Sign-based specification tests for martingale difference with conditional heteroscedasity

This article proposes Cramer-von Mises (CM) and Kolmogrove-Smirnov (KS) test statistics based on the signs of a time series to test the null hypothesis that the series is a martingale difference sequence (MDS) with conditional heteroscedasity. Both of test statistics allowing for heavy-tailedness, non-stationarity, and nonlinear serial dependence of unknown forms, are easy-to-implement. Unlike the sign-based variance-ratio test in Wright (2000), our sign-based CM and KS tests have no need to select the lag. Unlike other often used specification tests for MDS, our sign-based CM and KS tests are robust and have exact distributions which can be simulated easily. Simulation studies and applications further demonstrate the importance of our sign-based CM and KS tests.