computed by the square of the exact Fourier transform over the harmonic Ere-

quencies scaled by 2/T. More formally, the periodogram can be defined by:

Furthermore, Geweke and Porter-Hudak showed that the number of ordinates in the

periodogram regression should increase more slowly than the sample size, and subse-

quently a representative function was defined to be T p where 0 < p < 1. G PH argued

that the residuais of the

a standard distribution

important parameter in

periodogram regression are asymptotically iid so that they have r2

with a variance of -. The number of ordinates remains an 6

this kind of periodogram regression since too small a number

will decrease the degrees of heedom whereas too high a number will contaminate the

results due to the inclusion of low frequency ordinates. G P H showed that the sensitivity

of fractional differencing parameter to the choice of ordinates can be used to check for

possible A RMA contamination of the ffactionally differenced process. They suggest that

p should be as low as 0.5 or 0.55 to avoid contaminating the estimation. The asymp- 7r2

totic variance of the estimated residuals e.g. - is frequently imported in the regression 6

to increase the &ciency.

Periodogram regression has been subject of much recent debate. Hassler(l993a) ar-

gues that the residuals of the periodogram regression are not asymptotically independent,

as claimed by GPH. Thus, in their argument the regression range is asymptotically

unbounded. To ensure validity of the periodogram regression, they add normality of

residuals as a condition for the regression results to hold. This requires that the residuals

not to be contaminated so that the fractional difference parameter is insensitive to the

number of ordinates. Hassler(l993b) showed that chi-squared distributed residuals with

skewness of 8.5 and kurtosis of 12 will lead to a complete break down of the periodogram

regression. Cheung(1993) in a Monte Carlo experiment shows that GPH test is robust

to moderate ARMA components, ARCH effects, and shiRs in the variance. He founds

several factors which can lead to spurious rejection of the no fractional-integration hy-

pothesis. For example, GPH test is biased towards d < 0 alternatives in the presence

of negative MA components of larger than 10.71. Furthermore, when there are shifts in

the mean, the G P H test tends to yield spurious evidence for d > 0 alternatives. LI ore-

over, a large AR parameter, as expected, also biases the GPH test in favor of d > 0

alternatives. Sowd(1992a) compared his maximum likelihood estimator with the GPH

and found a comparable bias and significantly larger IIfSE in GPH estimator. These

arguments against the periodogram regression suggest care when choosing the number

of ordinates in the regression. The sensitivity of regressions to the number of ordinates

can be used as a guide in performing the periodogram regression. The application of the

periodogram regression has the advantage that it does not require the selection of the

ARMA parameters, as does the recent exact M L estimators with hactionally integrated

models first introduced by Sowell(l992a, b ) , in which the model selection is based on in-

formation criteria. As different models may be favored by different criteria, the exact iLlL

estimators may lead to different estimates. The periodogram regression is not efficient

as pointed out by Sowell(1992a) and emphasized by Hassler and Wolters(1995), but it

has the advantage of revealing its inefficiency. By varying the number of ordinates, it is

possible to detect an eventual bias due to ARMA parameters.

2.2.2 Modified R/S Test

The second statistic which is commonly used to test for long memory alternatives has

been introduced by Lo(1991), based on an extension oi' the classical re-scaled range

test discussed by Rosenblat t (1956) and extensively analyzed by Mandelbrot (lS72, 1977).

This statistic uses the notion of strong mixing which provides a measure of the decline

in statistical dependence between events separated by successively longer spans of time.

According to this terminology, a time series is strong mixing if the maximal dependence

between any two events becomes very small as the time span between those two dates

increases. This idea may provide a distinction between long-range and short-range sta-

tistical dependence, and enable us to develop a method for detecting long-term memory.

The R/S test takes strong-mixing a s an operational definition of short-range dependence

and builds it into the null hypothesis of short dependence. More formally, the R/S test

denoted by QT is defined as:

where;

In which:

The terms in the first equation are the maximum of the partial sums of the first

k deviations of Xt from the sample mean and the minimum of the same sequence of

partial sums. The difference between these two is called range for obvious reasons. In the

traditional rescaled range statistic, this range is scaled by the sample variance whereas

in the modified re-scaled range it is scaled by the sum of the variance and a weighted sum

of its auto-covariances. Therefore, the only difference between the traditional re-scaled

range analyzed extensively by Manderlbrot(l972,1977), is in the denominator. While the

traditional re-scaled range uses the Mziance to scale the range, the modified re-scaled

range (IIfRR), scales the range by ~ ; ( r ) , for which r provides the truncation lag. This

denominator is a heteroskedasticity- and autocorrelation-consistent variance estimator,

and includes the sample variance and a weighted sum of sample autcxovariances. The

MRR test is robust to short-range dependence due to the inclusion of this weighted

sum in the variance estimator. This variance estimator [Newey-West(l987)l yields a

strictly positive variance estimator and depends crucially on the truncation lags ( T ) .

Andrews(l991) and Lo and MacKinlay(l989), showed that when r becomes large relative

to sample size T, the finitesample distribution of the estimator can be radically different

horn its asymptotic distribution. However the truncation lag can not be too small since

the auto-covariances beyond lag r may be substantially different from zero. A data-

dependent truncation lag is introduced by Andrews(l991) and its performance has been

analyzed through a Monte Carlo study. This data-dependent formula is defined to be:

Here p is the estimated first order autocorrelation function. In this case the weighting

parameter introduced in the Newey- West variance-covariance matrix i .e. relation (8) will

change to 1 - I+l. It can be shown that under the null hypothesis of no long memory

the asymptotic distribution of the QT standardized by the square root of the sample size

can be established, i.e.:

Critical values for V are tabulated in Lo(1991). Usiry these critical values a test of

the ndl hypothesis may be performed at the 5 percent si&cance level by accepting or

rejecting according to whether VT is or is not in the interval [0.809,1.862] which assigns

equal probability to each tail. Clearly a value smaller than 0.809 provides evidence for

negative differencing, whereas a value greater than 1.862 provides evidence of positive

differencing, at a 95% level of confidence.

As we indicated earlier, the behavior of this dat a-dependent formula was studied

by Andrews(l99l) and Lo(1991) in a Monte Carlo experiment which showed that the

MRR statistic, unlike GPH estimator, has very strong power against autoregressive

alternatives. In fact, as indicated by Lo(1991) and verified by Cheung(1993), the IIfRR

is conservative for values of r that are not too large relative to the sample size and tends

to reject too frequently. Furthermore, it has been verified that this statistic has better

power against negative differencing alternatives than the positive differencing. However,

as in the case of GPH test, this statistic has low power against a &I A alternative when

the MA parameter is close to -0.9. Overall, the MRR statistics is robust to the moderate

ARMA contamination as was the GPH test. However this test performs better against

autoregressive alternatives.

Clearly, G P H might be a more dcient means of detecting long-range dependency,

if one is interested solely in &act ionally-differenced alternatives. However, the modified

R/S test is perhaps most useful for detecting departures into a broader class of alternative

hypotheses. In fact, Lo(1991) compares MRR to a portmanteau test statistic that can

complement a comprehensive analysis of long-range dependence. In this paper we use

GPH in combination with the Lo test, so as to first estimate the differencing parameter,

and have the assurance of a test which might be more robust to short run dependency.

More specifically, we use the MRR test since it is robust to autoregressive short run

dependence.

2.3 Fkactional Coint egrat ion

nactional cointegration as applied in this paper follows closely that of Cheung and

Lai(1993a) and is closely associated with the Engle and Granger(l988) two stage cointe-

gration procedure. Assuming two series of yt and xt, the OLS residuals horn a first stage

regression of xt on y, is used to check for kactional cointegration. More formally, assum-

ing Zt to be the matrix containing yt and xt so that h'Zt is the estimated error correction

term, the periodogram regression can be used to estimate the fractional differencing pa-

rameter (d)? Cheung and Lai(1993a) showed that if Zt is I (1) and d > 0, so that Zt

process is fkactionally cointegrated, the OLS estimator of 6 will be consistent with a rate

of convergence equal to TI-^. However as Engle and Graoger(l988) noticed and Cheung

and Lai(1993a) followed, the OLS seeks the 6 which minimizes the residual variance and

therefore is moat likely to be stationary, so that a t distribution for the estimated resid-

uals will reject the null too often. In other words, the estimated residuals will make the

estimated differencing parameter biased towards rejection of the null hypothesis. This

implies that the distribution of the spectral regression is non-standard. A Monte Carlo

simulation is used to tabulate the size of this non-standard distribution. As in Engle and

Granger(l988) two independent random walks with the same sample size of the real data

in a 10,000 Monte Carlo replication are used to estimate the size of the distribution4.

As can be noticed from Table-1, the distribution is negatively skewed implying a higher

rate of rejection of non-stationarity hypothesis than the standard t distribution. The

mean of the distribution is also negative, which confirms the hypothesis that in spectral

regressions the average differencing parameter tends to be under-estimated.

For the MRR test, as we indicated earlier, the size of the distribution can be easily

computed horn its known distribution function. The values most commonly used provide

a range of the 10.809, 1.8621 for the non-reject ion of the null hypothesis of short memory

'As Baillie et aL(1994) quote, if yc and xt are I(1) and a'& is i ( d ) , the error correction representation for Zt takes the form A(L)(l - L)Zt = -q[l - (1 - L) i-d](l - L)da'& + C ( L ) E ~ , where ~t is I(0) and for which only I(0) terms are involved.

4450 and 2500 are used as sample sizes for the weekly and daily series, respectively. The actual sample size might be different from what is used in the Monte Carlo experiment. We are reporting these two numbers in the Monte Cario experiment to avoid using too many tables unnecessarily. We have also simulated the distributions with the actual sample size (available upon request). The difference between the distributions is quite negligible. In fact, it seems that for sample sizes large enough(1arger than 1000) the simulated distribution will remain virtually unchanged. For smaller sample sizes, the distribution is only slightly different.

at the five percent signiscance level, and a range of [0.721,2.098] at the one percent signif-

icance level. We have repeated our Monte Carlo simulation for the MRR test and found

no sigdicant departure hom these values. In fact, the size distribution of the MRR test

under non-cointegration alternatives is calculated and reported in Table-2. For the non-

cointegration alternatives two independent random walks are generated and the M R R

test is performed on the differenced estimated residuals, whereas for the cointegration

alternative two cointegrated random wa&s are generated. The size distribution of the

Modified RIS statistic appears to be slightly different from the asymptotic distribution

tabulated in Lo(1991) and presented earlier.

The mean and the standard deviation of the of the modified R/S distribution as

presented in Table-:! are less than the theoretical mean e-g. fi -- 1.25 and standard

deviation 6 = 0.27, respectively. Using theoretical critical values, the size of the

modified RIS under a non-cointegrated alternative scceeds the nominal values with a

slight improvement as the number of observations increase. However, the bulk of the

excess rejection, specifically about 90 percent of the rejection comes from the left side.

This is in complete accordance with earlier results. As we indicated earlier. the fact

that the parameters of the model are unknown and that we are replacing them with

the estimated parameters will lead to an under-estimation of the differencing parameter.

Since the true differencing parameter is unity, the differenced series will quite often exhibit

negative differencing parameters.

Both of these tests are robust to strong conditional heteroskedasticity and moderate

ARMA contaminations, whereas the GPH test is biased towards accepting long memory

and MRR tests are biased towards rejecting long memory in the presence of strong AR

of order 0.7 or higher. Both of them are sensitive to strong negative M A contamination

of order - 0.9 arising from possible excess differencing, however. These characteristics

provide us with a good framework to test for kactional cointegration. Most importantly,

in the presence of strong iirst order autoregression of the magnitude 0.5 and higher, it

will be preferable to use differenced series, especially in the case of Ill R R tests, so that

the high order of autoregression can not bias the test.

2.4 Data and Empirical Results

One major criticism for using spot prices is that there is not an actual marketplace for

spot crude oil contracts. In facts, since there is a minimum three weeks window on

delivery of crude oil spot contracts, one may claim that the spot market for crude oil

does not scist. To avoid this problem we use the nearest futures contract as a proxy

for the spot price. This approximation should not alter the results significantly. In fact,

Borenstein et al. (1997) found Little change in their final results when the spot price was

replaced by the nearest futures contract.

Daily settlement prices for one- and three- months ahead futures prices to the last

trading day of the month are used. The three months ahead futures data and daily and

weekly series are used to investigate the robustness of the ha1 results. The weekly data

include the last trading day of the week and are computed for both one and three months

ahead futures prices. The heating oil futures market was launched in 1977, the market

for West Texas intermediate crude was launched in 1983, and unleaded gasoline opened

on the futures market in 1984. The number of observations used in the regressions cor-

responds to the available data. The data ends 10 September 1993. In total, 2610(501)

daily(week1y) number of observations for the one month ahead crude oil-heating oil re

gression and 21 l7(418) daily(week1y) number of observations for the one month ahead

regressions involving unleaded gasoline is used. The corresponding numbers for the three

month ahead regressions are 2451(498) and 2041(418) respectively. The data are Com-

modity Systems, Inc. (CSI) . All the variables are in logarith~ns.~

See Borenstein et aL(1997) for the difference in using levels and logarithm in interpreting the con- temporaneous relationship between crude oil and the corresponding product. Our results do not change qualitatively in either case.

2.4.1 Univariat e Analysis

In recent years, several papers have discussed evidence of long memory in financial series.

The interests rate futures, as in h g and Lo(1993), the exchange rate as in Fang, Lai,

and Lai(lW4), and the stock market returns as in Cheung and Lai(1995) have been the

main target. However, application of long memory tests to the commodity market is fairly

rare. Among the few exceptions, Helms, Kaen, and Rosenman(l984) find some evidence

of long memory in the commodity futures prices using classical R/S test. However

since the classical R/S test is not robust to short term dependency and conditional

heteroskedasticity, their conclusion does not come as a surprise. F'urthermore, Peterson,

Ma, and Ritchey(l992) find a memory component in cash prices of 17 commodities using

variance ratio analysis and Cheung and Lai(1993b) use MRR to test for long memory

in the gold market. They find that weekly gold spot prices exhibit an unstable long

memory process in which the evidence of long memory disappears if only a few sensitive

observations are dropped. Our paper provides the first attempt to analyze long memory

in the commodity futures using GPH and MRR tests, although our main concern is the

interdependence of the energy futures rather than the individual futures series themselves.

Tables 3 4 provide conventional unit-root tests on levels and k t differences of the

individual series for daily data. Tables 56 replicate the same tests for weekly data.

Clearly, a l l tests suggest a unit-root in levels and a stationary first difference. The

results are unanimous in the sense that whether the null hypothesis is non-stationarity as

in the Augmented Dickey Mer(ADF) tests or st ationarity as in Kwiatkowski, Phillips,

Schmidt, and Shin (KPSS)' , the data appear to be generated by a non-stationary process

whereas the first difference appears to be stationary. This is not surprising as many

other hancial series exhibit non-stationarity. However, to understand more about the

characteristics of the individual series and before analyzing the long memory behavior, in

% the conventional unit-root tests as ADF the unit-root is the null hypothesis to be tested. Since, in the classical hypothesis testing the null hypothesis is accepted unless there is strong evidence against it, the conventional unitroot tests such as ADF tend not to reject the null too often. The KPSS test, on the other hand, has a null of stationzuity, so that it does not reject the stationarity too often.

line with other empirical studies some descriptive statistics are presented in Tables 7-8.

The Kendd-Stuart (1958) statistic is used to test for the skewness and excess kurtosis.

Evidence of skewness is rare, whereas some evidence of excess kurtosis is present. More

specifically, for the weekly individual series no evidence of skewness is found, whereas

some evidence of skewness can be found for the daily series. However, evidence of excess

kurtosis is widespread in both weekly and daily series, although they are very weak for

the weekly series. In fact, the Jarque-Bera statistic for normality suggests strong non-

normality stemming from excess kurtosis. Whennore, the excess kurtosis is negative in

all series suggesting that the distributions underlying the series have a thinner tail than

the normal series. To test for short term linear dependency, the standard Ljung-Box

statistics is used for both diffmenced series and the squared differenced series. Evidence

of short term dependency in first moments is strong for the daily series and rather weak

for the weekly series: whereas evidence of short term Linear dependency in the higher

moments is strong in both weekly and daily series. This short term dependency par-

ticularly the higher order moments, provides evidence of possible short term nonlinear

dynamics generated by models such as autoregressiveconditional heteroskedasticity as in

ARCH(p) frameworks [Engle(1982)] or GARC H@, q ) [Bollerslev(l986)] models. These

family of models have been used extensively in recent years to capture the time-varying

short term volatility in hancial series and are the prime candidates to model the short

term dependency uncovered by Q statistics. A full discussion of the modelling procedure

is beyond the scope of this paper and can be obtained from recent survey papers such as

Bollerslev et a1. (1992)

Table-9 and Table-10 provide the estimated AR (1)-GARC H @, q) for the individual

differenced series together with the correspondent Q statistics. The order of the model has

been determined by Schwartz7s Bayesian Information Criterion. Q statistics have been

computed born estimated residuals standardized by estimated t ime-varying standard

errors. Significant evidence of conditional heteroskedasticity can be found in all series.

The corresponding Q st at ist ics provide evidence on the adequacy of aut oregressive and

conditional heteroskedasticity model to account for short-term dependency- and nonlinear

dynamics in the series. Except for some cases in the three months ahead daily energy

futures series, where some evidence of short term dependency in the first moment can

be traced, no evidence of short terrn dependency in first or second moments is found.

Fbrthermore, the estimated autoregressive coefficients in a l l series w e p t for the daily

gasoline futures are not significantly different £?om zero, and in no case are greater than

0.1 in absolute terms. This indicates a very low order of autoregression in the Merenced

series, if any. These findings are crucial to our long memory analysis since, as stated

earlier, both the GPH test and MRR tests are robust to conditional heteroskedasticity

and to moderate short term dependency.

Tables 11-14 provide results of the long memory tests in the energy future series. The

standard t statistic can be used to test for the estimated parameters in the periodogram

regression. The critical d u e s for the MRR test, as provided earlier, is [0.809, 18621 and

[0.721, 2.0981 for five percent and one percent significance level respectively.

Except for some relatively weak evidence of long memory in the one month ahead

crude futures provided by the G P H test, the long memory tests confirm the findings of

the traditional unit-root test. The estimated differencing parameter is not too sensitive

to the number of ordinates used in the G P H test. Hence the possible underlying ARMA

contamination is not causing a significant bias in the spectral regression. Furthermore,

the results are robust to the number of ordinates in rejecting the null hypothesis of long

memory: none of the series (differenced series) appear to have differencing parameters

different from one (zero), except for some cases in the one-month ahead crude futures.

The re-scaled range statistics(MRR) can no where reject the null hypothesis of short

memory in favor of long memory. In fact, the robustness of the MRR test corroborates

the G P H test, especially in cases where the estimated differencing parameter is sensitive

to the number of ordinates. Since MRR is relatively robust to AR contamination, and

since the corresponding autocorrelations do not show any strong MA representation, the

AdRR statistic strengthens the GPH results by providing no evidence for long memory.

The analysis provided in this section clearly suggests short-term dependency and

conditional heteroskedasticity as the main scurces of the dynamics of differenced energy

price series. The differenced series do not seem to follow any long memory process.

This means that the series follow a random walk process; a shock to the series tends to

take them out of equilibrium permanently. With the absence of long memory, the series

do not exhibit mean reversion, which is essential to any notion of long term equilibrium.

Instead, energy future returns seem to be driven by moderate short-term dependency and

strong nonlinear volatility represented by conditional heteroskedasticity. An interesting

feature of this analysis is that mean shifts in the series which are quite often are cited

as a source of bias in unit-root tests (see Schwan(1994)), are not sigdicant. Otherwise,

as suggested by Cheung(1993), the estimated differencing parameters should be biased

towards accepting long memory alternatives.

2.4.2 Multivariate Analysis

Having established the results for individual energy futures series, the next step is to

conduct fractional cointegration tests. To this end OLS regressions of heating oil and

unleaded gasoline prices on the crude oil prices are performed, and the results are pre-

sented in the Table-15. Autocorrelation coefficients for the levels and the first differences

of variables together with the autocorrelation coefficients of the estimated residuals are

presented in Tables 1619. Not surprisingly, we find a very slow pace of decrease for the

Levels and a very rapid decrease for the first differences. This confirms the presence of

unit-root in levels. However, the autocorrelation co&cients of the error correct ion terms

'As discussed by Madcinnon(l991) and cited by Booth and Tse(1995), adding a trend in the cointe- grating relation can make the resulting test invariant to the value of a possible drift term in the DGP. Furthermore, Booth and Choudhury(l991), in an attempt to provide an economic interpretation, ar- gue that the trend term can account for the possibility that the time-varying risk premia of market are different across markets. To this end, all the fist stage OLS regressions are conducted with and without trend. The results do not differ much, and we only report the results without trend. Further- more, to account for possible seasonality in the series, we have also added seasonal. Since we are not interested in persistence due to seasonality, the results reported hereafter correspond to regressions with monthly dummies. We have also repeated the regressions with seasonally adjusted series. The results are qualitatively similar.

die out more slowly than those of first differences and faster than those of levels. In fact,

the persistence of the autocorrelation coeflicients is in sharp contrast to an I (0 ) , process

which suggests possible evidence of long memory and mean reversion. Autocorrelation

coacients for the error correction terms reveal a relatively rapid rate of decay, which

is in sharp contrast to the I(1) properties of individual series. However, this decay still

exhibits significant persistence over time, which shows that the error correction tern re-

sponds more slowly than an I(0) process. Considering Figure-1, the correlogram for the

one daily one-month ahead error-correction t enns, one can ider long memory character-

istics. One can also infer that the autocorrelations do not show significant periodic cycles

as might be induced by seasonal variations. This seasonal persistence has been elimi-

nated by adding the seasonal durnmies. The reminder of persistence, therefore, should

be attributed to non-seasonal cemovement of the oil and the corresponding products.

Augmented Dickey-Wer(ADF), and Kwiatkowski, Phillips, Schmidt and Shin(K PSS)

unit-root tests are performed on the estimated residuals to check for the possible cointe-

gration. As indicated earlier, ADF is based on a non-stationary null hypothesis, whereas

K PSS assumes a st at ionary null hypothesis. Although AD F has low power against frac-

tionally integrated alternatives, Lee and Schmidt(l993) document rather strong power

for the KPSS test. Combining these two tests will certainly provide a better under-

standing of the possibly hactionally integrated alternatives before we proceed with the

long memory test. The results of these two tests are provided in Tables 20-21.

ADF tests reject the null hypothesis rather strongly in all cases, whereas KPSS can

not reject the null hypothesis of stationarity in at least some cases. More specifically, for

daily energy futures prices, the conventional ADF test consistently rejects the unit-root

null, whereas for the same series the K PSS test almost u d o d y rejects the stationary

null hypothesis. For the weekly data, however, the ADF tests also reject the unit-root

null, whereas the KPSS tests provide mixed results. One way to explain these codicting

results is to take the possible fractional integration alternative into consideration. Since

ADF is very weak in detecting fkactiondy cointegrated alternatives, its rejection of the

unit-root null might be accounted for by a fractional integration alternative. On the

other hand, since the KPSS test has rather good power against fractional cointegration,

its rejection of the stationarity null hypothesis might be attributed to a fractionally

integrated alternative.

These results cast doubt on the conventional unit-root tests and suggest the possi-

bility of a long memory process. To this end, Tables 22-23 provide the results of the

long memory tests on the error correction terms. The results provide strong evidence for

fkactional cointegration in both Heating Oil-Crude Oil and Unleaded Gasoline-Crude Oil

relations. For one month ahead energy futures, both Heating Oil-Crude Oil and Unleaded

Gasoline-Crude Oil error correction terms seem to exhibit long memory characteristics.

In both cases the GPH test is rather insensitive to the number of ordinates indicating in-

siguficant ARMA contamination."he estimated differencing parameter is si&cantly

different fkom both zero and one at 5 percent significance level. The estimated differ-

encing parameter is generally higher for the unleaded gasoline-crude oil relation than

for heating oil-crude oil. This indicates a higher memory for the former error correction

tern than the latter. The modified R/S(MRR) test rejects the null hypothesis of short

memory in the lower tail. This indicates a negative fkactiond differencing parameter,

which is consistent with the fact that we are performing the test on the differenced er-

ror correction term and that the original Werencing parameter is less that one. The

estimated differencing parameter, as expected, is higher in daily data than in the weekly

series. This is also confirmed by a strong rejection of the short-memory hypothesis in

the daily series.

This strong evidence of long memory may provide some directions as to how energy

futures are behaving over the time. In terms of Engle and Granger(l987), (fractional)

cointegration means that equilibrium between the series will occasionally occur, at least

'Our experiment on error correction terms for regressions without seasonal dummies were more sen- sitive to the number of ordinates implying a strong short term contamination. These results provide support for the hypothesis that seasonal factors can provide more short term and not long term persis- tence in the error correction term.

to a close approximation, whereas if (fkactiond) cointegration does not hold, the equilib

rium concept has no practical implications. Granger (1986) suggests a type of Granger

causalits' in at least one direction in the error correction models implying that at least one

of the series should be Granger-caused by the other. The fact that the error correction

term in the relationship between crude oil and products shows the characteristics of a

long-memory process suggests that the adjustment towards equilibrium between crude oil

and products is more time consuming than has been suggested by cointegration studies.

In other words, the process of adjustment in the energy futures market is not a short

term phenomenon but disequilibrium is frequent.

It remains, then, to provide a plausible explanation as to why the error correction

terms exhibit long memory. The cointegration hypothesis is based on the premise that

since crude oil is a main component of the refined products, any shock to the system

should fade out quickly. This suggests that there is a very strong causality link between

crude oil and its products. The evidence of long memory found in this paper suggests

that this link might not be as strong as commonly believed. In other words, long memory

implies that the market is treating crude oil and products as rather separate commodities.

Booth and Tse(1995) suggest that the existence of long memory may imply that futures

prices adjust because the spread between the series is out of equilibrium. In our case,

this means that the spread between crude oil and products indicates when the market

will readjust.

2.5 Conclusion

This paper has examined evidence of long memory in the energy futures market. Long

memory tests have been conducted on both individual series and the corresponding coin-

tegrating relations. The individual series seem not to exhibit long memory; neither GPH

nor MRR test can reject 1(1) hypothesis. However, evidence of generalized conditional

heteraskedasticity GP.RC H is strong in all individual series. This suggests that the indi-

vidual series are not predictable, although their volatility might be. F'urther research is

required to analyze the possibility of long memory in the volatility in light of recent work

of long memory models with generalized conditional heteroskedasticity, a s in Baillie et

al. (1996).

Although none of the series exhibit long-memory properties, they seem to be &action-

ally cointegrated. For the one month ahead series, all error correction terms exhibit long

memory. For the three month ahead series, however, the results are mixed. Specifically,

neither GPH nor MRR test can reject the null hypothesis of unit-root in the error cor-

rection term of the heating oil-crude oil relation. We have attributed this result to strong

seasonality in the heating oil series. We have repeated ow estimation by accounting for

seasonally. The corresponding results uniformly suggest fkact ional coint egrat ion between

crude oil and products.

These results imply that the response of energy market to shocks is not as fast as

it might be suggested by cointegration analysis. In other words, the mean-reverting

behavior of the error correction terms implied in the long-memory process indicates that

the divergence between crude oil and products may not be a short term phenomenon

as might be suggested by traditional cointegration tests; It may take a long time before

the spreads between heating oil and crude oil and unleaded gasoline and crude oil adjust

to their long run equilibrium. In fact, positive autocorrelation of the error correction

terms in the short-term and their negative autocorrelation over the long horizon points

to the existence a predictable component in the energy futures market, which can be

fully captured by a long-memory process with long-non periodic cycles.

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Table 2.1: Size of GPH Test for Cointegration(Null Hypothesis of d = 1) T = 450 T = 2500

p = 0.5 p = 0.55 p = 0.6 p = 0.5 p = 0.55 p = 0.6 Percentile

1% -3.113 -3.097 -3.046 -3.01 1 -2.993 -2.78 1 5% -2.223 -2.200 -2.147 -2.088 -2.053 -2.025 10% -1.752 -1.750 -1.721 -1.680 -1.635 -1.584 25% -1.068 -1.031 -1.053 -1.004 -0.984 -0.933 50% -0.344 -0.330 -0.317 -0.297 -0.292 -0.255 75% 0.310 0.333 0.334 0.345 0-355 0.410 90% 0.882 0.881 0.907 0.907 0.915 0.964 95% 1.210 1.211 1.242 1.246 1.269 1.295 99% 1.814 1.827 1.833 1.900 1.905 1.900 Mean -0.407 -0.373 -0.376 -0.349 -0.336 -0.285

Skewness -0.337 -0.372 -0.301 -0.264 -0.260 -0.254 Kurtosis 0.327 0.409 0.337 0.270 0.236 0.216

Notc: The number of ordinates for the periadogram regression is n = T p lor which T is the rirmplc she. The empirical size is obtained through 10,000 rcplicntionv of two independent random walks.

Table 2.2: Size of Modified R/S Test for Fractional Cointegration Non-Coint egrat ion Null T = 450 T = 2500

Percentile 1% 0.690 0.705 5% 0.776 0.800

95% 1.567 1.608 99% 1.773 1.821 Mean 1.127 1.151 S.D. 0.241 0.247

5lin./Max. 0.44/2.48 0.581235 Size %-Test 0.02 0.015 Size 596-Test 0.08 0.063

Size lO%Terst 0.14 0.121 The empirical size is obtained through 10,000 rcplicntious of two random walks. T h e MRR test iy pcrformcd o n thc

diffcrcnccd ortituntor rcuidunts. T h c last three rows indicatc thc pcrccntngc of occurarlcc of MRR tcit for countegration in tllc 1, 5, and 10 pcrccut critical areas obtained from thc classical R / S st;rtistics.

Table 2.3: Results of Unitroot Tests on Daily Energy F'utures(1) One Month Ahead

Crude Oil Heating Oil Unleaded Gasoline Level/ Difference Level/Difference Level/ Difference

KPSS(l2) 4.30**/0.061 3.97**/0.056 1.77**/0.060 ADF rcprorcrlts AugrucntMf Dickey-Fullcr tm, whcrciw KPSS stands for Kwintkowski, Phi l ip , Schmidt. and

YongcLcol(l992) test. " shows 1 pcrccnt lcvcl and s t a ~ d s for 5 ~ C K C C U L Y ~ ~ U ~ ~ ~ C ~ L U C C ICVCI. MI scrior ilrc iu logs. Nurnbcm in () represent lag uubgucntrrtious.

Table 2.4: Results of Unitroot Tests on Daily Energy Futures(2) Three Month Ahead

Crude Oil Heating Oil Unleaded Gasoline LevelIDifference LevelIDBerence LevelIDifference

ADF(2) -2.251-31.82** -2.331-30.85** -2.801-28.13** AD F(4) -2.131-23.76** -2.191-25.87** -2.591-20.76** AD F(8) -2.011-17.35** -2.01/-17.34** -2.361-15.85**

ADF(12) -2.321-12.63** -2.421-12.33** -2.681-1 1.54** KPSS(2) 18.501**/0.068 16.830**/0.60 8.401**/0.075 KPSS(4) 11.130**/0.072 lO.l2'P*/O.O63 5.063**/0.078 K PSS(8) 6.213**/0.079 5.654**/0.070 2.836**/0.085 KPSS(12) 4.320**/0.077 3.933**/0.067 1.978**/0.083

ADF represents Augmented Dickey-Futler tcu, whereas KPSS stands for Kwintkowski, Philip, Schmidt, nnd Yongcheol(I992) test. ** shows I percent Icvcl and stands for 5 percent significrmcc level. All serial arc in logs.

Numbers in () represent lag augmentations.

Table 2.5: Results of Unitroot Tests on Weekly Energy Futures(1) One Month Ahead

Crude Oil Heating Oil Unleaded Gasoline LevelIDifkrence Level/DSerence Level/DBerence

ADF(2) -2.511-1 1.59*8 -2.491-11.85** -2.731-Il.OT** ADF(4) -2.841-8.74** -2.831-8.92** -3.07/-8.57** ADF(8) -2.941-7.26** 3.271-6.84** -2.961-7.37**

AD F(12) -2.891-6.08** -3.211-5.78** -2.671-6.36** K PSS(2) 3.65**/0.058 3.39**/0.056 1.46**/0.053 KPSS(4) 2.22**/0.053 2.07**/0.051 0.899**/0.050 KPSS(8) 1.28**/0.048 1 .19**/0.045 0.659*/0.050 KPSS(12) 0.92**/0.048 0.86**/0.043 0.384*/0.056

ADF rcprcvcnts Augmented Dickey-hller tcs, whereas KPSS stands for Kwiatkowski. Philips, Schmidt, and Yongchcol(l992) tort. " shows 1 pcrcent level and stanch for 5 percent significance Icvcl. All scrici are in logs.

Numben in () rcprcrc~tt Iog ot~grncctztions.

Table 2.6: Results of Unitroot Tests on Weekly Energy Futures(2) Three Month Ahead

Crude Oil Heating Oil Unleaded Gasoline Level/DSerence Level/Difference Level/DSerence

KPSS(8) 1.340**/0.064 1.224**/0.056 0.603**/0.065 KPSS(12) 0.958**/0.062 0.879**/0.054 0.439**/0.068

ADF reprcuents Augmented Dickey-Fullcr tcu, whereov KPSS stands for Kwintkowski, Philips, Sch~nidt, nud Yongchcol(l992) tort. ** shows 1 percent lcvcl nnd stands for 5 percent significance levci. All scrics arc in logs.

Numbers in () represent lag augmcnt~rtiorw.

Table 2.7: Descriptive St at kt ics, One Month Ahead Energy Futures Mean S.D. Max kfin Skew Kurt Q Q Nob

Daily Crude Oil 3.06 0.25 3.68 2.34 0.02 -0.59* 313.2* 471.7* 2610

Heating Oil -0.53 0.23 0.07 -1.15 0.14* -0.80* 283.4* 280.1* 2610 Gasoline -0.56 0.20 0.01 -1.16 0.03 -0.12 179.1* 290.1* 2117

Weekly Crudeoil 3.06 0.25 3.68 2.36 0.02 -0.56* 38.6* 196.9* 501

Heating Oil -0.53 0.23 0.06 -1.13 0.15 -0.79* 43.6* 237.6* 501 Gasoline -0.56 0.20 -0.01 -1.16 0.03 -0.06 24.57" 112.0* 418

All scrics arc in logarithm. The descriptive statistics are for the rnenn. stimdnrd dcvirrtiou, mac&nt~m, minimum, cxccss skcwncw. kurtosis, Ljung-Box Q test for thc futures returns and thc squorcd returns reupcctivcly. The nurubcr of llrgs(m)

for the Q test is 24 and 120 for the w t ~ k l y and diuly scrics rcspcctivcly. ' indicatccl statistical sigdicnticc a t 5 pcrccnt levcl.

Table 2.8: Descriptive Statistics, Three Month Ahead Energy Futures Mean S.D. Max blin Skew Kurt Q Q Nob

Crude Oil 3.04 0.24 3.60 2.37 0.03 -0.62* 315.2* 949.3* 2451 Heating Oil -0.54 0.23 0.05 -1.10 0.17* -0.87* 141.0* 525.2* 2451

Gasoline -0.58 0.19 -0.6 -1.18 -0.15* -0.04 121.4" 465.0* 2041 Weekly

Crude Oil 3.04 0.24 3.6 2.38 0.02 -0.61* 36.6 215.8* 498 Heating Oil -0.54 0.23 0.04 -1.68 0.17 -0.85* 47.5* 249.8* 498

Gasoline -0.58 0.19 -0.06 -1.18 -0.15 -0.01 29.2 145.9" 418 A11 series are in logarithm. The descriptive statistics arc for the mean, standard dcvintion, maximum, minimum, excess

skewness, kurtosis, Ljung-Box Q test for thc futures returns and the squared rcturns respectively. Thc nunlbcr of lags(m) for the Q test is 24 and 120 for the weekly and daily series rcspcctivcly. indicated statistical significance a t 5 pcrccnt

Icvcl.

Table 2.9: Short Term Dependency and Conditional Heteroskedasticity in Energy h- tures. One Month Ahead Series

QO a1 PO $01 0 2 71 n Q(=) Qs(rn)

Daily Crude Oil -29E4 -14E-1 -15E-5 -13 .88 130.2 113.8

(.14) (-66) (3.75) (10.9) (90.1) Heating Oil -223-2 .97E3 .4635 .10 .89 121.3 87.3

(-76) (-46) (4.13) (9.0) (83.6) Gasoline -.28E3 .66E-1 45E5 .17 -.I0 -93 129.4 145.9

Crude Oil .26E3 -.41E1 .49M .14 1.37 -.51 34.5 23.5 (.19) ( - 8 1 (3.87) (4.89) (20.3) (-10.8)

Heating Oil -42E3 -.95E1 .19E3 .20 -72 33.5 32.6 (.25) (-1.83) (2.99) (4.16) (12.7)

Gasoline -.56E3 -.70E3 .30E-3 .13 .15 -. 18 .78 30.4 20.0 (-.30) (-1.33) (2.85) (3.74) (4.81) (-4.04) (21.4)

The cstimates GARCH(p,q) model for which the order, p,q irrC determined by Schwnrtz Baymion criterion is: - * - - . .

u: = ~o + C; + Cq Y , u ~ , The wyrxtptotic t arc iu parlmnthescs bcIow the cstiruntd parmetera. Ljung-Bm 4 test h performed on the standardized rcriduaIs (ztfit) and their squares Q,. The number of In@ (m) for the tcat is 24 i d 120 for the v n k l y and daily r e r i a

reupcctivcly.

Table 2.10: Short Term Dependency and Conditional Heteroskedasticity in Energy Fu- - -

tures, Three Month Ahead Series QO PO PI P2 71 % Q(m) Qs(m)

Daily Crude Oil .62E4 .79E3 .11E5 -08 -92 132 184*

(.32) (0.04) (5.79) (11.9) (170.7) Heating Oil .22E3 -0.3 .343%5 .06 .93 178* 215*

(.85) (-1.1) (8.16) (10.6) (161.4) Gasoline -.21E3 .60E1 .40E5 .86E1 .9 1 134 157*

(--68) (2.48) (3.89) (8.05) (88.6) weekly

Crude Oil -80E3 -.48E1 .56U .19 -79 28.8 20.0 (0.63) (-0.90) (3.00) (4.99) (24.4)

Heating Oil .15E1 -78E1 .15E-3 .52E1 .16 .70 38.5* 22.3 (1.06) (-1.67) (2.93) (1.19) (2-55) (10.8)

Gasoline -.51E3 -.35E1 . 9 8 M -15 .80 31.2 20.8 (-.30) (-.64) (2.58) (3.77) (17.8)

Thc estimates GARCH(p, q) model for which the order, p,q arc determined by Schwnrtz Bayesian criterion is: X L = Q O + Q L X C - L + U ~ , ut l t -~-N(O,~t ) .: = P o + p-a=;-j

The srymptotic t arc in pasanthues bdow the e t i m a t d parnmetera. Ljung-Bm Q t a t is performed on the standardbed residuals ( ; f i t ) and their squnrcs Q.. The number of lags (m) for the t a t is 24 and 120 for the weekly and ddly scrim

Table 2.11: Results of Long Memory Tests on Daily Energy Futures(1) One Month Ahead

Crude Oil Heating Oil Unleaded Gasoline G P H ( p = 0.5) 1.09(0.101) 1.13(0.101) 1.06(0.108)

G H P ( p = 0.55) 1.16(0.816) l.ll(0.816) 1.05(0.087) GHP(p = 0.6) 1.139(0.066) l.Og(0.066) 1.12(0.070)

MRR 1.3990(1) 1.2883(3) 1.3064(3) CPH rcprorenh Gcwekc and Porter-Hudak(l983) test whereas MRR represents tho modified R / S tort r1bcuszlc.d by

Lo(1991). For GPH t a t the ~iumber in paranthorcs arc the asymptotic standard errors whcrecw for thc M R R toit ttlcy arc thc numbcr of truncation llrgs. GPH t a t has a standard t distribution whereas M R R s h e a t five percent i.i

[O.809,1.8621.

Table 2.12: Results of Long Memory Tests on Daily Energy Futures(2) Three Month Ahead

Crude Oil Heating Oil Unleaded Gasoline GPH(p=0.5) 1.16(0.104) 1.160(0.104) 1.009(0.110)

GHP(p = 0.55) 1.150(0.083) 1.150(0.083) 1.102(0.878) G H P ( p = 0.6) 1.128(0.670) 1.082(0.067) 1.071 (0.071)

MRR 1.4421(2) 1.3217(1) 1.4397(4) G P H rcprcrcnts Gewekc and Portcr-Hudak(l983) t a t whereas M R R reprcicnts the modified R / S t a t discuvscd by

La(1991). For C P H t a t the number in paranthevcs a rc the uympto t i c standard errors whercnv for thc AIRR t a t thcy a r c thc number of truncation lags. G P H t a t has a s tandard t distribution whercas MRR s h c a t five pcrccnt is

(0.809,1.862].

Table 2.13: Results of Long Memory Tests on Weekly Energy Futures(l) One Month Ahead

Cmde Oil Heating Oil Unleaded Gasoline

GHP(p = 0.55) 0.98(0.14) 1.02(0.14) 0.85(0. 15) G H P ( p = 0.6) 1.04(0.11) 1.07(0.11) 0.87(0.12)

MRR 1.4395 1.2768(3) 1.2873(1) GPH rcprtuents Ccwckc and Porter-Hudak(l983) tc i t whcrcos IVRR rcprcsents tlic niodificd R / S t m t tliscnsscd by

Lo(l99 1). For GPH tc5t thc numhcr in plrranthcvcs arc rhc asymptotic ~ t i ~ ~ l r l z ~ d crrors whcrcits for thc MRR t a t thcp a rc the number of truncation lags. GPH tcst hns a standard t distribution whcrclrs MRR sizc at tivc pcrccnt is

[O.809,l.8W].

Table 2.14: Results of Long Memory Tests on Weekly Energy F'utures(2) Three Month Ahead

Crude Oil Heating Oil Unleaded Gasoline GPH(p = 0.5) 0.938(0.171) 0.969(0.l?1) 1.002(0.181) GHP(p = 0.55) 1.098(0.140) 1.240(1. 140) 0.972(0.149) G H P ( p = 0.6) 1.093(0.116) 1.118(0.116) 0.993(0.124)

MRR 1.4975(2) 1 .2768(2) 1.2873(1) GPH represents Gcwcke and Porter-Hudak(l983) t a t wherens MRR rcprmcnts thc modif id R / S test discusvcrl by

Lo(1991). For G P H tort the number in pnranthcscr a re the asymptotic standard errors whcrcau for thc MRR t a t they a rc the numbcr of truncation lags- G P H tort has a staudard t distribution whcrcas M R R skc a t fivc pcrccnt is

[O.809,1.862].

Table 2.15: Results of OLS Regression Between Energy Futures Dailv Data Weeklv Data " " -- -.

a B RL a p RL 1 Month Hail = a + PCoilf E -3.24 0.89 0.97 -3.23 0.89 0.971

Ahead Gasoil = a + PCoilf e -3.27 0.90 0.956 -3.27 0.90 0.928

3 Months Hoil = cu + DCdE + E -3.27 0.91 0.965 -3.27 0.91 0.982 Ahead Gasoil = cu fPCd1 + E -3.36 0.92 0.955 -3.34 0.92 0.95

Note: Coil, Hoil, and Gasoil represent Cmdc Oil, Hating Oil sad Unleaded Gasoline respectively.

Table 2.16: Autocorrelation Coefficients, p ( t ) , One Month Ahead Daily T Crude oil Heating oil Unleaded Gasoline ZHc kc

Level/Diff Level/Diff Level /DiE 1 0.996/'0.004 0.996/0.042 0.994/0.62 0.952 0.969 2 0.99210.000 0.991/0.009 0.987/0.039 0.914 0.939 3 0.988/0.005 0.9861-0.081 0.9801-0.049 0.875 0.911 4 0.984/0.005 0.982/-0.0 18 0.9731-0.011 0.838 0.887 5 0.9801-0.007 0.9781-0.053 0.967/-0.004 0.804 0.856 10 0.964/0.010 0.962/0.080 0.939/0.030 0.676 0.727 20 0.922/0.002 0.919/0.037 0.875/0.029 0.482 0.503 60 0.716/0.002 0.699/0.043 0.6271-0.025 0.164 0.209 240 0.4831-0.001 0.424/-0.015 0.3751-0.000 0.108 0.243

Notc: THC, FCC rcprment the cstimeted residuals of thc OLS cstirnrrtiou of Heating Oil ou Crudc Oil srrd Uulcud~d C.uoIiac ou Crude Oil roipc~tively.

Table 2.17: Autocorrelation Coefficients, p ( t ) , Three Months Ahead Daily T Crude oil Heating oil Unleaded Gasoline FHc Zcc

Levei /D ZF Level/Diff Level /Diff

Notc: FHC, TGC rcprcvcnt tbc cstirnatcd residuals of thc OLS estimation of Hcating Oil on Crudc Oil and Unlclrdcd Gwolinc on Crude Oil rcspcctivcly.

Table 2.18: Autocorrelation Coefficients, p ( t ) , One Month Ahead Weekly T Crude oil Heating oil Unleaded Gasoline BHC Zcc

Level/Diff LeveI/Diff Level/DX

40 0.3521-0.003 0.389/0.029 0.0971-0.033 0.024 0.190 Note: FHC, FCC represent the estimated residuals of the OLS ortimation of Heating Oil on Crude Oil and Unlcndtd

Gasoiinc on Crude Oil reupectively.

Table 2.19: Autocorrelation Coefficients, p(t ) , Three Month Ahead Weekly C

T Crude oil Heating oil Unleaded Gasoline EHc E G ~

Level/DifF Level/Diff Level/Diff 1 0.982/:0.002 0.981/:0.058 0.971/:0.031 0.838 0.891 2 0.965/0.011 0.965/0.120 0.945/0.071 0.737 0.788 3 0.946/0.004 0.943/0.029 0.91610.02'7 0.629 0.684 4 0.9241-0.006 0.921/0.085 0.884/0.081 0.539 0.592 5 0.9021-0.005 0.896/0.001 0.8481-0.047 0.490 0.520 10 0.7821-0.003 0.7521-0.075 0.6791-0.109 0.304 0.389 20 0.5771-0.010 0.5041-0.075 0.4131-0.034 0.170 0.289 30 0.4841-0.001 0.428I0.068 0.270/0.054 0.223 0.249 40 0.391/0.002 0.417/0.031 0.1 13/0.010 0.155 0.204

Note: FHc, TGc rcprcscnt thc estimated rcviduelv of the OLS atimntion of Heating Oil on Crudc Oil and Unlcudcd Gwolinc on Crude Oil rcspcctivety.

Table 2.20: Results of Unitroot Tests on Error Correction Terms, Daily Energy Fhtures One Month Ahead Three Months Ahead

LI

~ H C EGC ~ H C k c ADF(2) -7.42* -5.77** -6.53** -5.55** ADF(4) -7.11** -5.81** -6.OT** -5.18** ADF(8) -6.43** -5.88** -5.79** -5.46** ADF(12) -6.00** 5.60** -5.7TL* -5.46**

KPSS(12) 0.812** 1.929** 1.711** 1.5335** Notc: FHC, &-c represent the cltimated rcridunlv of the OLS crtirnntioa of Heating Oil ou Crude Oil nud Uulcndcd

GlroIinc ou Crudc Oil rwpectivcly. ADF reprcscnts augmented Dickey-Fultcr t w t whcrcuv KPSS s tands for Kwintkowski. Pbilips, Schmidt, and Shin(L0'32) t a t . The nu~nbers iu parunhcicr rcprcscnt oumbcr of lags. ** shows 1 percent

signific;mcc lcvel whcrens st;~nCfY for 5 pcrceut significance Icvct.

Table 2.21: Results of Unitroot Tests m Error Correction Tenns, Weekly Energy Futures One Month Ahead Three klonths Ahead

KPSS(8) 0.290 0.758* 0.613* 1.232** KPSS(12) 0.231 0.615* 0.483* 0.951**

Notc: THc, rcprmcnt the oltirnutcd rcsiduniu of thc OLS mtiruution of Hcating Oil ou Crrtdc Oil and Unlca~dcri Gasoline on Crudc Oil rcspcctivcly. ADF rcprcsentv uitgrrcutcd Dickcy-Fullcr t a t wltcrcas KPSS stlrnds for Kwiatkowski.

Philips. Schmidt, aud Shin(1'392) test. T h c numbcrs iu pnrunkolcs rcprescnt number of lags. *' shows I pcrccnt siguificlmcc lcvcl whereas stands for 5 pcrccut sibg~ificance lcvcl.

Table 2.22: Results of Fkactional Integration Using GPH and bfRR, Tests(1) One Month Ahead

ZHC ~ G C

Daily Series d H o : d = l & : d = 0 d & : d = l & : d = O

p = 0.5 0.65 -3.4** 6.4** 0.49 -4.P* 4.5** GPH p = 0.55 0.60 4 9 * * 7.4** 0.59 -4?* 6.8**

p = 0.6 0.74 4 0 * * 11.3** 0.80 -2.9** 11.4**

MRR 0.67*(5) 0.61**(1) PVeeklv Series

p = 0.5 0.48 -3.1** 2.8* 0.43 -3.2** 2.4* G P H p = 0.55 0.35 -4.6** 2.5* 0.49 -3.4** 3.3**

p = 0.6 0.50 -4.5** 4.5** 0.48 42** 3.9**

MRR 0.55**(1) 0.72* (4) Note: zHc, rcprcvcnt the estimated rcsidualv of the OLS estimation of Hcating Oil on Crude Oil and Unlcadd

Gnsolinc ou Crude Oil reupectively. Thc numbers in parantheses represent number of l a p . ** shows a 1 pcrccnt significnncc level wherenu * stanch for 5 percent significnncc level. Thc null hypothesis of d = l is tortcv against d#O iind

the null of d=O is tcvtccl against d#O.

Table 2.23: Results of Fractional Integration Using GPH and MRR Tests(2) Three Month Ahead

~ H C k c Daily Series

d E&,:d=l & : d = O d & : d = l & : d = O p = 0.5 0.85 -1.5 8.2** 0.63 -3.4** 5.8**

G P H p=0.55 0.73 -3.1** 8.4** 0.68 -3.?** 7.8** p = 0.6 0.77 -3.5** 1 L6** 0.82 -2.5* lf.5**

MRR 0.63 (6) 0.53*(3) Weekly Series

p = 0.5 0.47 -3.3** 3.3** 0.40 -3.3** 2.2 G P H p = 0.55 0.58 -2.2* 4.9** 0.40 -4.0** 2?*

p = 0.6 0.71 -2.5* 8.0** 0.56 -3.6** 4.5**

MRR 0.62*(5) 0.52*(1) Note: GC, ZGC rcpresexlt the cstimnted rcrrirlunls: of the OLS estimation of Heutiug Oil on Crudc Oil and Unlcrul~xl

Cieiolitlc on Crude Oil rcupcctivcly. Thc nrtmbcrs in paranthaor reprcvcnt number of lags. ** shows o 1 pcrce~rt significnncc lcvcl whcrcav s t u d s for 5 pcrccut sipllificance levcl. Thc null hypothcvis of d = l is tcutm against d#l iind

the null of d=O is tcvccd trgirinvt d#O.

Figure 2- 1 : Correlogram for the Coint egrat ing Residuals

Chapter 3

Convenience Yield, Long Memory,

and Mean Reversion in the

Petroleum Market

3.1 Introduction

The contemporaneous spot and futures prices of a storable commodity usually differ. To

interpret this difference two general approaches are advanced in the literature. Fama

and F'rench(1987) provide a summary of these two approaches, namely the capital asset

pricing(CAPM) and the theory of storage, for our analysis we focus on the latter. The

theory of storage as advanced by earlier works of Working (1949), Breman (1958), and

Telser (1958) explains this difference in terms of the interest rate foregone in storing

a commodity, the warehousing cost, and a convenience yield in holding the inventory.

This convenience yield consists of a stream of implicit benefits which the processors or

consumers of a commodity receive kom holding the inventories of the commodity. These

benefits my arise because the inventory may provide some productive value as when it

is an input to the production of other commodity or there may be a convenience yield

horn holding inventories to meet unexpected demand.

The main focus of the theory of storage relies on the interpretation of the convenience

yield and its implications for the behavior of the spot and futures prices. The standard

assumption in this literature relies on the fact that for the relevant range of inventory

the marginal warehousing cost can be considered roughly constant so that the variation

in the convenience yield dominates the variation in the wst-of-carry. Furthermore, it is

argued that the convenience yield is a hyperbolically decreasing function of the level of

inventory1. This implies that the marginal convenience yield declines with increases in

inventory but at a decreasing rate, and it allows to make predictions about the impact

of demand and supply shocks on spot and futures prices. For instance, a negative supply

shock will decrease the inventory and increase the convenience yield and causes the futures

price to decline relative to the spot price. A fair amount of discussion surrounding this

issue can be found in French(1986) and Fama and French(1987) and can be traced in

some recent work as in Ng and Pirrong(l994,1996) and Schwan and Szalanary(l994).

The theoretical Literature assumes that the convenience yield is stationary and mean

reverting, as pointed out by Ng and Pirrong(l996). The idea Lies in the fact that since

there is an equilibrium level of inventory and inventory is mean reverting and stationary,

then the convenience yield must be mean reverting and stationary as well. As good as this

analysis may sound it does not distinguish between stationarity and mean reversion and it

relies on the assumption of stationary inventory levels which should not be left untested.

In fact, it can be verified that the assumed hyperbolic relation between inventory and the

convenience yield may give rise to a stationary and mean reverting convenience yield even

when inventory is random walk2. Whennore, as will be discussed later mean reversion

does not necessarily imply stationarity, so that we may have a non-stationary but mean

reverting convenience yield.

This paper first establishes theoretically that a mean reverting convenience yield im-

.- . .

'See Fama and Fkench(l987) for further discussion. 2~ simple example can explain this issue further. Assuming that the relation between the convenience

yield and the level of inventory is exponential, one can easily find a model for which the random walk inventory gives rise to a stationary or non-stationary mean reverting convenience yield depending on the parameters of the exponential function.

plies that the market expects supply and demand to adjust and spot prices to mean

revert after a period of time which itself depends on the degree of mean reversion in the

inventory level. Otherwise, the contemporaneous spot and futures prices should move

together so that the convenience yield adjusts aRer a shod period of divergence. It

then uses the long memory concept or more formally the ARFIMA (Autoregressive-

Fkactiondy Int egrated-Moving Average) process to analyze empirically the properties of

convenience yield and subsequently relates it to the spot price behavior. Other implica-

tions of the theory of storage are also examined by extending the ARFIkIA process to

include the variance structure in the form of GARCH family. It is demonstrated that

this approach provides a unified £kamework for the analysis of the theory of storage.

The findings of this paper hints to strong evidence in favor of long memory process

in the convenience yield. The estimated differencing parameter, a measure of degree of

persistence, is found to be between 0.70 and 0.80, indicating a rather strong persistence.

We attribute this non-stationary and mean-revert ing convenience yields to expected mean

reversion of the spot prices. Furthermore, the volatility process and its relation with the

convenience yield is also estimated. It is found that the heating oil and unleaded gasoline

conform very well with the predictions of the theory of storage and their convenience yield

is more volatile for positive price shocks. While crude oil also c o h this prediction,

this type of asymmetric behavior for the volatility process fails to be sigmficant.

This paper is organized as follows. The first section examines the theoretical con-

siderations of the theory of storage based on a simple no-arbitrage condition and its

implications for convenience yield and price variability. The second section discusses

the empirical methodology, the estimation procedure, and related issues, and the third

section presents the data and discusses the empirical results. The find section evaluates

the results and provides a summary.

3.2 Theoret i d Considerat ions

The theory of storage can be formally discussed within the framework of a cost-of-carry

expression which relates the spot and futures prices. It is well known and well discussed

that the no-arbitrage condition between the spot price and the futures price based on a

wst-of-cany expression will lead to the following relation:

Here t represents the current period, T is the futures contract expiration time, ct is

the proportional warehousing cost minus the convenience yield, and rt is the risk-he

interest rate. St represents the spot price at time t , whereas is the futures price at

time t for a contract expiring at T. This relation implies that the return horn purchasing

a commodity at time t and selling it at delivery time T should equal the interest foregone

during the storage, plus the mit~ginal warehousing cost and minus the convenience yield.

Taking the logarithm of both sides of equation (3.1) yields a convenient relation

between the futures price and the spot price:

Where Ct represents the convenience yield for carrying the storable product over the t

to T interval, rt represents the interest rate, and w stands for the warehousing fee. Clearly,

the interest-adjusted basis (In - In St - r, (2'- t ) - w(T - t)) represents the convenience

yield3, and the degree of persistence of the convenience yield can be estimated from the

degree of persistence of the adjusted basis. In the case of commodity futures one may

In this simple formula we are ignoring the adjustment term for the socded marking-temarket feature of the futures contracts. As discussed in Brenner and Kroner(1995) this adjustment term depends on the volatility cf interest rate and spot price processes and declines as time to maturity decreases. In this paper, as in Brenner and Kroner(1995) we asurne this adjustment term to be nonstochastic.

expect higher persistence in the interest-adjusted basis due to the fact that shocks to the

cost-of-cany are more persistent. In other words, assuming a non-stochastic warehousing

fees, the persistence of the cost-of-cany can be attributed to the behavior of the interest

rate or the corresponding convenience yield. Whether the interest rate has a stochastic

trend or not is an open issue which is beyond the scope of this paper. However, due to

the cyclical or seasonal pat terns of their inventory, the convenience yields are supposedly

very persistent in agricultural and petroleum products and to a lower extent in metals.'

When the true spot price is not available, as in the case of crude oil" Gibson and

Schwartz(l991) argue and Bessembinder et aL(1995) follow that the slope of term struc-

ture of convenience yields can be used to infer the implied convenience yield. Rewriting

(3.1) and ignoring the warehousing cost yields:

c~ represents the spot convenience yield of a (T - to) futures contract at date toy Cta,lL

and represent one period ahead convenience yields applying now and ( t i - l ) period

from now, respectively. r~ represents the (T - to) period spot interest rate, rt,-,,t, is the

one period forward interest rate applying (tiJ periods from now, and finally t , stands

for the terminal period T. Clearly, this relation can be written for any adjacent maturity

futures contract so that using two consecutive contracts and dividing the two sides will

result in the following represent at ion for the one-mont h ahead forward convenience yield:

' See Fama and French(1987) and Brenner and Kroner(1995). "Gibson and Schwartz(l991) argue that even the crude oil contracts traded in the s*called spot

market are de f acto forward contracts with delivery occuring up to 30 days ahead.

This represents the T - l periods ahead me month forward convenience yield. There-

fore, either (3.2) or (3.4) can be used in analyzing the convenience yield depending on the

adabi l i ty of data. Bessembinder et aL(1995) approximate the foregone interest rate in

this relation by:

Where TT represent per-period, continuously compound, spot interest rate applicable

to the corresponding intervals t to T. Then CT-L,T represents the one month ahead one-

month forward convenience yield, whereas CT is the instantaneous convenience yield. In

an empirical analysis interest rates such as two T-bill yields with maturities as close as

possible to the futures contracts can be applied to equation (3.5), although for commod-

ity futures interest rate does not prove to be a sigmficant factor in the persistence of

convenience yield.

The behavior of the convenience yield, thus, is the cornerstone for the analysis of

the theory of storage. Theoretically, the theory of storage predicts a mean reverting

convenience yield, since this mean reversion will insure that the adjustments in demand

and supply will bring the spot and futures prices to their long run equilibrium. Any

departure from equilibrium should be transitory and mean reverting if the no-arbitrage

condition holds, however, it may exhibit sigdicant persistence. This provides a strong

tool to test the theory of storage by testing for cointegration between spot price, futures

price and interest rate. This approach has been pursued extensively as discussed and

summarized in Brenner and Kroner (1995). However, by following this line of analysis, one

may depart fiom the implications of the nearbitrage relation, since, as stated earlier, this

relation requires only a mean-reverting convenience yield. Furthermore, the stochastic

nature of the convenience yield and its persistence can provide significant information

about the behavior of prices in equilibrium which will be missed by a simple cointegration

analysis.

In fad, Bessernbinder et aL(1995) use the theory of storage to test whether investors

anticipate mean reversion in spot asset prices. Their analysis is focused on the slope of

the term structure of futures prices and the way it cova.ries with the spot price. The

persistence of the convenience yield may provide yet another device to test for mean

reversion in equilibrium asset prices. A long memory-mean reverting convenience yield

will imply mean reverting spot prices, since any departure born the equilibrium price

will take a considerable amount of time to be reverted, especially in markets with strong

seasonal fluctuations such as petroleum. In other words, a quick adjustment between

the spot and futures prices implies that individuals are not anticipating mean-reverting

spot prices so that spot prices will not be predictable in equilibrium. Slow adjustment

between spot and futures prices implies that the individuals anticipate a mean reversion

in spot prices so that an element of predictability exists in the equilibrium prices.

To illustrate more formally how this persistence in the convenience yield can be used

to infer mean reversion in the spot prices (Bessembinder et aL(1995)) one can define rt

the on*period bias in the period t futures price as a forecast of the future spot price

or in other words, the equilibrium risk premium earned by speculators for taking a long

posit ion:

Then, the relation between the expected spot price and the spot price at time t can

be d&ed as:

The equilibrium spot price is considered mean reverting when the percentage change

60

in the expected future spot price is less than the percentage change in the current spot

price. This also implies that if the market anticipates mean reversion the difference

between the current spot price and the expected future spot price should be persistent.

From the right hand side, one can infer that the persistence of spot priceexpected future

spot price adjustment should be accompanied by persistence in either the interest rate,

the risk premium, or in the convenience yield. Except for the risk premium6, the other

two sources of mean reversion are available. This implies that one can use the persistence

of the convenience yield and the interest rate to infer if the market is expecting the spot

price to mean revert and the corresponding source. In other words, one can study the

evidence of long memory in the basis and the interest-adjusted basis to infer if the market

is anticipating any mean reversion in spot prices, and if the potential mean reversion can

be attributed to the persistence in interest rate or the convenience yield.

Among other testable implication of this approach is the fact that part of the p r e

dictability in the price of storable commodities, such as agricultural products and petre

leum can be attributed to expected seasonal variations, so that the inclusion of seasonal

dummies in the estimation of adjusted basis should account for a si&cant portion of

their persistence. We expect interest rates to have very little power in explaining possible

mean reversion in the spot price in markets with si&cant fluctuations in the inventory,

such as petroleum.? Overall, in any commodity market where the inventory and therefore

the convenience yield is volatile, or equivalently, where the long run supply is substan-

tially more elastic than the short-run supply, the spot prices are expected to mean revert

and the extent of persistence in the convenience yield can be used as an indicator of this

'Risk Premium can also be a significant source of mean reversion in the asset prices. For furher discussion, see Fkench and Farna(1988).

?The no-arbitrage condition can be used to derive the corresponding cost-of-caq relations for the financial assets such as exchange rates and stock prices. A full discussion of this derivation can be found in Brenner and Kroner(1995). For these financial assets, the persistence in the basis(forward premium) can also be used to infer sources of mean reversion. For example, evidence of long memory in the foxward premium as found by BaiIlie and Bollerslev(l994) can be attributed to the interest rate differentials. In general, unlike storable commodities, for these financial assets one can expect a significant portion of the mean reversion in the spot price to be driven by the behavior of the interest rate.

mean reversion.

One important feature of equation (3.1) is the fact that the time to expiration of

the futures contract, T - t , is fixed, while the time of expiration, t , is changing. If the

frequency of the data is higher than the frequency of the futures contract, then (3.1) shows

that any regression of the In on the In St will have a variance which is converging to

zero as the time to expiration , T-t , converges to zero. This means that, theoretically, the

variance of residuals is changing over time. This also occurs when instead of dowing the

contract to expire, one rolls over the neaxest contract as the old contract expires. These

are two common features in the empirical attempt to test the unbiasednes hypothesis or

the market aciency theorem. Clearly, as Brenner and Kroner(l995) point out, in both

cases cointegration does not ackt unless one modifies the strict definition of cointegration

to permit time-varying variances in the cointegrating regression residuals. These issues

should also be accounted for in our empirical procedure.

Briefly, the no-arbitrage condition provides us with the tool to attribute the persis-

tence in the adjusted basis(convenience yield) to the mean reversion in equilibrium asset

prices. We distinguish between the short-term persistence and long-term persistence,

since issues such a s sampling periods may cause short-term dependency in the adjusted

basis. However, any long-term dependency should be theoretically attributed to the

cyclical nature of the underlying factors such as inventories, and should be interpreted

as evidence for mean reverting equilibrium asset prices. The extent of this long term

dependency can provide us with a measure of this mean reversion.

3.3 Econometrics and Estimation Procedure

To estimate the degree of persistence in the basis and the adjusted basis, we use a long

memory model and estimate the differencing parameter directly. A survey of long mem-

ory processes and the corresponding estimation procedures is provided by Baillie(1997).

Briefly, assuming the following process:

where d denotes the fractional differencing parameter. The process yr is said to be

integrated of order d, I ( d ) . For -0.5 < d < 0.5, the process is covariance stationary

while for 0.5 5 d < 1, the process exhibits mean reversion with an infinite variance. If

0 < d < 0.5, the process eventually exhibits a strong-positive dependency between distant

observations, whereas if 0 < d < -0.5, the process exhibits negative dependence between

distant observations. The kactional differencing operator (1 - L ) ~ can be expanded as a

binomial series:

00

The fractional differencing operator can be rewritten as C r(j - d ) D ' / r ( j + l ) r ( - d ) j=O

where r(.) denotes a gamma function. The process is said to exhibit long memory since its

aut ocorrelation function shows a hyperbolic decay unlike the standard A R MA processes

for which autocorrelations decay geometrically.

As indicated earlier, when the fkequency of the data is higher than that of the fu-

tures contract, theoretically a timevarying residual variance will render the commonly

used cointegration procedures unsuitable. In order to overcome this problem which is

widespread among the empirical papers, it is necessary to allow for the possibiliQ of time-

varying variances in the estimation of the Eract ional differencing parameter. Currently

the only a d a b l e procedure for this estimation is an approximate maximum likelihood

estimation as discwed by Chung and Baillie(1993) and used in Baillie, Chung, and

Tieslau(l996). This procedure fundamentally follows that of Bollersiev(1985,1987) for

the simple M C H - GARCH models and can be generalized to cover any proposed

GARCH parameterization. Formally, an ARFIMA(p, d, q ) - GARCH(P, Q) process in

its mast general case can be written as follows:

Here yt is the corresponding time series and XI t and Xzt are vectors of predetermined

variables. In this framework, the distribution of the residuals (D) can be normal or t

depending on the corresponding series, the variance of the residuals is timevarying and

can be affected by any proposed set of variables XZL The d in this model represents the

fractional differencing parameter with a zero value representing no long term persistence

while a unit value indicates unit root. Any intermediate value represents a long memory

process, with the persistence rate as an increasing function of the value of the differ-

encing parameter. The @(L) and 8 ( L ) i.e. the ARMA process controls for the short

term dynamics of the series. The importance of this characterization is the fact that

it incorporates long memory in the GARCH framework allowing for joint estimation

of the fkactional differencing parameter together with the parameters of the GARCH

process. This format is general enough to allow for testing some important implications

of the theory of storage. For example, by expanding this kamework to Exponential

GARCH(EGARCH), one can test for the implications of theory of storage concerning

volatility when the basis has different signs. The theory of storage predicts that the con-

venience yield will be more volatile when positive. We will return to this issue in some

more detail later.

As mentioned earlier, the differencing parameter will provide us with a measure of

persistence of the time series. For values of differencing parameter in the range of -0.5 <

d < 0.5 the process will be covariance stationary with finite variance, although it still

exhibits a higher rate of persistence than an invertible ARMA process. For values of d

in the range of 0.5 5 d < 1 the process will be covariance non-stationary and will have

infinite variance although it wil l still be mean-reverting. The impulse response function

for this process is also well defined and will be presented in the next section. This

impulse response can indicate how long a shock to the convenience yield may persist.

This is important, since it provides a method of measuring the mean reversion in the

prices.

The estimation of the above mentioned relations can be based on an approximate

likelihood estimation procedure. In fact depending on the residual distribution (D), the

Likelihood function is readily derived by BoUerslev(1986,1987). Assuming a normally

distributed residuals, the following will represent the log likelihood function:

in which:

and:

However, if the residuals have a t distribution, the log likelihood function will look

We:

Then, conditional sum of squares estimation(CSS) will maximize the corresponding

likelihood function. Clearly, this estimation is inferior to full maximum likelihood estirna-

tion with unconditionally normal residuals as proposed by Sowell(1992), since it depends

on initial values which may effectively change the results. However, in the absence of a

full MLE for an ARFIMA process with ARCH effects, the only estimation candidate

will be CSS. Besides, Monte Carlo simulations as provided by Chung and BaUe(l993)

show that this approximate likelihood procedure provides good results for models with

ARMA process as of order 0, 1, or 2 and for sample sizes greater than 100. Since our

sample size is large enough, we can reliably use the CSS procedure in the remainder of

the paper.

3.4 Data and Empirical Results

Daily closing futures prices for crude oil, heating oil, and unleaded gasoline traded on the

NYMEX are used, as provided by Commodity Systems Inc." One convenient feature

of this market is the fact that delivery can occur in any month of the year in a seven day

period following the first business day of each month. We use the closest delivery date

8The effect of contract switching or rollover has been discussed in the Literature. Ma, Mercer, and Walker(1992) report biases that can be generated from selection of different methods to rollover futures contracts. In the case of crude oil, last day of trading is the third business day prior to the 25th of the month preceeding the delivery month, whereas the last day of trading for beating oil and unleaded gasoline is the last business day in the month prior to delivery. Ng and Pirrong(l996) use all futures returns calculated using prices from that contract. They also include dummy variables equal to one on contract switch dates and to zero otherwise in order to correct for potential discontinuities, Schwarz and Szakmary(l994), on the other hand, in additon to the interest adjustment, choose rollover on the day preceding the last day of trading to allow for the considerable distance between the determination of trading and delivery. This is very significant in the crude oil market, since the considerable time interval between the expiration of a contract and the openning of the new one can cause significant bias in the results. We have eiiminated this problem by deleting the last tw*three days of each contract if the price swing is considerable.

to prwide the closest possible match between the spot price and the futures price. As

indicated earlier, a real spot price for crude oil may not exist. In fact, there is a three

week window for the delivery of the cash market which makes it almost indistinguishable

form the nearby futures price. Perhaps, that is the reason why this futures contract

is sometimes called spot contract. Therefore, the nearby futures price, as suggested by

French and Fama(1987) and applied by others, is used as a proxy for the spot price

and the subsequent futures price as the futures price. For the heating oil and unleaded

gasoline, the daily spot price exists for two cash markets, namely New York Harbor

for heating oil and Gulf of Mexico for unleaded gasoline. Ng and Pirrong(l996) argue

that these spot prices closely match the actual transactions prices. Furthermore, as they

argue the existence of an active refined market for which the traders use the quotes to

price arms-length transactions strongly suggests that they are reliable measures of spot

prices. The starting date for our data-set is the first day of futures trade on crude oil in

the New York market (September 1983 for crude oil and heating oil and February 1985

for unleaded gasoline). In general, 1568 observations for the crude oil market, 2059 for

unleaded gasoline, and 2422 for heating oil are available.

The convenience yield is constructed by both imposing and estimating the parameters

as pravided theoretically in relations (2) and (4). However, since the overall results do not

show any significant difference, those discussed hereafter refer to the model with imposed

parameters. Figures 1-3 present the evolution of the corresponding convenience yields.

Clearly, when the convenience yield is negative, it is persistently negative and when it is

positive it is persistently positive. As Fama and Fkench(1988) suggest this may represent

the fact that production does not adjust quickly enough to demand or supply shocks.

In fact, this persistence, with negative or positive signs, hints to a long memory process

as an alternative explanation for the time series properties of convenience yields. The

standard unit-root tests with non-stationarity as null hypothesis are supposedly in favor

of the series being I(O), since overall a.ll series exhibit strong mean reverting behavior. In

fact, this mean-reverting property is, one may claim, the prime reason behind the general

belief that convenience yields are I(0). However, as we will mention later, this may lead to

mis-specification in economic modelling and prevent some potentially profound analyses

provided by the behavior of convenience yield. To follow these issues more formally we

start our analysis with some standard unit-root test.

Table- 1 presents Phillips and Perron(1987) and Kwiatkowski, Phillips, Schmidt, and

Shin, KPSS: (1992) unit-root tests on the estimated convenience yield. The latter tests

the null hypothesis of stationarity while the former tests the null of non-stationarity.

Clearly, the results of the Phillips and Perron test are uniform in rejecting the null of

non-stationarity. This result has been used by others to infer stationarity of the con-

venience yield, aod to derive a vector auta-regressive representation of spot and futures

pricesg. However, the KPSS test casts some initial doubt on this conclusion. Indeed,

almost for aLI three convenience yields the null hypothesis of stationarity can be rejected

using KPSS test. Rejection of both stationarity and non-stationarity hints to I(d) as

a possible underlying representation. Considering the fact that the standard unit-root

tests with a null hypothesis of non-stationarity have low power against fractional integra-

tion, whereas KPSS test has a rather good power, the results suggest possibility of the

convenience yield to be fractionally integrated. For al l three series both stationarity and

non-stationarity can be rejected, providing us with reasons to believe that the stochastic

process underlying these series may not be I(1) or I(0) . This fact can be further analyzed

if we consider the corresponding correlograms a s depicted in Figures 3.43.6. Consistent

with our findings, the autocorrelation functions for all three cases seem to exhibits a slow

rate of decay, which is in contrast to the rapid rate of decay of an I (0 ) process and an I ( 1 )

process for which the corresponding autocorrelation shows much more persistence. The

interesting feature separating heating oil from the other two commodities is the cyclical

nature of its correlogram. This will be analyzed in some detail in the next section.

-- -

"or instance, see Ng and Pirrong(1994.1996) and Schwan and Szakmary(l994).

3.4.1 Estimating ARFIMA-GARCH Model

With the standard unit-root tests rejecting both stationarity and non-stationarity, the

ARFI MA representation provides us with a viable alternative to estimate the persistence

of convenience yields. However, as we indicated earlier, depending on the characteristics

of the time series, different GARCH represent ations can be used in estimating the model.

These G ARCH represent ations enable us to analyze the volatility in the convenience

yield. Persistence and volatility provide the basic characteristics of the time series under

consideration. Theoretically, since we are utilizing a daily data-set, we expect sigdcant

persistence and considerable volatility. However, we also expect the convenience yield to

be more volatile when positive, since presumably a negative supply shock or a positive

demand shock make spot prices more volatile.

The theory of storage predicts that when inventory is low the convenience yield rises

faster as inventory is used to meet increased demand. Since negative inventory is im-

possible but future consumption can be augmented by accumulating current stocks, the

convenience yield will be more volatile when positive to reflect the fact that spot prices

are more volatile when positive price shocks occur. Furthermore, the change in the ex-

pected spot price is smaller since the market anticipates a future demand and supply

adjustment. Consequently, when inventory is low, the spread between spot and futures

prices is more variable, resulting in a more volatile convenience yield. In other words,

the predictions of theory of storage that shocks produce more independent variation in

spot and forward prices when inventory is low implies that the interest-adjusted basis

is more volatile when it is negative, so that the convenience yield is more volatile when

positive. The standard approach taken by Fama and French(1988) through the inspec-

tion of standard deviation of daily changes in the interest-adjusted basis for industrial

metals confirms this assumption. Our approach allows a more comprehensive study of

the volatility of convenience yield.

Specifically, the possible asymmetric response nature of the convenience yield to pos

itive and negative shocks can be estimated using the Ekponential GARCH (EGARCH)

model as proposed by Nelson(l991) or the s c x d e d GJR model named after Glosten,

Jagannathan, and R~nkIe(1993)~O. In an EGARCH (I, 1) representation, the variance

will depend on both the size and the sign of lagged residuals:

Unlike the GARCH model where the nature of volatility is assumed to be symmetric,

the EGARCH model is asymmetric since the level of adjusted residuals ( E ~ - ~ / C T ~ - ~ ) is

included with a coefficient y. A negative value for y implies that, ceteris paribzrs, positive

shocks to the convenience yield generate less volatility than negative shocks, whereas a

positive value results in more volatile conveniences yield following positive shocks than

following negative shocks. The asymmetric nature of the EGARCH representation in-

sures that negative and positive shocks have a different impact on volatility.

On the other hand, the G J R representation allows for the GARCH coefficients to be

different when the shocks have different signs. In other word, for a G J R ( 1 , l ) model the

variance will be represented by:

This representation allows a quadratic response to shocks with different slope coef-

ficients for positive and negative shocks, but maintains the assertion that the minimum

volatility will result when there is no shock. A sigdicant non-zero value for 7 provides ev-

idence for an asymmetric effect on the volatility of convenience yield. A positive y results

lo For a detailed comparative discussion of models of volatility with assymmetric response see Engle and Ng(1993).

in the convenience yield to be more volatile when positive, and a negative 7 translates in

a more volatile convenience yield when negative. This insures that our approach provides

a reasonable alternative for analyzing persistence and volatiliw of the convenience yield

series.

We &st estimate the ARFI MA representation without considering the GARCH

representation. This will help us later to compare the results, and consider if signifi-

cant efEciency gains have been made by estimating the mean and the volatility process

simultaneously. The results appear in Table2. We used both standard normal and

t-distribution in our analysis, and the standard normal distribution appears to outper-

form the t-distribution. Although the undedying distribution might not be normal,

Q M L E is still asymptotically consistent, and the heteroskedasticity consistent- robust

standard errors as discussed by BoUerslev and Wooldridge(l992) correct for the possible

non-normality. These standard errors, although inacient for the case of an asymmetric

distribution, provide us with a measure of how s i d c a n t this departure &om normality

is. Furthermore, although nothing prevents us &om estimating the differencing parame-

ter when the underlying parameter is greater or equal to 0.5, since the likelihood function

in this case is not theoretically defined, we use first difference in the empirical estimation.

The estimated differencing parameters have a range of 0.70 to 0.80 indicating a high

degree of persistence in the convenience yields. The short-term dependency in the form of

the ARMA representation is found to be insigruficant in all three series. We have fitted

ARMA representations to the maximum of three lags and found little support for it. In

Table-2 we only report the moving-average representation which fits more favorably".

The estimated differencing parameters are s i d c a n t enough to assure existence of long-

memory behavior in the convenience yields. Before we analyze these findings let us

complete our discussion further, by estimating the ARF I MA - GARCH representation.

"The quasi maximum likelihood estimation for the differencing parameter for all three convenience yields appears to be in accordance with that of GPH estimation both qualititavely and quantitavely. In fact the GPH method gave resuits similar to those reported in Table-2 for almost ail range of coordinates, an indication of the fact that little short term dependency exists in the time series.

Table3 provides the empirical results for estimation of ARFIMA - GARCH(1,l)

model. The estimated mean process represented ma.i.nly by the differencing parameter

is comparable to the results of a simple ARFIMA process. The G M C H parameters

provide measurable evidence for the volatility of the convenience yield. The convenience

yield in the crude oil and heating oil markets seems to be much more volatile than that

of the unleaded gasoline. In fact, the sum of alpha and beta, providing a measure for

the degree of volatili@, is very close to unity in both these markets, implying a possible

IGARCH representation. However, as indicated earlier, the point of interest is the

implication of the theory of storage towards the direction of the volatility. The theory

of storage predicts that positive price shocks will lead to higher volatility in convenience

yields. Thus, asymmetric GA RC H representations should perform relatively better than

a simple GARCH. The two aforementioned asymmetric GARCH models (EGARCH

and GJR) were estimated and the results are reported in Table 3.4 and Table 3.5.

Starting from G J R model, and judging from the maximized likelihood function, the

GJR model seems to out-perform that of simple GARCH representation. In all three

cases the estimated 7 parameter is negative, providing evidence for the convenience yield

to be more volatile when shocks are positive. As stated earlier, this implies that negative

oil shocks (positive price shocks) will cause more volatility in the convenience yield.

Judging from significance levels and the maximized likelihood function, it seems that

this asymmetry is more transparent in heating oil, and that crude oil is less likely to

show this asymmetry. The EGARCH representation to a large extent confirms this

result. Looking at Table-5, one can infer that the crude oil convenience yield does not

show any sipficant asymmetry, whereas both unleaded gasoline and heating oil show

significant asymmetry in their response to negative and positive shocks. In fact, one can

observe that the EGARCH representation performs much better than GJR for both

heating oil and unleaded gasoline.

To determine if interest rate or seasonatity are the sources of mean reversion, we have

repeated our estimations on convenience yields constructed without accounting for the

interest rate and a seasonally adjusted convenience yield. The former did not provide

different results whereas the latter provided significantly lower persistence for the con-

venience yield of the heating oil. This indicates that in the petroleum market, as might

be expected, interest rate is not a significant factor in determining the level of storage.

Furthermore, it shows that the seasonality is an important source of mean reversion for

heating oil only. In fact, the estimated differencing parameter for the convenience yields

of crude oil and unleaded gasoline is negligibly different for the seasonally adjusted data,

whereas it is (0.72) for the convenience yield of the heating oil, sigdicantly lower than

(0 .go).

Judging from this evidence we offer the following conclusions concerning the volatility

of convenience yield in the energy market. First, the evidence overall suggests an asym-

metric nature for the volatility for which negative oil shocks will lead to a more volatile

convenience yield, confirming the predictions of the theory of storage. Second, the esti-

mate of the mean process does not change sigrdicantly over different sets of GARCH

parameters. In fact, the information matrix appears to be diagonal between the para-

meters in the conditional mean and those in the conditional variance. Third? the nature

of persistence in the convenience yield and therefore the mean reversion does not seem

to stem from the interest rate. In particular, the interest rate is not sigdicant at all,

whereas the seasonality accounts partially for the persistence in the convenience yield of

heating oil.

3.4.2 Impulse Responses

The analysis of persistence in a time series will inevitably invite the impulse response

analysis. The behavior of impulse responses can contribute to our understanding of na-

ture of the convenience yields, and subsequent price expectations. A mean reverting

convenience yield provides evidence for expected mean reversion in spot prices. Further-

more, the magnitude of persistence in the convenience yield also provides evidence on

the magnitude of any expected price reversion.

Chung(1994) has discussed the calculation of impulse responses and their asymptotic

standard errors for ARFI MA process in some detail. The basic idea is to use the first-

order generating functions with the help of theory of impulse responses in the general

vector autoregressive moving-average(VARMA) models as developed by blittnik and

Zadrozny(l993) to explicitly derive the impulse responses and their asymptotic standard

errors for an ARFIMA representation. They show that, as expected, the impulse re-

sponses and their corresponding standard errors decay at a very slow pace contrary to

those provided by an M M A process.

The impulse response for the ARFIMA(0, d, 1) can be stated as:

The corresponding r j coefficients represent the impulse responses and each rdects the

change in the process yt caused by a shock in previous 5 - j for which j = 1.2: . . . , n, .. .. A

simple recalculation using a power expansion of the fractional differencing portion shows

that these impulse responses truncated at n number of lags can be written as:

The impulse responses are non-linear tramdormations of the estimated parameters,

with a known impulse response generator function in the form of equation T . The stan-

dard procedure of transforming the variancecovariance matrix by the matrix of deriva-

tives of impulse responses with respect to the vector of original parameters can be used

to derive the asymptotic variance-covariance matrix of impulse responses.

Figures 7-9, plot the cumulative impulse responses for the corresponding convenience

yields derived fkom the ARFIMA estimation as presented in Table-3. One feature which

is in common in all three cases is the slow rate of decay of the impulse responses. Un-

like standard ARMA processes for which the impulse response die out very quickJy, in

ARFIMA representation the hyperbolic decay of impulse responses is a norm. Bail-

lie, Chung and Tieslau(1996) show how ARMA and ARIMA alternatives differ from

ARFIiLlA representation. In addition to the slow rate of decay, the AELFIILIA process

provides considerably tighter confidence bounds around cumulative impulse responses.

The ARMA process on the other hand gives a wider confidence interval which may even

include negative impulse responses.

This slow rate of decay demonstrates the very fact that a shock to the convenience

yield will have its impact even after a long period of time. For example, considering crude

oil, a shock to convenience yields will still retains around 30% of its impact even after

100 steps ahead. In fact, one may conclude that around 30% of shocks to convenience

yields will not disappear in foreseeable future. Considering the fact that each month

around fifteen observations for crude oil convenience yields are available, the conclusion

one can draw is that around 30% of shocks will still be present after seven months. As

we indicated earlier, the persistence of convenience yields provides a measure of market

expectations for future spot prices. Therefore, it appears that the market expects around

70% of shocks to the price to die out in the nsrt seven month. This finding coincides

with that of Bessembinder et al. (19%). Using elasticities of distant futures price changes

with respect to near price changes they conclude that the market expects around half of

shocks to crude oil to be transitory and to be mean-reverted in the next seven months.'*

The impulse responses of unleaded gasoline and heating oil match those of crude oil

'*The interest rate did not play any signiscant role in the persistence of the convenience yields. In fact, the estimated impulse responses are unaffected by the inclusion or evclusion of the interest rate. This raises questions for the Brenner and Kroaer(1995) results, at least for the commodity market.

rather closely. Perhaps, the mception may lie in the nature of heating convenience yield

which shows a quicker rate of decay when the seasonally adjusted convenience yield is

used. In fact, heating oil is the only case where the addition of seasonal dummies led

to rather significant changes in the estimated differencing parameter and subsequently

to the cumulative impulse response. This, as we hinted earlier, can be attributed to the

fact that the autocorrelation function in this case, unlike those of crude oil and unleaded

gasoline, shows a significant cyclical pattern so that any shock to the convenience yield

is expected to vanish within next 90 days, representing a cycle. This implies that market

expects cyclical heating oil prices but not crude oil or unleaded gasoline.

3.4.3 Spread, Volatility and the Direction of Causality

The theory of storage also provides some insight into the potential causality between

spread and volatility of the convenience yield. The EGARCH and GJR representations

allow us to test for any possible asymmetry in the evolution of the convenience yield.

However, one question which still remains unanswered is whet her the volatility of the con-

venience yield is the cause of higher (lower) convenience yield, or that the higher (lower)

convenience yield is causing increased volatility. The first quest ion, i.e. the asymmetrical

behavior of the convenience yield, has attracted considerable attention. Pindyck(l994)

suggests an increasing and convex relation between spreads, inventory, and anticipated

demand for heating oil. Fama and Fkench(1988) and Fkench(1986) explore the relation

between spreads and volatility. They argue that based on the theory of storage and the

pro-cyclical behavior of spot prices, positive demand shocks near a business cycle peak re-

duce inventories, raise convenience yields, and generate negative int erest-adjust ed bases.

This implies that the most negative interest-adjusted bases tend to occur near business-

cycle peaks. Ng and Pirrong(l996) reach similar results in a more general framework.

The second question which explores the direction of causality rather than the asym-

metric nature done, has remained rather intact, though one can still trace it in some

applied work. For example, &om Fama and Fkenh(1988), one can infer that the direc-

tion should run &om the mean process to the volatility. When the mean process, namely

the interest adjusted basis changes near a business-cycle peak, it becomes more volatile

to imply periods of adjustment of the prices to fundamental demand or supply shocks.

One significant exception is Ng and Pirrong(l996), who study this possible causality in

the fiamework of a bivariate GARCH representation as well as some univariate analysis.

They are more interested in the possible causaliw between the spot price and the ad-

justed spread, rather than the convenience yield and its volatility. They find that large

negative spreads are associated with large spot return variances, whereas large positive

spreads have a statistically significant but far smaller impact on spot return volatility.

Furthermore, they test for Keynes nonnal backwardation theory by regressing current

spread on the lagged spread as well as lagged spot and futures returns. Under the n e

arbitrage normal backwardation hypothesis, they argue, the co&cients on lagged shocks

should be negative as higher volatility, ceteris paribus, should depress the futures price

relative to the spot price. They find the coefficients on the lagged squared shocks to

be positive as well as negative, and not always significant, while the lagged spreads are

sigdicant for several lags.

Clearly, our approach provides a suitable kamework for further development of this

analysis. Theoretically, the causality should run fiom the mean process to the volatility

process. According to the theory of storage any variation in hndamentd supply and

demand is the main reason behind the volatility of spot prices. Following a demand or

supply shock, the adjusted spread-convenience yield will be in disequilibrium. Following

the discussion of the last section, we have strong reasons to believe that this disequilibrium

is persistent. If we regress the volatility measure on the lagged convenience yield, the

estimated coefficient should be sigdicant since this lagged value measures the expected

convenience yield when it is persistent. By the same token, if the volatility process

is asymmetric this causality should be asymmetric for the case of positive or negative

convenience yield.

Table-? provides results of the aforementioned causality tests. Four tests are con-

ducted: the &st tests the causality running from overall (positive and negative) conve

nience yield (mean process) to its volatility, the second tests the causality running &om

positive convenience yield to the volatility, the third from negative convenience yield to its

volatility, and the fourth tests the causality running horn volatility to the mean process.

Parameters, 6,6+, and 6- represent coeEcients of lagged mean, lagged positive mean,

and lagged negative mean (absolute values) in the mean equation. 6* is the coacient of

lagged variance in the rnean equation. Fkom Table-?, the results seem to conform with

the predictions of the theory of storage: the volatility process seems to be driven by the

mean process rather than the converse.

More specifically, except for crude oil in which the direction of causality seems to be

running from the variance process to the mean process, unleaded gasoline and heating

oil provide strong evidence against this direction for causality. In fact, for all three

commodities the null hypothesis of causality of the volatility process was strongly rejected

at five percent significance level. However, the null hypothesis of the mean process

causing volatility has strong support. Indeed, in all three cases, one car; aot reject the

causality of the rnean process for any reasonable signdicance level. However, the pattern

of this causality is asymmetric, in accordance with the fundamentals of the theory of

storage: a posit ive(negative) convenience y ield(adj usted spread) is more(1ess) likely to

cause volatility.

3.5 Conclusion

This paper uses an ARFIMA - GARCH framework to analyze the behavior of energy

convenience yields. Theoretically, it is argued that the persistence of daily convenience

yields provides market observations of the future expected spot price. A strongly per-

sistent but mean reverting convenience yield implies that the expected price does not

adjust quickly to the change in the current spot price, an indication of expectation of

mean reversion in the spot price. AR FI MA, a mean reverting but possibly very persis-

tent and non-stationary process, provides us with the required tool. Although, to ensure

generality of the theory of storage, convenience yield must be mean reverting, it might

well be non-stationary. Specifically, for a high-kequency data-set such as that used in

this paper, the issue of non-stationarity is very

Building on these ideas, the paper estimates persistence and volatility of convenience

yields for crude oil, unleaded gasoline, and heating oil in an ARFIMA - GARCH(1,I)

framework. We also consider two other representations for volatility, namely the EGARCH.

and the G J R representations to account for the possibie asymmetry of the convenience

yield as predicted by the theory of storage. The estimated differencing parameters are

0.77,0.72, and 0.79 for crude oil, unleaded gasoline, and heating oil respectively. This

implies non-stationarity but mean reversion for all three convenience yields. The daily

convenience yields for energy futures have infinite variance although they exhibit mean

reversion over long period. The impulse responses show a high degree of persistence and a

very slow rate of decay with about 20% - 30% of shocks still present after seven months.

Furthermore, it was found that the interest rate does not play any sipficant role in

the persistence of convenience yields whereas seasonality plays a significant role in the

persistence of the convenience yield of heating oil. In fact, heating oil is the only market

for which seasonal dummies can alter the est h a t ed differencing paraxnet er sigruficantly,

although in all three markets they are si&cant. The shape of the convenience yield

provides market expectation of future spot price. These results imply that the market

perceives about seventy percent of price shocks to crude oil, unleaded gasoline, and heat-

ing oil to be transitory, and to revert within next seven months. Furthermore: the market

expects some of the shocks to heating oil to be seasonal.

We have also examined the volatility process in our general fkarnework. It is found that

the heating oil and unleaded gasoline very well conform with the predictions of theory of

'"This has two important implications. First, stmdzrd unit root tests applied in the Literature can be misleading. Second, for an ARFIMA process the Granger representation of the error correction model is different from that of a simple ARMA process. These issues are normally neglected in the energy futures literature. Prime examples are Ng and Pirrong(l996) and Schwarz and Szakmary(l994).

storage and their convenience yield is more volatile for positive price shocks. While crude

oil also confirms this prediction, this type of asymmetric behavior for the volatility process

fails to be significant. Mhermore, consistent with the theory of storage, we found strong

evidence in favor of causality running from the convenience yield to the volatility process

rather than the converse. Interestingly, strong support for the asymmetric nature of this

causality was found. It is shown that positive convenience yields are more likely to lead

to further volatility.

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Table 3.1: Testing Order of Integration for Convenience Yields in the Petroleum Market Ho : I (1 ) Ha : I(0)

&a. Ztz V P V7

Crude Oil -7.65 -7.56 0.599 0.232 UnleadedGasoline -12.00 -12.24 0.498 0.322

Heating Oil -9.57 -9.65 0.341 0.229 ZL., and Z; rcprescnt Philips nnd Pcrron test statistics for a regression with constant ntld with constnnt and trcnd

rcupcctivcly. qP, ?T stand for KPSS t a t stntisticl for n regrcwioa with constant and with constant and trend respectively. All the t a t arc pcrfomed using Newy .and West(1987) adjustment lags. Thc number of lags is based on L12

= int[l2(~/100)?1 analyzed in Kwiatkoaski ct n1.(1992) for which T repracnts the ~azoplc sirc. L4 = inr[4(~/100)t] criterion wiw a190 wcd and provided simmihr qualitative results.

Table 3.2: &timation of ARFIMA Model Crude Oil Unleaded Gasoline Heating Oil

P -0.0016 -0.0009 -0.0026 (0.0064) (0.0049) (0 .0066) [0.0064] [0.0054] [0.0065]

91 0.0530 0.0915 0.0878 (0.10 14) (0.0726) (0.0865) [O. 04291 [O. 04041 [O. 03791

d 0.7935 0.7071 0.7917 (0.0842) (0.0454) (0 .0470) [o. 03641 [0.0315] (0.02941

Max. Logl. -1914.7 -3842.3 -3778.8 Skewness -0.02 0.92 0.25 Kurtosis 30.57 12.16 19.24 Q(10) 27.99 5.47 16.14 Q2 ( 10) 147.75 190.62 268.8

Thc following model is estimated.

The numbers in (.) rcprevcnt h c t c r o s k ~ ~ t i c i t y couiutent-robust standard errors ns diucusscd iu Bollcrulev nnd Wooldridge(l992) and the numberis in [.I give the standard errors CIIICUIL..~ from the inxctac of thc Hcssinn matrix

Table 3.3: Estimtion of ARFIMA - GARCH(1,l) Model Crude Oil Unleaded Gasoline Heating Oil

P -0.0053 -0.0025 -0.0066 (0.0035) (0.0087) (0.0063) [0.0026] [O-OOSl] [0 . 00451

9 1 -0.0722 0.0799 0.1336 (0.0580) (0.0583) (0.0590) [0.0415] [0.0505] [0.043 11

d 0.7533 0.7273 0.8012 (0.0580) (0.0379) (0.0399) [0 .0308] [0.0316] [0.0321]

X 0.0201 1.2768 0.0291 (0.0056) (0.3229) (0.041 5) [0.0029] [O. 11561 [0.0072]

a 0.6272 0.3883 0.1009 (0.1483) (0.1257) (0.0752) [0.069 11 [0.0603] [O. 0 1441

0 0.5422 0.1481 0.8907 (0.0709) (0.1325) (0.0959) [0.0318] [0.0618] [0.0152]

Max. Logl. -1206.5 -3655.5 -3442.4

Q(10)/Q2(10) 21.60/17.3 7.3712-35 4.50/3.36 Thc cstirnatcd modcl is;

Thc nnmbcrs in (.) rcprcsent hcteroskcdasticity cosistcnt-robust stamdard crrom as discaszld in Bollcrshv and Wooldridgc(l992) oud the numbcris in [.I givc thc standard errors culculated from thc invcrsc of thc Hmian matrix. Thc

Q oud Q2 arc tkc Lijung-E3au statistics on the standardized rcviduals and the squorc of stirndrrrdizcd roriduds rcupcct ivcly.

Table 3.4: Estimtion of ARFIMA - EGARCH(1,l) Model Crude Oil Unleaded Gasoline Heating Oil

P -0.0039 -0.0024 0.0017 (0.0037) [O. 00281

61 -0.0657 (0.0634) [o. 04301

d 0.7663 (0.0511) [0.0286]

A -0.0102 (0.0353) [0.0159]

Q! 0.5938 (0.1893) (0.06271

P 0.9326 (0. 0450) [om491

7' 0.0279 (0.0296)

(0.0092) [0.0059] 0.0536 (0.0530) [ 0.04631 0.7218 (0.0419) [0.0319] 0.4010 (0.1477) [0.0506] 0.4432 (0.0792) [o. 04241 0.5559 (0.1563) [0.0594] 0.1084 (0.0540)

(0.0053) [o. 00431 0.1382 (0.0655) [O. 04421 0.7948 (0.0468) [0.0328] 0.0373 (0.0176) [0.0053] 0.2550 (0.0902) [0.0228] 0.9672 (0.0252) [0 .OO6O] 0.0627 (0 .OZ88)

(0.01 951 [0.0277] [0.0107] Max. Logl. -1201.9 -3636.9 -3385.4

Q(l0)/Q2(l0) 14.97/15.85 6.5811.75 4.16/8.73 The cstimatcci model is;

Thc numbers in (.) rcprcscnt hetcroakdasticity cosktent-robust standard errors as diucuusmi in Bol lc~ lcv and Wooldridgc(l992) and thc uumbcris in [.] give the standard crrocs cnIculnted from the invcrvc of the Hessian matrix. Thc

Q and Q2 are the LijungBox statistics on the standardized rcviduals and the squarc of stirndurdizd rcriduuh rcupcct ively.

Table 3.5: Estimtion of ARFIMA - GJR(1,l) Model Crude Oil Unleaded Gasoline Heating Oil

P -0.0036 0.0009 -0.0027 (0 -0035) (0.0080) (0.0051) [0.0028] [0.0062] [0.0041]

0 1 -0.0669 0.0745 0.1169 (0.0569) (0.0716) (0.0602) [0.042 11 [O .0538] [0.0412]

d 0.7576 0.7247 0.7694 (0.0631) (0.0417) (0.0327) [O. 03081 [0.0332] [0.0029 11

X 0.0205 1.2773 0.0320 (0.0056) (0.3344) (0.0199) [0.0029] [O. 11711 [0.0053]

Q 0.7024 0.5539 0- 1599 (0.1662) (0.2322) (0.0199) [O. 08721 [O. l002] [0.0227]

P 0.5410 0.1525 0.8866 (0.0713) (0.1276) (0.0494) [0.0318] [0 .0603] [0.0125]

7 -0.1541 -0.3365 -0.1117 (0.1271) (0.2256) (0.0478) [O. 0806 ] [O. lOZO] [0.0204]

Mitx. Logl. -1204.6 -3648.6 -3418.9

Q(10)/Q2(10) 20.94/17.97 7.891225 3.88/5.61 Thc cstin~ated model b;

where D y = l if E C < O , DF = 0 othcrwkc

Thc numbers in (.) represent heteroskcdasticity couiutent-robust stntldlrrd errors as d i s c ~ ? ~ c d in Bollcrvlcv aad tVooldridgc(l992) uud the rru~uberb in [.] give tllc stnndnrd errors ccrlculutcct from the invcrsc of the Hcssinn ntntrix. Thc

Q and Q2 are the Lijung-Box statistics ou thc stirndiirdizcd rcsidrmlv and the squtrrc of stsndrrrdizd rcvidt~alzi respect ivcly.

Table 3.6: Result of Causality Test between the Mean and Volatility Process Crude Oil Unleaded Gasoline Heating Oil

6 0.0241 0.0806 0.0406 (0.0077) (0.0880) (0.021 1) [O .004 11 [0.0305] [0.0066]

6 = 0 62.07** 9 -22- 80.53** 6+ 0 -0479 0.1224 0.1597

(0.0169) (0.1065) (0.0582) [0.0089] [0.0275] [O. 02541

6+=0 68.58'" 14.03** 192.99**

Thc cstimiitcd model is;

6,6+, 6- rcprcscnt thc cocfiicicnts ou thc luggcrl menu process, lagged pasitivc trlcirrl proccss, and lagged ucgntivc proems in thc volatility cquartior~ rmpcctivclp. 61 rcprcscutv tLc coetficicut of thc volutility proc-s in thc mean cqtttrtion. Thc

uurnbcrs in (.) rcprcscntv hctcrwkcdusticity cosiutcnt-robust s t i ~ ~ ~ d a t d errors iw cii.rcuzrs;cd in Bollcrslcv and Wooldridgc(l992) and thc uumbcriv in [.I givc thc stiiudard crrors culculutcrl from tlic iuvcrae of thc Hessian matrix.

Figure % 1: Evolution of the Convenience Yield and the Conditional Het eroskedasticity, Crude Oil

Figure 3-2: Evolution of the Convenience Yield and the Conditional Heteroskedasticity, Unleaded Gassoline

Figure 3-3: Evolution of the Convenience Yield and the Conditional Het eroskedast icity. Heating Oil

I

w U M ~b(n n m nvr no? s ~ u t o ? na *tw

Figure 3-4: Correlogram for the Convenience Yield of Crude Oil

Figure 3-5: Correlogram for the Convenience Yield of Unleaded Gasoline

Figure 3-6: Correlogram for the Convenience Yield of Heating Oil

Figure 3-7: The Cumulative Impulse Response, ARFIMA(0,O. 753,l) , Crude Oil

Figure 3-8: The Cumulative Impluse Response, ARFIMA(O1O.727,1), Unleaded Gasoline

Figure 3-9: The Cumulative Impulse Response, ARFIMA(0,0.801,1), Heating Oil

Chapter 4

Long Memory and Conditional

Het eroskedast icity, A Monte Carlo

Investigation

4.1 Introduction

Long-memory processes have received considerable attention in recent years since they

were first introduced by Granger and Joyeux(l980) and Hosking(l981). A number of

procedures have been introduced to estimate the differencing parameter (d) in the frame-

work of an ARFI MA ( Autoregressive-Fractionally integrated-hIoving Average) process.

One can class^ these estimation procedures in two general categories: fkequency domain

estimations and time domain estimators. Exact maximum likelihood estimator (M L E)

as proposed by Sowell(1992) and quasi maximum likelihood estimator (QMLE) as sug-

gested by Baillie et aL(1994) are among the time domain estimators, while the GPH

estimator as proposed by Geweke and Porter-Hudak (1983) is among frequency domain

estimators.

The performance of these estimators has been studied extensively in the literature.

Cheung and Diebold(1994) studied the efficiency and bias of the exact MLE and the

fkquency-domain MLE when and when not the mean is known. Cheung(1994) studied

the performance of GPH and MRR under different realizations of short term depen-

dency, mean shift, and ARCH effects. One aspect which has received very limited effort

is the effect of different realizations of the variance process on the estimation of the differ-

encing parameter. There are two minor exceptions: Cheung(l994) who studied the effect

of ARCH on GPH and MRR, and Hauser et al. (1994) who investigate the effect of

GARCH(1,l) on GPH estimation. Hauser et aL(1994) argue that long run dependency

and heteroskedasticity can be severely misleading if one looks exclusively at the other.

However, their study is confined to GARCH models and the estimation procedure is

restricted to GPH. Cheung(1994) study, on the other hand, finds no significant distor-

tionary effect &om the ARCH factor on the estimated differencing parameter. These

studies shed some light on how mis-specification of the variance structure may affect the

estimation of the differencing parameter.

This paper extends this analysis to include different characterizations of the variance

process in addition to comparing the performance of different estimation procedures. Of

specific importance is the inclusion of an asymmetric response variance structure such

as that implied in EGARCH model. We extensively analyze the effect of asymmetry in

the variance structure on the estimation of the fractional differencing parameter. In fact,

for the empirical series studied in applied economics and finance, a wide range for the

volatility and asymmetry in the variance response is found. For instance, Cheung and

Ng(1992) found a very wide range of asymmetry and volatility in their study which in-

cludes 300 US stock returns. These empirical studies specifically call for broader analysis

of the ARCH effect on the estimation of the differencing parameter than those performed

by Cheung(1994) and Hauser et al. (1994) and justifies our empirical design.

This paper finds: First, different parametrization for the conditional heteroskedast ic-

ity will strongly affect the estimation of the differencing parameter. Most importantly, we

found strong bias for all estimators when the conditional heteroskedasticity is asymmet-

ric. The bias tends to increase when the variance response is more asymmetric. Second,

It is also found that the relative efficiency of MLE with respect to Q M L E disappear with

the introduction of heteroskedastic processes. Third, it is found that the robust standard

errors for the Q M L E procedure tend to over-estimate(under-estimate) the true standard

errors when the degree of asymmetry is high(1ow). This is more transparent when the

sample size is smaller.

The paper consists of three sections. The second section provides the analytical

hmework by introducing the estimation procedures. The third section provides the

simulation results. The last section concludes.

4.2 Analytical Framework

A pure ARFIMA process is characterized by:

for t = 1,2, . . .T where L is the lag operator and d is the differencing parameter which

satisfies, -$ < d < 1, so that the process is stationary and invertible. p represents

the finite mean. In this representation we allow for the variance to be time varying.

The characteristics of this timevarying variance can be represented in two most general

formulations. The first, a GARCH@, q) representation can be written as follows:

This model can be rearranged and can be represented in the following ARMA process:

Here $(L) = a(L) + P(L) . Thus, for a GARCH(1,l) process a necessary condition

for the variance to be positive is a1 +& $ 1. A al + PI = 1 will yield IGARCH(1,l)

model. This ARMA representation of the GARCH can also be extended to include the

newly introduced FIGARCH.

The second representation is the exponential conditional het eroskedasticity, EG ARC H,

as introduced by Nelson(l991):

Where

Here, rt = ~p;', Et-1 (2,) = 0, and Et-, (2:) = 1. In this model 7 captures the

magnitude effect as in spirit of GARCH models whereas 6 represents the sign effect

on the conditional heteroskedasticity. In other words, assuming for the moment that

7 = 0, and 0 # 0, a positive(negative) 0 implies that the innovation in the variance is

negative(p0sitive) when the process is positive(negative). The novelty of this model,

in addition to the possibility of asymmetric response to positive-negative shocks, is the

fact that unlike the GARCH model there is no inequality constraints whatsoever for the

variance to be positive. This will allow for a richer parameterization of the conditional

heteroskedast icity. Furthermore, in this model the AR representation can fully explain

the persistence of shocks to the variance. If ln (<) follows an AR(1) representation then

Ivl 1 < 1, where Ivl I is the AR(1) parameter, ensures strict stationarity and ergodicity of

In (a:). In a more general case, Nelson(l991) showed that 4 is strictly stationary and

ergodic if and only if all the roots of autoregression lies outside the unit circle. This

model can also be extended to include long memory.

T h e processes will enable us to study the effect of merent parametrization for the

variance process on the estimation of the differencing parameter. The parameter 0 in

the EGARCH process captures what is called asymmetric response of the variance to

positive and negative shocks. This parameter will play an important role in out Monte

Carlo simulation.

The estimation methods we consider in this simulation exercise are exact maximum

likelihood estimator (MLE) as suggested by Sowell(1992), quasi maximum likelihood

estimator (QMLE) as discussed by Chung and Baillie(1993), and the periodogram re-

gression (GPH) introduced by Geweke and Porter-Hudak(l983). The QMLE, which can

also be named conditional sum of squares estimator (CSS), can be regarded as being

a time domain alternative of the Fox and Taqqu(1986) bequency domain estimator of

the ARFIMA model. In what follows we briefly introduce each of the aforementioned

procedures.

The Gaussian likelihood function can be defined as,

where YT = (yl -/I, yz - p, ..., y~ - p)' . The MLE estimator for the differencing para-

meter d can be derived after replacing for the unknown mean p by consistent estimator

p. In other words:

MLE = argmax L(YT, d) d

is Sowell's exact maximum likelihood estimation. The construction of this likelihood

function relies on the (2' x T) Toeplitz covariance matrix C ( d ) which is a function of

the parameters, in this case d. For a pure ARFIMA process, Sowell(1992) derives the

zj element of this covariance matrix by:

here, h = li - jl and represents a gamma function. As long as the estimated mean

ii is consistent, Sowell's feasible MLE provides a consistent estimate of the differencing

parameter. This specifically holds when the sample size is large enough.

Assuming that the residuals are normally distributed, the second time domain esti-

mator, Q ML E, minimixes the following:

where

The QMLE, then, will be:

QMLE = argminS(d, p) (4.11) d,lr

Clearly, this estimation depends on the initial observations. Chung and Baillie(1993)

show that this effect is asymptotically negligible implying that QM L E is asymptotically

equivalent to MLE. More interestingly, this procedure, unlike the exact MLE, d o w s

for a direct estimation of the variance representation. It also allows for non-normal

density functions in the spirit of generalized density function (GED). In our Monte

Carlo simulation, we assume a constant variance to allow for the comparison. When the

assumption of normality is violated, the Hessian does not provide a consistent estimate

for the standard errors. Bollerslev and Woodridge(l992) foUowing White(1980) provide

a consistent estimator for the asymptotic standard errors. These asymptotic standard

errors can be computed using the diagonal elements of the matrix J- 'KJ- ' , where J

and K are the Hessian and the outer product of the gradient evaluated at the optimum,

respectively.

The frequency domain estimation of Geweke and Porter-Hudak (GPH) uses the con-

cept of periodogram at frequencies close to zero to estimate the differencing parameter.

Ln this procedure an ordinary least squares of the periodogram on fkequencies provides

the estimation:

where,

In this formulation I, is the periodogram of yt, XI, ..., X, are the spectral ordinates,

and f is the spectral density of E. When E~ is a white nose, vj - N(O,7?/6) and the

periodogram regression will provide a consistent estimator for the differencing parameter

d.

4.3 Simulation Results

The white-noise series as in (1) are generated by the IMSL subroutine D RNNOA, for

T = 100,200,500 in which T is the sample size.' The realization of the corresponding

DGP, is then generated using (3) or (5). All likelihood maximizations are performed

using Davidson-Fletcher-Powell algorithm as provided in library GQOPT.* The true

differencing parameter d is set to zero in a l l simulation^.^ The number of ordinates in

GPH estimation is set to the maximum, i.e. (T/2) - 1, where T is the sample size. This

is justified since the processes by design do not have ARMA contamination. The seed

value is set equal in all simulations. For each DGP, N = 1000 replications are performed

and the results are summarized in the corresponding tables.

The first three tables represent the simulation results for the G ARCH representat ion

corresponding to the G P H, MLE, and QIIlLE estimation procedures, respectively. The

parameters for the GARCH representation yield a wide spectra for volatility which

includes low, medium, and high volatility series. In summary, fifteen models for the

GARCH representation are generated for each estimation procedure. Furthermore, for

the EGARCH representation, in addition to the and P, y and 0 parameters are set to

different values to yield low, medium, and high asymmetric variance response. In total,

30 models are estimated for each estimation procedure. In the following two subsections

'The number of observations is chosen in line with other empirical studies and considering the fact that the MLE and QMLE estimators for larger sample sizes are very time consuming. Furthermore. we acknowlege the need for lurther research to address the power of the estimators.

I M S L and GQOPT are Fortran 77 libraries. 3The simulation is also repeated for daerent values of the merencing parameter. The results remain

comparable. These simulation results are suppressed to censerve space.

we summarize the result concerning the corresponding bias aod standard errors.

In line with other researches which study the relation between different estimators of

the differencing parameter such as that in Hauser et aL(1994) we expect the frequency

domain estimator to have a higher standard error and a lower bias. In addition, we expect

the fiequency domain estimator to be less efficient than the time domain estimators. As

far as, the comparison between the time domain estimators, it is very hard to reach any

inclusive conclusion. Cheung and Diebold(1994) found 911.1 L E to be less efficient relative

to MLE particularly for small sample sizes. For a sample size of T = 100 their findings

demonstrates Ad LE to be approximately 10 percent more ac i en t than QMLE. They

also found QMLE to be less biased than MLE. We are not sure if these results hold

if the underlying process is either GARCH or EGAIICH. In the subsequent sections

these questions will be addressed in some detail.

4.3.1 GARCH Model

The following conclusions can be reached4:

1) For aIl parameter sets the bias is smallest for GPH estimation. The bias for QMLE

is found to be slightly smaller than that of MLE. Within each estimation method, the

bias increases as cr parameter increases relative to 0. Considering the fact that keeping

(a + P ) constant a higher a parameter impLies a higher uncondtional kurtosis(Bollerslev,

1986), one can infer that higher unconditional kurtosis leads to higher bias in all three

est imat ion procedures.

2) The standard error, on the other hand, is highest for the GPH. For all three

procedures, the standard error increases for a larger a parameter. Furthermore, the

standard error for hdLE procedure is larger than the standard error for QMLE for a

small sample size such as T = 100. However for a larger sample size T = 500 the two

estimations provide ident ical standard errors.

'The IGARCH model was also considered, however the results were not materially different born the GARCH model therefore they are suppressed to conserve space.

The efliciency of the three estimators, namely, GPH, MLE, and QMLE in terms of

the relative MSE demonstrates that in general, as expected, GPH is the least efficient.

However, for a small sample size, T = 100, MLE seems to be more efficient whereas for

higher sample sizes, T = 500, these two estimators seem to converge with Q MLE being

marginally more efficient. The small difference between QMLE and MLE for T = 200

and T = 500 can be attributed to the Monte Carlo standard error. This implies that

overall when the underlying volatility is symmetric, the performance of G P H, QII1 L E,

and M L E in general and in terms of efficiency will be identical to the general finding

that MLE is the most a c i e n t whereas G PH is the least efficient. However, the relative

efficiency of MLE to QMLE is much smaller than those found in Diebold and Che-

ung(1994). Furthermore, we also found that the relative efficiency of GPH increases as

the degree of volatility increases or as a increases keeping a + ,O constant. This resdt

demonstrates that G PH is relatively more robust to GARCH contaminations.

3) For Q MLE, the robust standard errors consistently under-estimate the true stan-

dard errors. The degree of under-estimation increases as the unconditional kurtosis in-

creases. In other words, keeping a + ,l3 const ant, the degree of under-estimation of robust

standard errors increases as a increases. In our example, the this under-estimation

reaches its maximum when a = 0.9 and fl =O.

4.3.2 EGARCH Model

The EGARCH representation enables us to look at the effect of the asymmetry in the

variance response on the estimators of the differencing parameter in addition to the degree

of volatility. The bcus of this section will be primarily on this effect. Since no prior study

has addressed this issue, it is not possible to compare our results with others. However,

in general the combination of asymmetry and volatility provides a rich framework for the

analysis of the effect of the variance representation. This feature could not be studied in

the GARCH framework. Before we analyze the results in some more depth, we present

the following general conclusions:

1) For the EGARCH model, the GPH procedure provides the smallest bias followed

by QMLE procedure. As before, the MLE procedure has the largest bias. F'urthermore,

the bias increases with the degree of asymmetry in the variance response as represented

by 0. The magnitude degree as represented by 7 parameter also results in an increase

in the bias, however to a much lower extent. The degree of volatility of the model as

represented by ,8 parameter also has a si&ca.nt effect on the bias. The larger the degree

of volatility, the higher the value of bias. The relative bias of the M L E procedure to the

Q MLE bias increases as the degree of asymmetry, 101 increases. This implies that the

MLE procedure is relatively more sensitive to asymmetry in the variance response.

2) The standard error, on the other hand, is highest for the GPH. The MLE and the

QMLE procedures yield virtually the same standard errors. Overall, higher degrees of

volatility will lead to higher standard errors. Furthermore. a higher degree of asymmetry

in the variance response tends to increase the standard error.

3) The robust standard error for QMLE procedure under-estimates the true standard

error when the degrees of volatility as represented by P is low. This under-estimation in-

creases as the degree of asymmetry of the variance response increases. For higher degrees

of volatility, the robust standard errors under-estimate the true standard errors when the

degree of asymmetry is low, otherwise the robust standard error over-estimate. The de-

gree of under-estimation increases as the magnitude effect and the degree of asymmetric

variance response increases. Furthermore, the degree of over-estimation is highest when

the degree of asymmetry in the variance response is highest and the magnitude effect is

lowest.

We have used a rather wide range for 8, the parameter of asymmetric variance re-

sponse. In empirical applications 0 has a wide range. Engle and Ng(1994) found 0 to be

-0.145 for the Japanese stock index. However, 0 can range widely. Cheung and Ng(1992),

as indicated earlier, in a study for 300 US firms found B to vary widely. These results

justify the range of our simulation parameters. We have also conducted the same sim-

ulations with positive values for 8, and found no qualitative difference. To present the

aforementioned r d t s more carefully Figures 1-4 are provided to illustrate the relation

between the degree of asymmetry in the variance response and the efficiency of each

procedure.

Using Figures 1-2, one can compare relative &ciency of the three methods of esti-

mation. The figures depicts MSE for each method versus the absolute value of 8 , the

parameter of asymmetry. Clearly, for a sample size of T = 100 as in Figure 1, G PH is the

least &cient and MLE is the most efficient. In general, MLE is marginally more &-

cient than Q MLE. This is in contrast to the case found in the Literature such as Cheung

and Diebold(1994) in which MLE found to be relatively much more efficient. It is also

conceivable that the relative &ciency of GPH increases as the degrees of asymmetry

increases. We also have depicted the robust standard errors which for lower degrees of

asymmetry under-estimates the true M S E whereas for higher degrees of asymmetry it

over-estimates the true MSE by a wide margin. Figure 2 illustrates the same results for

T = 500. In this case, MLE and QMLE provide identical results. However the rela-

tive efficiency gain of GP H disappear. The robust standard errors, in this case, clearly

over-est imat e the true MS E for higher degrees of asymrnet ry.

Figures 3-4 illustrate the same question for EGARCH model when the volatility is

low. The major difference between high volatility model and low volatility model lies

in the fact that for the latter the robust srandard errors always under-estimate the true

standard errors. This is very transparent when the sample size is small T = 100. For

larger sample sizes T = 500, the robust standard errors under-estimate the true standard

error when the degree of asymmetry is high, otherwise for lower degrees of asymmetry

the difference is negligible.

4.4 Conclusion

The paper investigates performance of the estimation methods for ARFIMA models

when the underlying processes are not white noise or specificdy when they are con-

ditiondy heteroskedast ic. The differencing parameter for processes with realizations

of GARCH(1,l) and EGARCH(1,l) is estimated using GPH, MLE, and QMLE

procedures and the corresponding bias and the efficiency are analyzed. F'urthermore,

performance of the robust standard errors for QMLE is also discussed.

Overall the bias seems to be increasing with the increase in unconditional kurtosis

and degreg of asymmetry. The GPH has the smallest bias and QMLE has a sightly

lower bias than MLE. The standard error increases with the increase in the degree of

asymmetry. Although the difference between Q MLE and MLE procedures is marginal,

both outperform GPH procedure. In general, the GPH : a s expected, is the least

efficient and the M L E and the Q MLE are marginally Werent. This finding contradicts

the previous findings that MLE is more efficient than Q MLE.

It is also found that the robust standard errors tend to under-estimates when the

degree of volatility is low. For higher degrees of volatility, however, the robust standard

error under-estimate when the degree of asymmetry is low otherwise it over-estimate.

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Table 4.1: Distribution of ARFIMA(O,d,O) When Underlying Processes are GARCH(1,l) , GPH Estimation

C A R C W ( a . B ) Mean S . D . Mia M a r 1% 5% 95% 99% a B

T = 100 0.1 0-0 -0.005 0.129 -0.502 0.459 -0.313 -0.219 0.195 0.284 0-3 0.0 -0.003 0.138 -0.436 0.466 -0.355 -0.240 0.223 0.300 0.5 0.0 -0.003 0.153 -0.516 0.612 -0.372 -0.262 0.246 0.336 0.7 0.0 -0.006 0.173 -0.516 0.720 -0.304 -0.293 0.298 0.416 0.9 0.0 -0.012 0.199 -0.683 0.729 -0.449 -0.320 0.318 0.19n 0.2 0.7 -0.006 0.139 -0.44s 0.399 -0.346 -0.252 0.215 0.303 0-3 0.6 -0.006 0.113 -0.517 0.4SJ -0.390 -0.256 0.22'1 0.350 0.4 0.5 -0.008 0.159 -0.563 0.515 -0.307 -0.281 0.247 0.383 0.2 0.75 -0.005 0.140 -0.173 0.407 -0.359 -0.250 0.217 0.327 0.3 0.65 -0.006 0.151 -0.654 0.497 -0.386 -0.265 0.237 0.349 0.4 0.55 -0.007 0.160 -0.562 0.574 -0.103 -0.2s; 0.2.m 0.365 0.2 0.79 -0.002 0.139 -0.547 O..lL9 -0.357 -0.241 0.222 0.312 0.3 0.69 -0.003 0.151 -0.610 0.526 -0.386 -0.166 0.239 0.368 0.4 0.59 -0.005 0.161 -0.544 0.614 -0.394 -0.'182 0.247 0.398 0.3 0.70 -0.002 0.149 -0.556 0.526 -0.381 -0.256 0.240 0.367

T = 200 0.1 0.0 -0.001 O.081 -0.305 0.201 -0.211 4.137 0.13'1 0.167 0.3 0.0 -0.003 0.090 -0.382 0.248 -0.230 -0.160 0.145 0.198 0.5 0-0 -0.004 0.103 -0.365 0.373 -0.254 -0.ld3 0.161 0.246 0.7 0.0 -0.OOc) 0.126 -0.155 0.536 -0.307 -0.210 0.198 0.319 0.9 0.0 -0.012 0.154 -0.531 0.676 -0.400 4.247 0.249 0.429 0.2 0.7 -0.002 0.090 -0.336 0.260 -0.227 -0.160 0.113 0.199 0.3 0.G -0.003 0.099 -0.321 0.367 -0.250 -0.175 0.155 0.229 fl. t 0.5 -0.001 0.108 -0.335 0.439 -0.267 -0.lr13 0.170 0.255 0.2 0.75 -0.002 0.092 -0.31s 0.30s -0.230 -0.162 0.I4.l 0.211 0.3 0.65 +0.003 0.103 -0.313 0.439 -0.241 -0.IU1 0.160 0.247 ~ . . t 0.55 -0.005 0.112 -0.3.1; o..ts8 - 0 . 2 ~ -IJL'JO O.lfn o.%n 0.2 0.79 -0.002 0.095 -0.366 0.320 -0.234 -0.L66 0.LLf O.2Uf 0.3 0.69 -0.003 0.106 -0.3.i: U..L;O - u . ' ~ ~ a -0.1.4.1 0 LGG 0.240 0-1 0.59 -0.005 0.115 -0.364 0.518 -0.291 -0.194 0. la9 0.268

0.3 0.70 0.000 O.Or).l -0.271 0.423 -0.200 -0.137 0.132 0.237 Note: Thc processes satis& the folowing:

( 1 - LId(yt - p) = Et

E L - iid N(0 , u;) u: = w + aedl + PUT-^

Table 4.2: Distribution of AR.FMA(O,d,O) When Underlying Processes are G ARCH (1,1), MLE Estimatign

CARCH(a.8) .\lean S.D. )[in M a x 1% 5% 95% 93% Q f.3

Table 4.3: Distribution of ARFIPI/LA(O,dlO) When Underlying Processes are GARCH (1.1) QMLE Estimation

CARC?f(a .P) Lieen S.D. Robust S.E. M i a 1 % 1% 5 % 95% 99% Q

0-3 0.70 -0.011 0.074 0.070 -0.215 0.363 -0.160 -0.132 0.111 0.176 Note: The proccssev satisfy the folowing:

(1 - q d ( y t - P ) = st ~t - iid N(0 , uf )

U: = w + + P Q ~ , ~

Table 4.4: Distribution of ARFIMA(O,d,O) When Underlying Processes are EGARCH(1,1), GPH Estimation

ECARCH(a.8) Mean 5 . D . 51i11 Slrx 1% 95% 99% - e

0.8 -0.9 -0.090 0.235 -0.769 U.id9 -0.607 -0.445 0.342 0.550 T = 2 W

0 . 4 -0.1 -0.003 U.Ur); -0.329 U.262 -0.221) -0.162 11.131 0.175 U. .I -0.3 -0.005 U.093 -0.336 U.314 -0.2.1L -0. IG6 0.141 0.201 U.3 -0.5 -0.007 0.103 -0.350 0.336 - 2 -0. Id2 0.162 0.227 0.4 -0-7 -0.011 U . l i d -0.101 0.423 -0.270 -0.213 0. ld7 0.280

Model 1 0.4 -0.9 -0.016 0.134 -0.454 0.490 -0.319 -0.231 0.206 0.323 0.b -0.1 -0.004 0.099 -0.343 0.318 -0.252 -0.171 0 . 1 7 11.?l4 0. d -0.3 -0.007 0.107 -0.107 0.354 O f -0.ld6 0.161 0.256 0.d -0.5 -0.011 0.121 -a..13t) O..W -0.279 -0.219 0.186 0.269 0.8 -0.7 -0.016 u .nn -0.31.1 0.325 -0.314 -0.233 0.215 0 mi 0 . d -0.9 -0.013 0.152 -0.603 0.579 -0.335 -0.250 U.212 0.395 U.4 -0.1 -0.005 0-096 4.317 0.316 -0.250 -0.171 0.149 0.225 0.4 -0.3 -0.021 0.112 -0.196 0.338 -0.295 -0.203 0.176 0.213 0. t 4 . 5 -0.046 0.135 -0.706 0.411 -0.360 -0.247 0.175 11 315 0.4 -0.7 -0.075 0.150 -0.519 0.470 -0.412 -0.230 O.ld5 0.341

hIudcl 2 0.1 -0.9 -0.091 0. lG0 -0.607 0.506 -0.440 -0.346 0.184 0.37-1 0.6 -0.1 -0.010 0.131 -0.515 0.571 -0.320 -0.217 U.208 0.314 0.3 -0.3 -0.023 U . -0.769 0.641 -0.370 -0 .22 0.211 0.345 O.d - 4 1 .? -0.044 0.156 -0.524 0.642 I . - 0 . 9 2 2 0.382 0.3 -0.7 -0.064 0.171 -0.606 0.631 -0:131 -0.336 0.21 I 0 . 129 O.d 4 . 9 -0.Od3 0.163 -0.A70 0.591 -U.S2G -0.36a 0.217 0.IfX U:I -U 1 -U.OU.t 0.093 -0.LU5 0.259 -d.?Sd - 1 f i 0112 U.207 0. I -0 3 - 0 . 0 0.106 4.3S4 0.37 1 - 0 . 2 -0.187 0.147 0.213 0. ,l 4 5 0 . 0 . 3 2 -0.669 0.49 -U.S23 -0.231 rl.177 11.296 0.4 -0-7 -0.~3U 0.158 -0.560 0.596 -0.,(08 -0.2r19 0213 0.254

UOJCI 3 0.4 -0.9 -0.073 0.160 -0.605 0.668 -0. I52 -0.340 0.241 0.43d O.d -0 1 -0.007 0.121 -0.44d 0.424 -0.299 -0.206 0.190 0.304 0.6 -0.3 -0.OLd 0.136 -0.720 0.572 4.363 -0.239 0.210 0 3-19 0.8 -0.5 -0.030 0.161 -0.714 0.677 -0.433 -U.290 0.220 0.100 0.8 -0.7 -0.064 0.182 -0.740 0.750 -0.497 -0.328 0.249 04-56 0.8 -0.9 -0.073 0,102 -0.703 0.765 -0.602 -0.371 0.266 0.514

E C A R C H ( a . 0 ) Mean S.D. Min Max 1% 5% 95% 99% 7 9

T = 500 0.4 -0.1 -0.t)OO 0.053 -0.199 0.160 -0.tW -0.092 0.064 0.i19 0.4 -0.3 -0.002 0.056 -0.179 0.174 -0.127 -0.097 0.086 0.127 0.4 -0.5 -0.004 0.062 -0.205 0.199 -0.143 -0.104 0.100 0.150 0.4 -0.7 -0.005 0.072 -0.219 0.233 -0.160 -0.116 0.116 0.178

LCodcl 1 0.4 -0.9 -0.007 0.083 -0.141 0.306 -0.180 -0.135 0.133 0.206 0.8 -0.1 -0.001 0.061 -0.230 0.200 -0.134 -0.101 0.097 0.136 0.8 -0.3 -0.003 0.065 -0.204 0.228 -0.111 -0.103 0.LU5 0.175 0.8 -0.5 -0.005 0.074 -0.226 0.276 -0.165 -0.121 0. l l b 0. 192 0.d -0.7 -0,007 O.Od6 -0.291 0.360 -0.165 -0.136 0.LIO 0.222 0.8 -0.9 -0.UO'J 0.100 -0.337 0.413 -0.215 -0.159 0.163 0.272 U.4 -0.1 -0.UW 0.065 -0.213 0.27 1 -0.155 -0.107 0.099 0.165 0. .l -0.3 -0.013 0.011 -0.335 0.369 -0.Ld4 -0.13d 0.121 0.273 0.4 -0.5 -0.035 0.107 -0.172 0.419 -0.256 -0.189 U.152 0.273 0.4 -0.7 -0.063 0.130 4.506 0.485 -0.343 -0.251 0.170 0.303

Model 2 0.4 -0.3 -0.089 0.148 -0.70L 0.560 -U:106 -0.304 1 0.347 i1.d -U 1 -0.005 0.102 -0.425 0.446 -0.239 -0.162 0.160 0.256 0.8 -0.3 -0.017 0.115 -0.616 0.515 -0.266 -0.199 0.166 0.276 0.6 -0.5 -0.037 0.133 -0.619 0.572 -6.310 -0.239 0.169 0.317 0.A -0.7 -0.U60 0.152 4.691 0.651 -0.408 -0.294 0 . 1 0.350 U. J -0.9 -0.060 0.166 -0.737 0.731 -0.504 -0.323 0.209 0.395 0.4 -0. I -0.001 0.059 -0.207 0.240 -0.127 -0.100 0 . W l 0.135 0.4 -0.3 -0.007 0.071 9 0.337 -0.159 -0.116 0.115 0.178 0.4 -0.5 -0.017 0.046 -0.169 0.424 -0.215 -0.160 0.119 0.257 0.4 4 . 7 -0.034 0.125 -0.483 0.566 -0.303 -0.215 0.Ld.l 0.352

MoJcl3 0.4 -0.9 -0.063 0.151 -0.485 0.680 -0.376 -0.267 0.223 0.394 0.3 -0.1 -0.004 0.062 -0.-134 0.349 -0.190 -0.129 0.135 0.200 0.d -0.3 -0.011 0.101 -0.628 0.495 -0.222 -0.lGU U.l.5i 0.270 0.S -0.5 -U.025 0.126 -0.455 0.672 -0.296 -0.210 0.167 0.34* 0.8 -0.7 4J.0.11 0.15.1 -0.596 U.bl7 -0.373 -0.255 0.220 0.132 0. .% -0 I) -U.OCU U.Lfri -0.674 0.922 -0 154 -0 318 0.255 J..iUfi

Xotc: Thc proccssm satisfy the folowing: ( 1 - LId(yt - P) = Et

_ct - iid N(o,Q:) ln (u;) = J + (1 - O(L))-' (1 + a(L1) g(a-11

d z t ) = Bzt + 7 [Izrl - E (I=tl)l Mode1 1 refers to the cusc where a = O and /3 = 0.5 hIodcl 2 refers to the case where a = 0 and P = 0.95

hlodcl 3 rcfers to the cnsc where a = 0.15 .and P = 0.85

Table Distribution when Underlying Processes EGARCH(1,1), PILE Estimation

ECARCH(a.@) Mcrn S.U. M i n Mmx 1% 5% 95% 99% - A

- -

0 .4 0.4 0.4

Madel 2 0 . 1 O.J 0.d 0.8 0.8 0.n 0:l 0.4 0.4 0.4

Model 3 0.1 0.n 0.d 0. d 0.d

are

0.4 -0.3 -0.038 0.093 -0.404 0.450 -0.262 -0.262 0.101 0.186 0 .4 -0.5 -0.057 0.11a -0.630 0.450 -0.319 -0.240 0.139 U.225 0 .4 -0.7 -0.081 0.139 -0.506 O:I?5 -0.372 -0.293 0.146 0.309

model 3 0.4 -0.9 -0.102 0.161 -0.560 0.486 -0.418 -0.329 0.173 0.396 0.8 -0. I -0.034 0.104 -0.464 0.351 -0.268 -0.209 0.135 0.222 0.8 4 . 3 -0.044 0.122 -0.657 0.469 -0.326 -0.232 0.155 0.292 0.8 -0.5 -0.063 0.140 -0.596 0.586 -0.380 -0.192 0.165 0.334 0.8 -0.7 -0.083 0.159 -0.590 OA89 -0:l42 -0.321 0.166 0.396 0.8 -0.9 -0.101 0.176 -0.672 0.489 -0.504 -0.355 0.201 0.450

E C A R C H ( o . 8 ) Mean S.D. Uin A 1% JX 95% 99% 7 9

T = 500 0.1 -0.1 -0.010 0.044 -0.180 0.115 -0.118 -0.U65 0.059 0.093 0.4 -0.3 -0.011 0.047 -0.171 0.135 -0.124 -0.093 0.066 0.106 0.1 -0.5 4.013 0.055 -0.173 0.166 -0.133 -0.101 0.077 0.13.1 0.4 -0.7 -0.016 0.065 -0.166 0.204 -0.158 -0.116 0.093 0.159

Model 1 0 4 -0.9 -0.018 0.076 -0.210 0.251 -0.t78 -0.135 0.115 0.ial 0.8 -0.1 *0.011 0.053 -0.187 0.157 -0.134 -0.101 0.070 0.11.1 0.8 -0.3 -0.012 0.057 -0.173 U.1911 -0.139 -0.106 0.078 0.136 0.b -0.5 -0.014 0.067 -0.191 0.238 -0.153 -0.119 0.098 0.166 0.8 -0.7 -0.017 0.U78 4.222 0.2b6 -0.LLIO -0.137 0.1 18 IJ.199 0.B -0.9 -0.020 0 090 -0.261 0.359 -0.262 -0.151 0.135 0.130 0.4 -0. L -0.013 0.056 -U.199 0.210 -0.141 -0.lU.l U.077 0.130 0:l -0.3 -0.026 0.012 -0.292 0.302 -0.180 -0.136 0.09rl 0.179 0:l -0 Tr -0.019 0.097 -0.419 0.159 -0.256 -0.185 0.110 0.235 0. I -0.7 -0.077 0.120 4.456 U:160 -0.307 -0.250 0.128 U.ZG5

0 1 2 0.4 -0.9 -0.103 0.136 -0.640 0.,196 -0.39'1 -0.291 0.134 0.323 0.d -0.1 -0018 0.092 -0.391 0.375 -0.224 -0.163 0.L.lt 0.232 0.d -0.3 -0.031 0.105 -0.549 0.430 *0.250 -0. I&& U. 148 0.2r)3 0.8 -0.5 -0.051 0.124 -0.545 0.49'1 -0.304 -0.235 0.153 0.296 0.8 -0.7 -0.073 0.140 -0.552 0.496 -c].362 -0.2r)S U. 163 0.336

U . S -0 3 -0.073 U.1SS I 0. t92 -0:lUfi -0 292 0.199 0.157 Note: Thc pracrssucs satisfy tho folowing:

(1 - ~ ) ~ ( y t - P) = €1 Er - iid N(o,c:)

In (c:) = u + (1 - @(L))-I (1 + a ( L ) ) g ( a - 1 )

9 (z t 1 = 02, + Y [ I 4 - E ( lztl) l hIodcl 1 rcfcn to thc w c whcrc a = 0 and 0 = 0.5 hIodcl 2 refers ta the c u e whcre a = 0 and P = 0.95

Modcl 3 refcrv to the case where a = 0.15 and @ = 0.85

Table 4.6: Distribution of ARFIMA(O,d,O) When Underlying Processes are EGARCH(1,1), QMLE Estimation

ECARCH(a.0) Slcrn S.D. Robust S.E. Yin hlrx 1% 5 9C 96% 99% - a

0.1 -0.3 -0.065 0.128 0.127 -0.635 0.449 -0.375 -0.261 0.140 0 2GL UA -0.5 -0.087 0.1.15 0.154 -0.569 0.605 -0.408 -0.306 0.166 U ? 9 % 0.4 -0.7 -0.112 0.163 0. l n l -05.12 0.709 -0.445 -0.364 O.lr10 0337

Model 3 0 . I -0.9 -0.136 0.ldU 0.20-1 -0.69.; 1 . 7 -0.509 -0.101 0.199 0.3;L 0.6 -0. I -0.U.55 0.147 0.135 -0.543 0.460 -11.399 -0.290 0. Id0 0.304 O.a -0.3 -0.068 0.157 0.150 -0.661 0..555 -0.416 - 0 . 3 1 0 I91 I) 360 0.b -0.5 -0.0dB 0.171 0.172 -0.797 0.690 - 0 . 5 -0.157 0.191 037.5 0.h -0.7 -0.110 0.185 0.195 -0.593 U.756 - 0 . 5 7 -0.397 U.230 0119 0.3 -0.9 -0.129 0.199 0.21 1 -U.d35 2 -0.5G-I -0.419 0.231 0. I23

T = 200 0.4 -0.1 -0.023 0.071 U.070 -0.266 0. 162 -0.197 -0.146 U.UU.1 0.129 0.4 -0.3 -0.026 0.077 0.016 -0.29'2 0.18.; -0.215 - 0 . 1 5 0.096 0.1.12 0.4 -0.5 -0.029 0.087 0.085 -0.334 0 . 7 1 -0.229 -0.169 0.115 0.161 0:l -0.7 -0.034 0.10 i 0.097 -0.351 0.359 -0.2.16 -0.193 0.136 U.213

Model 1 0.4 -0-9 -0.039 0.115 0.114 -0.357 U.433 -0.267 -0.116 O.LSb 0.264 O.d -0.1 -0.0?5 0.083 O.OS2 0 2 7 0.245 -0.22-1 -0.165 0.108 0.163 0.N -0.3 -0.028 0.090 0 .0d9 -0.324 0.291 -0.233 -0 .12 0.124 6.197 0.H -0.5 -0.032 0.103 0.098 -0.303 0.3tlG -0.251 -0.197 0.I.I.I 0 .2 1.1 0.6 -0.7 -0.036 0.117 0.112 -0.42B 0.450 - 0 . 2 -0.220 0.164 0.Jl 1 0.3 -0.9 -0.041 0.131 0.116 -0.521 I).;L0 -0.302 -U.231 0 . 1 0.346 0. ,l -0.1 -0.026 0.061 0.063 -0.301 0.261 - 2 -0.lG.i 0.099 Oi5': 0.4 -0.3 -0.045 0.096 0.107 -0:IZd 0.322 -0.295 -0.201 0.115 0 2 i i 0.4 -0. .i - 0 . 0 2 0.1 I3 0.139 -0.60G 0:15d -0.346 - 0 . 2 - 0.132 0.2.50 U:l -0.7 -0.096 0.135 0 . 170 -U.,l94 U . G U 1 - 0 . 1 1 . 2 9 U 156 0.26 l

Model 2 0:i -0.9 -0.120 0.148 0.193 -41.561 0.A6; - 1 -0 .32 UL33 0.356 0.8 -0.1 4 0 3 5 0.117 0 I12 -0.500 0 . 1 I -0.325 -0.222 U.152 0.276 0.S 4 . 3 -0.049 0.130 0.132 -0.639 0.573 -0.345 -0.249 0.156 1).2**.i 0.A -0.5 -0.069 0.144 0.156 -0.189 0.649 -0.111 -0 .23 0.139 0.:151 0.8 -0.7 -0.0A9 0.156 0 181 -0.551 0.796 -0.422 -0.316 0 17R 0.409 11.6 -0.9 -0.106 0.172 0.197 -0.696 U.964 -0.481 -0.357 0.164 0433 0.4 -0.1 -0.026 0.077 0.078 -0.263 0.182 -0.221 -0.143 0.099 0.143 0.4 -0.3 -0.037 0.092 0.096 -0.363 0.315 -0.262 -0.16-1 0.106 0.190 U.4 -0.5 4.055 0.1 L5 0.12; -0.579 0.461 -0.314 -0.232 0.143 0.229 0.4 -0.7 -0.07; 0.138 0.152 -0.511 0.599 -0.351 -0.276 0 . 1 3 o.3u.1

Model 3 U.4 -0.9 -0.U97 0.156 0.177 -0.550 0.710 -0.406 -0.306 0.176 0.374 0.n -0.1 -0.032 0.104 0.103 -0.474 U.372 -0.262 -0.20.1 0.137 0.220 0.6 -0.3 -O.0,12 0.120 0.120 -0.595 0.576 -0.311 -0.226 0.155 0.297 0.3 -0.5 -0.060 0.142 0.146 -0.5.15 0.740 -0.364 -0.276 0. 176 O.J,lI 0.8 -0.7 -0.077 0.161 0.169 -0.517 0.636 -0.12d 4.305 0.192 U.110 0.H -0.9 -0.094 0. 178 0.189 -0.5Sl 0.665 -0. td7 -0.346 U.102 0.466

EGARCH(a.8) Mean S . D . itobast S.E. Mia 1 1% 5 5€ 95 'l 99 TC 7 e

I). 1 0. I 0.4

Model 2 0.4 0.8 O.d 0.* 0.8

0.178 0.069 0.093 0.114 0.145 0.170 0.10; 0.122 I). 15.1 0.171

O.?i -0.9 4 .068 0.161 I). 154 - 0 . 5 1 1.135 -0.399 -0.283 2 0.19'1

Note: The procwes satisfy thc folowing: ( 1 - ~ ) ~ ( y t - p) = Et

=t .r iid N(0,uf) In (of) = u + (1 - 4 ( ~ ) ) - l ( 1 + Q(L)) g(zt-1)

d z t ) = e=t + Y [lztl - E( Iz t l ) l hlodeI 1 refem to the case where ct = 0 and i3 = 0.5 XIodcI 2 refers to the case where a = 0 and fl = 0.95

hIodel 3 refcry to the cave where a = 0.15 and P = 0.85

Figure 41: Effciency of Methods for Estimation of ARFIMA When the Underlying Processes are EGARCH, High Volatiliw T=100

Figure 4 2 : Efticieny of Methods for Estimation of ARFrnLA Processes are EGARCH, High Volatility T=500

When the Underlying

Figure 4 3 : Efficiency of Methods for Estimation of ARFLM When the Underlying Processes are EG ARCH, L,ow Volatility T= 100

Figure 4-4: Efficiency of Methods for Estimation of ARFIhLA When the Underlying Processes are EGARCH, Low Volatility T=500

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