Amélie Charles

Olivier Darné

Variance ratio tests of random walk: An overview

Keywords: Random walk hypothesis, Variance ratio tests, Stock market efficiency. JEL Classification: G14, G15, C14

published or not. The documents may come

Introduction

There exists a long tradition in the literature concerning the test of the random walk and martingale hypothesis, both in macroeconomics and finance. For instance, the random walk hypothesis [RWH] provides a mean to test the weak-form efficiency -and hence, non-predictability -of financial markets [START_REF] Fama | Efficient capital markets: A review of theory and empirical work[END_REF]1991), and to measure the longrun effects of shocks on the path of real output in macroeconomics [START_REF] Campbell | Permanent and transitory components in macroeconomic fluctuations[END_REF][START_REF] Cochrane | How big is the random walk in GNP?[END_REF][START_REF] Cogley | International evidence on the size of the random walk in output[END_REF].

Given a time series {y t } T t=1 , the RWH correspond to φ = 1 in the first-order autoregressive model

y t = µ + φy t-1 + ε t
where µ is an unknown drift parameter and the error terms ε t are, in general, neither independent nor identically distributed (i.i.d.) 1 .

Many statistical tests 2 were designed to test the RWH but a class of test, based on the variance-ratio [VR] methodology, has gained tremendous popularity in the recent years (see, e.g., [START_REF] Campbell | Permanent and transitory components in macroeconomic fluctuations[END_REF][START_REF] Cochrane | How big is the random walk in GNP?[END_REF][START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF]; [START_REF] Poterba | Mean reversion in stock prices: Evidence and implications[END_REF]. The VR methodology consists of testing the RWH against stationary alternatives, by exploiting the fact that the variance of random walk increments is linear in all sampling intervals, i.e., the sample variance of k-period return (or k-period differences), y ty t-k , of the time series y t , is k times the sample variance of one-period return (or the first difference), y ty t-1 . The VR at lag k is then defined as the ratio between (1/k)th of the k-period return (or the kth difference) to the variance of the one-period return (or the first difference). Hence, for a random walk process, the variance computed at each individual lag interval k (k = 2, 3, . . . ) should be equal to unity.

The use of the VR statistic can be advantageous when testing against several interesting alternatives to the random walk model, most notably those hypotheses associated with mean reversion. In fact, a number of authors (e.g., [START_REF] Lo | The size and power of the variance ratio test in finite samples: A Monte Carlo investigation[END_REF][START_REF] Faust | When are variance ratio tests for serial dependence optimal?[END_REF][START_REF] Richardson | Tests of financial models in the presence of overlapping observations[END_REF] found that the VR statistic had optimal power against such alternative. However, while the intuition behind the VR test is rather simple, conducting a statistical inference using the VR test is less straightforward. What makes thing complicated is that the VR test typically uses overlapping data in computing the variance of long-horizon returns. The use of overlapping was suggested by [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] because it can potentially improve power of the VR test, but the use of overlapping data also adds to the difficulties of analyzing the exact distribution of the VR test statistic. However, virtually nothing is often known about the exact distribution of the VR test statistic that uses overlapping data, and not even its moments are known 3 . In practice, asymptotic distribution instead of exact distribution is often used for conducting statistical inference on the VR test, for fixed k and the sample size T increasing to infinity. [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] proposed two statistics for testing an individual VR estimate which are robust under homoscedasticity and heteroscedasticity. In practice, it is customary to examine the VR statistics for several k values. The null is rejected if it is rejected for some k value. However, as stressed by [START_REF] Chow | A simple multiple variance ratio test[END_REF], this sequential procedure leads to an oversized testing strategy. In this context, multiple VR tests have been suggested such as multiple comparison tests [START_REF] Chow | A simple multiple variance ratio test[END_REF] and Wald-type joint tests (Richardson and Smith, 1989;[START_REF] Cecchetti | Variance ratio tests: Small-sample properties with an application to international output data[END_REF]. Even though the individual Lo-MacKinlay and multiple VR tests are quite powerful testing for homoscedastic or heteroscedastic nulls, it is critical to note that these tests are asymptotic tests in that their sampling distributions are approximated by their limiting distributions. Indeed, the sampling distribution of the VR statistic can be far from normal in finite sample, showing severe bias and right skewness. These finite sample deficiencies may give rise to serious size distortions or low power, which can lead to misleading inferences. This is especially true when the sample size is not large enough to justify asymptotic approximations [START_REF] Cecchetti | Variance ratio tests: Small-sample properties with an application to international output data[END_REF]. To circumvent this problem, some alternatives 4

Individual variance ratio tests

The VR test is often used (see [START_REF] Cochrane | How big is the random walk in GNP?[END_REF][START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF][START_REF] Poterba | Mean reversion in stock prices: Evidence and implications[END_REF]; among others) to test the hypothesis that a given time series or its first difference (or return), x t = y ty t-1 , is a collection of i.i.d. observations or that it follows a martingale difference sequence. Define the VR of k-period return as

V (k) = Var(x t + x t-1 + • • • + x t-k+1 )/k Var(x t ) = Var(y t -y t-k )/k Var(y t -y t-1 ) = 1 + 2 k-1 ∑ i=1 (k -i) k ρ i
where ρ i is the i-th lag autocorrelation coefficient of {x t }. V (k) is a particular linear combination of the first (k -1) autocorrelation coefficients, with linearly declining weights. The central idea of the variance ratio test is based on the observation that when returns are uncorrelated over time, we should have Var(

x t + • • • + x t-k+1 ) = kVar(x t ), i.e. V (k) = 1.
One can therefore think of VR test as a specification test of

H 0 : ρ 1 = • • • = ρ k = 0, i.e.
, returns are serially uncorrelated.

A test can be constructed by considering statistic based on an estimator of

V (k) V R(k) = σ2 (k) σ2 (1)
where σ2 (1) is the unbiased estimator of the one-period return variance, using the one-period returns x t , and is defined as

σ2 (1) = (T -1) -1 T ∑ t=1 (x t -μ) 2 = (T -1) -1 T ∑ t=1 (y t -y t-1 -μ) 2 (1)
with μ = T -1 ∑ T t=1 x t is the estimated mean. For the estimator of k-period return variance σ2 (k), using k-period returns (x t + • • • + x t-k+1 ), there are many ways to do it. Due to limited sample size and the desire to improve the power of the test, this estimator is often performed using overlapping long-horizon returns (k-period), as advocated by [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] 5 , and it is defined as

σ2 (k) = m -1 T ∑ t=k (x t + x t-1 + • • • + x t-k+1 -k μ) 2 = m -1 T ∑ t=k (y t -y t-k -k μ) 2 where m = k(T -k + 1)(1 -kT -1
). The value of m is chosen such as σ2 (k) is an unbiased estimator of the k-period return variance when σ 2 t is constant over time. Following [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF], the VR statistic can be written as

V R(x; k) = (T k) -1 T ∑ t=k (x t + • • • + x t-k+1 -k μ) 2 ÷ T -1 T ∑ t=1 (x t -μ) 2 (2)
Moreover, [START_REF] Cochrane | How big is the random walk in GNP?[END_REF] showed that the estimator of V (k) can be interpreted in terms of the frequency domain. This estimator which uses the usual consistent estimators of variance is asymptotically equivalent to 2π the normalized spectral density estimator at the zero frequency which uses the Bartlett kernel. Formally, we have

V R f = 2π f ∆y (0) σ2 (1) (3)
where f ∆y (0) represents the estimator of the spectrum evaluated at frequency 0 with

2π f ∆y (0) = 2π m ∑ j=1 W k (λ j )I ∆y (λ j ) I ∆y = (2πT ) -1 d y (λ j ) 2 d y = T -1 ∑ t=1 [y t -y t-1 -μ] e -iλ j t W k (λ j ) = ∑ | j|≤k (1 -| j|/k) exp(-i jλ) = k -1 sin(kλ j /2) sin(λ j /2) 2 (4)
where λ j = 2π jT -1 , j = 1, . . . , T -1, I ∆y denotes the periodogram, d y is the discrete Fourier transform, μ is an estimate of the mean of ∆y t , and W (λ j ) is the Bartlett window.

If the data-generating process of time series is a random walk, the expected value of V R(x; k) should be equal to unity for all horizons k. If returns are positively (negatively) autocorrelated, the VR should be higher (lower) than unity. Time series (in level) is said to be mean-reverting if V R(x; k) is significantly lower than unity at long horizons k. On the contrary, time series is mean averting, i.e. explosive, if V R(x; k) is significantly higher than unity at long horizons [START_REF] Poterba | Mean reversion in stock prices: Evidence and implications[END_REF].

We describe the most popular individual VR tests developed by [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] as well as some of its improvements.

Lo and MacKinlay (1988) tests

Lo and MacKinlay (1988) proposed the asymptotic distribution of V R(x; k) by assuming that k is fixed when T → ∞. They showed that if x t is i.i.d., i.e. under the assumption of homoscedasticity, then under the null hypothesis that V (k) = 1, the test statistic M 1 (k) is given by

M 1 (k) = V R(x; k) -1 φ(k) 1/2 (5)
which follows the standard normal distribution asymptotically. The asymptotic variance, φ(k), is given by

φ(k) = 2(2k -1)(k -1) 3kT (6)
To accommodate x t 's exhibiting conditional heteroscedasticity, [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] proposed the heteroscedasticity6 robust test statistic M 2 (k)

M 2 (k) = V R(x; k) -1 φ * (k) 1/2 (7)
which follows the standard normal distribution asymptotically under null hypothesis that V (k) = 1, where

φ * (k) = k-1 ∑ j=1 2(k -j) k 2 δ( j) δ( j) = T ∑ t= j+1 (x t -μ) 2 (x t-j -μ) 2 ÷    T ∑ t=1 (x t -μ) 2 2   
The M 2 (k) test is applicable to x t 's generated from a martingale difference time series (see Appendix for a discussion on the assumptions). The usual decision rule for the standard normal distribution is applied to both tests.

The finite-sample properties of the VR test were studied by [START_REF] Lo | The size and power of the variance ratio test in finite samples: A Monte Carlo investigation[END_REF], who found that the two-sided test has size generally quite close to the nominal level, as long as the test is robustified against any conditional heteroscedasticity.

The VR statistic has been found by several authors (e.g., [START_REF] Richardson | Tests of financial models in the presence of overlapping observations[END_REF][START_REF] Faust | When are variance ratio tests for serial dependence optimal?[END_REF] to be particularly powerful when testing against mean reverting alternatives to the random walk model, particularly when k is large. Ng and Zhang (2004) suggested a modified VR statistic and proposed to approximate the small-sample distribution of this statistic using a beta distribution that matches the exact mean and the asymptotic variance. 8 Richardson and Stock (1989) showed that the VR statistic, without any normalization, converges to a functional of Brownian motion.

properties under the [START_REF] Richardson | Drawing inferences from statistics based on multiyear asset returns[END_REF] framework, and characterized the maximal possible power by taking a continuous-time limit given a fixed data span T .

However, [START_REF] Deo | On the asymptotic power of the variance ratio test[END_REF] argued that the VR statistic is inconsistent against an important class of mean reverting alternatives under this framework when the horizon k is increasing proportional to the sample size, i.e. k/T → δ > 0. Chen and Deo (2006) also showed that the k/T → δ > 0 asymptotic distribution cannot approximate the finite-sample distribution of the VR statistic when k/T is small and is sensitive to conditional heteroscedasticity.

Chen and Deo (2006) suggested a simple power transformation of the VR statistic that, when k is not too large 9 , provides a better approximation to the normal distribution in finite samples and is able to solve the well-known right skewness problem. They showed that the transformed VR statistic leads to significant gains in power against mean reverting alternatives. Furthermore, the distribution of the transformed VR statistic is shown, both theoretically and through simulations, to be robust to conditional heteroscedasticity 10 .

First, they defined the VR statistic based on the periodogram as

V R p (k) = 1 (1 -k/T ) 4π T σ2 (T -1)/2 ∑ j=1 W k (λ j )I ∆y (λ j ) (8)
where I ∆y (λ j ) and W k (λ j ) are defined as in (4). This expression of the VR statistic is precisely the normalized discrete periodogram average estimate of the spectral density of a stationary process at the origin [START_REF] Brockwell | Time Series: Theory and Methods[END_REF]. To obtain their transformed VR statistic, noted V R β p (k), they applied the following power transformation 11

to V R p (k) β = 1 - 2 3 ∑ [(T -1)/2] j=1 W k (λ j ) ∑ [(T -1)/2] j=1 W 3 k (λ j ) ∑ [(T -1)/2] j=1 W 2 k (λ j ) 2 (9)

Wright (2000) tests

As already noted, the Lo-MacKinlay tests, which are asymptotic tests whose sampling distribution is approximated based on its limiting distribution, are biased and rightskewed in finite samples. In this respect, [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] proposed a nonparametric 9 Deo and Richardson (2003) advocated that large values of k should not be used when testing for the mean reversion using the VR statistics. 10 To adjust for conditional heteroscedasticity, [START_REF] Chen | The variance ratio statistic at large horizons[END_REF] proposed a modified version of the standard deviation of the transformed VR statistic (even for standard VR statistic).

11 See Chen and Deo (2004) for a discussion on power transformations.

alternative to conventional asymptotic VR tests using signs and ranks. [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] tests have two advantages over Lo-MacKinlay test when sample size is relatively small: (i) as the rank (R 1 and R 2 ) and sign (S 1 and S 2 ) tests have exact sampling distribution, there is no need to resort to asymptotic distribution approximation, and (ii) the tests may be more powerful than the conventional VR tests against a wide range of models displaying serial correlation, including fractionally integrated alternatives. The tests based on ranks are exact under the i.i.d. assumption, whereas the tests based on signs are exact even under conditional heteroscedasticity. Moreover, [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] showed that ranks-based tests display low size distortions, under conditional heteroscedasticity.

Given T observations of first differences of a variable, {x 1 , . . . , x T }, and let r(x) be the rank of x t among (x 1 , . . . , x T ). Under the null hypothesis that x t is generated from an i.i.d. sequence, r(x) is a random permutation of the numbers of 1, . . . , T with equal probability. [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] suggested the R 1 and R 2 statistics, defined as

R 1 (k) = (T k) -1 ∑ T t=k (r 1,t + • • • + r 1,t-k+1 ) 2 T -1 ∑ T t=k r 2 1,t -1 × φ(k) -1/2 (10) R 2 (k) = (T k) -1 ∑ T t=k (r 2,t + • • • + r 2,t-k+1 ) 2 T -1 ∑ T t=k r 2 2,t -1 × φ(k) -1/2 (11)
where the standardized ranks r 1,t and r 2,t are given by

r 1,t = r(x t ) -T +1 2 (T -1)(T + 1)/12 r 2,t = Φ -1 r(x) T + 1
where φ(k) is defined in [START_REF] Brockwell | Time Series: Theory and Methods[END_REF], and Φ -1 is the inverse of the standard normal cumulative distribution function. The R 1 and R 2 statistics follow the same exact sampling distribution. The critical values of these tests can be obtained by simulating their exact distributions.

The tests based on the signs of first differences are given by

S 1 (k) = (T k) -1 ∑ T t=k (s t + • • • + s t-k+1 ) 2 T -1 ∑ T t=k s 2 t -1 × φ(k) -1/2 (12) S 2 (k) = (T k) -1 ∑ T t=k (s t (μ) + • • • + s t-k+1 (μ)) 2 T -1 ∑ T t=k s t (μ) 2 -1 × φ(k) -1/2 (13)
where φ(k) is defined in (6), s t = 2u(x t , 0), s t (μ) = 2u(x t , µ), and

u(x t , q) = 0.5 if x t > q -0.

Correction of size distorsions

As pointed out by [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] • They computed the Sidack-adjusted p-value for each test j as p(S)

i j = 1 -(1 -p i j ) m i = 1, . . . , m
where p i j is the p-value corresponding to the VR test j computed for an individual k value, and m is the number of k values.

• They also employed the Hochberg (1988) adjusted p-values which are obtained as p(H)

i j = min{[k -R(p i j ) + 1]p i j , 1}
Given a significance level α, the decision rule states that, using the VR test j, the null is rejected if p(S) j = min 1≤i≤m p(S) i j ≤ α or p(H) j = min 1≤i≤m p(H) i j ≤ α. However, these methods assume that the test statistics computed at different intervals are uncorrelated. In order to take into account possible correlations among the statistics, Belaire-Franch and Contreras (2004) and Belaire-Franch and Opong (2005) suggested to compute bootstrap-adjusted p-values as described in [START_REF] Psaradakis | p-Value adjustments for multiple tests for nonlinearity[END_REF]. The goal of the procedure is to obtain an approximation to the null sampling distribution of min 1≤i≤m p i j by resampling with replacement from the original returns 12 .

Choi (1999) test

When implementing the VR tests, the choice of holding period k is important.

However, this choice is usually rather arbitrary and ad hoc. To overcome this issue, [START_REF] Choi | Testing the random walk hypothesis for real exchange rates[END_REF] proposed a data-dependent procedure to determinate the optimal value of k. Choi (1999) suggested a VR test based in frequency domain since [START_REF] Cochrane | How big is the random walk in GNP?[END_REF] showed that the estimator of V (k) which uses the usual consistent estimators of variance is asymptotically equivalent to 2π the normalized spectral density estimator at the zero frequency which uses the Bartlett kernel. However, Choi (1999) employed rather the Quadratic Spectral [QS] kernel because this kernel is optimal in estimating the spectral density at the zero frequency [START_REF] Andrews | Heteroskedasticity and autocorrelation consistent covariance matrix estimation[END_REF]. The VR estimator is defined as

V R(k) = 1 + 2 T -1 ∑ i=1 h(i/k) ρ(i), ( 14
)
where ρ(i) is the autocorrelation function, and h(x) is the QS window defined as

h(x) = 25 12π 2 x 2 sin(6πx/5) 6πx/5 -cos(6πx/5)
The standardized statistic is

V R f = V R(k) -1 (2) 1/2 (T /k) -1/2 (15)
Under the null hypothesis the test statistic V R f follows the standard normal distribution asymptotically 13 . Note that it is assumed that T → ∞, k → ∞ and T /k → ∞.

Various methods for optimally selecting the truncation point for the spectral density at the zero frequency are available [START_REF] Andrews | Heteroskedasticity and autocorrelation consistent covariance matrix estimation[END_REF] These multiple VR tests consider the following joint null hypothesis H 0 : V (k i ) = 1 for all i = 1, . . . , m, against the alternative H 1 : V (k i ) = 1 for some k i . To test the joint null hypothesis, Chow-Denning's (1993) test statistic is defined as

Chow and Denning (1993) tests

MV 1 = √ T max 1≤i≤m |M 1 (k i )| (16)
where M 1 (k i ) is defined in [START_REF] Belaire-Franch | Some evidence of random walk behaviour of Euro exchange rates using ranks and signs[END_REF]. This is based on the idea that the decision regarding the null hypothesis can be obtained from the maximum absolute value of the individual VR statistics. In order to control the size of the multiple VR test and because the limit distribution of these statistics is complex, they applied the Sidak Similarly, the heteroscedasticity-robust version of the Chow-Denning test MV 2 can be written as

MV 2 = √ T max 1≤i≤m |M 2 (k i )| (17)
where M 2 (k i ) is defined in [START_REF] Cajueiro | Testing for predictability in equity returns for European transition markets[END_REF]

MV T = √ T g N (x 1 , . . . , x T ) (18)
where

g t (x 1 , . . . , x T ) = max 1≤i≤m |M r (k i )| with M r (k i ) = V R(x; k i )-1, and V R(x; k) is as defined in (2)
. The sampling distribution function for the MV T statistic is written as

G T (x) = P √ T g t (x 1 , . . . , x T ) ≤ x
Whang and [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] showed that the asymptotic null distribution of the statistic is that of a maximum of a multivariate normal vector with unknown covariance matrix, which is complicated to estimate. Therefore, they proposed to approximate the null distribution by means of the subsampling approach. m) is rejected at the level of significance α if the observed MV T is greater than this critical value or if the p-value is less than α. To implement the subsampling technique, a choice of block length b should be made.

hypothesis that V (k i ) = 1 (i = 1, . . . ,
Whang and [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] recommended that a number of block lengths from an equally spaced grid in the interval of [2.5T 0.3 , 3.5T 0.6 ] be taken. However, they found that the size and power properties of their test are not sensitive to the choice of the block length.

CD (R 1 ) = max 1≤i≤m |R 1 (k i )| CD (R 2 ) = max 1≤i≤m |R 2 (k i )| CD (S 1 ) = max 1≤i≤m |S 1 (k i )| CD (S 2 ) = max 1≤i≤m |S 2 (k i )|
The ranks-based procedures are exact under the i.

RS(k) = T (VR -1 k ) ′ Φ -1 (VR -1 k )
where VR is the (k × 1) vector of sample k variance ratios, 1 k is the (k × 1) unit vector, and Φ is the covariance matrix of VR.

S(k) = [VR(k) -E [VR(k)]] ′ Σ -1 (k) [VR(k) -E [VR(k)]]
where E is the expectation operator, VR a column vector sequence of VR statistics

VR(k) = [V R(2), . . . ,V R(q)],
and Σ(k) is a measure of the covariance matrix of VR.

tests in the definition of [START_REF] Richardson | Tests of financial models in the presence of overlapping observations[END_REF] procedure but found that these tests were inferior to rank and sign-based CD (.) tests.

The joint VR S(k) statistic follows a χ 2 distribution with k degrees of freedom.

Cecchetti and Lam (1994) studied the empirical distribution of S(k) using Monte Carlo techniques. For each simulation, they computed a value for the statistic S(k), using the mean vector and covariance matrix Ṽ R(k) and Σ(k) taken as the true population values, and tabulate the distribution 16 .

However, as suggested by [START_REF] Cecchetti | Variance ratio tests: Small-sample properties with an application to international output data[END_REF], the empirical distributions of the VR have large positive skewness, suggesting that inference based on the χ 2 distribution will be misleading.

QP(k) = V p,β (k) -µ β ′ Σ -1 β V p,β (k) -µ β where V p,β a column vector sequence of VR statistics V p,β (k) = [V R β p (2), . . . ,V R β p (k)] with V R β p (
k) the power transformed VR as in ( 9), µ β and Σ(k) β are a measure of the expectation and covariance matrix of V p,β , respectively. The joint VR QP(k) statistic follows a χ 2 distribution with k degrees of freedom. Moreover, [START_REF] Chen | The variance ratio statistic at large horizons[END_REF] showed from Monte Carlo simulations that their joint VR test displayed much higher power.

Note that the [START_REF] Chen | The variance ratio statistic at large horizons[END_REF] test is a joint test with one-sided alternative (H 1 :

V (k i ) < 1, for some k i ) while the [START_REF] Richardson | Tests of financial models in the presence of overlapping observations[END_REF] and [START_REF] Cecchetti | Variance ratio tests: Small-sample properties with an application to international output data[END_REF] tests are joint tests with two-sided alternative (H 1 : V (k i ) = 1). Therefore, at α level of significance, the null hypothesis that V (k i ) = 1 is rejected if the test statistic is greater than the upper 2α critical value of a χ 2 distribution for the [START_REF] Chen | The variance ratio statistic at large horizons[END_REF] test and than the upper α critical value of a χ 2 distribution for the [START_REF] Richardson | Tests of financial models in the presence of overlapping observations[END_REF] and [START_REF] Cecchetti | Variance ratio tests: Small-sample properties with an application to international output data[END_REF] tests.

Bootstrapping variance ratio tests

As already noted, [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF], based on ranks and signs, and [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF], using the subsampling method, proposed the VR tests which do not rely asymptotic approximations in order to overcome the difficulties due to using VR tests based on asymptotic approximations (severe bias and right skewness). As an alternative, some researchers proposed to employ a bootstrap method, which is a resampling method that approximates the sampling distribution of a test statistic [START_REF] Efron | Bootstrap methods: Another look at the jackknife[END_REF], to the VR test statistic. The bootstrap is a distribution-free randomization technique, which can be used to estimate the sampling distribution of the VR statistic, when the distribution of the original population is unknown. We describe the two most used bootstrapping VR tests, i.e. those suggested by [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] in a theoretical framework and by [START_REF] Malliaropulos | Mean reversion in Southeast Asian stock markets[END_REF] in an empirical framework 17 .

Kim (2006) test

Kim (2006) used the wild bootstrap which is a resampling method that approximates the sampling distribution of the VR statistic, and is applicable to data with unknown forms of conditional and unconditional heteroscedasticity (see [START_REF] Mammen | Bootstrap and wild bootstrap for high dimensional linear models[END_REF][START_REF] Davidson | The wild bootstrap, tamed at last[END_REF]. [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] applied the wild bootstrap to Lo-MacKinlay, M 2 (k), and Chow-Denning, MV 2 (k i ), VR tests. The wild bootstrap test based on MV 2 (k i ) can be conducted in three stages as below (i) Form a bootstrap sample of T observations X * t = η t X t (t = 1, . . . , T ) where η t is a random sequence with E(η) = 0 and E(η 2 ) = 1.

(ii) Calculate MV * = MV 2 (X * ; k i ), the MV 2 (X * ; k i ) statistic obtained from the bootstrap sample generated in stage (i).

(iii) Repeat (i) and (ii) sufficiently many, say m, times to form a bootstrap distribution of the test statistic {MV 2 (X * ; k j ; j)} m j=1 .

The p-value of the test can be obtained as the proportion of {MV 2 (X * ; k j ; j)} m j=1 greater than the sample value of MV 2 (k i ). The wild bootstrap version of M 2 (k)

test can be implemented in a similar manner as a two-tailed test, where we obtain

M * = M 2 (X * ; k) in stage (ii) and {M 2 (X * ; k j ; j)} m j=1 in stage (iii). Conditionally on X t , X *
t is a serially uncorrelated sequence with zero mean and variance X 2 t . As such, M * and MV * have the same asymptotic distributions as M 2 (k) and MV 2 (k i ), respectively. Since X * t is a serially uncorrelated sequence, wild bootstrapping approximates the sampling distributions under the null hypothesis, which is a desirable property for a bootstrap test.

To implement the wild bootstrap test, a specific form of η t should be chosen. [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] recommends using the standard normal distribution for η t since he reports that other choices provided qualitatively similar small sample results. Note that the wild bootstrap is valid and the test statistics being bootstrapped are pivotal asymptotically under the condition that X t follows a martingale difference sequence 18 . [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] showed that the sub-sampling test of [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] displays small sample properties far inferior to the wild bootstrap test under a small sample size.

Malliaropulos and Priestley (1999) test

Malliaropulos and Priestley (1999) used a weighted bootstrap method proposed by [START_REF] Wu | Jakknife, bootstrap and other resampling methods in regression analysis[END_REF] which is robust to the presence of heteroscedasticity, which is done by resampling normalized returns instead of actual returns. Basically, the returns are normalized by multiplying each observation of actual returns, for each one of the time series of returns, by a corresponding random factor and resample from these normalized returns 19 . The bootstrap scheme can be summarized with the following algorithm (i) For each t, draw a weighting factor z * t (t = 1, . . . , T ) with replacement from the empirical distribution of normalized returns z t = (r tr)/σ(r), where r = T -1 ∑ T t=1 r t is the mean and σ(r 2 is the standard error of return.

) = T -1 ∑ T t=1 (r t -r)
(ii) Form the bootstrap sample of T observations r * t = z * t r t (t = 1, . . . , T ) by multiplying each observation of actual returns with its corresponding random weighting factor.

(iii) Calculate the VR statistic V R * (k) from the pseudo data r * t for k = 1, . . . , K.

(iv) Repeat steps (i) and (ii) M times, obtaining V R * (k; m) (m = 1, . . . , M) and calculate the relevant quantiles, mean, median and standard deviation of 18 See MacKinnon (2002) for the advantages of bootstrapping asymptotical pivotal statistics. Note that there are other possible choices of two-point distributions for the wild bootstrap, which potentially outperform the standard normal distribution when the sample size is small [START_REF] Davidson | Implementing the wild bootstrap using a two-point distribution[END_REF]. 19 See Malliaropulos and Priestley (1999) and [START_REF] Cajueiro | Testing for predictability in equity returns for European transition markets[END_REF] for a detailed discussion on the weighted bootstrap method proposed by [START_REF] Wu | Jakknife, bootstrap and other resampling methods in regression analysis[END_REF].

the sampling distribution of V R * (k) under the null hypothesis of serially uncorrelated returns.

Using this procedure, resampling from normalized returns instead from actual returns, the weighted bootstrap method accounts for the possible non-constancy of the variance of returns. The strongest difficulty with resampling schemes, such as bootstrap, is that they may generate data that is less dependent than the original data. The main idea of the weighted bootstrap scheme is to overcome this difficulty 20 .

Malliaropulos and Priestley (1999) and [START_REF] Cajueiro | Testing for predictability in equity returns for European transition markets[END_REF] used this bootstrap method to approximate the sampling distribution of the Lo-MacKinlay VR statistics as well as the Wald statistic of the Cecchetti and Lam (1994) test.

Note that Malliaropulos and Priestley (1999) bootstrapped the VR statistics, which not asymptotically pivotal, under heteroscedasticity in an empirical framework. Their bootstrap tests are not supported by any asymptotic theory or Monte Carlo evidence to evaluate their properties in contrast to the bootstrap tests proposed by [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF].

Empirical applications

The VR tests have been widely used and their applications have often covered There have been many studies that tested efficiency of Latin American stock markets. However, the results are overall mixed and scattered over studies that employ different sample periods, methods and data frequencies. [START_REF] Urrutia | Tests of random walk and market efficiency for Latin American emerging markets[END_REF], using the Lo-MacKinlay VR test, rejected the RWH for the Latin American emerging equity markets of Argentina, Brazil, Chile and Mexico, whereas the runs test indicated weak form efficiency over the period 1980:3-1988:12. In contrast, Ojah and Karemera (1999) 20 Cribari-Neto and Zarkos (1999) evaluated the performance of this bootstrap methodology by comparing the weighted with the unweighted bootstrap. Their results suggested that weighted bootstrap estimators perform very well, outperforming others estimators, even in the case of heteroscedastic errors and non-normality (fat tails).

found that the Latin American equity returns follow a random walk and were generally weak-form efficient. [START_REF] Grieb | Random walk tests for Latin American equity indices and individual firms[END_REF] formation criterion. This particular specification of heteroscedasticity was motivated by the observation that in many financial time series, the magnitude of residuals appeared to be related to the magnitude of recent residuals. The LM [START_REF] Campbell | Exact nonparametric tests of orthogonality and random walk in the presence of a drift parameter[END_REF] indicates clearly that all stocks show strong conditional heteroscedasticity.

Tables 2 and3 The holding periods (k) considered are [START_REF] Andrews | Heteroskedasticity and autocorrelation consistent covariance matrix estimation[END_REF][START_REF] Belaire-Franch | Some evidence of random walk behaviour of Euro exchange rates using ranks and signs[END_REF][START_REF] Campbell | Exact nonparametric tests of orthogonality and random walk in the presence of a drift parameter[END_REF][START_REF] Fleming | Bootstrap tests of multiple inequality restrictions on variance ratios[END_REF]. As advocated by [START_REF] Deo | On the asymptotic power of the variance ratio test[END_REF], we use relatively short holding periods when testing for the mean reversion using VR tests. For the wild bootstrap test (MV * ), as suggested by Kim (2006) 23 , the number of bootstrap replications m is set to 1000. As recommended by [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF], we take a number of block lengths from an equally spaced grid in the interval of [2.5T 0.3 , 3.5T

Conclusion

This paper reviewed the recent developments in the field of the variance-ratio tests of random walk and martingale hypothesis. In particular, we presented the conventional individual and multiple VR tests as well as their improved modifications based on power-transformed statistics, rank and sign tests, subsampling and bootstrap methods, among others.

We also re-examined the weak-form efficiency for five emerging equity markets in Latin America. We found that Argentina, Brazil, Chile and Mexico follow reject the random walk hypothesis and, consequently, these four Latin American markets are not weak-form efficient. We do not conclude for Ecuador because the results are mixed.

We did not deal with the possible presence of structural breaks, due to financial or economic events, which can affect the VR tests. We left this issue to further research which can be conducted even by applying VR tests using a moving subsample window [START_REF] Yilmaz | Martingale property of exchange rates and central bank intervention[END_REF]Kim and Shansuddin, 2007) or by modifying VR tests to take into account structural changes [START_REF] Lee | Spurious rejections by variance ratio tests in the presence of a break in drift[END_REF].

Appendix: Assumptions on the VR tests

We present some of the main assumptions for the underlying time series which drive the VR tests.

The [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF], [START_REF] Chow | A simple multiple variance ratio test[END_REF] and [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] tests are driven by Assumption H * of [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] which assumes H * 1: For all t, E(ε t ) = 0, and E(ε t ε t-τ ) = 0 for any τ = 0. H * 2: {ε t } is φ-mixing with coefficients φ(m) of size r/(2r -1) or is α-mixing with coefficients α(m) of size r/(r -1), where r > 1, such that for all t and for any τ ≥ 0, there exists some δ > 0 for which

E|ε t ε t-τ | 2(r+δ) < ∆ < ∞ H * 3: lim T →∞ T -1 ∑ T t=1 E(ε 2 t ) = σ 2 0 < ∞ H * 4: For all t, E(ε t ε t-j ε t ε t-i ) = 0
T →∞ T -1 T ∑ t=1 E(ε 2 t ) = σ 2 0
Their test is then robust to violations of this assumption.

The [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] tests are driven by Assumption A in which it is considered that x t = µ + z t and z t = σ t ε t . Letting I t = {x t , x t-1 , x t-2 , . . . }, the assumptions are A1: z t is i.i.d.

A2: σ t and ε t are independent, conditional on I t-1 .

A3: E(ε t |I t-1 ) = 0 and 1(ε t > 0) is an i.i.d. binomial variable that is 1 with probability 1 2 and 0 otherwise.

Assumption A1 stipulates that the first-differences are i.i.d. while the combination of Assumptions A2 and A3 is sufficient, but not necessary, for x t to be a martingale difference sequence. Moreover, Assumption A2 is satisfied in a GARCH model and also by a stochastic volatility model in which the innovations to volatility are independent of ε t . Assumption A3 allows to ε t to be t-distributed with time-varying degrees of freedom. The rank-based tests of [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF] and Belaire-Franch and Contreras (2004) on Assumption A1 while their sign-based tests are based on Assumptions A2 and A3.

Finally, the Chen and Deo's (2006) assumptions on the martingale difference sequence are also different B1: {ε t } is ergodic and E(ε t |F t ) = 0 for all t, where F t is a sigma field, ε t is F t measurable, and F t-1 ⊂ F t for all t.

B2: E(ε 2 t ) = σ 2 < ∞.

B3: For any integer q, 2 ≤ q ≤ 8, and for q nonnegative integers s i , E ∏ q i=1 ε s i t i = 0 when at least one s i is exactly one and ∑ q i=1 s i ≤ 8. B4: For any integer r, 2 ≤ r ≤ 4, and for r nonnegative integers s i , E ∏ r i=1 ε s i t i |F t = 0 when at least one s i is exactly one and ∑ r i=1 s i ≤ 4, for all t < t i , i = 1, 2, 3, 4.

B5: lim

T →∞

Var E ε 2 t+T ε 2 t+T + j |F t = 0 uniformly in j for every j > 0.

B6: lim

T →∞ E ε 2 t ε 2 t-T = σ 4 .
Assumptions B1-B6 allow the innovations ε t to be a martingale difference sequence with conditional heteroscedasticity. [START_REF] Chen | The variance ratio statistic at large horizons[END_REF] showed that the stochastic volatility and GARCH models satisfy Assumptions B1-B6. Assumptions B3-B4 state that the series {ε t } shows product moment behavior similar to that of an independent white noise process. Assumptions B5-B6 state that ε t and ε t-T are roughly independent for large lags T .

  have been proposed, such as Chen and Deo (2006) with a power-transformed VR statistic, Wright (2000) with exact VR tests based on rank and sign, Whang and Kim (2003) with subsampling method, and Kim (2006) with bootstrap method, among others. Therefore, because of the important literature on the VR tests we propose an overview on this subject. The rest of this paper is organized as follows. The individual and multiple VR tests are presented in Section 2 and 3, respectively. Section 4 discusses bootstrapping VR tests. An empirical illustration is proposed in Section 5. Section 6 concludes.

  [START_REF] Chow | A simple multiple variance ratio test[END_REF] proposed to using[START_REF] Hochberg | Some conservative generalizations of the T-method in simultaneous inference[END_REF] procedure for the multiple comparison of the set of VR estimates with unity, which allow us to examine a vector of individual VR tests while controlling for overall test size. For a set of m test statistics, the RWH is rejected if any one of the estimated VRs is significantly different from one.

  (1967) probability inequality and give an upper bound to the critical values taken in the studentized maximum modulus [SMM] distribution. Indeed, the statistic follows the SMM distribution with m and T degrees of freedom, i.e. SMM(α, m, T ), where m is the number of k values14 . The null hypothesis is rejected at α level of significance if the MV 1 statistic is greater than the [1 -(α * /2)]th percentile of the standard normal distribution where α * = 1 -(1 -α) 1/m .14 The critical values of the test are tabulated in[START_REF] Hahn | A table of percentage points of the distribution of the largest value of K Student T variates and its applications[END_REF] and[START_REF] Stoline | Tables of the studentized maximum modulus distribution and an application to multiple comparisons among means[END_REF]. It should be noted that when T is large, the critical values of the test can be calculated from the limiting distribution of the statistic.

  Consider a subsample (x t , . . . , x t-b+1 ) of size b for t = 1, . . . , Tb + 1. The statistic MV T calculated from the subsample is denoted as g T,b,t = g b (x t , . . . , x t-b+1 ). Then,G T (x)is approximated by the distribution function obtained by the collection of g T,b,t 's calculated from all individual subsamples. It can be written as ĜT,b (x) = (Tb + 2) b,t ≤ x where l(.) is the indicator function that takes 1 if the condition inside the bracket is satisfied and 0 otherwise. The 100(1 -α)% critical value for the test can be calculated as the (1 -α)th percentile of ĜT,b , while the p-value of the test is estimated as 1 -ĜT,b (MV T ). The null

3. 3

 3 Belaire-Franch and Contreras (2004) tests Recently, Belaire-Franch and Contreras (2004) proposed to substitute the standard VR tests by Wright's rank and sign-based tests, in the definition of Chow and Denning (1993) procedure to create a multiple rank and sign VR tests. The statistics are defined as

  i.d. assumption whereas the signsbased procedures are exact under both the i.i.d. and martingale difference sequence assumption. Belaire-Franch and Contreras (2004) showed that the ranks-based tests CD (R 1 ) and CD (R 2 ) are more powerful than their signs-based counterparts, CD (S 1 ) and CD (S 2 ) 15 . Moreover, Colletaz (2005) and Kim and Shansuddin (2007) also proposed an extension to the Wright's VR methodology following Chow-Denning, but only for the rank (R 1 )and sign (S 1 ) tests, respectively.

3. 4 Wald-type tests 3 . 4 . 1

 4341 [START_REF] Richardson | Tests of financial models in the presence of overlapping observations[END_REF] test Richardson and Smith (1991) suggested a joint test based on the following Wald statistic

3. 4 . 3

 43 Chen and Deo (2006) test Chen and Deo (2006) also proposed a joint VR test based on their individual power transformed VR statistic. They define the following Wald statistic

  emerging markets: Asian markets (Kim and Shamsuddin, 2008; Hoque et al., 2007), Eastern European markets (Smith and Ryoo, 2003), African markets (Smith et al., 2002; Al-Khazali et al., 2007; Lagoarde-Segot and Lucey, 2008) and Latin American markets (Chaudhuri and Wu, 2003; Chang et al., 2004). In this section we propose an illustration by examining the RWH for five emerging markets in Latin American, including Argentina, Brazil, Chile, Ecuador and Mexico. We use daily market prices spanning 03 August 1993 to 22 May 2007. All data are collected from Thomson Financial Datastream.

  5 otherwiseSimilarly to R 1 and R 2 tests, the critical values of the S 1 and S 2 tests can be obtained by simulating its exact sampling distribution.

	Note that S 1 assumes a zero drift value. In a recent work, Luger (2003) used ranks
	and signs to extend the Campbell and Dufour (1997) nonparametric approach to
	test for random walk with unknown drift. Belaire-Franch and Contreras (2004) and
	Belaire-Franch and Opong (2005a) applied the procedure described in Luger (2003) to
	compute S 2 .

  The Lo-MacKinlay test is an individual test where the null hypothesis is tested for an individual value of k. The question as to whether or not a time series is meanreverting requires that the null hypothesis hold true for all values of k. In view of this, it is necessary to conduct a joint test where a multiple comparison of VRs over a set of different time horizons is made. However, conducting separate individual tests for a number of k values may be misleading as it leads to over rejection of the null hypothesis of a joint test, above the nominal size. As stressed by[START_REF] Chow | A simple multiple variance ratio test[END_REF], this sequential procedure leads to an oversized testing strategy. Thus, the weakness of Lo-MacKinlay's test is that it ignores the joint nature of testing for the RWH.We present some multiple VR tests based on multiple comparison tests(Chow and Denning, 1993; Whang and Kim, 2003; Belaire-Franch and Contreras, 2004) and Wald-type tests (Richardson and Smith, 1991; Cecchetti and Lam, 1993; Chen and Deo, 2006) that combine the information contained in statistics at several horizons.

; Andrews and Monahan, 1992; Newey and West, 1994; among others).

[START_REF] Choi | Testing the random walk hypothesis for real exchange rates[END_REF]

employed the Andrews's (1991) methods to select the truncation point optimally and compute the VR test.

Note that the small sample properties of this automatic VR test under heteroscedasticity are unknown and have not investigated properly.

3 Multiple variance ratio tests

  , and it has the same critical values as MV 1 . However, with finite-sample sizes it may be preferable to use critical values obtained by simulations as done by Chow and Denning themselves. Nevertheless, as pointed out by Fong et al. (1997), Hochberg's approach is valid only if the vector of test statistics is multivariate normal. This condition is satisfied by VRs if there is little overlap in the data, i.e. if k/T is small.

3.2 Whang and Kim (2003) test

Whang and

[START_REF] Whang | A multiple variance ratio test using subsampling[END_REF]

developed a multiple VR test which uses a subsampling technique of Politis, Romano and Wolf (1997), which is a data-intensive method of approximating the sampling distribution. It can show better properties than the conventional VR tests when the sample size is relatively small. The Monte Carlo experiment results reported in

[START_REF] Whang | A multiple variance ratio test using subsampling[END_REF]

confirmed that their new VR test show excellent power in small samples, coupled with little or no serious size distortions.

To test the joint null hypothesis,

[START_REF] Whang | A multiple variance ratio test using subsampling[END_REF]

considered the statistic

Table 1

 1 presents summary statistics for the stock returns calculated as the first differences in the logs of the stock price indexes. The data are all leptokurtic as might be expected from daily stocks returns. Three series (Argentina, Brazil and Ecuador) are skewed. To check for nonlinear dependencies, we apply the Lagrange Multiplier test for autoregressive conditional heteroscedasticity (ARCH) on the residuals of the ARMA model, where the lag length is selected based on the Akaike and Schwarz in-

	reexamined the random walk prop-

  report the results of various individual and multiple VR tests21 , respectively, for the five Latin American markets. Since these stock returns exhibit conditional heteroscedasticity, we do not consider theLo-MacKinlay M 1 (k) and Chow-Denning MV 1 tests. More precisely, for individual VR tests we apply the Lo-MacKinlay M 2 (k) test as well as the Wright's R 1 , R 2 and S 2 tests. For multiple VR tests, we apply the Chow-Denning MV 2 test, the Richardson-Smith RS test, the 21 Some computational resources for VR tests are available from the different authors. Kim (2007) provides various VR tests written in R (http://www.r-project.org/) available from http://wwwpersonal.buseco.monash.edu.au/ jaekim/vrtest.htm. Moreover, a Stata module for the Lo-MacKinlay test written by Baum (2006) is available from http://fmwww.bc.edu/repec/bocode/l/lomackinlay.ado. Whang-Kim subsampling MV T test, the Belaire-Contreras rank-based CD(R 1 ) and CD(R 2 ) tests as well as the Kim's bootstrap MV * test 22

  0.6 ] for the subsampling test (MV T ). Some convergence amongst the individual and multiple VR tests is observed for Chile and Mexico. Indeed, the individual and multiple VR tests reject the RWH for these two Latin American markets, indicating that these markets have not been weak-Finally, we found mixed results from the various VR tests for Ecuador. Indeed, the Lo-MacKinlay M 2 (k) test is not significant, whereas the rank and sign-based tests as well as Chen-Deo V R β p are significant. Furthermore, the Kim's bootstrap and the Whang-Kim's subsampling tests do not reject the RWH while the others multiple VR tests show that Ecuador follows a random walk. Consequently, it is impossible to conclude on the weak-form efficiency for Ecuadorian market.

	form efficient.

For Argentina, some divergence amongst the individual tests is observed. The statistics of individual tests do not provide the same results. Nevertheless, as shown by

[START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF]

, the rank-based VR tests are more powerful than the conventional Lo-MacKinlay and the sign-based VR tests. Thus, in this context, it seems that Argentina does not follow a random walk. Furthermore, when the RWH is generally rejected under the multiple VR tests. Consequently, the exchange rate market of Argentina seems to be inefficiency.

Applying the individual VR tests shows that Brazil follows a random walk.

However, this result changes when the multiple VR tests are employed (except for Chen-Deo QP(k) test), showing that, as found by

[START_REF] Chang | Testing predictability in emerging equity markets[END_REF]

, the Brazilian exchange rate market is not an efficient market. As already noted, conducting individual tests for a number of k values may be misleading as it leads to over rejection of the null hypothesis of a joint test, above the nominal size.

Table 2 :

 2 Results of individual VR tests

	VR tests	k Argentina Brazil Chile Ecuador Mexico
			2	3.00 *	2.22 *	8.74 *	-1.41	3.60 *
	M 2 (k)	5 10	1.63 1.13	1.61 1.13	9.87 * 10.11 *	-1.99 * -1.17	2.05 * 1.56
			30	1.45	3.78 * 10.66 *	0.10	1.44
			2	3.34 *	1.50	16.96 *	1.33	7.31 *
	R 1 (k)	5 10	2.45 * 1.91 *	1.28 -2.24 * 16.91 * 17.29 *	5.11 * 8.61 *	4.64 * 3.53 *
			30	2.93 *	0.56	17.13 *	14.55 *	4.42 *
			2	3.10 *	1.61	15.87 *	1.10	7.04 *
	R 2 (k)	5 10	2.23 * 1.61	-1.14 16.31 * -2.30 * 15.73 *	4.14 * 7.73 *	4.16 * 3.02 *
			30	2.17 *	0.31	15.34 *	13.85 *	3.57 *
			2	2.09 *	0.90	12.79 *	7.34 *	5.95 *
	S 2 (k)	5 10	0.96 0.13	0.01 -0.01 12.92 * 12.79 *	14.98 * 21.45 *	3.58 * 2.37 *
			30	1.15	0.41	13.15 *	34.78 *	2.68 *
			2	1.63 * *	0.71	0.71	0.31	2.83 *
	V R	β p (k)	5 10	1.45 * * 0.95	-0.25 -1.27 -1.46 * * -0.54	0.29 1.43 * *	1.28 0.63
			30	1.06	-0.88	-0.12	3.31 *	0.60

Table 3 :

 3 Results of multiple VR tests Significant at the 5% level. The p-value are given for the MV * (Kim, 2006) and MV T (Belaire and Contreras, 2004) tests whereas the VR statistic is reported for the others tests.

	VR tests	Block length Argentina Brazil	Chile	Ecuador Mexico
	MV 2		3.00 *	3.78 *	10.66 *	1.99	3.60 *
	RS		29.94 *	83.91 * 380.74 * 46.94 *	43.87 *
	QP(k)		3.81	6.17	9.63 *	20.16 *	9.40 *
	MV *		0.01 *	0.00 *	0.00 *	0.10	0.00 *
	CD(R 1 )		3.69 *	4.85 *	18.14 *	13.65 *	7.84 *
	CD(R 2 )		4.19 *	6.42 *	18.31 *	11.63 *	7.89 *
		93	0.00 *	0.00 *	0.00 *	0.01 *	0.00 *
		157	0.00 *	0.00 *	0.00 *	0.05	0.00 *
	MV T	221 285	0.00 * 0.00 *	0.00 * 0.00 *	0.00 * 0.00 *	0.26 0.32	0.00 * 0.00 *
		349	0.00 *	0.00 *	0.00 *	0.32	0.00 *
		413	0.00 *	0.00 *	0.00 *	0.28	0.00 *
	*

When the error terms are not an i.i.d. sequence, the random walk process is denominated martingale process, whereas the sequence {ε t } T t=1 is the so-called martingale difference sequence (m.d.s.).Campbell et al. (1997) refers to the "random walk 3".

[START_REF] Daniel | The power and size of mean reversion tests[END_REF] explores a wider range of possible test statistics.

Recently,[START_REF] Kan | Exact variance ratio test with overlapping data[END_REF] provided analytical formulas for the moments of the sample variance ratio under both the null and the alternatives. See also[START_REF] Shively | An exact invariant variance ratio test[END_REF] for the case of mean.

[START_REF] Hoque | A comparison of variance ratio tests of random walk: A case of Asian emerging stock markets[END_REF] proposed a comparison of several variance ratio tests.

[START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] andCampbell et al. (1997) argued that using overlapping data in estimating the variances allowed to obtain a more efficient estimator and hence a more powerful test.

It has been argued that misleading conclusions may be obtained with VR statistics when time-varying volatility is present in the data. See, for example,Kim, Nelson and Startz (1991, 1998) andKim and Nelson (1998) who also proposed a solution based on a Bayesian approach and the use of a Gibbs sampler.

Note that the p-value adjustments can be applied to other VR tests for a joint hypothesis.

The estimator V R f based on Bartlett window as in[START_REF] Cochrane | How big is the random walk in GNP?[END_REF] also has a limiting normal distribution. However,[START_REF] Cogley | International evidence on the size of the random walk in output[END_REF] showed that this estimator seems inappropriate because it is right skewed. Therefore,[START_REF] Cogley | International evidence on the size of the random walk in output[END_REF] proposed to approximate this estimator by a multiple of chi-square variate, giving improvement over the normal.

[START_REF] Belaire-Franch | Ranks and signs-based multiple variance ratio tests[END_REF] also suggested to substitute the Wright's rank and sign-based

Note that[START_REF] Cecchetti | Variance ratio tests: Small-sample properties with an application to international output data[END_REF] showed that the empirical distribution of the statistic S(k) is numerically identical to the quadratic sum of the deviations of the first (k -1) autocorrelations from their population values, weighted by their covariance matrix.

Recently, Fleming et al. (2006) developed a bootstrap method for testing multiple inequality restrictions on VRs.

We do not apply the S 1 test suggested byWright (2001) since it assumes a zero drift which need not be satisfied in practice as well as the multiple signs-based tests developed by Belaire-Franch and Contreras (2004) since the rank-based tests are more powerful.

Following[START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF], we use the standard normal distribution for η t to implement the wild bootstrap test. He reports that other choices provided qualitatively similar sample results.

* Means significant at 1% level, respectively. LM[START_REF] Campbell | Exact nonparametric tests of orthogonality and random walk in the presence of a drift parameter[END_REF] indicates the lagrange multiplier test for conditional heteroscedasticity with 10 lags.

* and * * Significant at the 5% and 10% level, respectively. We report the VR statistic for each test.


Amélie Charles
email: acharles@audencia.com

Olivier Darné
email: olivier.darne@univ-nantes.fr

The random walk hypothesis for Chinese stock markets: Evidence from variance ratio tests *

Keywords: Chinese stock markets, market efficiency, random walk hypothesis, variance ratio test. JEL Classification: G14, G15, C14

This study examines the random walk hypothesis for the Shanghai and Shenzhen stock markets for both A and B shares, using daily data over the period 1992-2007. The hypothesis is tested with new multiple variance ratio tests -Whang-Kim subsampling and Kim's wild bootstrap tests -as well as the conventional multiple Chow-Denning test. We find that Class B shares for Chinese stock exchanges do not follow the random walk hypothesis, and therefore are significantly inefficient. The Class A shares seem more efficient.

Introduction

China's stock market is a relatively new but increasingly important part of the Chinese financial system which is undergoing a structural shift from a heavilyregulated and almost exclusively bank-based system to one with much greater diversity of institutions including a vigorous and increasingly sophisticated stock market.

The Chinese stock exchanges consist of two official markets 1 : the Shanghai Stock Exchange (SSE), located in the coastal city of Shanghai -the largest city and one of the financial centres in China, and Shenzhen Stock Exchange (SZE), situated in the southern city of Shenzhen which is the first and most important special economic zone in China neighboring Hong Kong. SSE was established on 26 November 1990 and started trading on 19 December of the same year, and SZE was established on 1 December 1990 and started trading on 3 July 1991. There are some differences between these two stock exchanges. First, the characteristics of companies listed in these two stock exchanges are different. While most companies listed on the SSE are large and state-owned, those on the SZE are small, joint ventures, and export-oriented.

Second, the relative size of these two markets has been undergoing major changes, such as government policy shift 2 . SSE and SZE trade principally two classes of shares which are A and B shares 3 .

A shares are denominated and traded in the local currency, Renminbi [RMB], and designed for domestic investors 4 , and B shares are denominated in RMB but are subscribed and traded in foreign currencies, either the US dollars in the SSE or the HK dollars in the SZE. Since February 2001, B shares have been opened to domestic Chinese investors holding US or HK dollars. This policy change has greatly stimulated the trading of B-shares and accelerated the market integration process of A-share markets with B-share markets and the international stock markets. There are substantial different characteristics in the market for these two types of shares. Of 1 See Ma and Chung (1993) and [START_REF] Xia | The Re-emerging Securities Market in China[END_REF], among others, for a review on the history and development of China's stock market.

2 See [START_REF] Xu | The microstructure of the Chinese stock market[END_REF] for a discussion on the microstructure of the Chinese stock market. 3 Additional types of shares include C-shares, H-shares and N-shares. C-shares can only be traded among Chinese state institutions, enterprises and departments with a legal person status. H-shares and N-shares are available only to foreign investors, and represent the shares listed on the Hong Kong Stock

Exchange and the New York Stock Exchange, respectively. See [START_REF] Seddighi | The Chinese stock exchange market: Operations and efficiency[END_REF] for a discussion on the different types of shares in the Chinese stock market. 4 Rules governing the ownership of A-shares have also recently been relaxed. Specifically, foreign institutional investors who are approved by the Chinese government (known as Qualified Foreign Institutional Investors) are now permitted to trade A-shares. However, any capital invested and any related income cannot be repatriated for 3 years (Wang, Burton and Hanah, 2004).

these two types of shares listed on the domestic exchanges, A share trades dominate, both in terms of size and level of activity. Indeed, the average volume traded in Class

A is approximately 40 times the average volume traded in Class B, therefore the Class A shares should be more liquid. Furthermore, the main investors in Class A shares are individuals while Class B shares are mainly owned by large foreign institutional investors 5 . Moreover, foreign investors in the Chinese stock markets may be at an informational disadvantage relative to domestic Chinese investors (information asymmetry) due to language barriers, different accounting standards and the lack of reliable information about local firms [START_REF] Chakravarty | Information asymmetry, market segmentation and the pricing of cross-listed shares: Theory and evidence from Chinese A and B shares[END_REF][START_REF] Chan | Are domestic investors better informed than foreign investors? Evidence from the perfectly segmented market in China[END_REF].

Chinese companies that are listed as B shares are subject to more strict disclosure requirements and they are generally more financially stable than A share companies.

Finally, the returns to A and B shares have behaved quite differently, for example, [START_REF] Chen | Foreign ownership restrictions and market segmentation in ChinaŠs stock markets[END_REF] argued that A shares are over-priced and that the returns to B shares move more closely with market fundamentals than do those for A share prices. All these factors may imply different efficiency behavior between the A and B shares.

Since their establishment, the two exchanges expanded rapidly and operated in a continually changing regulatory environment. China's stock market is now the second largest in Asia, behind only Japan. The speculation is that China's securities market has the potential to rank among the top four or five in the world within the coming decade [START_REF] Ma | China's financial liberalization agenda[END_REF].

In this paper we analyze the efficiency of the Chinese stock markets6 . The literature on market efficiency and stock market predictability is vast, as researchers have been discussing this theme in depth for the past decades (see [START_REF] Fama | Efficient capital markets: A review of theory and empirical work[END_REF]1991;Fama andFrench, 1988, Lo and[START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF]among others). A capital market is considered as efficient if stock prices at any time fully reflect all available and relevant information. Therefore, given only past price and return data, the current price is the best predictor of the future price, and the price change or return is expected to be zero.

This the essence of the weak-form efficient market hypothesis [EMH], which implies a random walk. It is this random-walk implication of the EMH which is most commonly 5 Wang, [START_REF] Wang | Return and risk interaction in the Chinese stock market[END_REF] and [START_REF] Qiao | Policy change and lead-lag relations among China's segmented stock markets[END_REF]

investigated interactions between

Chinese A shares and B shares traded on SSE and SZE. They find evidence of causal relations between shares of the same type, i.e., between the two A (B) share markets, than those between shares of different types. Furthermore, [START_REF] Qiao | Policy change and lead-lag relations among China's segmented stock markets[END_REF] showed that since the implementation of liberal policy that allowed domestic investors to invest in B-share markets A-share markets tend to lead their B-share counterparts in the same stock exchange.

tested in the empirical literature.

Various methodologies have been used to test the EMH for Chinese stock markets and obtained mixed results (see Table 1). [START_REF] Wu | The analysis of the efficient of securities market in our country[END_REF], [START_REF] Laurence | Weak-form efficiency and causality tests in Chinese stock markets[END_REF] and [START_REF] Mookerjee | An empirical analysis of the equity markets in China[END_REF] applied serial correlation tests while [START_REF] Liu | Are Chinese stock markets efficient? A cointegration and causality analysis[END_REF], [START_REF] Groenewold | The efficiency of the Chinese stock market and the role of banks[END_REF] and [START_REF] Seddighi | The Chinese stock exchange market: Operations and efficiency[END_REF] employed unit root tests. However, [START_REF] Lo | The size and power of the variance ratio test in finite samples: A Monte Carlo investigation[END_REF] showed that unit root and serial correlation tests are less powerful than variance-ratio [VR] test, especially in the presence of heteroscedasticy. Thus, [START_REF] Long | Information transmission in the Shanghai equity market[END_REF], [START_REF] Darrat | On testing the random-walk hypothesis: A model comparison approach[END_REF], [START_REF] Ma | Are China's stock markets really weak-form efficient?[END_REF], [START_REF] Lee | Stock returns and volatility on China's stock markets[END_REF], [START_REF] Lima | Tests of the random walk hypothesis for equity markets: Evidence from China, Hong Kong and Singapore[END_REF] and [START_REF] Fifield | Further evidence on the efficiency of the Chinese stock markets: A note[END_REF] [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] and [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] proposed two alternatives which do not rely asymptotic approximations by suggesting subsampling and wild bootstrap version of the Chow-Denning test, respectively.

We contribute to the literature by re-examining the weak form of the EMH, using daily Chinese stock market data. While numerous studies addressed issues in this area in the recent past, we extend the existing studies in two ways. First, this study is based on a much more extensive sample. We study daily data for the Shanghai and Shenzhen stock markets for both A and B shares over the period 1992-2007. This large sample provides us with a greater variety of information and should reflect the dramatic changes that have taken place in China's securities sector in the past decade.

We also investigate the EMH over various sub-periods in order to analyze the effects of the important changes in the relationship between the banks and the stock market in 1996 and 2000 as well as those of the implementation of the new policy allowing domestic citizens to invest in B-share markets in 2001. Second, the weak form of the EMH is evaluated from new VR tests, which are more powerful than those applied in the previous studies. More precisely, we adopt multiple VR tests suggested by [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] and [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF], in addition to the conventional [START_REF] Chow | A simple multiple variance ratio test[END_REF] test, to examine the RWH for the Shanghai and Shenzhen stock markets.

The rest of this paper is organized as follows. Section 2 summarizes the characteristics of the data on Chinese stock markets. Section 3 discusses the variance ratio tests. Section 4 reports the empirical results. The conclusion is drawn in Section 5.

Insert Table 1 here 2 Data and summary statistics Table 2 presents summary statistics for the stock returns calculated as the first differences in the logs of the stock price indexes. For the SSE, as shown in previous studies (e.g., [START_REF] Chakravarty | Information asymmetry, market segmentation and the pricing of cross-listed shares: Theory and evidence from Chinese A and B shares[END_REF]Groenewold et al., 2001), the A shares have higher returns than B shares. In contrast, the A-shares returns are slightly smaller than the B-shares returns for the SZE. All the returns are highly non-normal, i.e. showing evidence of significant positive skewness and excess kurtosis, as might be expected from daily stocks returns. The Lagrange Multiplier test for the presence of the ARCH effect indicates clearly that all stocks show strong conditional heteroscedasticity, which is a common feature of financial data.

Insert Table 2 here 3 The variance ratio tests

Since the seminal work of Lo andMacKinlay (1988, 1989) and [START_REF] Poterba | Mean reversion in stock prices: Evidence and implications[END_REF], the standard variance ratio [VR] test or its improved modifications have been widely used for testing market efficiency8 .

The VR methodology consists of testing the random walk hypothesis [RWH] against stationary alternatives, by exploiting the fact that the variance of random walk increments is linear in all sampling intervals, i.e., the sample variance of k-period return is k times the sample variance of one-period return. The VR at lag k is then defined as the ratio between (1/k)th of the k-period return to the variance of the oneperiod return. Hence, for a random walk process, the VR should be equal to 1 for all values of k.

Let x t be an asset return at time t, where t = 1, . . . , T . It is assumed that x t is a realization of the underlying stochastic process X t , which follows a martingale difference sequence. This means that X t 's are serially uncorrelated, but are allowed to be conditionally or unconditionally heteroscedastic. Following [START_REF] Wright | Alternative variance-ratio tests using ranks and signs[END_REF], the VR statistic be written as

V R(x; k) = (T k) -1 T ∑ t=k (x t + • • • + x t-k+1 -k μ) 2 ÷ T -1 T ∑ t=1 (x t -μ) 2 (1)
where μ = T -1 ∑ T t=1 x t . This is an estimator for the unknown population VR, denoted as V (k), which is the ratio of 1/k times the variance of the k-period return to the variance of the one-period return. If the stock return follows a random walk, the expected value of V R(x; k) should be equal to unity for all horizons k. If this ratio is less than one at long horizons, then we have indications of negative serial correlation (mean-reversion) and ratios greater than one at long horizons implies positive serial correlation (mean-aversion). [START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] proposed the asymptotic distribution of V R(x; k) by assuming that k is fixed when T → ∞. They show that under the assumption of conditional heteroscedasticity, then under the null hypothesis that V (k) = 1, the test statistic M(x; k) is given by9

M(x; k) = V R(x; k) -1 φ * (k) 1/2 (2)
follows the standard normal distribution asymptotically, where [START_REF] Chow | A simple multiple variance ratio test[END_REF] tests [START_REF] Chow | A simple multiple variance ratio test[END_REF] provide a multiple VR test for the joint null hypothesis

φ * (k) = k-1 ∑ j=1 2(k -j) k δ( j) δ( j) = T ∑ t= j+1 (x t -μ) 2 (x t-j -μ) 2 ÷    T ∑ t=1 (x t -μ) 2 2    3.1
V (k i ) = 1 for i = 1, .
. . , m against the alternative that V (k i ) = 1 for some holding period k i . The (heteroscedasticity-robust) test statistic can be written as

MV (x; k i ) = √ T max 1≤i≤m |M(x; k i )| (

Whang and Kim (2003) test

Whang and Kim (2003) develop a multiple VR test which uses a subsampling technique of [START_REF] Politis | Subsampling for heteroskedastic time series[END_REF], which is a data-intensive method of approximating the sampling distribution.

To test the joint null hypothesis that [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] consider the statistic

V (k i ) = 1 (i = 1, . . . , m) against V (k i ) = 1 (for some k i ),
MV T = √ T g N (x 1 , . . . , x T ) (4)
where

g t (x 1 , . . . , x T ) = max 1≤i≤m |M r (x; k i )| with M r (x; k i ) = V R(x; k i ) -1,

and

V R(x; k) is as defined in (1). The sampling distribution function for the MV T statistic is written as

G T (x) = P √ T g t (x 1 , . . . , x T ) ≤ x
They show that the asymptotic null distribution of the statistic is that of a maximum of a multivariate normal vector with unknown covariance matrix, which is complicated to estimate. Therefore, they propose to approximate the null distribution by means of the subsampling approach.

Consider a subsample (x t , . . . , x t-b+1 ) of size b for t = 1, . . . , Tb + 1. The statistic MV T calculated from the subsample is denoted as g T,b,t = g b (x t , . . . , x t-b+1 ). Then, G T (x) is approximated by the distribution function obtained by the collection of g T,b,t 's calculated from all individual subsamples. It can be written as

ĜT,b (x) = (T -b + 2) -1 T -b+1 ∑ t=0 l √ bg T,b,t ≤ x
where l(.) is the indicator function that takes 1 if the condition inside the bracket is satisfied and 0 otherwise.

The 100(1 -α)% critical value for the test can be calculated as the (1 -α)th percentile of ĜT,b , while the p-value of the test is estimated as 1 -ĜT,b (MV T ). The null hypothesis that V (k i ) = 1 (i = 1, . . . , m) is rejected at the level of significance α if the observed MV T is greater than this critical value or if the p-value is less than α.

To implement the subsampling technique, a choice of block length b should be made. [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] recommend that a number of block lengths from an equally spaced grid in the interval of [2.5T 0.3 , 3.5T 0.6 ] be taken. However, they find that the size and power properties of their test are not sensitive to the choice of the block length.

3.3 [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] test [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] uses the wild bootstrap which is a resampling method that approximates the sampling distribution of the VR test statistic, and is applicable to data with unknown forms of conditional and unconditional heteroscedasticity [START_REF] Mammen | Bootstrap and wild bootstrap for high dimensional linear models[END_REF].

The wild bootstrap is applied to Chow-Denning, MV (x; k i ), VR test. The wild bootstrap test based on MV (x; k i ) can be conducted in three stages as below (i) Form a bootstrap sample of T observations x * t = η t x t (t = 1, . . . , T ) where η t is a random sequence with E(η) = 0 and E(η 2 ) = 1.

(ii) Calculate MV * = MV (x * ; k i ), the MV (x * ; k i ) statistic obtained from the bootstrap sample generated in stage (i).

(iii) Repeat (i) and (ii) sufficiently many, say m, times to form a bootstrap distribution of the test statistic {MV ( j)} m j=1 .

The bootstrap distribution {MV ( j)} m j=1 is used to approximate the sampling distribution of the MV (x; k i ) statistic. The p-value of the test can be obtained as the proportion of {MV ( j)} m j=1 greater than the MV (x; k i ) statistic calculated from the original data. Following [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] we use the standard normal distribution for η t to implement the wild bootstrap test 10 . (2,5,10,20,40).

Empirical findings

As advocated by [START_REF] Deo | On the asymptotic power of the variance ratio test[END_REF], we use relatively short holding periods when testing for the mean reversion using VR tests. For the wild bootstrap test (MV * ), the number of bootstrap replications m is set to 1000.

The W K test shows rejection of RWH for the four Chinese stock markets, as the pvalues are less than 1%. The MV and MV * confirm this result but only for B shares in the SSE and the SZE. Therefore, the Class B shares do not follow the RWH, which suggests that institutional investors do not play a role in explaining results of weak form efficiency tests. This result is similar to those obtained by [START_REF] Lima | Tests of the random walk hypothesis for equity markets: Evidence from China, Hong Kong and Singapore[END_REF].

Contrary to the W K test, the MV and MV * tests accept the null hypothesis of efficiency for the Class A shares. Moreover, [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] shows that the MV * test gives higher power than the W K test, except when the sample size is very small 11 . This result allows to conclude, with caution, that the Class A shares seem to follow the RWH. In this case, liquidity and market capitalization may play a role in explaining results of weak form efficiency tests. The results on the Class A shares contrast with those obtained by [START_REF] Darrat | On testing the random-walk hypothesis: A model comparison approach[END_REF], [START_REF] Ma | Are China's stock markets really weak-form efficient?[END_REF] and [START_REF] Lee | Stock returns and volatility on China's stock markets[END_REF] who reject the RWH for Class A shares in the SSE and the SZE. This can be explained by the fact that these authors employed the individual Lo-MacKinlay VR test rather than multiple VR tests 12 . Consequently, Class B shares seem to be less 10 Kim (2006) reports that other choices provided qualitatively similar sample results. 11 The Monte Carlo experiment results reported in [START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] confirm that their VR test show excellent power in very small samples.

12 Our results are confirmed when we run the multiple VR tests over the same periods used by [START_REF] Darrat | On testing the random-walk hypothesis: A model comparison approach[END_REF], [START_REF] Ma | Are China's stock markets really weak-form efficient?[END_REF] and [START_REF] Lee | Stock returns and volatility on China's stock markets[END_REF]. The results are available upon request.

efficient than Class A shares. One possible explanation for the greater inefficiency of Class B shares is information asymmetry. In particular, foreign investors in Class B shares may have an information disadvantage relative to domestic investors trading Class A shares due to language barriers, different accounting standards and the lack of reliable information about local firms [START_REF] Chakravarty | Information asymmetry, market segmentation and the pricing of cross-listed shares: Theory and evidence from Chinese A and B shares[END_REF][START_REF] Chan | Are domestic investors better informed than foreign investors? Evidence from the perfectly segmented market in China[END_REF].

Insert Table 3 here

As shown by [START_REF] Groenewold | The efficiency of the Chinese stock market and the role of banks[END_REF], the banks have a traditional importance in the Chinese financial system, and thus the important changes in the relationship between the banks and the stock market in 1996 and 2000 could have significant effects on the efficiency of the Chinese stock markets. More precisely, until 1996 banks had a dominant influence on the stock market. In 1996, regulations were further tightened by preventing banks from offering loans for stock transactions. The aim of these was to encourage independent competitive firms as brokers and sources of funds such as mutual funds independent of the banks. In early 2000 the 1996 regulations were reversed and banks resumed their positions as important sources of funds for stock investment [START_REF] Surry | Fair shares: China adjusts rules to boost its stock markets[END_REF]. Moreover, in February 2001 the Chinese government adopted a more liberal policy that allowed domestic investors to invest in B-share markets, which were only available to foreign investors. This change could also affect the efficiency of the Chinese stock markets for B shares.

We investigated these questions by re-running the multiple VR tests for the Shanghai A and B shares and the Shenzhen A and B shares for the following subperi- 4567.

The results for Class A in both the Shanghai and Shenzhen stock exchanges show efficiency. The multiple VR tests find the non-rejection of RWH over the four subperiods 14 . Therefore, the changes in the relationship between the banks and the stock market do not affect the efficiency of Class A for Chinese stock markets. This result suggests that the exclusion of the banks have not improved the efficiency of the stock market. This is consistent with the liquidity explanation, that the banks are important sources of liquidity and that excluding them from involvement in the stock market may make for a more competitive and diversified brokerage industry in the long 13 The first subperiod begins in 03/01/1992 and 24/02/1992 for Shanghai A and B shares, respectively. 14 Only the W K test reject RWH on the sub-period 2001-2007 for the two Class A shares and on the sub-period 1992-1996 for the Shanghai A share but the MV * test gives higher power than the W K test [START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF].

run but in the short run reduces the amount of trading in the market and so slows the information diffusion [START_REF] Groenewold | The efficiency of the Chinese stock market and the role of banks[END_REF].

The results for Class B in both the Shanghai and Shenzhen stock exchanges are mixed. The multiple VR tests reject the RWH over the sub-periods 1992-1996 and 1996-1999

  The data used in this study are daily closing price stock indices for Chinese stock exchanges in Shanghai and Shenzhen for A and B shares. The data comes from Thomson Financial Datastream. The time period covered is from January 3 and February 24, 1992 to July 6, 2007 for the Shanghai A and the Shanghai B, respectively, and from October 5, 1992 to July 6, 2007 for the Shenzhen A and B.

  3) This is based on the idea that the decision regarding the null hypothesis can be made based on the maximum absolute value of the individual VR statistics. The statistic follows the studentized maximum modulus [SMM] distribution with m and T degrees of freedom, i.e. SMM(α, m, T ), where m is the number of k values, whose critical values are tabulated in Stoline and Ury (1979). When T is large, the null hypothesis is rejected at α level of significance if the MV (x; k i ) statistic is greater than the [1 -(α * /2)]th percentile of the standard normal distribution where α * = 1 -(1 -α) 1/m .

  while this hypothesis is not rejected over the sub-period2000- 2001, showing that the efficiency improved when banks permitted to re-enter the stock market in 2000. However, the RWH seems to be again rejected over the subperiod 2001-2007 after the B-shares opening to domestic Chinese investors, implying inefficiency during this period. This result should be taken with caution because the VR tests are close to their 5% significant level. Nevertheless, the test statistics of the multiple VR tests decreased after 2001 compared to before 2000, suggesting that the relaxation of restrictions on domestic investor participation in Class B shares has had a positive impact on pricing efficiency in the B-share market.Insert Tables4-7here5 ConclusionThis study has examined the weak-form efficiency of the Chinese stock markets, especially for the Shanghai and Shenzhen stock markets for both A and B shares, using new and conventional multiple variance ratio statistics. These tests, which are robust to heteroscedasticity, are the[START_REF] Whang | A multiple variance ratio test using subsampling[END_REF] subsampling test and[START_REF] Kim | Wild bootstrapping variance ratio tests[END_REF] bootstrap test, which do not rely asymptotic approximations, as well as the[START_REF] Chow | A simple multiple variance ratio test[END_REF] test. In addition, the paper investigated the impact on Chinese stock market efficiency following the changes in the relationship between the banks and the stock market as well as the regulatory change that widened the B-share market to include domestic investors.The results suggest that Class A shares appear more efficient than Class B shares, implying that liquidity, market capitalization and information asymmetry can play a role in explaining the weak-form efficiency. Class B shares for Chinese stock exchanges do not follow the random walk hypothesis and therefore are significantly inefficient, but they become efficiency after the re-entry of banks in the stock market. However, Class B shares seems to appear inefficient after the B-shares opening to domestic Chinese investors. Nevertheless, the entry of Chinese investors to the B-share market have positively impacted the B-share market efficiency. Further research should investigate the effects of the re-entry of banks and the entry of domestic investors on the efficiency of Class B shares.

  7 used the individual Lo-MacKinlay VR test but they did not find consensus on the weak-form efficiency for Chinese stock markets.

Nevertheless, it is preferable to employ multiple VR tests rather than individual VR tests. Indeed, in practice, it is customary to examine the VR statistics for several holding periods (k) to test the random-walk hypothesis

[RWH]

. The null is rejected if it is rejected for some k value. As stressed by

[START_REF] Chow | A simple multiple variance ratio test[END_REF]

, this sequential procedure leads to size distortions and therefore they suggested multiple VR tests to avoid this problem. However, a well-known problem with the VR test is that the standard (individual and multiple) VR tests, which are tests based on asymptotic approximations, are biased (severe size distortions and low power) and right-skewed in finite samples, resulting in misleading statistical inference. Recently,

Whang and

  Table 3 reports the multiple VR test results. The test statistic is displayed for the Chow-Denning (MV ) test, while the p-values are reported for the Kim (MV * ) and Whang-Kim (W K) tests. The holding periods (k i 's) considered are

Table 1 :

 1 Selected studies on the EMH for Chinese stock markets.

	Studies	Sample	Methodologies	Comments
	Wu (1996)

Table 2 :

 2 Summary statistics of stock returns(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007).Notes: The skewness and kurtosis statistics are standard-normally distributed under the null of normality distributed returns. ARCH(10) indicates the Lagrange multiplier test for conditional heteroscedasticity with 10 lags. * and * * means significant at 1% and 5% level, respectively. 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * Shanghai B 5.834 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.004 * 0.004 * 0.000 * Shenzhen B 5.646 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.007 * 0.034 * MV : Chow-Denning test; MV * : Kim test; W K: Whang-Kim test; * means significant at 1% level. The 1% critical value for the Chow-Denning test is 3.022. Following[START_REF] Whang | A multiple variance ratio test using subsampling[END_REF], block lengths are chosen from an equally spaced grid in the interval [2.5T 0.3 ,3.5T0.6 ].

	Stock market Obs. Mean (% daily) SD (% daily) Skewness Kurtosis ARCH(10)
	Shanghai A	4047	0.0643	2.7442	6.0618 *	151.9544 * 23.0674 * *
	Shanghai B	4011	0.0180	2.1851	0.3612 *	8.5989 *	545.2869 *
	Shenzhen A	3850	0.0355	2.2605	0.9071 *	21.9627 *	323.9831 *
	Shenzhen B	3850	0.0401	2.1536	0.3086 *	10.6953 *	657.5540 *

* *

Table 4 :

 4 Multiple VR test results (1992[START_REF] Wu | The analysis of the efficient of securities market in our country[END_REF]. 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * Shanghai B 5.006 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * 0.000 * * 0.001 * 0.000 * 0.000 * 0.000 * 0.000 * 0.032 * 0.060 Whang-Kim test; * and * * mean significant at 1% and 5% level, respectively. The 1% and 5% critical values for the Chow-Denning test are 3.022 and 2.491, respectively. Following[START_REF] Whang | A multiple variance ratio test using subsampling[END_REF], block lengths are chosen from an equally spaced grid in the interval [2.5T 0.3 ,3.5T0.6 ].

	MV	MV *		W K
			Block lengths
		51	82	113	144	175	206
	Shanghai A 1.920 0.000 Shenzhen A 0.718 0.129 0.817 0.989	0.811	0.732	0.712	0.691 0.709
	Shenzhen B 2.881

* * MV : Chow-Denning test; MV * : Kim test; W K:

See Chan, Fung and Thapa (2007) for a review and synthesis of the financial research on China.

[START_REF] Lima | Tests of the random walk hypothesis for equity markets: Evidence from China, Hong Kong and Singapore[END_REF] and[START_REF] Fifield | Further evidence on the efficiency of the Chinese stock markets: A note[END_REF] also employed the Cecchetti-Lam VR and the non-parametricWright's VR tests, respectively.

See Hoque, Kim and Pyun (2007) and Charles and Darné for a review.

[START_REF] Lo | Stock market prices do not follow random walk: Evidence from a simple specification test[END_REF] also propose a test statistic under the assumption of homoscedasticity. We focus only on VR statistic which is robust under heteroscedasticity since, as shown in Section 3, all the data display heteroscedasticity.

Serial correlation EMH for Shanghai and Shenzhen (individual shares) Laurence et al. (1997Laurence et al. ( ) 1993Laurence et al. ( -1996 (D) (D) Serial correlation EMH for A-shares but not for B-shares Liu et al. (1997Liu et al. ( ) 1992Liu et al. ( -1995 (D) (D) Dickey-Fuller unit root EMH for Shanghai and Shenzhen Mookerjee and Yu (1999) 1990-1993 (M) Serial correlation and run not EMH for Shanghai and Shenzhen Long et al. (1999) 1992-1994 (W) Lo-MacKinlay VR EMH for Shanghai (A and B-shares) Darrat and Zhong (2000) 1990-1998 (W) Lo-MacKinlay VR not EMH for Shanghai and Shenzhen (A-shares) Ma and Barnes (2001) 1990-1998 Serial correlation, run not EMH for Shanghai (A and B-shares) (D, W, M) and Lo-MacKinlay VR and Shenzhen (B-shares) Lee, Chen and Rui (2001) 1990-1997 (D) Lo-MacKinlay VR not EMH for Shanghai and Shenzhen (A and B-shares) Groenewold et al. (2003) 1992-2001 (D) Dickey-Fuller and EMH for Shanghai and Shenzhen Phillips-Perron unit root (A and B-shares) Seddighi and Nian (2004) 2000-2000 (D) Serial correlation and not EMH for Shanghai Dickey-Fuller unit root Lima and Tabak (2004) 1992-2000 (D) Lo-MacKinlay and EMH for A-shares but not for B-shares Cecchetti-Lam VR Fifield and Jetty (2008) 1996-2005 (D) Lo-MacKinlay and EMH for A-shares but not for B-shares Wright VR Notes: D: daily; W: weekly; M: monthly. EMH: efficiency market hypothesis.