Testing for Serial Correlation in the Presence of Stochastic Volatility

Asia-Pacific Financial Markets7: 321–337, 2000.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

321

Testing for Serial Correlation in thePresence of Stochastic Volatility

MANABU ASAIFaculty of Economics, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu Shiga 525-8577, Japan

Abstract. This paper develops a regression-based testing procedure for serial correlation in thepresence of stochastic volatility. The asymptotic distribution of the test is derived, and the finitesample properties are investigated. Monte Carlo results shows that the test is reliable in terms ofboth size and power performances, when the underlying process is a log-linear stochastic volatility.Moreover, the test is superior to Woolridge’s (1991) robust LM tests in terms of size in finite sample.Serial correlation tests were conducted for nominal returns of ten exchange rates, and indicated thatthere is a strong evidence of serial correlation for Yen/Dollar exchange rates.

Key words: robust LM tests, serial correlation tests, stochastic volatility.

1. Introduction

Changes in asset return variance or volatility over time, may be modeled usingeither the GARCH class models pioneered by Engle (1982), or stochastic volatility(SV) models; see, e.g., Taylor (1994) and Ghysels, Harvey, and Renault (1996).A test for the presence of serial correlation is routinely carried out as a test forefficiency in financial markets. Most of the existing tests of serial correlation,however, require homoskedastic errors and hence may not be robust to hetero-skedasticity. There also exist tests of serial correlation literature that take accountof heteroskedasticity, such as Diebold (1986), Wooldridge (1991), Bollerslev andWooldridge (1992), and Whang (1998). These tests are, however, designed to havegood performance only under conditional heteroskedasticity. The purpose of thispaper is to develop a regression-based testing procedure for serial correlation in thepresence of stochastic volatility.

The organization of this paper is as follows, Section 2 develops a testing proced-ure in the presence of stochastic volatility. Section 3 shows finite sample propertyof the test. Section 4 investigates some empirical results of serial correlation forten exchange rates, including four major rates and those of Asian-Pacific countries.Section 5 concludes the paper.

322 MANABU ASAI

2. The Testing Procedure

2.1. THE MODEL AND ASSUMPTIONS

Consider the model

yt = xtβ + ut , t = 1, . . . , T , (1)

whereyt is a dependent variable,xt = [x1t , . . . , xkt ] is a 1×K vector of variables,which may include stochastic and non-stochastic variables, lagged regressors andlagged values ofyt , andβ is aK×1 vector of unknown parameters.ut follows thestationary AR(p) process

ut = ρ1ut−1+ · · · + ρput−p + et t = 1, . . . , T , (2)

whereρ1, . . . , ρp are unknown autoregressive parameters. Sinceut is stationary,in its MA(∞) representation,

ut =∞∑k=0

ψ0 = 1 ,

ψk ’s satisfy that∑∞

k=0 |ψk| < ∞. The termet is assumed to follow a simplestationary log-linear SV process

et =√htηt , ηt ∼ i.i.d. N(0,1) , (3a)

ln ht = ω + φ ln ht−1+ σννt , νt ∼ i.i.d. N(0,1), |φ| < 1 , (3b)

whereνt is generated independently ofηt . Moments ofet are as follows:

mr = E[ert ] =r!

2r/2(r/2)! exp

(r

2µh + r

2

8σ 2h

),

whereµh = ω/(1−φ) andσ 2h = σ 2

ν /(1−φ2). If r = 2, thenm2 = exp(µh+σ 2h /2).

All the odd moments are zero and autocovariances ofet are given byE[et et−1] = 0(i = 1,2, . . .). Autovariances ofe2c

t are

E[e2ct c

2ct−i] = (E[η2c

t ])2E[hct hct−i] =((2c)!c!2c m2

)2

exp(c2σ 2hφ

i) . (4)

For the volatility processht , E[hct ] = exp(cµh + c2σ 2h /2) and covariances ofhct

andhdt−i are

E[hct hdt−i] = E[hct ]E[hdt ]exp(cdσ 2hφ

i) . (5)

The Models (1)–(3) can be estimated by GMM, Asai’s (1998) QML methodvia a log-GARCH approach, or Bayesian Markov chain Monte Carlo technique byJacquier, Polson, and Rossi (1994).

TESTING FOR SERIAL CORRELATION IN THE PRESENCE OF STOCHASTIC VOLATILITY 323

The null hypothesis to be tested isH0: ρ1 = · · · = ρp = 0, against thealternative hypothesis H1: Not all ρj = 0, (j = 1, . . . , p).

Before stating assumptions for the model, some notations are made. Let theparameter spacesB ⊆ RK , P ⊆ Rp, and M ⊆ R3, whereβ ∈ B, ρ =(ρ1, . . . , ρp) ∈ P , andθ = (ω, φ, σν) ∈ M, be compact and have nonemptyinteriors. And let(βo, ρo, θo) be the true parameter vector andβo ∈ int(B),ρo ∈ int(P ), θo ∈ int(M). Sincem2 is finite given compact parameter spaces,Var[µt ] = m2

∑∞k=0ψ

2k < σ 2

u < ∞, whereσ 2u is a positive constant. Define the

OLS estimatorβT = (X′X)−1X′y, whereX is theT ×K matrix with rowsxt andy is theT × 1 vector with elementsyt . Also,

µt ≡ ρ1(yt−1− xt−1β)+ · · · + ρp(yt−p − xt−pβ) ,λt ≡ ∂µt

∂ρ ′= [yt−1− xt−1β, . . . , yt−p − xt−pβ] : 1× p ,

andλ ≡ λt(βT ) = (ut−1, . . . , ut−p), whereut = yt − xt βT . Note that the Models(1)–(3) can be written as

yt = xtβ + ρ1(yt−1− xt−1β)+ · · · + ρp(yt−p − xt−pβ)+ et , et ∼ SV.

Thus,ut = 0 corresponds to the null hypothesis.The following assumptions are made for the model:

A1. xt is a sequence of independent not (necessarily) identically distributed 1×K random vector.yt is generated by Models (1)–(3) with true parameters(βo, ρo, θo). et , ut andht are unobservable whileyt andxt are observable.

A2. (a) There exist positive constantsδ and1 such that, for allt ,E[|xit xjt |1+δ] <1 (i, j = 1, . . . , K); (b) CT = T −1

∑Tt=1E[x′t xt ] is nonsingular for (all)T

sufficiently large,1 such that det(CT ) > δ > 0.A3. E[(yt − xtβo)2λo′t λot ] is nonsingular, whereλot ≡ λt(β

o), if the expectationexists.

With respect to Assumption A2, the first assumption ensures that the elements ofthe average covariance matrix of the regressors are uniformly bounded. The secondpart ensures the eventual nonsingularity of the average covariance matrix of theregressors and the uniform boundedness of the elements of the inverse. Turingto Assumption A3, it corresponds to nonsingularity of White’s (1980) covariancematrix estimator in an auxiliary regression ofut on ut−1, . . . , ut−p. In the nextsection, a testing procedure under stochastic volatility is proposed.

2.2. ROBUST LM TEST WITH LOG-ARCH CORRECTION

Wooldridge’s (1991) LM test for AR(p) serial correlation is robust to failure ofthe functional form of the (conditional or unconditional) variance,σ 2

t , of the errorterm. When one believes an ARCH(m) model is a better approximation to the

324 MANABU ASAI

conditional variance than a constant variance model, Wooldridge (1991) suggestedthe following procedure. The first step is to obtainσ 2

t as the fitted values fromthe regression of squared least squares residuals on constant and its lags up tom,and the second step is to construct the robust LM statistic by usingσ 2

t . Althoughthis test has correct asymptotic size regardless of the form of heteroskedasticitybecause it is a robust version of the LM test, the power performance relies heav-ily on the choice of he parametric form ofσ 2

t to approximateσ 2t . In the context

of SV models, unconditional variance,ht , and conditional variance,E[ht |κt ] areunobservable, whereκt = (xt , yt−1, xt−1, . . . , y1, x1). It follows that the test isinapplicable to SV models straightforwardly. In this section, a testing procedurebased on Wooldridge’s robust LM test is proposed.

Asai (1998) showed that a simple log-linear SV process inet can be interpretedas a log-GARCH(1,1) model inet ;

et = σtzt , (6a)

ln σ 2t = α0+ α1 ln e2

t−1+ α2 ln σ 2t−1 , (6b)

whereα0 = ω + (1− φ)cη + (1− φ∗) ln c2z , α1 = φ − φ∗, α2 = φ∗, andcη =

E(ln η2t ) ' −1.27,

φ∗ = 1

2φ

1+ φ2+ 2σ 2ν

π2−√(

1− φ2+ 2σ 2ν

π2

)2

+ 8φ2σ 2ν

π2

,

cz = exp

[σ 2ν

4(1− φ∗2) −cη

2− (φ

∗ − φ)φ∗2(1− φ∗) ψ

(1

2

)+

+ 1

2

∞∑i=1

[ln0

((φ∗ − φ)φ∗i + 1

2

)− ln0

(1

2

)]].

zt is a weak stationary process, odd-order moments of which are all zero, andE(z2

t ) = 1 andE(ztzt−j ) = 0 for j > 1. The forth order moment exists if|φ| < 1andσν > 0, and is larger than three.σt is measurable with respect to the timet − 1information set. This representation enables us to test for serial correlation in thepresence of log-linear SV, using the framework of Wooldridge’s robust LM tests.

The construction of the robust LM statistic involves the following steps:

1. Obtain the fitted value denoted here by lnσ 2t , t = 1, . . . , T from the linear

regression

ln u2t = a0 + a1 ln u2

t−1+ · · · + am ln u2t−m + εt , t = 1, . . . , T .

2. Definex∗t = xt/σt andut = ut /σt , t = 1, . . . , T .3. Save the 1× p vector of residuals, sayrt , from the regression of each of the

λt onx∗t , whereλt = (ut−1, . . . , ut−p).


4. Compute(T − SSR), where SSR is the sum of squared residuals from theregression of 1 onut rt . (T − SSR) ∼ χ2(p) asymptotically under H0.

The following Proposition provides asymptotic property of the test statisticunder the null hypothesis, when the underlying process is a log-linear SV.

PROPOSITION 1.Given Assumptions A1–A3 and H0,

TR2u

d→ χ2(p),

whereR2u is the uncentered r-squared from the regression

1 on [ut /σt ](λt − xt BT )/σt , t = 1, . . . , T ,

and

BT ≡[

T∑t=1

(x′t xt /σ2t )

]−1 T∑t=1

(x′t λt /σ2t ) : K × p.

Note that, because the dependent variable in the above regression is unity,TR2u =

T − SSR, where SSR is defined in step 4.The remaining problem is how to choosem. Although the resulting test has

correct asymptotic size regardless of the form of heteroskedasticity, the powerperformance relies on the choice of the parametric form ofσ 2

t to approximateσ 2t .

Indeed,m = 10 is enough in empirical analysis. A log-GARCH(1,1) process inetcan be interpreted as an ARMA(1,1) process in lne2

t , and thus an AR(∞) processin ln e2

t ;

ln e2t = ((1− φ)cη + ω)/(1− φ∗)+ (φ − φ∗)

∞∑j=1

φ∗j ln e2t−j + εt . (7)

If, for example,φ = 0.90 andσν = 0.363, which were used in Monte Carlo simu-lations of Jacquire, Polson, and Rossi (1994), thenφ∗ = 0.817 and the coefficientof ln y2

t−11 is 0.009.2 Although the approximation will be improved if one selectsm larger than 10, LM test corrected for log-ARCH(10) is considered to assess therobustness of the test statistic.

In Section 3, the finite sample property of the test is investigated. To compare itwith other statistics, Wooldridge’s ARCH-corrected LM test is used. If the trueetfollows an ARCH(m) process instead of the SV model, then theσ 2

t are obtained asthe fitted values from the regression

u2t on 1, u2

t−1, . . . , u2t−m .

Silvapulle and Evans (1998) investigated the finite sample property of ARCH(2)-corrected robust LM tests when the true GDP is ARCH(1), ARCH(2), andGARCH(1,1). They compared it with conventional Durbin–Watson, Box–Pierce,

326 MANABU ASAI

Ljung–Box, LM tests, ARCH(2)-corrected versions of these tests, and Diebold’s(1986) corrected test. They report that the ARCH(2)-corrected robust LM test hasgood performance even when the underlying process is GARCH(1,1), while othersnot. Thus, ARCH(2) and ARCH(10) corrected robust LM tests are used in Sec-tion 3 to compre the performance when the underlying process is a log-linear SV.It should be noted that Whang (1998) proposed a test using a nonparametric kernelestimate of the unknown conditional heteroskedasticity. That test is not consideredin this study, since it is not regression-based.

3. Size and Power in Finite Samples

This section examines finite-sample properties of the robust LM test with log-ARCH correction (RLML, hereafter) through simulation experiments. All the workdescribed subsequently was conducted using the GAUSS programming language.

The true model in the experiments is given by a linear regression

yt = xtβo + ut for t = −20, . . . , T

with xt = (1, x1t ) and the true vectorβo = (1,1)′. The expression{x1t : t =−20, . . . , T } is generated from a uniform distribution, and{ut : t = −19, . . . , T }is generated from an AR(1) with a log-linear SV process

ut = ρut−1+ et with u−20 = 0 ,

where et is generated by Equations (3) and (4). The data are creatingT + 20observations, and discarding the first 20 observations to remove the effect of theinitial conditions. Samples of sizeT = 250, 500, 1000, 2000, and 4000 are usedin the experiments. A sample of 1000–4000 is not uncommon in studies usingdaily or weekly data. As for the parameters in Equations (3) and (4), we take(ωo, φo, σ oν ) ∈ {(−0.7360, 0.90, 0.3629), (−0.3680, 0.95, 0.2600), (−0.1472,0.98, 0.1657)}. These parameter values are also used by Jacquire, Polson, and Rossi(1994), among others. As mentioned in the previous section,m = 10 is chosen.

First, the size of RLML tests is considered. The null hypothesis specifiesρ = 0.For comparison, three more test statistics are calculated; Wooldridge’s LM test withARCH(2)-correction (RLM2 hereafter) and with ARCH(10)-correction (RLM10hereafter), the conventional portmanteau test by Ljung and Box (1978) (LB here-after). Table I indicates that RLML and RLM2 have approximately correct sizeeven forT = 250, while LB over-rejects the null ofρ = 0. Rejection frequencyof LB increases as the value ofφ increases. Table II shows sizes of RLML, RLM2and RLM10 tests for joint hypothesis H0: ρ1 = · · · = ρ10 = 0. Sizes of these testsincrease asT increase. While the size of the RLML seems indifferent to the valueof φ, the size of RLM2 and RLM10 tends to decrease asφ increase. Therefore,RLM2 and RLM10 suffers from the size distortion whenφ is larger than 0.95.

Next, powers of RLML, RLM2 and RLM10 tests are considered. Table IIIpresents simulation results for the powers of these tests forρ = 0.1, 0.2, 0.3,


Table I. Size of the tests (p = 1, 5000 replications)

T 5 percent level 1 percent level

φ = 0.90 φ = 0.95 φ = 0.98 φ = 0.90 φ = 0.95 φ = 0.98

RLML 250 0.043 0.045 0.048 0.0060 0.0068 0.0088

500 0.050 0.048 0.049 0.0088 0.0060 0.0062

1000 0.051 0.051 0.045 0.0068 0.0088 0.0064

2000 0.052 0.052 0.056 0.0092 0.0104 0.0098

4000 0.054 0.045 0.048 0.0096 0.0094 0.0092

RLM2 250 0.046 0.045 0.042 0.0070 0.0064 0.0068

500 0.048 0.050 0.046 0.0090 0.0098 0.0058

1000 0.052 0.053 0.046 0.0082 0.0086 0.0082

2000 0.050 0.049 0.052 0.0088 0.0112 0.0116

4000 0.049 0.047 0.051 0.0102 0.0098 0.0098

RLM10 250 0.046 0.041 0.039 0.0072 0.0074 0.0040

500 0.045 0.047 0.041 0.0084 0.0076 0.0048

1000 0.047 0.052 0.039 0.0078 0.0082 0.0066

2000 0.049 0.051 0.049 0.0084 0.0110 0.0096

4000 0.049 0.048 0.051 0.0090 0.0098 0.0082

LB 250 0.123 0.202 0.384 0.0454 0.1036 0.2578

500 0.136 0.219 0.473 0.0530 0.1106 0.3532

1000 0.147 0.202 0.514 0.0574 0.1062 0.3956

2000 0.156 0.191 0.543 0.0610 0.0850 0.4300

4000 0.156 0.175 0.501 0.0624 0.0770 0.3832

Note: ‘φ = 0.90’ denotes the SV parameter setting(ω, φ, σν) = (−0.7360, 0.90, 0.3629), and soon.

indicating that these tests have appreciable power except whenρ = 0.1. Forρ = 0.1, power increases atT increases. WhenT is larger than 1000, powersfor 5 percent level are large enough.

The Monte Carlo results show that the RLML test of 5 percent level is robustto stochastic volatility in terms of both size and power performances for samplesizes in studies using daily or weekly data. Moreover, the RLML is superior to therobust LM test with ARCH(2)-correction and with ARCH(10)-correction, and LBin terms of size in finite sample.

Since the log-GARCH model is a special case of EGARCH, estimatingσ 2t

using a log-ARCH model provides a better approximation of EGARCH than usingARCH models. Thus, the RLML test is reliable when the underlying process isa log-linear SV or an EGARCH, while Wooldridge’s robust LM tests can be usedwhen the underlying process is an ARCH or a GARCH. In applications to financialtime series, using both tests will provide reliable results.

328 MANABU ASAI

Table II. Size of the tests (p = 10, 5000 replications)

T 5 percent level 1 percent level

φ = 0.90 φ = 0.95 φ = 0.98 φ = 0.90 φ = 0.95 φ = 0.98

RLML 250 0.0350 0.0330 0.0282 0.0042 0.0050 0.0032

500 0.0374 0.0384 0.0344 0.0080 0.0050 0.0042

1000 0.0428 0.0484 0.0380 0.0076 0.0072 0.0050

2000 0.0486 0.0398 0.0406 0.0088 0.0080 0.0062

4000 0.0444 0.0398 0.0436 0.0066 0.0066 0.0062

RLM2 250 0.0344 0.0322 0.0162 0.0042 0.0046 0.0006

500 0.0406 0.0378 0.0198 0.0066 0.0066 0.0018

1000 0.0386 0.0430 0.0264 0.0070 0.0070 0.0018

2000 0.0522 0.0376 0.0356 0.0088 0.0062 0.0064

4000 0.0454 0.0380 0.0374 0.0068 0.0058 0.0068

RLM10 250 0.0286 0.0282 0.0202 0.0044 0.0032 0.0016

500 0.0388 0.0298 0.0192 0.0064 0.0048 0.0018

1000 0.0364 0.0408 0.0230 0.0054 0.0066 0.0018

2000 0.0526 0.0412 0.0256 0.0080 0.0088 0.0038

4000 0.0448 0.0394 0.0246 0.0074 0.0076 0.0012

Note ‘φ = 0.90’ denotes the SV parameter setting(ω, φ, σν) = (−0.7360, 0.90, 0.3629), and soon.

4. Empirical Analysis

The present data set consists of daily observations on the exchange rates overthe period 24 November 1989 through 20 February 1998, for a total of 2151observations.3 This section deals with four major exchange rates, Pound/Dollar,Deutsche-Mark/Dollar, Yen/Dollar, and Swiss-Franc/Dollar. Adding to these rates,the focus is on six exchange rates concerning Asian-Pacific region: Hong-Kong-Dollar/Dollar, Thai-Baht/Dollar, Australian-Dollar/Dollar, Malaysian-Ringgit/Dollar, Philippine-Peso/Dollar, and Singapore-Dollar/Dollar. In this section, weinvestigte serial correlation of the nominal returns; i.e.,rit ≡ ln sit − ln sit−1, wheresit denotes the spot exchange rate for the currency of countryi to U.S. dollar atday t .

Table IV shows the autocorrelation tests results. The null hypothesis is H0:ρ1 = · · · = ρ10 = 0. As noted in the previous section, the RLML test isrelible when the underlying process ofrit is a log-linear SV or an EGARCH,while RLM2 and RLM10 tests can be used when the underlying process isan ARCH or a GARCH. LB rejects the null for all exchange rates except forDeutsche-Mark/Dollar, Swiss-Franc/Dollar and Australian-Dollar/Dollar, but theresults are likely to be caused by over-rejections. Serial correlations are foundin five exchange rates; Yen/Dollar, Thai-Baht/Dollar, Malaysian-Ringgit/Dollar,


Table III. Rejection frequencies of the tests (p = 1, 5000 replications)

T ρ 5 percent level 1 percent levelφ = 0.90 φ = 0.95 φ = 0.98 φ = 0.90 φ = 0.95 φ = 0.98

RLML 250 0.1 0.224 0.237 0.211 0.075 0.081 0.0690.2 0.702 0.692 0.677 0.449 0.436 0.4170.3 0.961 0.959 0.946 0.856 0.862 0.832

RLM2 250 0.1 0.248 0.232 0.165 0.085 0.084 0.0480.2 0.742 0.684 0.499 0.491 0.434 0.2620.3 0.971 0.947 0.794 0.896 0.854 0.596

RLM10 250 0.1 0.228 0.208 0.158 0.079 0.072 0.0460.2 0.695 0.608 0.500 0.462 0.371 0.2710.3 0.940 0.888 0.783 0.846 0.769 0.604

RLML 500 0.1 0.421 0.423 0.409 0.206 0.206 0.1910.2 0.933 0.943 0.932 0.819 0.829 0.8040.3 0.999 0.999 0.997 0.994 0.995 0.989

RLM2 500 0.1 0.450 0.416 0.291 0.226 0.196 0.1180.2 0.954 0.936 0.765 0.853 0.823 0.5660.3 1.000 0.997 0.946 0.997 0.992 0.876

RLM10 500 0.1 0.431 0.393 0.290 0.221 0.183 0.1180.2 0.933 0.893 0.743 0.831 0.763 0.5580.3 0.993 0.973 0.920 0.987 0.952 0.849

RLML 1000 0.1 0.705 0.734 0.691 0.476 0.489 0.4480.2 0.998 0.999 0.996 0.990 0.992 0.9860.3 1.000 1.000 1.000 1.000 1.000 0.999

RLM2 1000 0.1 0.742 0.731 0.548 0.523 0.495 0.3180.2 0.998 0.998 0.955 0.995 0.994 0.8960.3 1.000 1.000 0.994 1.000 0.999 0.984

RLM10 1000 0.1 0.740 0.696 0.510 0.524 0.465 0.2820.2 0.994 0.971 0.910 0.989 0.952 0.8330.3 0.998 0.995 0.978 0.997 0.990 0.959

RLML 2000 0.1 0.942 0.944 0.943 0.832 0.841 0.8430.2 1.000 1.000 1.000 1.000 1.000 1.0000.3 1.000 1.000 1.000 1.000 1.000 1.000

RLM2 2000 0.1 0.953 0.953 0.884 0.870 0.859 0.7390.2 1.000 1.000 0.995 1.000 0.999 0.9910.3 1.000 1.000 0.999 1.000 1.000 0.998

RLM10 2000 0.1 0.958 0.929 0.749 0.876 0.826 0.5800.2 1.000 0.997 0.969 0.999 0.993 0.9450.3 1.000 0.998 0.987 1.000 0.997 0.979

RLML 4000 0.1 0.998 0.999 0.998 0.990 0.995 0.9910.2 1.000 1.000 1.000 1.000 1.000 1.0000.3 1.000 1.000 1.000 1.000 1.000 1.000

RLM2 4000 0.1 0.999 0.999 0.991 0.994 0.995 0.9740.2 1.000 1.000 0.999 1.000 1.000 0.9990.3 1.000 1.000 1.000 1.000 1.000 0.999

RLM10 4000 0.1 0.999 0.993 0.904 0.995 0.984 0.8270.2 1.000 0.999 0.986 1.000 0.998 0.9770.3 1.000 0.999 0.994 1.000 0.999 0.990

Note: ‘φ = 0.90’ denotes the SV parameter setting(ω, φ, σν ) = (−0.7360, 0.90, 0.3629), and so on.

330 MANABU ASAI

Table IV. Autocorrelation tests(p = 10)

RLML RLM2 RLM10 LB

Pound/$ 13.32 13.05 15.68 25.57∗DM/$ 11.55 13.33 14.66 15.95

Yen/$ 20.19∗ 20.74∗ 21.90∗ 25.57∗SF/$ 7.676 5.582 7.493 8.306

Hong-Kong$/$ 14.43 13.48 12.02 95.53∗Thai-Baht/$ 12.76 20.69∗ 15.52 152.2∗Australian$/$ 11.14 11.01 10.06 9.334

Malaysian-Ringgit/$ 15.01 18.96∗ 8.526 166.4∗Philippine-Peso/$ 7.120 25.39∗ 20.86∗ 76.86∗Singapore$/$ 20.04∗ 10.34 9.645 45.99∗

Note: The critical value is given byχ20.05(10) = 18.30.

Philippine-Peso/Dollar, Singapore-Dollar/Dollar. For Yen/Dollar rates, all RLML,RLM2 and RLM10 rejects the null hypothesis, indicating a strong evidence ofserial correlations. One of the reasons for this is that Japanese government oftenintervenes in markets to keep the rates stable.

5. Conclusion

This paper develops a regression-based testing procedure for serial correlation inthe presence of log-linear stochastic volatility. The asymptotic distribution of thetest is derived, and the finite sample properties are investigated. Monte Carlo resultsshows that the test is reliable in terms of both size and power performances, whenthe underlying process is a stochastic volatility. Moreover, the test is superior toWooldridge’s robust LM tests in terms of size in finite sample. Serial correlationtests were conducted for nominal returns of ten exchange rates, and indicated thatthere is a strong evidence of serial correlation for Yen/Dollar exchange rates.

It should be noted that it is open to question whether the test works under long-memory stochastic volatility or conditional heteroskedasticity processes. The teststatistics can be extended to applying long-memory stochastic volatility models.This task awaits further research.

Mathematical Appendix

Some remarks and definitions are made to begin with Mathematical Appendix. Bydefinition, {λt(κt , β) : β ∈ B} is a sequence such thatλt(κt , β) is measurable


with respect toκt for eachβ ∈ B, λt (κt , β) is differentiable on int(B) for allκt , t = 1,2, . . . , that is,

∂λt

∂β= [−x′t−1, . . . , −x′t−p] : K × p .

Let

BT ≡(

T∑t=1

x′t ct xt

)−1∑Tt=1 x

′t ctλt ,

BoT =(

T∑t=1

E[x′t ctxt ])−1∑T

t=1E[x′t ctλt ] ,

ξ ≡ T −1/2T∑

t−=1

(λt − x′t BT )′ct (yt − xt βT ) ,

wherect satisfies the following condition.

CONDITION 1. ct is a sequence of weighting values such that 0< ct < c < ∞and limt→∞ T −1∑∞

t=1 ct <∞.

Note that Assumption A2(b) ensures existence ofBoT , and thatBoT = O byAssumption A1.

For convenience, I include Lemma A.1 of Wooldridge (1990) which is re-peatedly used in the proof of Proposition 1.

LEMMA 1. Assume that the sequence of random functions{QT (wT , θ) : θ ∈2,T = 1,2, . . .}, whereQT (wT , ·) is continuous on2 and2 is a compact subsetof Rq, and the sequence of nonrandom functions{QT (θ) : θ ∈ θ, T = 1,2, . . .},satisfy the following conditions:

(i) supθ∈2 |QT (wT , θ)− QT (θ)| p→ 0;(ii) {QT (θ) : θ ∈ 2,T = 1,2, . . .} is continuous on2 uniformly inT . Let θT be

a sequence of random vectors such thatθT − θoTp→ 0 where{θoT } ∈ 2.

ThenQT (wT , θT )− QT (θoT )

p→ 0.Proof. See Wooldridge (1990, Lemma A.1, p. 40). 2

LEMMA 2. For a simple stationary stochastic volatility model ((3a), (3b)),

E[hct hdt−ihft−j ] = E[hct ]E[hdt ]E[hft ]exp(cdσ 2hφ

i + cf σ 2hφ

j + df σ 2hφ|j−i|).

332 MANABU ASAI

Proof. Let ςt = (lnht, ln ht−i , ln ht−j )′. Then ςt has multivariate normaldistribution with the mean vectorµ3 = (µh, µh, µh)′ and the covariance matrix

63 = σ

2h σ 2

hφi σ 2

hφj

σ 2hφ

i σ 2h σ 2

hφ|j−i|

σ 2hφ

j σ 2hφ|j−i| σ 2

h

.

Hence, the moment generating function ofςt is given by

E[eθ ′ς ] = exp(θ ′µ3+ θ ′63θ/2).

Letting θ = (c, d, f )′, we obtain the desired result. 2LEMMA 3. Given Assumptions A1 and A2,βT exists almost surely for (all)T

sufficiently large andβTp→ βo.

Proof. βT exists almost surely for (all)T sufficiently large provided(X′X/T )is nonsingular almost surely forT sufficiently large. When this is true, Assump-tion A1 allows us to write

βT = (X′X/T )−1(X′y/T ) = βo + (X′X/T )−1(X′u/T ) .

Sincext satisfies Assumptions 1 and 2 of White (1980), by the proof of Lemma 1of White (1980),(X′X/T ) is nonsingular almost surely forT sufficiently large,so thatβT exists. Thus, the elements ofC−1

T are uniformly bounded forT suf-

ficiently large, so that|(X′X/T )−1 − C−1T | a.s.→ 0, and thus plim(X′X/T )−1 =

C−1T . Next, sincexit (i = 1, . . . , K) are independent ofut by Assumption A1,E[T −1∑T

t=1 xitut ] = T −1∑Tt=1E[xitut ] = 0. By Assumption A2(a) and Jensen’s

inequality, we haveE[|xit |2] 6 11/(1+δ) and thus, by Assumption, A1,

Var

[T −1

T∑t=1

xitut

]= T −2

T∑t=1

e[x2it ]E[u2

t ] < T −1σ 2u1

1/(1+δ).

Hence, we obtain lim Var[T −1∑Tt=1 xitut ] = 0, and plimT −1∑T

t=1 x′t ut = 0.

Therefore,βTp→ βo. 2

LEMMA 4. Given Assumptions A1 and A2 and H0,

(i)

∣∣∣∣T −1T∑t=1

x′t ct xt − T −1T∑t=1

e[x′t ctxt ]∣∣∣∣ a.s.→ O;

(ii)T∑t=1

x′t ct (yt − xtβo) a.s.→ O;


(iii)T∑t=1

λo′t ct xt

p→ O;

(iv)T∑t=1

(∂λt

∂β

)′ct (yt − xtβo) a.s.→ O.

Proof. (i) For any i, j = 1, . . . , K, {ctxit xjt } is a sequence of independentrandom variables withE[ctxit xjt ] = ctE[xitxjt ] and

∞∑t=1

E[|ctxit xjt − ctE[xitxjt ]|1+δ]/t1+δ <∞∑t=1

11(c/t)1+δ <∞ ,

where11 is a positive constant, by Assumptions A1 and A2(a). Hence, we have (i)by Markov strong law of large numbers.

(ii) Given H0, for any i (i = 1, . . . , K), {ctxit (yt − xtβo)} is a martingaledifference sequence with

∞∑t=1

e|ctxit (yt − xtβo)|2/t2 <∞∑t=1

c212σ2u /t

2 <∞ ,

where12 is a positive constant, by Assumptions A1 and A2(a). Hence, we have(ii) by a strong law of large numbers for martingale differences; see, for example,Davidson (1994, Theorem 20.11).

(iii) For any i (i = 1, . . . , p) andj (j = 1, . . . , K), {(yt−i−xt−iβo)ctxjt } is asequence of random variables with the mean 0 and the variancec2

t E[x2j t ] < c212.

Since

E

[∣∣∣∣T −1T∑t=1

(yt−i − xt−iβo)ctxj t∣∣∣∣2]6 T −1σ 2

u c212→ 0 asT →∞ ,

T −1∑∞

t=1(yt−i − xt−iβo)ctxjt converges in mean square to zero. Hence, we have(iii) by Chebyshev’s inequality.

(iv) Given H0, for anyi (i = 1, . . . , p) andj (j = 1, . . . , K), {xj,t−ict (yt −xtβ

o)} is a martingale difference sequence with

∞∑t=1

E|xj,t−ict (yt − xtβo)|2/t2 <∞∑t=1

c212σ2u /t

2 <∞ ,

by Assumptions A1 and A2(a). Hence, we have (iv) by a strong law of largenumbers of martingale differences. 2LEMMA 5. Given Assumptions A1–A3 and H0,

(i) E[c2t (yt − xtβo)2(λot − xtBoT )′(λt − xtBoT ) = 4ot exists and is a finite positive

definite matrix;

334 MANABU ASAI

(ii) E[c4t (yt − xtβo)4λoitλojtλoktλolt ] <∞ for all i, j, k, l, and t;

(iii) |T −1∑Tt=1 c

2t (yt − xtβo)2(λot − xtBoT )′(λt − xtBoT )−4oT | a.s.→ O, where4oT =

T −1∑T

t=14ot ;

(iv) 4o−1/2T T −1/2∑T

t=1(λot − xtBoT )′ct (yt − xtβo) d→ N(0, Ip).

Proof. (i) Note that, under H0, yt − xtβo = et andλoit = yt−i − xt−iβo =et−i for i = 1, . . . , p. It follows that (i, j)the element of4ot can be written asc2t E[e2

t et−iet−j ] if 4ot exists. Fori = j , E[e2t e

2t−i] = m2

2 exp(σ 2hφ

i) by Equation

(4), while E[e2t et−iet−j ] = E[η2

t ]E[ηt−i]E[ηt−j ]E[hth1/2t−ih

1/2t−j ] = 0 for i 6= j .

Thus,4ot exists and is a finite positive definite matrix by Assumption A3 andBoT =O.

(ii) Noting

E[e4t e

4t−i] = E[h2

t h2t−i]E[η4

t ]E[η4t−i] = 9m2

2 exp(4σ 2hφ

i) ,

E[e4t e

3t−iet−j ] = E[h2

t h1/2t−ih

1/2t−j ]E[η4

t ]E[η3t−i]E[ηt−j ] = 0 for i 6= j ,

E[e4t e

2t−ie

2t−j ] = E[h2

t ht−iht−j ]E[η4t ]E[η2

t−i]E[η2t−j ]

= 3 exp(4µh + 3σ 2h )exp(σ 2

h (2φi + 2φ2

j + φj−i ) for i < j, by Lemma 2,

E[e4t e

2t−iet−j et−k] = E[h2

t ht−ih1/2t−jh

1/2t−k]E[η4

t ]E[η2t−i]E[ηt−j ]E[ηt−k] = 0

for i 6= j, i 6= k, andj 6= k ,E[e4

t et−iet−j et−ket−l] = E[h2t h

1/2t−ih

1/2t−jh

1/2t kh

1/2t−l]

× E[η4t ]E[ηt−i]E[ηt−j ]E[ηt−k]E[ηt−k] = 0

for i 6= j, i 6= k, i 6= l, j 6= k, j 6= l, k 6= l ,we have (ii) given H0:

(iii) Given H0, {c2t (yt − xtβo)2λoitλojt − c2

t E[(yt − xtβo)2λoitλojt ]} is a martingaledifference sequence. By Lemma 5(ii), we have Var[c2

t (yt−xtβo)2λoitλojt ] < c213 <

∞ for i, j = 1, . . . , p, under H0, where13 is a positive constant. It follows that

∞∑t=1

Var[c2t (yt − xtβo)2λoitλojt ]/t2 <

∞∑t=1

c213/t2 <∞ .

Therefore, (iii) holds by a strong law of large numbers for martingale differencesandBoT = O.

(iv) Since{(λot −xtBoT )′ct (yt−xtβ0)} is a martingale difference sequence underH0, we can apply a central limit theorem for martingale differences (see White(1984, Corollary 5.25) and Hamilton (1994, Proposition 7.9)) by using Lemma5(i)–(iii):

4o−1/2T T −1/2

T∑t=1

(λot − xtBoT )′ct (yt − xtβo) d→ N(0, Ip) . 2


PROPOSITION 1.Given Assumptions A1–A3 and H0,

TR2u

d→ χ2(p) ,

whereR2u is the uncentered r-squared from the regression

1 on [ut /σt ](λt − xt BT )/σt , t = 1, . . . , T ,

and

BT ≡[

T∑t=1

(x′t xt /σ2t )

]−1 T∑t=1

(x′t λt /σ2t ) : K × p .

Proof. Lemmas 1 and 3 yield thatBT − B0T

p→ 0. Therefore,

ξ = T −1/2T∑t=1

(λt − x′tB0T )′ct (yt − xt βT )−

− (BT − B0T )′T −1/2

T∑t=1

x′t ct (yt − xt βT ) .

Taylor series expansion forT −1/2∑Tt=1 x

′t ct (yt − xt BT ) aboutβ0 and Lemma 1

yield

T −1/2T∑t=1

x′t ct (yt − xt βT )

= T −1/2T∑t=1

x′t ct (yt − xtβ0)− T −1T∑t=1

x′t ct xtT1/2(βT − β0)+ op(1) .

(8)

Along with BT − BoTp→ O, this establishes that

ξT = T −1/2T∑t=1

(λt − xtBoT )′ct (yt − xt βT )+ op(1) ,

under H0. A mean value expansion and Lemma 1 yield

ξT = T −1/2T∑t=1

(λot − xtBoT )′ct (yt − xtβo)−

− T −1T∑t=1

(λot − xtBoT )′ctxt · T 1/2(βT − βo)+

+ T −1T∑t=1

(∂λt

∂β

)′ct (yt − xtβo) · T 1/2(βT − βo)+ op(1) .

336 MANABU ASAI

But, byBoT = O and Lemmas 4(iii) and (iv), we have

ξT = T −1/2T∑t=1

(λot − xtBoT )′ct (yt − xtβo) .

The asumptotic covariance matrix ofξT is positive definite by Lemma 5(i), and

4o−1/2T ξT

d→ N(0, Ip). Lemmas 1 and 5(iii) ensures that

4T ≡ T −1T∑t=1

c2t (yt − xtβo)2(λot − xtBoT )′(λt − xtBoT )

is a consistent estimator of4oT . It is easy to see that

ξ ′T 4−1T ξT = TR2

u ,

and TR2u has asumptotically chi-squared distribution with degree of freedomp.

Setting the weighting valuesct asσ−2t , the desired result is obtained. 2

Acknowledgements

I wish to express my gratitude to Professor Jin-Chuan Duan and two referees forhelpful suggestions and comments, and I would like to acknowledge financialsupports from Grant-in-Aid for Scientific Research 11303004 of the Ministry ofEducation.

Notes

1. See White (1980). In what follows the qualifier ‘all’ will be implicitly understood.2. For other values used in Jacquire, Polson, and Rossi (1994), the coefficient of lny2

t−11 is 0.017if φ = 0.95 andσν = 0.260 (φ∗ = 0.877), and it is 0.023 ifφ = 0.98 andσν = 0.166(φ∗ = 0.924).

3. The data were obtained through Datastream.

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