21
Journal of Econometrics 98 (2000) 107}127 A test for constant correlations in a multivariate GARCH model Y.K. Tse* Department of Economics, National University of Singapore, Singapore 119260, Singapore Received 1 April 1998; received in revised form 1 December 1998; accepted 1 October 1999 Abstract We introduce a Lagrange Multiplier (LM) test for the constant-correlation hypothesis in a multivariate GARCH model. The test examines the restrictions imposed on a model which encompasses the constant-correlation multivariate GARCH model. It requires the estimates of the constant-correlation model only and is computationally convenient. We report some Monte Carlo results on the "nite-sample properties of the LM statistic. The LM test is compared against the Information Matrix (IM) test due to Bera and Kim (1996). The LM test appears to have good power against the alternatives considered and is more robust to nonnormality. We apply the test to three data sets, namely, spot-futures prices, foreign exchange rates and stock market returns. The results show that the spot-futures and foreign exchange data have constant correlations, while the correlations across national stock market returns are time varying. ( 2000 Elsevier Science S.A. All rights reserved. JEL classixcation: C12 Keywords: Constant correlation; Information matrix test; Lagrange multiplier test; Monte Carlo experiment; Multivariate conditional heteroscedasticity 1. Introduction The success of the autoregressive conditional heteroscedasticity (ARCH) model and the generalized ARCH (GARCH) model in capturing the time-varying * Tel.: #65-7723954; fax: #65-7752646. E-mail address: ecstseyk@nus.edu.sg (Y.K. Tse). 0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 8 0 - 9

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Page 1: A test for constant correlations in a multivariate GARCH model

Journal of Econometrics 98 (2000) 107}127

A test for constant correlations ina multivariate GARCH model

Y.K. Tse*

Department of Economics, National University of Singapore, Singapore 119260, Singapore

Received 1 April 1998; received in revised form 1 December 1998; accepted 1 October 1999

Abstract

We introduce a Lagrange Multiplier (LM) test for the constant-correlation hypothesisin a multivariate GARCH model. The test examines the restrictions imposed on a modelwhich encompasses the constant-correlation multivariate GARCH model. It requires theestimates of the constant-correlation model only and is computationally convenient. Wereport some Monte Carlo results on the "nite-sample properties of the LM statistic. TheLM test is compared against the Information Matrix (IM) test due to Bera and Kim(1996). The LM test appears to have good power against the alternatives considered andis more robust to nonnormality. We apply the test to three data sets, namely, spot-futuresprices, foreign exchange rates and stock market returns. The results show that thespot-futures and foreign exchange data have constant correlations, while the correlationsacross national stock market returns are time varying. ( 2000 Elsevier Science S.A. Allrights reserved.

JEL classixcation: C12

Keywords: Constant correlation; Information matrix test; Lagrange multiplier test;Monte Carlo experiment; Multivariate conditional heteroscedasticity

1. Introduction

The success of the autoregressive conditional heteroscedasticity (ARCH) modeland the generalized ARCH (GARCH) model in capturing the time-varying

*Tel.: #65-7723954; fax: #65-7752646.E-mail address: [email protected] (Y.K. Tse).

0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 8 0 - 9

Page 2: A test for constant correlations in a multivariate GARCH model

variances of economic data in the univariate case has motivated many re-searchers to extend these models to the multivariate dimension. There are manyexamples in which empirical multivariate models of conditional heteroscedastic-ity can be used fruitfully. An illustrative list includes the following: model thechanging variance structure in an exchange rate regime (Bollerslev, 1990),calculate the optimal debt portfolio in multiple currencies (Kroner and Claes-sens, 1991), evaluate the multiperiod hedge ratios of currency futures (Lien andLuo, 1994), examine the international transmission of stock returns and volatil-ity (Karolyi, 1995) and estimate the optimal hedge ratio for stock index futures(Park and Switzer, 1995).

Bollerslev et al. (1988) provided the basic framework for a multivariateGARCH model. They extended the GARCH representation in the univariatecase to the vectorized conditional-variance matrix. While this so-called vechrepresentation is very general, empirical applications would require furtherrestrictions and more speci"c structures. A popular member of the vech-repres-entation family is the diagonal form. Under the diagonal form, each vari-ance}covariance term is postulated to follow a GARCH-type equation withthe lagged variance}covariance term and the product of the correspond-ing lagged residuals as the right-hand-side variables in the conditional-(co)variance equation. An advantage of this formulation is that the intuition ofthe GARCH model, which has been found to be very successful, is formallyadhered to.

It is often di$cult to verify the condition that the conditional-variance matrixof an estimated multivariate GARCH model is positive de"nite. Furthermore,such conditions are often very di$cult to impose during the optimisation of thelog-likelihood function. However, if we postulate the simple assumption that thecorrelations are time invariant, these di$culties nicely disappear. Bollerslev(1990) pointed out that under the assumption of constant correlations, themaximum likelihood estimate (MLE) of the correlation matrix is equal to thesample correlation matrix. When the correlation matrix is concentrated out ofthe log-likelihood function further simpli"cation is achieved in the optimisation.As the sample correlation matrix is always positive de"nite, the optimisationwill not fail as long as the conditional variances are positive.

Recently, Engle and Kroner (1995) proposed a class of multivariate condi-tional heteroscedasticity models called the BEKK (named after Baba, Engle,Kraft and Kroner) model. The motivation is to ensure the condition of a posit-ive-de"nite conditional-variance matrix in the process of optimisation. Engleand Kroner provided some theoretical analysis of the BEKK model and relatedit to the vech-representation form. Another approach examines the conditionalvariance as a factor model. The works by Diebold and Nerlove (1989), Engel andRodrigues (1989) and Engle et al. (1990) were along this line. One disadvantageof the BEKK and factor models is that the parameters cannot be easilyinterpreted, and their net e!ects on the future variances and covariances are not

108 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

Page 3: A test for constant correlations in a multivariate GARCH model

readily seen. In other words, the intuitions of the e!ects of the parameters ina univariate GARCH equation are lost.

Due to its computational simplicity, the constant-correlation GARCH modelis very popular among empirical researchers. Empirical research that uses thismodel includes: Bollerslev (1990), Kroner and Claessens (1991), Kroner andSultan (1991, 1993), Park and Switzer (1995) and Lien and Tse (1998). However,the following problems often seem to be overlooked in empirical applications.First, the assumption of constant correlation is often taken for granted andseldom analysed or tested. A notable exception, however, is the work by Beraand Kim (1996). Bera and Kim suggested an Information Matrix (IM) test forthe constant-correlation hypothesis in a bivariate GARCH model and appliedthe test to examine the correlation across national stock markets. Second, thee!ects of the assumption on the conditional-variance estimates are rarely con-sidered. In other words, Are the estimates of the parameters in the conditional-variance equations robust with respect to the constant-correlation assumption?In this paper we focus on the "rst question. Our objective is to providea convenient test (without having to estimate an encompassing model) for theconstant-correlation assumption and examine the properties of the test in smallsamples.

Bollerslev (1990) suggested some diagnostics for the constant-correlationmultivariate GARCH model. He computed the Ljung}Box portmanteau statis-tic on the cross products of the standardised residuals across di!erent equations.Critical values were based on the s2 distribution. Another diagnostic was basedon the regression involving the products of the standardised residuals. It was,however, pointed out by Li and Mak (1994) that the portmanteau statistic is notasymptotically a s2 and the use of a s2 approximation is inappropriate. For theresidual-based diagnostics, there are usually no su$cient guidelines as to thechoice of regressors in the arti"cial regression. Furthermore, the optimality ofthe portmanteau and residual-based tests is not established.

We propose a test for the constant-correlation hypothesis based on theLagrange Multiplier (LM) approach. We extend the constant-correlation modelto one in which the correlations are allowed to be time varying. When certainkey parameters in the extended model are imposed to be zero, the constant-correlation model is obtained. We consider the LM test for the zero restrictionson the key parameters. Finite-sample properties of the LM test are examinedusing Monte Carlo methods. The LM test is compared against the IM testdue to Bera and Kim (1996). We "nd that the LM test has good approxim-ate nominal size in sample sizes of 1000 or above. It is powerful againstthe alternative models with time-varying correlations considered. On the otherhand, while the IM test has good approximate nominal size, it lackspower. Empirical illustrations using real data, however, show that the IMtest rejects the constant-correlation hypothesis vehemently with very lowp values. To explain this anomaly, we examine the behaviour of the tests under

Y.K. Tse / Journal of Econometrics 98 (2000) 107}127 109

Page 4: A test for constant correlations in a multivariate GARCH model

nonnormality. It is found that while the Monte Carlo results show that the IMtest leads to gross over-rejection when the errors are nonnormal, the LM test isrelatively robust against nonnormality.

The plan of the rest of the paper is as follows. In Section 2 we derive the LMstatistic. Some Monte Carlo results on the "nite-sample distributions of the LMand IM tests are reported in Section 3. Section 4 describes some illustrativeexamples using real data. In Section 5 we examine the e!ects of nonnormality onthe tests. Finally, we give some concluding remarks in Section 6.

2. The test statistic

Consider a multivariate time series of observations MytN, t"1,2,¹, with

K elements each, so that yt"(y

1t,2, y

Kt)@. To focus on the conditional hetero-

scedasticity of the time series, we assume that the observations are of zero (orknown) means. This assumption simpli"es tremendously the discussions with-out straining the notations.

The conditional variance of ytis assumed to follow the time-varying structure

given by

Var(ytDU

t~1)"X

t,

where Utis the information set at time t. We denote the variance elements of

Xt

by p2it, for i"1,2,K, and the covariance elements by p

ijt, where

1)i(j)K. Following Bollerslev (1990), we consider the constant-correlationmodel in which the conditional variances of y

itfollow a GARCH process, while

the correlations are constant. Denoting C"MoijN as the correlation matrix, we

have

p2it"u

i#a

ip2i,t~1

#biy2i,t~1

, i"1,2,K (1)

pijt"o

ijpitpjt, 1)i(j)K. (2)

We assume that ui, a

iand b

iare nonnegative, a

i#b

i(1, for i"1,2,K and

C is positive de"nite. Although the conditional variances in the above equationsare assumed to follow low-order GARCH(1, 1) processes, the test derived belowcan be extended to the general GARCH(p, q) models without di$culties.

As pointed out by Bollerslev (1990), the constant-correlation model is com-putationally attractive. Speci"cally, the MLE of the correlation matrix is equalto the sample correlation matrix of the standardised residuals, which is alwayspositive de"nite. The correlation matrix can be further concentrated out fromthe log-likelihood function, resulting in a reduction in the number of parametersto be optimised. Furthermore, it is relatively easy to control the parameters ofthe conditional-variance equations during the optimisation so that p2

itare

110 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

Page 5: A test for constant correlations in a multivariate GARCH model

always positive. On the other hand, it is very di$cult to control a matrix ofparameters to be positive de"nite during the optimisation.

To test for the validity of the constant-correlation assumption, we extend theabove framework to include time-varying correlations. Under some restrictionson the parameter values of the extended model the constant-correlation model isderived. The LM test can then be applied to test for the restrictions. Thisapproach only requires estimates under the constant-correlation model, and canthus conveniently exploit the computational simplicity of the model.

To allow for time-varying correlations, we consider the following equationsfor the correlations:

oijt"o

ij#d

ijyi,t~1

yj,t~1

, (3)

where dij

for 1)i(j)K are additional parameters in the extended model.Thus, the correlations are assumed to respond to the products of previousobservations. From (3) the conditional covariances are given by

pijt"o

ijtpitpjt. (4)

Note that there are N"K2#2K parameters in the extended model withtime-varying correlations. The constant-correlation hypothesis can be tested byexamining the hypothesis H

0: d

ij"0, for 1)i(j)K. Under H

0, there are

M"K(K!1)/2 independent restrictions.It should be pointed out that (3) is speci"ed as a convenient alternative that

encompasses the constant-correlation model. To ensure that the alternativemodel provides well-de"ned positive-de"nite conditional-variance matrices, fur-ther restrictions have to be imposed on the parameters d

ij. As in the case of the

general vech speci"cation, such restrictions are very di$cult to derive. Indeed,empirical research using the vech speci"cation often leaves the issue as anempirical problem to be resolved in the optimisation stage. As our interest is inthe model under the null H

0, we shall not pursue the issue of searching for the

necessary restrictions. Thus, we assume that within a neighbourhood of dij"0,

the optimal properties of the LM test (such as its asymptotic e$ciency againstlocal alternatives) hold under some regularity conditions as stated in, forexample, Godfrey (1988).

As correlations are standardised measures, it might be arguable to allow thecorrelations to depend on the products of the lagged standardised residualsinstead. Thus, if we de"ne e

it"y

it/p

itas the standardised residual, an alternative

model might be written as

oijt"o

ij#d@

ijei,t~1

ej,t~1

. (5)

As eit

depends on other parameters of the model through pit, analytic derivation

of the LM statistic is intractable. If indeed (5) describes the true model, theremay be some loss in power in using (3) as the alternative hypothesis. In return,however, we obtain analytical tractability. Of course, there is no apriori reason

Y.K. Tse / Journal of Econometrics 98 (2000) 107}127 111

Page 6: A test for constant correlations in a multivariate GARCH model

that (5) would provide a better alternative than (3). Indeed, whether using (3)as the encompassing model would provide a test with good power is anempirical question. We shall examine its performance using Monte Carlomethods. Now we shall proceed to derive the LM statistic of H

0under the above

framework.We denote D

tas the diagonal matrix with diagonal elements given by p

it, and

Ct"Mo

ijtN as the time-varying correlation matrix. Hence the conditional-vari-

ance matrix of ytis given by X

t"D

tCtD

t. Under the normality assumption the

conditional log-likelihood of the observation at time t is given by (the constantterm is ignored)

lt"!

1

2ln DD

tCtD

tD!

1

2y@tD~1

tC~1t

D~1t

yt

"!

1

2ln DC

tD!

1

2

K+i/1

ln p2it!

1

2y@tD~1

tC~1

tD~1

tyt,

for t"1,2,¹, and the log-likelihood function l is given by l"+Tt/1

lt. For

simplicity, we have assumed that yi0

and p2i0

are "xed and known. Thisassumption has no e!ects on the asymptotic distributions of the LM statistic.Note that D~1

tyt

represents the standardised observations with unit variance.We denote D~1

tyt"e

t"(e

1t,2,e

Kt)@.

We now de"ne the following derivatives of p2it

with respect to ui, a

iand b

ifor

i"1,2,K,

dit"Lp2

it/Lu

i, e

it"Lp2

it/La

i, f

it"Lp2

it/Lb

i.

To calculate these derivatives, the following recursions may be used:

dit"1#a

idi,t~1

,

eit"p2

i,t~1#a

iei,t~1

,

fit"a

ifi,t~1

#y2i,t~1

, (6)

where the starting values are given by di1"1, e

i1"p2

i0and f

i1"y2

i0.

The "rst partial derivatives of lt

with respect to the model parameters aregiven by

Llt

Lui

"

(eHiteit!1)d

it2p2

it

,

Llt

Lai

"

(eHiteit!1)e

it2p2

it

,

112 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

Page 7: A test for constant correlations in a multivariate GARCH model

Llt

Lbi

"

(eHiteit!1) f

it2p2

it

,

Llt

Loij

"eHiteHjt!oij

t,

Llt

Ldij

"(eHiteHjt!oij

t)y

i,t~1yj,t~1

, (7)

where eHt"(eH

1t,2,eH

Kt)@"C~1

tet, and C~1

t"Moij

tN. Thus, if we denote the para-

meters of the model as h"(u1, a

1, b

1, u

2,2, b

K, o

12, o

13,2,o

K~1, K,

d12

,2,dK~1, K

)@, we can calculate Llt/Lh from the above equations. These

analytic derivatives can facilitate the evaluation of the MLE of the extendedmodel if desired. Note that on H

0, C

t"C for all t, so that eH

t"C~1e

tand

oijt"oij. In this case, e

tare just the standardised residuals calculated from the

algorithm suggested by Bollerslev (1990). We shall denote hK as the MLE ofh under H

0.

If we denote s as the N-element score vector given by s"Ll/Lh and < as theN]N information matrix given by <"E(!L2l/LhLh@), where E(.) denotes theexpectation operator, the LM statistic for H

0is given by s( @<K ~1s( , where the hats

denote evaluation at hK . In practice, < may be replaced by the (negative of the)Hessian matrix. Alternatively, we may use the sum of the cross products of the"rst derivatives of l

t. To this e!ect, we denote S as the ¹]N matrix the rows of

which are the partial derivatives Llt/Lh@, for t"1,2,¹. Thus, the LM statistic

for H0, denoted as LMC, can be calculated using the following formula (see, for

example, Godfrey, 1988)

LMC"s( @(SK @SK )~1s( (8)

"l@SK (SK @SK )~1SK @l, (9)

where l is the ¹]1 column vector of ones and SK is S evaluated at hK . Under theusual regularity conditions LMC is asymptotically distributed as a s2

M. Eq. (9)

shows that LMC can be interpreted as ¹ times R2, where R2 is the uncenteredcoe$cient of determination of the regression of l on SK . It is well-known thatother forms of the LM statistic are available. For example, further simpli"cationcan be obtained by making use of the fact that in SK @l the elements correspondingto the unrestricted parameters are zero. Eq. (9), however, is a convenient formand will be used throughout this paper.

We now discuss some extensions of the above framework. First, whenGARCH(p, q) models are considered we need to augment the parameters of theconditional-variance equations. Thus, we denote the coe$cients of the laggedconditional variances as a

ih, h"1,2, p, and the coe$cients of the lagged

y2it

terms as bik, k"1,2, q. We write the conditional-variance equations as

Y.K. Tse / Journal of Econometrics 98 (2000) 107}127 113

Page 8: A test for constant correlations in a multivariate GARCH model

(Note that we may allow the orders of the GARCH processes to vary with i.Thus, (p, q) should be regarded as the generic order.)

p2it"u

i#

p+h/1

aihp2i,t~h

#

q+k/1

biky2i,t~k

, i"1,2, K.

The partial derivatives Llt/Lu

i, Ll

t/La

ihand Ll

t/Lb

ikare required. We "rst

calculate the partial derivatives of p2it

with respect to ui, a

ihand b

ik, which will

be denoted as dit, e

ihtand f

ikt, respectively. These partial derivatives can be

calculated by recursions constructed as in (6). Speci"cally, we have

dit"1#

p+h/1

aihdi,t~h

,

eiht"p2

i,t~h#

p+

h{/1

aih{

eih,t~h{

,

fikt"

p+h/1

aih

fik,t~h

#y2i,t~k

.

The "rst partial derivatives of lt

with respect to ui, a

ihand b

ik(p#q#1

derivatives altogether) can be calculated using (7), with eiht

and fikt

replacingeit

and fit, respectively.

Second, when the conditional mean of ytdepends on some unknown para-

meters, S would have to be augmented by the partial derivatives of lt

withrespect to the parameters appearing in the conditional-mean equations. If theunknown means are linear in the parameters, the partial derivatives are straight-forward to obtain. Indeed, in such cases the asymptotic variance matrix of theMLE are block diagonal with respect to the conditional-mean and conditional-variance parameters (see Bera and Higgins, 1993). Thus, under such circumstan-ces, LMC calculated from (9) using only the conditional-variance parameters isasymptotically valid.

3. Monte Carlo results

In this section we report some Monte Carlo results on the empirical size andpower of LMC in "nite samples. In addition to the LM test, we examine the IMtest suggested by Bera and Kim (1996). The IM statistic, denoted by IMC, testsfor the hypothesis of constant correlation in a bivariate GARCH model. Denot-ing o( as the MLE of the (constant) correlation coe$cient in the bivariateGARCH model, and e(

itfor i"1,2 as the estimated standardised residuals, IMC

is given by

IMC"

[+Tt/1

(m21t

m22t!1!2o( 2)]2

4¹(1#4o( 2#o( 4),

114 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

Page 9: A test for constant correlations in a multivariate GARCH model

where

m1t"

e(1t!o( e(

2tJ1!o( 2

m2t"

e(2t!o( e(

1tJ1!o( 2

.

Bera and Kim (1996) showed that under the null hypothesis of constant correla-tion and the assumption that e

tare normally distributed, IMC is asymptotically

distributed as a s21.

Six experiments are considered. Experiments E1 through E4 are based onK"2, while Experiments E5 and E6 are based on K"3. The parameter valuesof the experiments are summarised in Panel A of Table 1. E1, E3 and E5represent models with high correlations, while E2, E4 and E6 represent modelswith low correlations. Also, E1, E2, E5 and E6 represent models with relativelyhigh persistence in the conditional variance (i.e., a#b is near to 1), while E3 andE4 represent models with relatively low persistence. Unlike LMC, which isapplicable for all six experiments, the IMC test as developed by Bera and Kim(1996) is only available for the bivariate experiments E1 through E4. Although itshould be possible to extend the IMC test to beyond the bivariate case, suchextension will not be pursued in this paper.

For each experiment we generate 2000 samples of MytN based on the normality

assumption, with sample size ¹ taken to be 300, 500, 1000 and 2000. Weestimate the parameters of the model using Bollerslev's (1990) algorithm andcalculate the LMC and IMC statistics. Panel B of Table 1 summarises theempirical sizes of the LM and IM tests assuming a nominal size of 5%. ForLMC there are signs of over-rejection in small samples. Over-rejection, however,seems to have satisfactorily reduced when the sample size reaches 1000. Exceptfor E5, the point estimates of the empirical size when ¹ is 1000 or above neverexceed the nominal size by more than 1.5%. The correlations seem to play a rolein determining the rate of convergence to the nominal size. Models with lowcorrelations are less subject to over-rejection in small samples. On the otherhand, the persistence of the conditional variance does not have much e!ecton the degree of over-rejection. The IM test appears to have very good approx-imate nominal sizes. There are signs of slight under-rejection in small samples.This problem, however, has largely disappeared when the sample size reaches1000.

Panel C reports the Monte Carlo sample means of the LMC and IMCstatistics. For E1 through E4, these values are expected to converge to 1. For E5and E6, the mean of the LMC statistic is expected to converge to 3. As can beobserved, the results re#ect the over-rejection of LMC and the under-rejectionof IMC. In summary, for sample sizes of 1000 or above, the LM and IM tests

Y.K. Tse / Journal of Econometrics 98 (2000) 107}127 115

Page 10: A test for constant correlations in a multivariate GARCH model

Table 1Parameter values of the Monte Carlo experiments and the estimated rejection probabilities andsample means of LMC and IMC!

Experiment

E1 E2 E3 E4 E5 E6

Panel A: True parameter valuesu

10.40 0.40 0.40 0.40 0.30 0.30

a1

0.80 0.80 0.40 0.40 0.80 0.80b1

0.15 0.15 0.30 0.30 0.10 0.10u

20.20 0.20 0.20 0.20 0.40 0.40

a2

0.70 0.70 0.50 0.50 0.60 0.60b2

0.20 0.20 0.20 0.20 0.25 0.25u

3* * * * 0.50 0.50

a3

* * * * 0.80 0.80b3

* * * * 0.15 0.15o12

0.80 0.20 0.80 0.20 0.60 0.20o13

* * * * 0.70 0.20o23

* * * * 0.80 0.20

Panel B: Estimated probabilities (%) of the Type-1 errors of the tests

LMC ¹"300 7.75 6.90 8.05 6.90 10.20 9.75¹"500 6.85 6.40 7.65 6.25 8.05 8.05¹"1000 6.40 5.20 6.45 5.40 7.35 6.30¹"2000 6.15 4.90 4.95 5.95 7.10 5.55

IMC ¹"300 3.65 4.20 3.75 4.50 * *

¹"500 4.20 4.10 4.40 3.70 * *

¹"1000 4.20 5.20 4.15 5.10 * *

¹"2000 5.30 4.20 5.30 4.50 * *

Panel C: Monte Carlo sample means of the test statistics

LMC ¹"300 1.287 1.193 1.249 1.235 3.868 3.774¹"500 1.182 1.094 1.214 1.133 3.633 3.453¹"1000 1.148 1.059 1.129 1.050 3.430 3.162¹"2000 1.071 1.005 1.025 1.071 3.328 3.114

IMC ¹"300 0.941 0.982 0.908 0.994 * *

¹"500 0.983 0.915 0.989 0.929 * *

¹"1000 0.927 1.028 0.971 0.963 * *

¹"2000 1.037 0.896 1.023 0.957 * *

!Note: Panel A gives the true parameter values of the Monte Carlo experiments. Panel B recordsthe estimated probabilities of rejecting the constant-correlation hypothesis using the LMC and IMCtests at the nominal size of 5%. Panel C summarises the Monte Carlo sample means of the teststatistics. For E1 through E4, the asymptotic expected value of LMC and IMC is 1. For E5 and E6,the asymptotic expected value of LMC (IMC not applicable) is 3. The Monte Carlo sample size is2000.

116 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

Page 11: A test for constant correlations in a multivariate GARCH model

provide reliable nominal sizes. In the case of the LM test, the reliability isenhanced when the correlations are low.

To examine the power of the LM and IM tests we conduct some experimentswith data generated from models with time-varying correlations. Two types ofmultivariate GARCH models are considered. In the "rst type, we assume thatthe data are generated from a BEKK model. This model has been applied in theliterature by, among others, Baillie and Myers (1991) and Karolyi (1995). It hasbeen found to "t the data satisfactorily. An advantage of the BEKK model isthat the conditional-variance matrices are always positive de"nite. Undoubted-ly, this is an important advantage in simulation studies. Following Engle andKroner (1995), the conditional-variance matrix X

tof the model is written as

Xt"X#BX

t~1B@#Cy

t~1y@

t~1C@,

where X"MpijN, B"Mb

ijN and C"Mc

ijN are K]K parameter matrices.

In the second type of models, we assume that the conditional variances followa GARCH process given in (1), while the conditional correlations follow theprocess speci"ed in (5). A di$culty with this model is that the simulatedcorrelation matrix is not guaranteed to be positive de"nite. This problem has tobe controlled by setting the parameters d@

ijto be su$ciently small. It should be

noted that it is not our intention to maintain that these two types of modelsrepresent processes that are descriptive of real data. Our objective is to examinethe power of the tests under some alternatives.

We consider two bivariate BEKK models denoted by P1 and P2, and onetrivariate BEKK model denoted by P3. The model parameters are summarisedin Panel A of Table 2. We consider sample sizes ¹ of 300, 500 and 1000. Basedon Monte Carlo samples of 1000 each, we estimate the power of LMC and IMCagainst the time-varying conditional-correlation models at nominal size of 5%.As a measure of the variability of the conditional correlation coe$cients, wecalculate the range (i.e., maximum!minimum) of the conditional correlationcoe$cients in each simulated sample of ¹ observations. Panel B summarises themaximum and minimum ranges of the conditional correlation coe$cients in theMonte Carlo samples of 1000. It can be seen that P2 has larger variability incorrelations than P1. The LM test is found to have high power in all experi-ments. It is quite remarkable that for Experiment P1, for which the range ofMo

12tN appears to be quite small, LMC has yet very good power. For example,

for P1 when ¹"1000, the maximum range of Mo12t

N is only 0.396 and yet theempirical power of LMC is 91.3%. In contrast, IMC has very weak power in allcases. Indeed, the empirical power of IMC is less than 10% in all cases except forP2 with ¹"1000.

For alternative models of the second type we consider two bivariate modelsdenoted by Q1 and Q2, and one trivariate model denoted by Q3. The modelparameters are presented in Panel A of Table 3. The values of d@

ijare set after

some experimentation to generate well-de"ned conditional-correlation matrices

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Table 2Estimated power of LMC and IMC when the true model is BEKK!

Experiment

P1 P2 P3

Panel A: True parameter values

p11

0.20 0.20 0.80p12

0.10 0.04 0.20p13

* * 0.20p22

0.20 0.20 0.80p23

* * 0.20p33

* * 0.80b11

0.60 0.40 0.50b12

0.20 0.20 0.20b13

* * 0.20b22

0.60 0.40 0.50b23

* * 0.20b33

* * 0.50c11

0.30 0.40 0.30c12

0.10 0.20 0.10c13

* * 0.10c22

0.30 0.40 0.30c23

* * 0.10c33

* * 0.30

Panel B: Maximum/minimum ranges of the correlation coezcients in the simulated samples and theempirical relative frequency (in %) of rejecting the constant-correlation hypothesis of LMC and IMC

Sample size ¹ 300 500 1000 300 500 1000 300 500 1000

Range of Moijt

NMo

12tN Max 0.498 0.401 0.396 0.907 0.833 0.861 0.349 0.365 0.347

Min 0.174 0.188 0.206 0.489 0.528 0.573 0.121 0.155 0.166Mo

13tN Max * * * * * * 0.349 0.362 0.356

Min * * * * * * 0.129 0.153 0.165Mo

23tN Max * * * * * * 0.346 0.367 0.340

Min * * * * * * 0.136 0.148 0.169

Power of tests (%)LMC 56.7 73.5 91.3 86.8 97.7 100.0 56.0 49.9 52.8IMC 6.0 5.9 9.0 7.6 9.6 16.0 * * *

!Note: Panel A gives the true parameter values of the Monte Carlo experiments. P1 and P2 arebivariate models, while P3 is trivariate. The parameter matrices B and C are both symmetric. PanelB records the maxima and mimina of the ranges of the conditional correlation coe$cients Mo

ijtN in

the simulated samples. It also gives the estimated probabilities of rejecting the constant-correlationhypothesis using the LMC and IMC statistics at the asymptotic nominal size of 5%. The MonteCarlo sample size is 1000.

118 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

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Table 3Estimated power of LMC and IMC when the correlation coe$cients are generated from Eq. (5)!

Experiment

Q1 Q2 Q3

Panel A: True parameter valuesu

10.20 0.20 0.20

a1

0.80 0.80 0.60b1

0.10 0.10 0.20u

20.20 0.20 0.20

a2

0.80 0.80 0.60b2

0.10 0.10 0.20u

3* * 0.20

a3

* * 0.60b3

* * 0.20o12

0.40 0.10 0.20o13

* * 0.20o23

* * 0.20d@12

0.03 0.06 0.03d@13

* * 0.03d@23

* * 0.03

Panel B: Maximum/minimum ranges of the correlation coezcients in the simulated samples and theempirical relative frequency (in %) of rejecting the constant-correlation hypothesis of LMC and IMC

Sample Size ¹ 300 500 1000 300 500 1000 300 500 1000

Range of Moijt

NMo

12tN Max 0.514 0.609 0.615 1.007 1.171 1.068 0.566 0.517 0.548

Min 0.149 0.194 0.214 0.308 0.394 0.424 0.159 0.179 0.217Mo

13tN Max * * * * * * 0.534 0.547 0.573

Min * * * * * * 0.155 0.186 0.212Mo

23tN Max * * * * * * 0.467 0.483 0.597

Min * * * * * * 0.163 0.197 0.228

Power of tests (%)LMC 15.2 23.1 29.4 21.4 28.7 47.1 14.7 19.3 30.3IMC 4.6 4.0 5.4 3.7 6.7 5.0 * * *

!Note: Panel A gives the true parameter values of the Monte Carlo experiments. Q1 and Q2 arebivariate models, and Q3 is a trivariate model. Panel B records the maximum and miminum ofthe ranges of the conditional correlation coe$cients Mo

ijtN in the simulated samples. It also gives the

estimated probabilities of rejecting the constant-correlation hypothesis using the LMC and IMCstatistics at the asymptotic nominal size of 5%. The Monte Carlo sample size is 1000.

in all simulated samples. From Panel B we can see that the variability of thecorrelation coe$cients are generally higher than that of the BEKK models.However, the power of the LM test is lower against this second type of models.For example, the maximum power achieved in the experiments is for Q2 with

Y.K. Tse / Journal of Econometrics 98 (2000) 107}127 119

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Fig. 1. Conditional correlations of a sample of P1, ¹"300.

¹"1000, which has an estimate of only 47.1%. As for the IM test, the empiricalpower is again found to be very low. Indeed, among the cases considered themaximum empirical power achieved is only 6.7%.

To compare the characteristics of the time-varying correlations in the twotypes of models, we examine the paths of the correlation coe$cients generated.Figs. 1 and 2 present the plots of two simulated paths of conditional correlationcoe$cients based on P1 and Q1, respectively, for ¹"300. It is obvious that thetwo models produce very di!erent time-varying correlation structures. Fig. 2exhibits a sample path of serially uncorrelated conditional correlation coe$-cients. This is due to the fact that e

1,t~1e2,t~1

are serially uncorrelated. On theother hand, Fig. 1 is characteristic of a sample path of a serially correlated timeseries. The Box}Pierce Q statistics based on the "rst 10 lagged serial correlationcoe$cients of the sample paths are 72.95 for the path in Fig. 1 and 8.89 for thepath in Fig. 2. Indeed, the autocorrelation coe$cients of the sample path ofot

generated from P1 are declining very slowly, which is characteristic of along-memory time series. Thus, while Q1 generates serially uncorrelated time-varying correlations, P1 generates series of correlation coe$cients with longmemory. As y

i,t~1yj,t~1

are speci"ed as the covariates for tracking the vari-ations in the correlation coe$cients in the LM test, this may help to explain thedi!erence in the power of the LM test found in the two types of time-varyingcorrelation models.

It should be noted that our Monte Carlo results have not, by all means,presented a comprehensive study of the power of the tests. Like other Monte

120 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

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Fig. 2. Conditional correlations of a sample of Q1, ¹"300.

Carlo studies, generalization to other models has to be done with care. Nonethe-less, our "ndings provide some evidence in support of the power of the LM test.

4. Some illustrative examples

We now consider the application of LMC to some real data sets. The data setsselected are described as follows (all returns, in percentage, are measured aslogarithmic di!erences):

Data Description of variables Data period and frequencies

DS1 Return observations of the spot index (S) Daily data from 89/1 to 96/8,and futures price (F) of the Nikkei with 1861 observationsStock Averge 225

DS2 Return observations of 3 Asian currencies, 5-Daily data from 78/1 to 94/6,namely, Japanese yen (J), Malaysian with 812 observationsringgit (M) and Singapore dollar (S). Therates are quoted against the US dollar

DS3 Return observations of 3 Asian stock Daily data from 90/1 to 94/8,markets, namely, Hong Kong (H), Japan with 1057 observations(J) and Singapore (S)

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Table 4Estimation results of constant-correlation models!

Data K Variable u a b Correlations LMC IMC

DS1 2 S 0.065 0.873 0.090 oSF

"0.941 0.893 518.00H(¹"1861) (0.011) (0.014) (0.011) (0.003) * *

F 0.055 0.896 0.076 * * *

(0.009) (0.012) (0.009) * * *

DS2 2 J 0.575 0.671 0.111 oJM

"0.467 0.004 267.75H(¹"812) (0.239) (0.115) (0.038) (0.028) * *

M 0.074 0.598 0.247 * * *

(0.025) (0.105) (0.061) * * *

2 J 0.648 0.631 0.124 oJS"0.537 0.186 180.53H

(0.226) (0.108) (0.040) (0.025) * *

S 0.051 0.714 0.153 * * *

(0.016) (0.067) (0.034) * * *

2 M 0.093 0.536 0.274 oMS

"0.611 3.235 205.18H(0.029) (0.115) (0.068) (0.022) * *

S 0.058 0.688 0.162 * * *

(0.016) (0.065) (0.032) * * *

3 J 0.605 0.647 0.126 oJM

"0.468 1.273 *

(0.217) (0.105) (0.039) (0.028) * *

M 0.076 0.595 0.245 oJS"0.538 * *

(0.024) (0.096) (0.055) (0.025) * *

S 0.046 0.741 0.139 oMS

"0.612 * *

(0.013) (0.051) (0.027) (0.022) * *

DS3 2 H 0.399 0.619 0.215 oHJ

"0.254 2.427 103.32H(¹"1057) (0.141) (0.111) (0.063) (0.029) * *

J 0.148 0.807 0.147 * * *

(0.037) (0.029) (0.025) * * *

2 H 0.259 0.752 0.132 oHS

"0.436 16.547H 30.98H(0.146) (0.115) (0.054) (0.025) * *

S 0.208 0.543 0.257 * * *

(0.041) (0.067) (0.045) * * *

DS1 was used by Lien and Tse (1998) in studying the hedging e!ectiveness ofthe Nikkei Stock Average futures. DS2 was documented in Tse and Tsui (1997)and Tse (1998). DS3 was used by Tse and Zuo (1996) to study the volatility of theAsia-Paci"c stock markets. The details of the description of these data sets canbe found in the above references, in which some summary statistics of these datasets are also reported. A bivariate GARCH(1,1) model with constant correlationis "tted to DS1. For DS2 and DS3, both trivariate and bivariate (pairwise)models are "tted. The results are summarised in Table 4. We present the

122 Y.K. Tse / Journal of Econometrics 98 (2000) 107}127

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Table 4. Continued

Data K Variable u a b Correlations LMC IMC

2 J 0.142 0.816 0.139 oJS"0.345 15.362H 22.31H

(0.036) (0.028) (0.023) (0.027) * *

S 0.229 0.474 0.326 * * *

(0.044) (0.073) (0.055) * * *

3 H 0.212 0.787 0.112 oHJ

"0.258 17.665H *

(0.104) (0.085) (0.043) (0.029) * *

J 0.145 0.818 0.135 oHS

"0.437 * *

(0.036) (0.028) (0.023) (0.025) * *

S 0.217 0.535 0.253 oJS"0.347 * *

(0.042) (0.068) (0.044) (0.027) * *

!Note: DS1 consists of the daily returns of the spot (S) and futures (F) of the Nikkei Stock Average.DS2 consists of the 5-day returns of the Japanese yen (J), Malaysian ringgit (M) and Singaporedollar (S). DS3 consists of the stock market returns of Hong Kong (H), Japan (J) and Singapore (S).LMC is the Lagrange multiplier statistic for constant correlations. It is asymptotically distributed asa s2

M, where M"1 for K"2 and M"3 for K"3.

HFor the LMC and IMC indicates statistical signi"cance at the 5% level.The "gures in the parentheses are standard errors.

estimated constant-correlation models as well as the computed LMC and IMCstatistics.

For DS1, the estimated correlation is 0.941, re#ecting a high degree ofco-movements between the spot and futures. This is perhaps not surprising,given the arbitrage-free pricing in an e$cient market. The LMC statistic hasa low value of 0.893. Thus, not only is the correlation high, the stability of therelationship is also veri"ed (the evidence provided by IMC will be discussedbelow).

For DS2, the correlations between the currency exchanges are much lower, allin the region of 0.45 to 0.65. The LMC for the pairwise models and the trivariatemodel are insigni"cant at the 5% level. Thus, there is no evidence againsttime-invariant correlations among the selected Asian currencies.

The results for DS3 are, however, quite di!erent. We observe that thecorrelations across the three Asian stock markets are quite low, all below 0.45.There is strong evidence (signi"cance at the 1% level) of time-varying correla-tions between Hong Kong/Singapore, Japan/Singapore and the trivariate case.Stock market performance is dependent, to a large extent, on domestic factors. Ifthese factors have low correlations and/or their relationships are notstable over time, we would expect the correlations of market returns to betime varying. To further illustrate the use of the test for K"4, we appendthe Australian market onto DS3. The results of the four Asia-Paci"c marketsare recorded in Table 5. Not surprisingly, we "nd evidence against constant

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Table 5Estimation results of the constant-correlation model of four Asia}Paci"c stock markets!

Market

Parameter A H J S

u 0.049 0.196 0.144 0.222(0.025) (0.081) (0.035) (0.043)

a 0.900 0.803 0.822 0.537(0.038) (0.066) (0.027) (0.070)

b 0.045 0.110 0.131 0.241(0.044) (0.034) (0.022) (0.043)

oAi

1.000 * * *

(**) * * *

oHi

0.381 1.000 * *

(0.026) (**) * *

oJi

0.306 0.259 1.000 *

(0.028) (0.029) (**) *

oSi

0.385 0.439 0.348 1.000(0.026) (0.025) (0.027) (**)

!Note: The data set consists of the daily returns (1057 observations) of four Asia}Paci"c stockmarkets, where A"Australia, H"Hong Kong, J"Japan and S"Singapore. The Lagrangemultiplier statistic, LMC, for testing constant correlations is 20.265. The LMC, which is asymp-totically distributed as a s2

6, is statistically signi"cant at the 5% level. The "gures in the parentheses

are standard errors.

correlations. The LMC statistic is 20.265, which is statistically signi"cant at the1% level.

In summary, depending on the data sets analysed, we have found evidence forand against the assumption of constant correlations based on the LMC statistic.In particular, we have found evidence against constant correlations acrossthe selected Asia}Paci"c stock markets. In view of this evidence it would beimportant to study models that admit time-varying correlations. In any case, thehypothesis of constant correlations should be tested before the empirical multi-variate GARCH models can be used for inference and its economic implicationsare drawn.

In contrast, the results for the IMC statistic are very di!erent. The constant-correlation assumption is vehemently rejected with very low p values. As theMonte Carlo results in Section 3 show that IMC has very low power in rejectingthe constant-correlation hypothesis, this result is rather unusual. One possibleexplanation for this anomaly is that the rejection of IMC for the real data sets isdue to other assumptions imposed in the derivation of the IM test. As normalityis imposed by Bera and Kim (1996) in deriving the moment conditions for IMC,the nonnormality of the data could well be the reason behind the rejection of theIM test. Indeed, the data used in Table 4, as in many other "nancial data, exhibit

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Table 6Empirical size of LMC and IMC when residuals are nonnormal!

Error distribution Sample size ¹ Experiment

Test E1 E2 E3 E4

LMC t8

300 11.1 8.0 9.9 8.4500 8.7 6.3 9.8 7.4

1000 7.4 6.1 9.2 5.9t12

300 10.2 9.2 10.1 10.2500 9.6 9.1 7.6 6.2

1000 6.5 7.3 7.7 5.7IMC t

8300 61.4 17.3 58.3 14.8500 78.7 17.9 76.4 16.7

1000 97.3 22.4 96.1 25.5t12

300 35.2 10.1 37.4 11.9500 48.6 10.8 47.6 9.1

1000 74.8 16.9 77.2 13.3

!Note: The "gures are the empirical relative frequencies (in %) of rejecting the constant-correlationhypothesis. The parameters of the experiments can be found in Table 1. The Monte Carlo samplesize is 1000.

kurtosis higher than that of a normal distribution (see the references cited abovefor the details). As the derivation of the LMC statistic also depends onthe normality assumption, it would be important to examine the robustness ofthe tests with respect to nonnormality. In the next section, we report someMonte Carlo results on this issue.

5. The e4ects of nonnormality

To examine the e!ects of nonnormality on the LM and IM tests, we conductfurther Monte Carlo experiments. We use the experimental setups for thebivariate models in Table 1, namely, Experiments E1 through E4. Errors arethen generated from nonnormal distributions. We consider t distributions with8 and 12 degrees of freedom. The sample size ¹ is taken to be 300, 500 and 1000.Based on Monte Carlo samples of 1000 runs, the empirical sizes of the LM andIM tests are reported in Table 6.

It is obvious that the IM test over-rejects the null hypothesis to a great extent.The problem of over-rejection increases with the sample size ¹. Also, over-rejection appears to be more serious for E1 and E3 as compared against E2 andE4. Thus, models with higher levels of correlation are more prone to over-rejection. When the two nonnormal distributions are compared, over-rejectionis less serious for t

12as compared against t

8.

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The LM test also demonstrates over-rejection under nonnormality in smallsamples. The e!ects of nonnormality, however, are much reduced. In particular,the following points can be observed. First, the problem of over-rejectiondiminishes as the sample size ¹ gets bigger. This is in contrast to the e!ects of¹ on over-rejection for the IM test. Second, there is no signi"cant di!erence inthe degree of over-rejection for models with di!erent levels of correlation. Third,as expected, models with errors generated from t

12are less susceptible to

over-rejection than models with t8

errors. Overall, for sample size of 1000 theLM test is quite robust against nonnormality.

6. Conclusions

We have introduced a LM test for the constant-correlation hypothesis ina multivariate GARCH model. The test requires only estimates of the constant-correlation model and is computationally convenient. We examine the "nite-sample properties of the test and compare it against a recent test suggested byBera and Kim (1996) for the bivariate case. Our Monte Carlo experiments showthat the tests have the appropriate size for sample size of 1000 or above, which isoften available for studies involving "nancial data. While the LM test has goodpower against the alternative models considered, the IM test is found to havevery low power. In applications to real data sets, the IM test rejects theconstant-correlation hypothesis with very low p values. We attribute this "ndingto the nonnormality in the real data. As our Monte Carlo results show, the IMtest leads to serious over-rejection when the error distribution has thick tails.In contrast, the LM test is quite robust against nonnormality. Though the LMtest is also found to over-reject the null hypothesis in smaller samples, theproblem of over-rejection appears to be diminishing with increases in the samplesize.

Acknowledgements

This research was supported by the National University of Singapore Aca-demic Research Grant RP-3981003. Comments from Anil Bera, Albert Tsui andthe anonymous referees are gratefully acknowledged. Any remaining errors andshortcomings are, of course, mine only.

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