Spectral based testing of the martingale hypothesis *

This paper proposes a method of testing whether a time series is a martingale. A general asymptotic theory is developed for the spectral distribution function of the first differences. Under the null hypothesis, the spectral distribution function is shaped as a straight line. Several tests are developed which determine whether the sample spectral distribution function possesses this shape. These tests are consistent against all MA alternatives. Additional tests are developed to analyze subsets of frequencies, which can enhance power against particular alternatives. Application of the test to stock prices finds evidence against the random walk theory.