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tinuously compounded (log) returns and simple returns are reported, although the analysis is based on the result of the first one.

B. Methods

B.1. Autocorrelation Tests

One of the most intuitive and simple tests of random walk is to test for serial dependence, i.e. autocorrelation. The autocorrelation is a time-series phenomenon, which implies the serial correlation between certain lagged values in a time series. The first-order autocorrelation, for instance, indicates to what extent neighboring observations are correlated. The autocorrelation test is always used to test RW3, which is a less restrictive version of random walk model, allowing the existence of dependent but uncorrelated increments in return data. The formula of autocorrelation at lag k is given by:

(1) where is the autocorrelation at lag ; is the log-return on stock at time; and is the log-return on stock at time. A greater than zero indicates a positive serial correlation whereas a less than zero indicates a negative serial correlation. Both positive and negative autocorrelation represent departures from the random walk model. If is significantly different from zero, the null hypothesis of a random walk is rejected.

The autocorrelation coefficients up to 5 lags for daily data and 3 lags for monthly data are reported in our test. Results of the Ljung-Box test for all lags up to the above mentioned for both daily and monthly data are also reported. The Ljung-Box test is a more powerful test by summing the squared autocorrelations. It provides evidence for whether departure for zero autocorrelation is observed at all lags up to certain lags in either direction. The Q-statistic up to a certain lag m is given by:

(2)

B.2. Variance Ratio Tests

We follow Lo and MacKinlay's (1988) single variance ratio (VR) test in our study. The test is based on a very important assumption of random walk that variance of increments is a linear function of the time interval. In other words, if the random walk holds, the variance of the qth differed value should be equal to q times the variance of the first differed value. For example, the variance of a two-period return should be equal to twice the variance of the one-period return. According to its definition, the formula of variance ratio is denoted by:

(3) where q is any positive integer. Under the null hypothesis of a random walk, VR(q) should be equal to one at all lags. If VR(q) is greater than one, there is positive serial correlation which indicates a persistence in prices, corresponding to the momentum effect. If VR(q) is less than one, there is negative serial correlation which indicates a reversal in prices, corresponding to the mean-reverting process.

Note that the above two test are also tests of how stock prices react to publicly available information in the past. If market efficiency holds true, information from past prices should be immediately and fully reflected in the current stock price. Therefore, future stock price change conditioned on past prices should be equal to zero.

B.3. Griffin-Kelly-Nardari DELAY Tests

As defined by Griffin, Kelly and Nardari (2007), 'delay is a measure of sensitivity of current returns to past market-wide information'.[2] Speaking differently, delay measures how quickly stock returns can react to market returns. The logi