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st for the null hypothesis of a random walk under both homoskedasticity and heteroskedasticity. Since the violation of a random walk can result either from changing variance, i.e. heteroskedasticity, or autocorrelation in returns, the test can help to discriminate reasons for deviation to some extent. The lag orders are 2, 4, 8 and 16. In Table VII, the variance ratio (VR(q)), the homoskedastic-consistent statistics (Z(q)) and the heteroskedastic-consistent statistics (Z*(q)) are presented for each lag.

As is pointed out by Lo and MacKinlay (1988), the variance ratio statistic VR(2) is equal to one plus the first-order correlation coefficient. Since all the autocorrelations are zero under RW1, VR(2) should equal one. The conclusion can be generalised further to state that for all q, VR(q) should equal one.

According to the first Panel in Table VII, of all stocks and indices, only LION and NAN D1 have variance ratios that are significantly different from one at all lags. Therefore, the null hypothesis of a random walk under both homoskedasticity and heteroskedasticity is rejected for LION and NAN D1, and thus they are not weak-form efficient because of autocorrelations. In terms of FARO, the null hypothesis of a homoskedastic random walk is rejected, while the hypothesis of a heteroskedastic random walk is not. This implies that the rejection of random walk under homoskedasticity could partly result from, if not entirely due to heteroskedasticity. On the other hand, both FEIC and NAN D10 follow random walk and turn out to be efficient in weak form, corresponding exactly to the autocorrelation results reached before in Table III.

Panel B shows that when monthly data are used, the null hypothesis under both forms of random walk can only be rejected for FARO. As for FEIC, the random walk null hypothesis is rejected under homoskedasticity, but not under heteroskedasticity, indicating that rejection is not due to changing variances because Z*(q) is heteroskedasticity-consistent.

As is shown in Panel A for daily data, all individual stocks have variance ratios less than one, implying negative autocorrelation. However, the autocorrelation for stocks is statistically insignificant except for LION. On the other hand, variance ratios for NAN D1 are greater than one and increasing in q. The above finding provides supplementary evidence to the results of autocorrelation tests. As Table III shows, NAN D1 has positive autocorrelation coefficients in all lags, suggesting a momentum effect in multiperiod returns. Both findings appear to be well supported by empirical evidence. While daily returns of individual stocks seem to be weakly negatively correlated (French and Roll (1986)), returns for best performing market indices such as NAN D1 show strong positive autocorrelation (Campbell, Lo, and MacKinlay (1997)). The fact that individual stocks have statistically insignificant autocorrelations is mainly due to the specific noise contained in company information, which makes individual security returns unpredictable. On the contrary, while the positive serial correlation for NAN D1 violates the random walk, such deviation provides investors with confidence to forecast future prices and reliability to make profits.

C. Griffin, Kelly and Nardari DELAY Tests

The results of delay test for the three stocks and two decile indices over the January 2000 to December 2005 peri