Spectral tests of the martingale hypothesis under conditional heteroscedasticity

We study the asymptotic distribution of the sample standardized spectral distribution function when the observed series is a conditionally heteroscedastic martingale difference. We show that the asymptotic distribution is no longer a Brownian bridge but another Gaussian process. Furthermore, this limiting process depends on the covariance structure of the second moments of the series. We show that this causes test statistics based on the sample spectral distribution, such as the Cramér von-Mises statistic, to have heavily right skewed distributions, which will lead to over-rejection of the martingale hypothesis in favour of mean reversion. A non-parametric correction to the test statistics is proposed to account for the conditional heteroscedasticity. We demonstrate that the corrected version of the Cramér von-Mises statistic has the usual limiting distribution which would be obtained in the absence of conditional heteroscedasticity. We also present Monte Carlo results on the finite sample distributions of uncorrected and corrected versions of the Cramér von-Mises statistic. Our simulation results show that this statistic can provide significant gains in power over the Box–Ljung–Pierce statistic against long-memory alternatives. An empirical application to stock returns is also provided.