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).
variances of economic data in the univariate case has motivated many re-searchers to extend these models to the multivariate dimension. There are many examples in which empirical multivariate models of conditional heteroscedastic-ity 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 Claes-sens, 1991), evaluate the multiperiod hedge ratios of currency futures (Lien and Luo, 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 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-repres-entation family is the diagonal form. Under the diagonal form, each vari-ance}covariance term is postulated to follow a GARCH-type equation with the 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 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.
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 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 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 statis-tic on the cross products of the standardised residuals across di!erent equations. Critical values were based on thes2distribution. 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 as2and the use of as2approximation 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 constant-correlation 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 approxim-ate 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
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 My
tN, t"1,2,¹, with Kelements each, so thaty
t"(y1t,2,yKt)@. To focus on the conditional
hetero-scedasticity of the time series, we assume that the observations are of zero (or known) means. This assumption simpli"es tremendously the discussions with-out straining the notations.
The conditional variance ofy
tis assumed to follow the time-varying structure
given by
Var(y
tDUt~1)"Xt,
whereU
t is the information set at timet. We denote the variance elements of
X
t by p2it, for i"1,2,K, and the covariance elements by pijt, where
1)i(j)K. Following Bollerslev (1990), we consider the constant-correlation
model in which the conditional variances ofy
itfollow a GARCH process, while
the correlations are constant. DenotingC"Mo
ijNas the correlation matrix, we
have
p2
it"ui#aip2i,t~1#biy2i,t~1, i"1,2,K (1)
p
ijt"oijpitpjt, 1)i(j)K. (2)
We assume thatu
i, ai andbi are nonnegative,ai#bi(1, fori"1,2,Kand
Cis 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$culties.
As pointed out by Bollerslev (1990), the constant-correlation model is com-putationally attractive. Speci"cally, the MLE of the correlation matrix is equal to the sample correlation matrix of the standardised residuals, which is always positive de"nite. The correlation matrix can be further concentrated out from the log-likelihood function, resulting in a reduction in the number of parameters to be optimised. Furthermore, it is relatively easy to control the parameters of the conditional-variance equations during the optimisation so that p2
always positive. On the other hand, it is very di$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:
oijt"o
ij#dijyi,t~1yj,t~1, (3)
wheredij for 1)i(j)K are additional parameters in the extended model. Thus, the correlations are assumed to respond to the products of previous observations. From (3) the conditional covariances are given by
p
ijt"oijtpitpjt. (4)
Note that there are N"K2#2K parameters in the extended model with time-varying correlations. The constant-correlation hypothesis can be tested by examining the hypothesis H
0: dij"0, for 1)i(j)K. Under H0, 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 alternative model provides well-de"ned positive-de"nite conditional-variance matrices, fur-ther restrictions have to be imposed on the parametersdij. As in the case of the general vech speci"cation, such restrictions are very di$cult to derive. Indeed, empirical research using the vech speci"cation often leaves the issue as an empirical problem to be resolved in the optimisation stage. As our interest is in the model under the null H
0, we shall not pursue the issue of searching for the
necessary restrictions. Thus, we assume that within a neighbourhood ofd
ij"0,
the optimal properties of the LM test (such as its asymptotic e$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"nee
it"yit/pitas the standardised residual, an alternative
model might be written as
oijt"o
ij#d@ijei,t~1ej,t~1. (5)
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
0under the above
framework. We denoteD
tas the diagonal matrix with diagonal elements given bypit, and
C
t"MoijtNas the time-varying correlation matrix. Hence the
conditional-vari-ance matrix ofy
tis given byXt"DtCtDt. Under the normality assumption the
conditional log-likelihood of the observation at timetis given by (the constant term is ignored)
simplicity, we have assumed that y
i0 and p2i0 are "xed and known. This
assumption has no e!ects on the asymptotic distributions of the LM statistic. Note that D~1
t yt represents the standardised observations with unit variance.
We denoteD~1
t yt"et"(e1t,2,eKt)@.
We now de"ne the following derivatives ofp2
itwith respect toui,aiandbifor
i"1,2,K,
d
it"Lp2it/Lui, eit"Lpit2/Lai, fit"Lp2it/Lbi.
To calculate these derivatives, the following recursions may be used:
d
where the starting values are given byd
i1"1,ei1"p2i0andfi1"y2i0.
The"rst partial derivatives ofl
t with respect to the model parameters are
Ll
analytic derivatives can facilitate the evaluation of the MLE of the extended model if desired. Note that on H
0, Ct"C for all t, so that eHt"C~1et and
oijt"oij. In this case,e
t are just the standardised residuals calculated from the
algorithm suggested by Bollerslev (1990). We shall denote hK as the MLE of hunder H
0.
If we denotesas theN-element score vector given bys"Ll/Lhand<as the
N]Ninformation matrix given by<"E(!L2l/LhLh@), where E(.) denotes the expectation operator, the LM statistic for H
0is given bys(@<K ~1s(, where the hats
denote evaluation athK. 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 ofl
t. To this e!ect, we denoteSas the¹]Nmatrix the rows of
which are the partial derivativesLl
t/Lh@, fort"1,2,¹. Thus, the LM statistic
for H
0, 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)
wherelis the¹]1 column vector of ones andSK isSevaluated athK. Under the usual regularity conditions LMC is asymptotically distributed as as2
M. Eq. (9)
shows that LMC can be interpreted as¹timesR2, whereR2is the uncentered coe$cient of determination of the regression of l onSK. It is well-known that other forms of the LM statistic are available. For example, further simpli"cation can be obtained by making use of the fact that inSK@lthe elements corresponding to the unrestricted parameters are zero. Eq. (9), however, is a convenient form and will be used throughout this paper.
We now discuss some extensions of the above framework. First, when GARCH(p,q) models are considered we need to augment the parameters of the conditional-variance equations. Thus, we denote the coe$cients of the lagged conditional variances as a
ih, h"1,2,p, and the coe$cients of the lagged y2
(Note that we may allow the orders of the GARCH processes to vary with i. Thus, (p,q) should be regarded as the generic order.)
p2
The partial derivativesLl
t/Lui, Llt/Laih and Llt/Lbik are required. We "rst
calculate the partial derivatives ofp2
itwith respect toui,aih andbik, which will
be denoted as d
it, eiht and fikt, respectively. These partial derivatives can be
calculated by recursions constructed as in (6). Speci"cally, we have
d
The "rst partial derivatives of l
t with respect to ui, aih and bik (p#q#1
derivatives altogether) can be calculated using (7), with e
iht and fikt replacing e
itSecond, when the conditional mean ofandfit, respectively. y
t depends on some unknown
para-meters, S would have to be augmented by the partial derivatives of l
t with
respect to the parameters appearing in the conditional-mean equations. If the unknown means are linear in the parameters, the partial derivatives are straight-forward to obtain. Indeed, in such cases the asymptotic variance matrix of the MLE are block diagonal with respect to the mean and conditional-variance parameters (see Bera and Higgins, 1993). Thus, under such circumstan-ces, LMC calculated from (9) using only the conditional-variance parameters is asymptotically valid.
3. Monte Carlo results
In this section we report some Monte Carlo results on the empirical size and power of LMC in"nite samples. In addition to the LM test, we examine the IM test suggested by Bera and Kim (1996). The IM statistic, denoted by IMC, tests for the hypothesis of constant correlation in a bivariate GARCH model. Denot-ing o( as the MLE of the (constant) correlation coe$cient in the bivariate GARCH model, ande(
itfori"1,2 as the estimated standardised residuals, IMC
is given by
IMC"[+Tt/1(m21tm22t!1!2o(2)]2
where
m
1t"
e(
1t!o(e(2t J1!o(2
m2t"e(2t!o(e(1t
J1!o(2.
Bera and Kim (1996) showed that under the null hypothesis of constant correla-tion and the assumpcorrela-tion thate
tare normally distributed, IMC is asymptotically
distributed as as21.
Six experiments are considered. Experiments E1 through E4 are based on
K"2, while Experiments E5 and E6 are based onK"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#bis 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 ofMy
tNbased on the normality
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 when¹is 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 approx-imate nominal sizes. There are signs of slight under-rejection in small samples. This problem, however, has largely disappeared when the sample size reaches 1000.
Table 1
Parameter values of the Monte Carlo experiments and the estimated rejection probabilities and sample means of LMC and IMC!
Experiment
E1 E2 E3 E4 E5 E6
Panel A:True parametervalues u
1 0.40 0.40 0.40 0.40 0.30 0.30
a1 0.80 0.80 0.40 0.40 0.80 0.80 b1 0.15 0.15 0.30 0.30 0.10 0.10 u2 0.20 0.20 0.20 0.20 0.40 0.40 a2 0.70 0.70 0.50 0.50 0.60 0.60 b2 0.20 0.20 0.20 0.20 0.25 0.25 u3 * * * * 0.50 0.50 a3 * * * * 0.80 0.80 b3 * * * * 0.15 0.15 o12 0.80 0.20 0.80 0.20 0.60 0.20 o13 * * * * 0.70 0.20 o23 * * * * 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 9.75
¹"500 6.85 6.40 7.65 6.25 8.05 8.05
¹"1000 6.40 5.20 6.45 5.40 7.35 6.30
¹"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 3.774
¹"500 1.182 1.094 1.214 1.133 3.633 3.453
¹"1000 1.148 1.059 1.129 1.050 3.430 3.162
¹"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 * *
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. Undoubted-ly, this is an important advantage in simulation studies. Following Engle and Kroner (1995), the conditional-variance matrixX
t of the model is written as
X
t"X#BXt~1B@#Cyt~1y@t~1C@,
whereX"Mp
ijN, B"MbijNandC"McijNareK]K parameter matrices.
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$culty with this model is that the simulated correlation matrix is not guaranteed to be positive de"nite. This problem has to be controlled by setting the parametersd@
ij to be su$ciently small. It should be
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 sizes¹of 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 of¹observations. Panel B summarises the maximum and minimum ranges of the conditional correlation coe$cients in the Monte Carlo samples of 1000. It can be seen that P2 has larger variability in correlations than P1. The LM test is found to have high power in all experi-ments. It is quite remarkable that for Experiment P1, for which the range of Mo
12tNappears to be quite small, LMC has yet very good power. For example,
for P1 when¹"1000, the maximum range ofMo
12tNis only 0.396 and yet the
empirical power of LMC is 91.3%. In contrast, IMC has very weak power in all cases. Indeed, the empirical power of IMC is less than 10% in all cases except for P2 with¹"1000.
For alternative models of the second type we consider two bivariate models denoted by Q1 and Q2, and one trivariate model denoted by Q3. The model parameters are presented in Panel A of Table 3. The values ofd@
ij are set after
Table 2
Estimated power of LMC and IMC when the true model is BEKK! Experiment
P1 P2 P3
Panel A:True parametervalues
p11 0.20 0.20 0.80
Panel B:Maximum/minimum ranges of the correlation coezcients in the simulated samples and the empirical relative frequency(in%)of rejecting the constant-correlation hypothesis of LMC and IMC Sample size¹ 300 500 1000 300 500 1000 300 500 1000 Range ofMoijtN
Mo12tNMax 0.498 0.401 0.396 0.907 0.833 0.861 0.349 0.365 0.347 Min 0.174 0.188 0.206 0.489 0.528 0.573 0.121 0.155 0.166
Mo13tNMax * * * * * * 0.349 0.362 0.356
Min * * * * * * 0.129 0.153 0.165
Mo23tNMax * * * * * * 0.346 0.367 0.340
Min * * * * * * 0.136 0.148 0.169
Power of tests (%)
LMC 56.7 73.5 91.3 86.8 97.7 100.0 56.0 49.9 52.8
IMC 6.0 5.9 9.0 7.6 9.6 16.0 * * *
Table 3
Estimated power of LMC and IMC when the correlation coe$cients are generated from Eq. (5)! Experiment
Q1 Q2 Q3
Panel A:True parametervalues
u1 0.20 0.20 0.20
a
1 0.80 0.80 0.60
b1 0.10 0.10 0.20
u2 0.20 0.20 0.20
a2 0.80 0.80 0.60
b2 0.10 0.10 0.20
u3 * * 0.20
a3 * * 0.60
b3 * * 0.20
o12 0.40 0.10 0.20
o13 * * 0.20
o23 * * 0.20
d@12 0.03 0.06 0.03
d@
13 * * 0.03
d@
23 * * 0.03
Panel B:Maximum/minimumranges 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 Size¹ 300 500 1000 300 500 1000 300 500 1000 Range ofMoijtN
Mo12tNMax 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 0.217
Mo13tNMax * * * * * * 0.534 0.547 0.573
Min * * * * * * 0.155 0.186 0.212
Mo23tNMax * * * * * * 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 30.3
IMC 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 are bivariate models, and Q3 is a trivariate model. Panel B records the maximum and miminum of the ranges of the conditional correlation coe$cientsMo
ijtNin the simulated samples. It also gives the estimated probabilities of rejecting the constant-correlation hypothesis using the LMC and IMC statistics at the asymptotic nominal size of 5%. The Monte Carlo sample size is 1000.
Fig. 1. Conditional correlations of a sample of P1,¹"300.
¹"1000, which has an estimate of only 47.1%. As for the IM test, the empirical
power is again found to be very low. Indeed, among the cases considered the maximum empirical power achieved is only 6.7%.
To compare the characteristics of the time-varying correlations in the two types of models, we examine the paths of the correlation coe$cients generated. Figs. 1 and 2 present the plots of two simulated paths of conditional correlation coe$cients based on P1 and Q1, respectively, for¹"300. It is obvious that the two models produce very di!erent time-varying correlation structures. Fig. 2 exhibits a sample path of serially uncorrelated conditional correlation coe$-cients. This is due to the fact thate
1,t~1e2,t~1are serially uncorrelated. On the
other hand, Fig. 1 is characteristic of a sample path of a serially correlated time series. The Box}PierceQstatistics 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
t generated from P1 are declining very slowly, which is characteristic of a
long-memory time series. Thus, while Q1 generates serially uncorrelated time-varying correlations, P1 generates series of correlation coe$cients with long memory. As y
i,t~1yj,t~1 are speci"ed as the covariates for tracking the
vari-ations in the correlation coe$cients in the LM test, this may help to explain the di!erence in the power of the LM test found in the two types of time-varying correlation models.
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 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) Daily data from 89/1 to 96/8, and futures price (F) of the Nikkei with 1861 observations Stock 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 observations
ringgit (M) and Singapore dollar (S). The rates are quoted against the US dollar
Table 4
Estimation 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 o
JM"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) * * *
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 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 as2M, whereM"1 forK"2 andM"3 forK"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.
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 0.196 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 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 asymp-totically distributed as as26, 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 multi-variate GARCH models can be used for inference and its economic implications are drawn.
Table 6
Empirical size of LMC and IMC when residuals are nonnormal! Error distribution Sample size¹ Experiment
Test E1 E2 E3 E4
LMC t
8 300500 11.18.7 8.06.3 9.99.8 8.47.4
1000 7.4 6.1 9.2 5.9
t
12 300500 10.29.6 9.29.1 10.17.6 10.26.2
1000 6.5 7.3 7.7 5.7
IMC t
8 300500 61.478.7 17.317.9 58.376.4 14.816.7
1000 97.3 22.4 96.1 25.5
t
12 300 35.2 10.1 37.4 11.9
500 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-correlation hypothesis. The parameters of the experiments can be found in Table 1. The Monte Carlo sample size is 1000.
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 considertdistributions with 8 and 12 degrees of freedom. The sample size¹is 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, over-rejection 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 over-rejection. When the two nonnormal distributions are compared, over-rejection is less serious fort
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 size¹gets bigger. This is in contrast to the e!ects of
¹on 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
12 are less susceptible to
over-rejection than models witht
8 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 constant-correlation model and is computationally convenient. We examine the "nite-sample 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 lowpvalues. 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 Aca-demic 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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