Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol99.Issue2.Dec2000:
Journal of Econometrics 99 (2000) 291}315
Spectral tests of the martingale hypothesis
under conditional heteroscedasticity
Rohit S. Deo*
New York University, 8-57 KMEC, 44, West 4th Street, NY 10012, USA
Abstract
We study the asymptotic distribution of the sample standardized spectral distribution
function when the observed series is a conditionally heteroscedastic martingale di!erence. We show that the asymptotic distribution is no longer a Brownian bridge but
another Gaussian process. Furthermore, this limiting process depends on the covariance
structure of the second moments of the series. We show that this causes test statistics
based on the sample spectral distribution, such as the CrameH r von-Mises statistic, to have
heavily right skewed distributions, which will lead to over-rejection of the martingale
hypothesis in favour of mean reversion. A non-parametric correction to the test statistics
is proposed to account for the conditional heteroscedasticity. We demonstrate that the
corrected version of the CrameH r von-Mises statistic has the usual limiting distribution
which would be obtained in the absence of conditional heteroscedasticity. We also
present Monte Carlo results on the "nite sample distributions of uncorrected and
corrected versions of the CrameH r von-Mises statistic. Our simulation results show that
this statistic can provide signi"cant gains in power over the Box}Ljung}Pierce statistic
against long-memory alternatives. An empirical application to stock returns is also
provided. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C12
Keywords: Sample spectral distribution function; Martingale di!erence; Conditional
heteroscedasticity; CrameH r von-Mises statistic
* Tel.: #1-212-998-0469; fax: #1-212-995-4003.
E-mail address: [email protected] (R.S. Deo).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 2 7 - 0
292
R.S. Deo / Journal of Econometrics 99 (2000) 291}315
1. Introduction
Economic theory suggests that many "nancial and economic time series like
stock returns and exchange rate returns are uncorrelated. More speci"cally, the
concept of market e$ciency leads one to believe that future values of such series
should be unpredictable given the past. This is the famous martingale di!erence
hypothesis and it is a more serious restriction than mere absence of correlation.
It implies that there is no non-trivial function of past data, linear or non-linear,
which can be used to predict future values. Testing such a general hypothesis is
however practically impossible, since it encompasses too many possibilities.
A more realistic approach towards testing the martingale hypothesis has been
through testing for the absence of correlation under various data-generating
mechanisms.
Two popular tests for uncorrelatedness are the variance ratio test of
Cochrane (1988) and the spectral based tests of Durlauf (1991). The spectralbased tests exploit the fact that under the null hypothesis of a martingale
di!erence, the spectral distribution function is a straight line. Thus, deviations
of the sample spectral distribution from the straight line may be used to test
for the presence of correlation. Durlauf showed that under the null hypothesis
of a martingale di!erence, a normalized version of the di!erence between
the sample and theoretical standardized spectral distribution function converges
to a Gaussian process. The asymptotic distributions of various functionals
of this di!erence can then be obtained and tests for departure from the null
of no correlation may be obtained. Durlauf (1991) obtained his limiting
distribution under conditions which ruled out conditional heteroscedasticity.
However, it is a well-accepted fact that most "nancial and economic series
which are hypothesized to be martingale di!erences show conditional heteroscedasticity. Thus, it is important to take this conditional heteroscedasticity into
account when studying the behaviour of the spectral-based tests of the martingale di!erence hypothesis.
In this paper, we show that the spectral-based tests no longer have the usual
limiting distribution when there is conditional heteroscedasticity. As a matter of
fact, we show that, in general, the limiting distribution is heavily right skewed,
with the amount of skewness depending on the degree of persistence in the
second moments. This fact may explain why such tests tend to reject the
martingale di!erence null in favour of mean reversion. We also suggest a way to
correct these tests in a non-parametric way to account for the conditional
heteroscedasticity. For one such corrected test, we prove that the asymptotic
distribution is the same as what would be obtained in the absence of conditional
heteroscedasticity. We provide Monte Carlo simulations for some of the uncorrected and corrected tests and also provide an empirical application.
The layout of the paper is as follows. In Section 2, we state the assumptions we
require for our results and discuss some popular models which satisfy these
R.S. Deo / Journal of Econometrics 99 (2000) 291}315
293
assumptions. In Section 3, we derive the limiting distributions of spectral-based
tests for the martingale di!erence hypothesis. Modi"ed versions of the test
statistics to account for the conditional heteroscedasticity are proposed and the
limiting distribution for one of the modi"ed test statistics is obtained. In Section
4, we present Monte Carlo simulation results for some of the uncorrected and
corrected test statistics and in Section 5 apply these tests to real data. We "nish
the paper with a technical appendix containing the proofs of all our results.
2. Assumptions
We will derive the limiting behaviour of various spectral distribution tests
under the hypothesis that the time series of interest, X , satis"es
t
X "k#e ,
(1)
t
t
where Me N is a martingale di!erence sequence and k is some real number. Thus,
t
X may be the "rst di!erence of a random walk with martingale di!erence
t
innovations. Our speci"cation allows the random walk to have a possible drift,
implying a non-zero mean for the observed series X . The assumptions we make
t
about the martingale di!erence series Me N in (1) are as follows:
t
Condition A. (i) E(e DF )"0, where F "pMe , e ,2N is the sigma
t t~1
t~1
t~1 t~2
"eld generated by Me , e ,2N.
t~1 t~2
(ii) E(e2)"p2(R.
t
(iii) lim
n~1+n E(e2DF )"p2'0 almost surely.
j/1 j j~1
n?=
(iv) There exists a random variable ; with E(;4)(R such that
P(De D'u))cP(D;D'u) for some 0(c(R and all t, all u*0.
t
(v) E(e2e e )"p4q is "nite and uniformly bounded for all t, r*1, s*1.
t t~r t~s
rs
(vi) lim
n~1+n e e E(e2DF )"p4q almost surely for any r*1,
rs
t/1 t~r t~s t t~1
n?=
s*1.
(vii) For any integer q, 2)q)8, and for q non-negative integers
s , E(
Spectral tests of the martingale hypothesis
under conditional heteroscedasticity
Rohit S. Deo*
New York University, 8-57 KMEC, 44, West 4th Street, NY 10012, USA
Abstract
We study the asymptotic distribution of the sample standardized spectral distribution
function when the observed series is a conditionally heteroscedastic martingale di!erence. We show that the asymptotic distribution is no longer a Brownian bridge but
another Gaussian process. Furthermore, this limiting process depends on the covariance
structure of the second moments of the series. We show that this causes test statistics
based on the sample spectral distribution, such as the CrameH r von-Mises statistic, to have
heavily right skewed distributions, which will lead to over-rejection of the martingale
hypothesis in favour of mean reversion. A non-parametric correction to the test statistics
is proposed to account for the conditional heteroscedasticity. We demonstrate that the
corrected version of the CrameH r von-Mises statistic has the usual limiting distribution
which would be obtained in the absence of conditional heteroscedasticity. We also
present Monte Carlo results on the "nite sample distributions of uncorrected and
corrected versions of the CrameH r von-Mises statistic. Our simulation results show that
this statistic can provide signi"cant gains in power over the Box}Ljung}Pierce statistic
against long-memory alternatives. An empirical application to stock returns is also
provided. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C12
Keywords: Sample spectral distribution function; Martingale di!erence; Conditional
heteroscedasticity; CrameH r von-Mises statistic
* Tel.: #1-212-998-0469; fax: #1-212-995-4003.
E-mail address: [email protected] (R.S. Deo).
0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 2 7 - 0
292
R.S. Deo / Journal of Econometrics 99 (2000) 291}315
1. Introduction
Economic theory suggests that many "nancial and economic time series like
stock returns and exchange rate returns are uncorrelated. More speci"cally, the
concept of market e$ciency leads one to believe that future values of such series
should be unpredictable given the past. This is the famous martingale di!erence
hypothesis and it is a more serious restriction than mere absence of correlation.
It implies that there is no non-trivial function of past data, linear or non-linear,
which can be used to predict future values. Testing such a general hypothesis is
however practically impossible, since it encompasses too many possibilities.
A more realistic approach towards testing the martingale hypothesis has been
through testing for the absence of correlation under various data-generating
mechanisms.
Two popular tests for uncorrelatedness are the variance ratio test of
Cochrane (1988) and the spectral based tests of Durlauf (1991). The spectralbased tests exploit the fact that under the null hypothesis of a martingale
di!erence, the spectral distribution function is a straight line. Thus, deviations
of the sample spectral distribution from the straight line may be used to test
for the presence of correlation. Durlauf showed that under the null hypothesis
of a martingale di!erence, a normalized version of the di!erence between
the sample and theoretical standardized spectral distribution function converges
to a Gaussian process. The asymptotic distributions of various functionals
of this di!erence can then be obtained and tests for departure from the null
of no correlation may be obtained. Durlauf (1991) obtained his limiting
distribution under conditions which ruled out conditional heteroscedasticity.
However, it is a well-accepted fact that most "nancial and economic series
which are hypothesized to be martingale di!erences show conditional heteroscedasticity. Thus, it is important to take this conditional heteroscedasticity into
account when studying the behaviour of the spectral-based tests of the martingale di!erence hypothesis.
In this paper, we show that the spectral-based tests no longer have the usual
limiting distribution when there is conditional heteroscedasticity. As a matter of
fact, we show that, in general, the limiting distribution is heavily right skewed,
with the amount of skewness depending on the degree of persistence in the
second moments. This fact may explain why such tests tend to reject the
martingale di!erence null in favour of mean reversion. We also suggest a way to
correct these tests in a non-parametric way to account for the conditional
heteroscedasticity. For one such corrected test, we prove that the asymptotic
distribution is the same as what would be obtained in the absence of conditional
heteroscedasticity. We provide Monte Carlo simulations for some of the uncorrected and corrected tests and also provide an empirical application.
The layout of the paper is as follows. In Section 2, we state the assumptions we
require for our results and discuss some popular models which satisfy these
R.S. Deo / Journal of Econometrics 99 (2000) 291}315
293
assumptions. In Section 3, we derive the limiting distributions of spectral-based
tests for the martingale di!erence hypothesis. Modi"ed versions of the test
statistics to account for the conditional heteroscedasticity are proposed and the
limiting distribution for one of the modi"ed test statistics is obtained. In Section
4, we present Monte Carlo simulation results for some of the uncorrected and
corrected test statistics and in Section 5 apply these tests to real data. We "nish
the paper with a technical appendix containing the proofs of all our results.
2. Assumptions
We will derive the limiting behaviour of various spectral distribution tests
under the hypothesis that the time series of interest, X , satis"es
t
X "k#e ,
(1)
t
t
where Me N is a martingale di!erence sequence and k is some real number. Thus,
t
X may be the "rst di!erence of a random walk with martingale di!erence
t
innovations. Our speci"cation allows the random walk to have a possible drift,
implying a non-zero mean for the observed series X . The assumptions we make
t
about the martingale di!erence series Me N in (1) are as follows:
t
Condition A. (i) E(e DF )"0, where F "pMe , e ,2N is the sigma
t t~1
t~1
t~1 t~2
"eld generated by Me , e ,2N.
t~1 t~2
(ii) E(e2)"p2(R.
t
(iii) lim
n~1+n E(e2DF )"p2'0 almost surely.
j/1 j j~1
n?=
(iv) There exists a random variable ; with E(;4)(R such that
P(De D'u))cP(D;D'u) for some 0(c(R and all t, all u*0.
t
(v) E(e2e e )"p4q is "nite and uniformly bounded for all t, r*1, s*1.
t t~r t~s
rs
(vi) lim
n~1+n e e E(e2DF )"p4q almost surely for any r*1,
rs
t/1 t~r t~s t t~1
n?=
s*1.
(vii) For any integer q, 2)q)8, and for q non-negative integers
s , E(