A sequential procedure for testing the existence of a random walk model in finite samples

Given the random walk model, we show, for the traditional unrestricted regression used in testing stationarity, that no matter what the initial value of the random walk is or its drift or its error standard deviation, the sampling distributions of certain statistics remain unchanged. Using Monte Carlo simulations, we estimate, for different finite samples, the sampling distributions of these statistics. After smoothing the percentiles of the empirical sampling distributions, we come up with a new set of critical values for testing the existence of a random walk, if each statistic is being used on an individual base. Combining the new sets of critical values, we finally suggest a general methodology for testing for a random walk model.