A Simple Specification Test of the Random Walk Hypothesis

We propose a simple test for the random walk hypothesis using variance estimators derived from data sampled at different frequencies. This Hausman--type specification-test exploits the linearity of the variance of random walk increments in the observation interval by comparing the (per unit time) variance estimates obtained from distinct sampling intervals. Test statistics are derived for both the i.i.d. Gaussian random walk and the more general uncorrelated but possibly heteroscedastic random walk. Monte Carlo experiments indicate that although the finite-sample behavior of our specification test is comparable to that of the Dickey-Fuller t-test and the Box-Pierce Q-statistic under the i.i.d. null, our test is more reliable than either of these tests under a heteroscedastic null. We also perform simulation experiments to compare the power of all three tests against two interesting alternative hypotheses: a stationary mean-reverting Markov process which has been interpreted as a ‘fads’ model of asset prices, and an explosive non-Markovian process which exhibits essentially the opposite time series properties. By choosing the sampling frequencies appropriately, the variance ratio test is shown to be as powerful as the Dickey-Fuller and Box--Pierce tests against both alternatives. As an empirical illustration, we perform our test on weekly stock market data from 1962 to 1985 and strongly reject the random walk hypothesis for several stock indexes.