A test for constant correlations in a multivariate GARCH model

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: [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

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variances of economic data in the univariate case has motivated many researchers to extend these models to the multivariate dimension. There are many examples in which empirical multivariate models of conditional heteroscedasticity can be used fruitfully. An illustrative list includes the following: model the changing variance structure in an exchange rate regime (Bollerslev, 1990), calculate the optimal debt portfolio in multiple currencies (Kroner and Claessens, 1991), evaluate the multiperiod hedge ratios of currency futures (Lien and Luo, 1994), examine the international transmission of stock returns and volatility (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 multivariate GARCH model. They extended the GARCH representation in the univariate case to the vectorized conditional-variance matrix. While this so-called vech representation is very general, empirical applications would require further restrictions and more speci"c structures. A popular member of the vech-representation family is the diagonal form. Under the diagonal form, each variance}covariance term is postulated to follow a GARCH-type equation with the lagged variance}covariance term and the product of the corresponding lagged residuals as the right-hand-side variables in the conditional(co)variance equation. An advantage of this formulation is that the intuition of the GARCH model, which has been found to be very successful, is formally adhered to. It is often di$cult to verify the condition that the conditional-variance matrix of an estimated multivariate GARCH model is positive de"nite. Furthermore, such conditions are often very di$cult to impose during the optimisation of the log-likelihood function. However, if we postulate the simple assumption that the correlations are time invariant, these di$culties nicely disappear. Bollerslev (1990) pointed out that under the assumption of constant correlations, the maximum likelihood estimate (MLE) of the correlation matrix is equal to the sample correlation matrix. When the correlation matrix is concentrated out of the log-likelihood function further simpli"cation is achieved in the optimisation. As the sample correlation matrix is always positive de"nite, the optimisation will not fail as long as the conditional variances are positive. Recently, Engle and Kroner (1995) proposed a class of multivariate conditional heteroscedasticity models called the BEKK (named after Baba, Engle, Kraft and Kroner) model. The motivation is to ensure the condition of a positive-de"nite conditional-variance matrix in the process of optimisation. Engle and Kroner provided some theoretical analysis of the BEKK model and related it to the vech-representation form. Another approach examines the conditional variance as a factor model. The works by Diebold and Nerlove (1989), Engel and Rodrigues (1989) and Engle et al. (1990) were along this line. One disadvantage of the BEKK and factor models is that the parameters cannot be easily interpreted, and their net e!ects on the future variances and covariances are not

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readily seen. In other words, the intuitions of the e!ects of the parameters in a univariate GARCH equation are lost. Due to its computational simplicity, the constant-correlation GARCH model is very popular among empirical researchers. Empirical research that uses this model includes: Bollerslev (1990), Kroner and Claessens (1991), Kroner and Sultan (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 and seldom analysed or tested. A notable exception, however, is the work by Bera and Kim (1996). Bera and Kim suggested an Information Matrix (IM) test for the constant-correlation hypothesis in a bivariate GARCH model and applied the test to examine the correlation across national stock markets. Second, the e!ects of the assumption on the conditional-variance estimates are rarely considered. In other words, Are the estimates of the parameters in the conditionalvariance equations robust with respect to the constant-correlation assumption? In this paper we focus on the "rst question. Our objective is to provide a convenient test (without having to estimate an encompassing model) for the constant-correlation assumption and examine the properties of the test in small samples. Bollerslev (1990) suggested some diagnostics for the constant-correlation multivariate GARCH model. He computed the Ljung}Box portmanteau statistic on the cross products of the standardised residuals across di!erent equations. Critical values were based on the s distribution. Another diagnostic was based on the regression involving the products of the standardised residuals. It was, however, pointed out by Li and Mak (1994) that the portmanteau statistic is not asymptotically a s and the use of a s approximation is inappropriate. For the residual-based diagnostics, there are usually no su$cient guidelines as to the choice of regressors in the arti"cial regression. Furthermore, the optimality of the portmanteau and residual-based tests is not established. We propose a test for the constant-correlation hypothesis based on the Lagrange Multiplier (LM) approach. We extend the constant-correlation model to one in which the correlations are allowed to be time varying. When certain key parameters in the extended model are imposed to be zero, the constantcorrelation model is obtained. We consider the LM test for the zero restrictions on the key parameters. Finite-sample properties of the LM test are examined using Monte Carlo methods. The LM test is compared against the IM test due to Bera and Kim (1996). We "nd that the LM test has good approximate nominal size in sample sizes of 1000 or above. It is powerful against the alternative models with time-varying correlations considered. On the other hand, while the IM test has good approximate nominal size, it lacks power. Empirical illustrations using real data, however, show that the IM test rejects the constant-correlation hypothesis vehemently with very low p values. To explain this anomaly, we examine the behaviour of the tests under

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nonnormality. It is found that while the Monte Carlo results show that the IM test leads to gross over-rejection when the errors are nonnormal, the LM test is relatively robust against nonnormality. The plan of the rest of the paper is as follows. In Section 2 we derive the LM statistic. Some Monte Carlo results on the "nite-sample distributions of the LM and IM tests are reported in Section 3. Section 4 describes some illustrative examples using real data. In Section 5 we examine the e!ects of nonnormality on the tests. Finally, we give some concluding remarks in Section 6.

2. The test statistic Consider a multivariate time series of observations +y ,, t"1,2, ¹, with R K elements each, so that y "(y ,2, y ). To focus on the conditional heteroR R )R scedasticity of the time series, we assume that the observations are of zero (or known) means. This assumption simpli"es tremendously the discussions without straining the notations. The conditional variance of y is assumed to follow the time-varying structure R given by Var(y "U )"X , R R\ R where U is the information set at time t. We denote the variance elements of R X by p , for i"1,2, K, and the covariance elements by p , where R GR GHR 1)i(j)K. Following Bollerslev (1990), we consider the constant-correlation model in which the conditional variances of y follow a GARCH process, while GR the correlations are constant. Denoting C"+o , as the correlation matrix, we GH have p "u #a p #b y , GR G G GR\ G GR\ p "o p p , 1)i(j)K. GHR GH GR HR

i"1,2, K

(1) (2)

We assume that u , a and b are nonnegative, a #b (1, for i"1,2, K and G G G G G C is positive de"nite. Although the conditional variances in the above equations are assumed to follow low-order GARCH(1, 1) processes, the test derived below can be extended to the general GARCH(p, q) models without di$culties. As pointed out by Bollerslev (1990), the constant-correlation model is computationally attractive. Speci"cally, the MLE of the correlation matrix is equal to the sample correlation matrix of the standardised residuals, which is always positive de"nite. The correlation matrix can be further concentrated out from the log-likelihood function, resulting in a reduction in the number of parameters to be optimised. Furthermore, it is relatively easy to control the parameters of the conditional-variance equations during the optimisation so that p are GR

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always positive. On the other hand, it is very di$cult to control a matrix of parameters to be positive de"nite during the optimisation. To test for the validity of the constant-correlation assumption, we extend the above framework to include time-varying correlations. Under some restrictions on the parameter values of the extended model the constant-correlation model is derived. The LM test can then be applied to test for the restrictions. This approach only requires estimates under the constant-correlation model, and can thus conveniently exploit the computational simplicity of the model. To allow for time-varying correlations, we consider the following equations for the correlations: o "o #d y y , (3) GHR GH GH GR\ HR\ where d for 1)i(j)K are additional parameters in the extended model. GH Thus, the correlations are assumed to respond to the products of previous observations. From (3) the conditional covariances are given by p "o p p . (4) GHR GHR GR HR Note that there are N"K#2K parameters in the extended model with time-varying correlations. The constant-correlation hypothesis can be tested by examining the hypothesis H : d "0, for 1)i(j)K. Under H , there are  GH  M"K(K!1)/2 independent restrictions. It should be pointed out that (3) is speci"ed as a convenient alternative that encompasses the constant-correlation model. To ensure that the alternative model provides well-de"ned positive-de"nite conditional-variance matrices, further restrictions have to be imposed on the parameters d . As in the case of the GH general vech speci"cation, such restrictions are very di$cult to derive. Indeed, empirical research using the vech speci"cation often leaves the issue as an empirical problem to be resolved in the optimisation stage. As our interest is in the model under the null H , we shall not pursue the issue of searching for the  necessary restrictions. Thus, we assume that within a neighbourhood of d "0, GH the optimal properties of the LM test (such as its asymptotic e$ciency against local alternatives) hold under some regularity conditions as stated in, for example, Godfrey (1988). As correlations are standardised measures, it might be arguable to allow the correlations to depend on the products of the lagged standardised residuals instead. Thus, if we de"ne e "y /p as the standardised residual, an alternative GR GR GR model might be written as o "o #d e e . (5) GHR GH GH GR\ HR\ As e depends on other parameters of the model through p , analytic derivation GR GR of the LM statistic is intractable. If indeed (5) describes the true model, there may 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

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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 an empirical question. We shall examine its performance using Monte Carlo methods. Now we shall proceed to derive the LM statistic of H under the above  framework. We denote D as the diagonal matrix with diagonal elements given by p , and R GR C "+o , as the time-varying correlation matrix. Hence the conditional-variR GHR ance matrix of y is given by X "D C D . Under the normality assumption the R R R R R conditional log-likelihood of the observation at time t is given by (the constant term is ignored) 1 1 l "! ln "D C D "! y D\C\D\y R R R R R R R 2 2 R R 1 1 ) 1 "! ln "C "! ln p ! y D\C\D\y , R R R R GR 2 R R 2 2 G for t"1,2, ¹, and the log-likelihood function l is given by l" 2 l . For R R simplicity, we have assumed that y and p are "xed and known. This G G assumption has no e!ects on the asymptotic distributions of the LM statistic. Note that D\y represents the standardised observations with unit variance. R R We denote D\y "e "(e ,2,e ). R R R R )R We now de"ne the following derivatives of p with respect to u , a and b for GR G G G i"1,2, K, d "*p /*u , GR GR G

e "*p /*a , f "*p /*b . GR GR G GR GR G

To calculate these derivatives, the following recursions may be used: d "1#a d , GR G GR\ e "p #a e , GR GR\ G GR\ f "a f #y , GR G GR\ GR\

(6)

where the starting values are given by d "1, e "p and f "y . G G G G G The "rst partial derivatives of l with respect to the model parameters are R given by *l (eHe !1)d R " GR GR GR , *u 2p G GR *l (eHe !1)e R " GR GR GR , *a 2p G GR

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*l (eHe !1) f R " GR GR GR , *b 2p G GR *l R "eHeH !oGH, GR HR R *o GH *l R "(eHeH !oGH)y y , GR HR R GR\ HR\ *d GH

(7)

where eH"(eH ,2,eH )"C\e , and C\"+oGH,. Thus, if we denote the paraR R )R R R R R meters of the model as h"(u , a , b , u ,2, b , o , o ,2,o ,     )   )\ ) d ,2,d ), we can calculate *l /*h from the above equations. These  )\ ) R analytic derivatives can facilitate the evaluation of the MLE of the extended model if desired. Note that on H , C "C for all t, so that eH"C\e and  R R R oGH"oGH. In this case, e are just the standardised residuals calculated from the R R algorithm suggested by Bollerslev (1990). We shall denote hK as the MLE of h under H .  If we denote s as the N-element score vector given by s"*l/*h and < as the N;N information matrix given by <"E(!*l/*h*h), where E(.) denotes the expectation operator, the LM statistic for H is given by s( 
(8) (9)

where l is the ¹;1 column vector of ones and SK is S evaluated at hK . Under the usual regularity conditions LMC is asymptotically distributed as a s . Eq. (9) + shows that LMC can be interpreted as ¹ times R, where R is the uncentered coe$cient of determination of the regression of l on SK . It is well-known that other forms of the LM statistic are available. For example, further simpli"cation can be obtained by making use of the fact that in SK l the elements corresponding to the unrestricted parameters are zero. Eq. (9), however, is a convenient form and will be used throughout this paper. We now discuss some extensions of the above framework. First, when GARCH(p, q) models are considered we need to augment the parameters of the conditional-variance equations. Thus, we denote the coe$cients of the lagged conditional variances as a , h"1,2, p, and the coe$cients of the lagged GF y terms as b , k"1,2, q. We write the conditional-variance equations as GR GI

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(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.) N O p "u # a p # b y , i"1,2, K. GR G GF GR\F GI GR\I F I The partial derivatives *l /*u , *l /*a and *l /*b are required. We "rst R G R GF R GI calculate the partial derivatives of p with respect to u , a and b , which will GR G GF GI be denoted as d , e and f , respectively. These partial derivatives can be GR GFR GIR calculated by recursions constructed as in (6). Speci"cally, we have N d "1# a d , GR GF GR\F F N e "p # a e , GFR GR\F GFY GFR\FY FY N f " a f #y . GIR GF GIR\F GR\I F The "rst partial derivatives of l with respect to u , a and b (p#q#1 R G GF GI derivatives altogether) can be calculated using (7), with e and f replacing GFR GIR e and f , respectively. GR GR Second, when the conditional mean of y depends on some unknown paraR meters, S would have to be augmented by the partial derivatives of l with R respect to the parameters appearing in the conditional-mean equations. If the unknown means are linear in the parameters, the partial derivatives are straightforward to obtain. Indeed, in such cases the asymptotic variance matrix of the MLE are block diagonal with respect to the conditional-mean and conditionalvariance parameters (see Bera and Higgins, 1993). Thus, under such circumstances, LMC calculated from (9) using only the conditional-variance parameters is asymptotically valid.

3. Monte Carlo results In this section we report some Monte Carlo results on the empirical size and power of LMC in "nite samples. In addition to the LM test, we examine the IM test suggested by Bera and Kim (1996). The IM statistic, denoted by IMC, tests for the hypothesis of constant correlation in a bivariate GARCH model. Denoting o( as the MLE of the (constant) correlation coe$cient in the bivariate GARCH model, and e( for i"1,2 as the estimated standardised residuals, IMC GR is given by [ 2 (m m !1!2o( )] IMC" R R R , 4¹(1#4o( #o( )

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where e( !o( e( R m " R R (1!o(  e( !o( e( R . m " R R (1!o(  Bera and Kim (1996) showed that under the null hypothesis of constant correlation and the assumption that e are normally distributed, IMC is asymptotically R distributed as a s .  Six experiments are considered. Experiments E1 through E4 are based on K"2, while Experiments E5 and E6 are based on K"3. The parameter values of the experiments are summarised in Panel A of Table 1. E1, E3 and E5 represent models with high correlations, while E2, E4 and E6 represent models with low correlations. Also, E1, E2, E5 and E6 represent models with relatively high persistence in the conditional variance (i.e., a#b is near to 1), while E3 and E4 represent models with relatively low persistence. Unlike LMC, which is applicable 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 it should be possible to extend the IMC test to beyond the bivariate case, such extension will not be pursued in this paper. For each experiment we generate 2000 samples of +y , based on the normality R assumption, with sample size ¹ taken to be 300, 500, 1000 and 2000. We estimate the parameters of the model using Bollerslev's (1990) algorithm and calculate the LMC and IMC statistics. Panel B of Table 1 summarises the empirical sizes of the LM and IM tests assuming a nominal size of 5%. For LMC there are signs of over-rejection in small samples. Over-rejection, however, seems to have satisfactorily reduced when the sample size reaches 1000. Except for E5, the point estimates of the empirical size when ¹ is 1000 or above never exceed the nominal size by more than 1.5%. The correlations seem to play a role in determining the rate of convergence to the nominal size. Models with low correlations are less subject to over-rejection in small samples. On the other hand, the persistence of the conditional variance does not have much e!ect on the degree of over-rejection. The IM test appears to have very good approximate nominal sizes. There are signs of slight under-rejection in small samples. This problem, however, has largely disappeared when the sample size reaches 1000. Panel C reports the Monte Carlo sample means of the LMC and IMC statistics. For E1 through E4, these values are expected to converge to 1. For E5 and E6, the mean of the LMC statistic is expected to converge to 3. As can be observed, the results re#ect the over-rejection of LMC and the under-rejection of IMC. In summary, for sample sizes of 1000 or above, the LM and IM tests

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Table 1 Parameter values of the Monte Carlo experiments and the estimated rejection probabilities and sample means of LMC and IMC Experiment E1 Panel A: True parameter values u 0.40  a 0.80  b 0.15  u 0.20  a 0.70  b 0.20  u *  a *  b *  o 0.80  o *  o *  Panel B: Estimated probabilities (%) of

E2

E3

E4

0.40 0.80 0.15 0.20 0.70 0.20 * * * 0.20 * *

0.40 0.40 0.30 0.20 0.50 0.20 * * * 0.80 * *

0.40 0.40 0.30 0.20 0.50 0.20 * * * 0.20 * *

E5

E6

0.30 0.80 0.10 0.40 0.60 0.25 0.50 0.80 0.15 0.60 0.70 0.80

0.30 0.80 0.10 0.40 0.60 0.25 0.50 0.80 0.15 0.20 0.20 0.20

the Type-1 errors of the tests

LMC

¹"300 ¹"500 ¹"1000 ¹"2000

7.75 6.85 6.40 6.15

6.90 6.40 5.20 4.90

8.05 7.65 6.45 4.95

6.90 6.25 5.40 5.95

10.20 8.05 7.35 7.10

9.75 8.05 6.30 5.55

IMC

¹"300 ¹"500 ¹"1000 ¹"2000

3.65 4.20 4.20 5.30

4.20 4.10 5.20 4.20

3.75 4.40 4.15 5.30

4.50 3.70 5.10 4.50

* * * *

* * * *

Panel C: Monte Carlo sample means of the test statistics LMC

¹"300 ¹"500 ¹"1000 ¹"2000

1.287 1.182 1.148 1.071

1.193 1.094 1.059 1.005

1.249 1.214 1.129 1.025

1.235 1.133 1.050 1.071

IMC

¹"300 ¹"500 ¹"1000 ¹"2000

0.941 0.983 0.927 1.037

0.982 0.915 1.028 0.896

0.908 0.989 0.971 1.023

0.994 0.929 0.963 0.957

3.868 3.633 3.430 3.328 * * * *

3.774 3.453 3.162 3.114 * * * *

Note: Panel A gives the true parameter values of the Monte Carlo experiments. Panel B records the estimated probabilities of rejecting the constant-correlation hypothesis using the LMC and IMC tests at the nominal size of 5%. Panel C summarises the Monte Carlo sample means of the test statistics. 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 is 2000.

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provide reliable nominal sizes. In the case of the LM test, the reliability is enhanced when the correlations are low. To examine the power of the LM and IM tests we conduct some experiments with data generated from models with time-varying correlations. Two types of multivariate GARCH models are considered. In the "rst type, we assume that the data are generated from a BEKK model. This model has been applied in the literature by, among others, Baillie and Myers (1991) and Karolyi (1995). It has been found to "t the data satisfactorily. An advantage of the BEKK model is that the conditional-variance matrices are always positive de"nite. Undoubtedly, this is an important advantage in simulation studies. Following Engle and Kroner (1995), the conditional-variance matrix X of the model is written as R X "X#BX B#Cy y C, R R\ R\ R\ where X"+p ,, B"+b , and C"+c , are K;K parameter matrices. GH GH GH In the second type of models, we assume that the conditional variances follow a GARCH process given in (1), while the conditional correlations follow the process speci"ed in (5). A di$culty with this model is that the simulated correlation matrix is not guaranteed to be positive de"nite. This problem has to be controlled by setting the parameters d to be su$ciently small. It should be GH noted that it is not our intention to maintain that these two types of models represent processes that are descriptive of real data. Our objective is to examine the power of the tests under some alternatives. We consider two bivariate BEKK models denoted by P1 and P2, and one trivariate BEKK model denoted by P3. The model parameters are summarised in Panel A of Table 2. We consider sample sizes ¹ of 300, 500 and 1000. Based on Monte Carlo samples of 1000 each, we estimate the power of LMC and IMC against the time-varying conditional-correlation models at nominal size of 5%. As a measure of the variability of the conditional correlation coe$cients, we calculate the range (i.e., maximum!minimum) of the conditional correlation coe$cients in each simulated sample of ¹ observations. Panel B summarises the maximum and minimum ranges of the conditional correlation coe$cients in the Monte Carlo samples of 1000. It can be seen that P2 has larger variability in correlations than P1. The LM test is found to have high power in all experiments. It is quite remarkable that for Experiment P1, for which the range of +o , appears to be quite small, LMC has yet very good power. For example, R for P1 when ¹"1000, the maximum range of +o , is only 0.396 and yet the R empirical power of LMC is 91.3%. In contrast, IMC has very weak power in all cases. Indeed, the empirical power of IMC is less than 10% in all cases except for P2 with ¹"1000. For alternative models of the second type we consider two bivariate models denoted by Q1 and Q2, and one trivariate model denoted by Q3. The model parameters are presented in Panel A of Table 3. The values of d are set after GH some experimentation to generate well-de"ned conditional-correlation matrices

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Table 2 Estimated power of LMC and IMC when the true model is BEKK Experiment P1

P2

P3

Panel A: True parameter values p 0.20 0.20 0.80  p 0.10 0.04 0.20  p * * 0.20  p 0.20 0.20 0.80  p * * 0.20  p * * 0.80  b 0.60 0.40 0.50  b 0.20 0.20 0.20  b * * 0.20  b 0.60 0.40 0.50  b * * 0.20  b * * 0.50  c 0.30 0.40 0.30  c 0.10 0.20 0.10  c * * 0.10  c 0.30 0.40 0.30  c * * 0.10  c * * 0.30  Panel B: Maximum/minimum ranges of the correlation coezcients in the simulated samples and the empirical 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 +o , GHR +o , Max R Min +o , Max R Min +o , Max R Min

0.498 0.174 * * * *

0.401 0.188 * * * *

0.396 0.206 * * * *

0.907 0.489 * * * *

0.833 0.528 * * * *

0.861 0.573 * * * *

0.349 0.121 0.349 0.129 0.346 0.136

0.365 0.155 0.362 0.153 0.367 0.148

0.347 0.166 0.356 0.165 0.340 0.169

Power of tests (%) LMC 56.7 IMC 6.0

73.5 5.9

91.3 9.0

86.8 7.6

97.7 9.6

100.0 16.0

56.0 *

49.9 *

52.8 *

Note: Panel A gives the true parameter values of the Monte Carlo experiments. P1 and P2 are bivariate models, while P3 is trivariate. The parameter matrices B and C are both symmetric. Panel B records the maxima and mimina of the ranges of the conditional correlation coe$cients +o , in GHR the simulated samples. It also gives the estimated probabilities of rejecting the constant-correlation hypothesis using the LMC and IMC statistics at the asymptotic nominal size of 5%. The Monte Carlo sample size is 1000.

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Table 3 Estimated power of LMC and IMC when the correlation coe$cients are generated from Eq. (5) Experiment Q1

Q2

Q3

Panel A: True parameter values u 0.20 0.20 0.20  a 0.80 0.80 0.60  b 0.10 0.10 0.20  u 0.20 0.20 0.20  a 0.80 0.80 0.60  b 0.10 0.10 0.20  u * * 0.20  a * * 0.60  b * * 0.20  o 0.40 0.10 0.20  o * * 0.20  o * * 0.20  d 0.03 0.06 0.03  d * * 0.03  d * * 0.03  Panel B: Maximum/minimum ranges of the correlation coezcients in the simulated samples and the empirical 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 +o , GHR +o , Max 0.514 R Min 0.149 +o , Max * R Min * +o , Max * R Min *

0.609 0.194 * * * *

0.615 0.214 * * * *

1.007 0.308 * * * *

1.171 0.394 * * * *

1.068 0.424 * * * *

0.566 0.159 0.534 0.155 0.467 0.163

0.517 0.179 0.547 0.186 0.483 0.197

0.548 0.217 0.573 0.212 0.597 0.228

Power of tests (%) LMC 15.2 IMC 4.6

23.1 4.0

29.4 5.4

21.4 3.7

28.7 6.7

47.1 5.0

14.7 *

19.3 *

30.3 *

Note: Panel A gives the true parameter values of the Monte Carlo experiments. Q1 and Q2 are bivariate models, and Q3 is a trivariate model. Panel B records the maximum and miminum of the ranges of the conditional correlation coe$cients +o , in the simulated samples. It also gives the GHR estimated probabilities of rejecting the constant-correlation hypothesis using the LMC and IMC statistics 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 the correlation 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

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¹"1000, which has an estimate of only 47.1%. As for the IM test, the empirical power is again found to be very low. Indeed, among the cases considered the maximum empirical power achieved is only 6.7%. To compare the characteristics of the time-varying correlations in the two types of models, we examine the paths of the correlation coe$cients generated. Figs. 1 and 2 present the plots of two simulated paths of conditional correlation coe$cients based on P1 and Q1, respectively, for ¹"300. It is obvious that the two models produce very di!erent time-varying correlation structures. Fig. 2 exhibits a sample path of serially uncorrelated conditional correlation coe$cients. This is due to the fact that e e are serially uncorrelated. On the R\ R\ other hand, Fig. 1 is characteristic of a sample path of a serially correlated time series. The Box}Pierce Q statistics based on the "rst 10 lagged serial correlation coe$cients of the sample paths are 72.95 for the path in Fig. 1 and 8.89 for the path in Fig. 2. Indeed, the autocorrelation coe$cients of the sample path of o generated from P1 are declining very slowly, which is characteristic of a R long-memory time series. Thus, while Q1 generates serially uncorrelated timevarying correlations, P1 generates series of correlation coe$cients with long memory. As y y are speci"ed as the covariates for tracking the variGR\ HR\ ations in the correlation coe$cients in the LM test, this may help to explain the di!erence in the power of the LM test found in the two types of time-varying correlation 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

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

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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. Nonetheless, 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 sets selected are described as follows (all returns, in percentage, are measured as logarithmic di!erences):

Data

Description of variables

Data period and frequencies

DS1

Return observations of the spot index (S) and futures price (F) of the Nikkei Stock Averge 225 Return observations of 3 Asian currencies, namely, Japanese yen (J), Malaysian ringgit (M) and Singapore dollar (S). The rates are quoted against the US dollar Return observations of 3 Asian stock markets, namely, Hong Kong (H), Japan (J) and Singapore (S)

Daily data from 89/1 to 96/8, with 1861 observations

DS2

DS3

5-Daily data from 78/1 to 94/6, with 812 observations

Daily data from 90/1 to 94/8, with 1057 observations

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DS1 was used by Lien and Tse (1998) in studying the hedging e!ectiveness of the 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 the Asia-Paci"c stock markets. The details of the description of these data sets can be found in the above references, in which some summary statistics of these data sets are also reported. A bivariate GARCH(1,1) model with constant correlation is "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 Table 4 Estimation results of constant-correlation models Data

K

Variable

u

a

b

Correlations

LMC

IMC

DS1 (¹"1861)

2

S

0.065 (0.011) 0.055 (0.009)

0.873 (0.014) 0.896 (0.012)

0.090 (0.011) 0.076 (0.009)

o "0.941 1$ (0.003) * *

0.893 * * *

518.00H * * *

0.575 (0.239) 0.074 (0.025)

0.671 (0.115) 0.598 (0.105)

0.111 (0.038) 0.247 (0.061)

o "0.467 (+ (0.028) * *

0.004 * * *

267.75H * * *

0.648 (0.226) 0.051 (0.016)

0.631 (0.108) 0.714 (0.067)

0.124 (0.040) 0.153 (0.034)

o "0.537 (1 (0.025) * *

0.186 * * *

180.53H * * *

0.093 (0.029) 0.058 (0.016)

0.536 (0.115) 0.688 (0.065)

0.274 (0.068) 0.162 (0.032)

o "0.611 +1 (0.022) * *

3.235 * * *

205.18H * * *

0.605 (0.217) 0.076 (0.024) 0.046 (0.013)

0.647 (0.105) 0.595 (0.096) 0.741 (0.051)

0.126 (0.039) 0.245 (0.055) 0.139 (0.027)

o "0.468 (+ (0.028) o "0.538 (1 (0.025) o "0.612 +1 (0.022)

1.273 * * * * *

* * * * * *

0.399 (0.141) 0.148 (0.037)

0.619 (0.111) 0.807 (0.029)

0.215 (0.063) 0.147 (0.025)

o "0.254 &( (0.029) * *

2.427 * * *

103.32H * * *

0.259 (0.146) 0.208 (0.041)

0.752 (0.115) 0.543 (0.067)

0.132 (0.054) 0.257 (0.045)

o "0.436 &1 (0.025) * *

16.547H * * *

30.98H * * *

F DS2 (¹"812)

2

J M

2

J S

2

M S

3

J M S

DS3 (¹"1057)

2

H J

2

H S

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Table 4. Continued Data

K

Variable

u

a

b

Correlations

LMC

IMC

2

J

0.142 (0.036) 0.229 (0.044)

0.816 (0.028) 0.474 (0.073)

0.139 (0.023) 0.326 (0.055)

o "0.345 (1 (0.027) * *

15.362H * * *

22.31H * * *

0.212 (0.104) 0.145 (0.036) 0.217 (0.042)

0.787 (0.085) 0.818 (0.028) 0.535 (0.068)

0.112 (0.043) 0.135 (0.023) 0.253 (0.044)

o "0.258 &( (0.029) o "0.437 &1 (0.025) o "0.347 (1 (0.027)

17.665H * * * * *

* * * * * *

S 3

H J S

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 Singapore dollar (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 as a s , 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 IMC statistics. For DS1, the estimated correlation is 0.941, re#ecting a high degree of co-movements between the spot and futures. This is perhaps not surprising, given the arbitrage-free pricing in an e$cient market. The LMC statistic has a low value of 0.893. Thus, not only is the correlation high, the stability of the relationship is also veri"ed (the evidence provided by IMC will be discussed below). For DS2, the correlations between the currency exchanges are much lower, all in the region of 0.45 to 0.65. The LMC for the pairwise models and the trivariate model are insigni"cant at the 5% level. Thus, there is no evidence against time-invariant correlations among the selected Asian currencies. The results for DS3 are, however, quite di!erent. We observe that the correlations 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 correlations between Hong Kong/Singapore, Japan/Singapore and the trivariate case. Stock market performance is dependent, to a large extent, on domestic factors. If these factors have low correlations and/or their relationships are not stable over time, we would expect the correlations of market returns to be time varying. To further illustrate the use of the test for K"4, we append the Australian market onto DS3. The results of the four Asia-Paci"c markets are recorded in Table 5. Not surprisingly, we "nd evidence against constant

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

A

H

J

S

u

0.049 (0.025) 0.900 (0.038) 0.045 (0.044) 1.000 (**) 0.381 (0.026) 0.306 (0.028) 0.385 (0.026)

0.196 (0.081) 0.803 (0.066) 0.110 (0.034) * * 1.000 (**) 0.259 (0.029) 0.439 (0.025)

0.144 (0.035) 0.822 (0.027) 0.131 (0.022) * * * * 1.000 (**) 0.348 (0.027)

0.222 (0.043) 0.537 (0.070) 0.241 (0.043) * * * * * * 1.000 (**)

a b o G  o G & oG ( oG 1

Note: The data set consists of the daily returns (1057 observations) of four Asia}Paci"c stock markets, where A"Australia, H"Hong Kong, J"Japan and S"Singapore. The Lagrange multiplier statistic, LMC, for testing constant correlations is 20.265. The LMC, which is asymptotically distributed as a s , 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 the 1% level. In summary, depending on the data sets analysed, we have found evidence for and against the assumption of constant correlations based on the LMC statistic. In particular, we have found evidence against constant correlations across the selected Asia}Paci"c stock markets. In view of this evidence it would be important to study models that admit time-varying correlations. In any case, the hypothesis of constant correlations should be tested before the empirical multivariate GARCH models can be used for inference and its economic implications are drawn. In contrast, the results for the IMC statistic are very di!erent. The constantcorrelation assumption is vehemently rejected with very low p values. As the Monte Carlo results in Section 3 show that IMC has very low power in rejecting the constant-correlation hypothesis, this result is rather unusual. One possible explanation for this anomaly is that the rejection of IMC for the real data sets is due to other assumptions imposed in the derivation of the IM test. As normality is 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 the IM test. Indeed, the data used in Table 4, as in many other "nancial data, exhibit

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

5. The e4ects of nonnormality To examine the e!ects of nonnormality on the LM and IM tests, we conduct further Monte Carlo experiments. We use the experimental setups for the bivariate models in Table 1, namely, Experiments E1 through E4. Errors are then generated from nonnormal distributions. We consider t distributions with 8 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 and IM 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, overrejection appears to be more serious for E1 and E3 as compared against E2 and E4. Thus, models with higher levels of correlation are more prone to overrejection. When the two nonnormal distributions are compared, over-rejection is less serious for t as compared against t .   Table 6 Empirical size of LMC and IMC when residuals are nonnormal Error distribution

Sample size ¹

Test LMC

t  t



IMC

t  t



300 500 1000 300 500 1000 300 500 1000 300 500 1000

Experiment E1

E2

E3

E4

11.1 8.7 7.4 10.2 9.6 6.5 61.4 78.7 97.3 35.2 48.6 74.8

8.0 6.3 6.1 9.2 9.1 7.3 17.3 17.9 22.4 10.1 10.8 16.9

9.9 9.8 9.2 10.1 7.6 7.7 58.3 76.4 96.1 37.4 47.6 77.2

8.4 7.4 5.9 10.2 6.2 5.7 14.8 16.7 25.5 11.9 9.1 13.3

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

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The LM test also demonstrates over-rejection under nonnormality in small samples. The e!ects of nonnormality, however, are much reduced. In particular, the following points can be observed. First, the problem of over-rejection diminishes as the sample size ¹ gets bigger. This is in contrast to the e!ects of ¹ on over-rejection for the IM test. Second, there is no signi"cant di!erence in the degree of over-rejection for models with di!erent levels of correlation. Third, as expected, models with errors generated from t are less susceptible to  over-rejection than models with t errors. Overall, for sample size of 1000 the  LM test is quite robust against nonnormality.

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

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

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