Generalized Spectral Tests for High Dimensional Multivariate Martingale Difference Hypotheses

This study proposes new generalized spectral tests for multivariate Martingale Difference Hypotheses, especially suitable for high-dimensionality situations. The new tests are based on the martingale difference divergence covariance (MDD) proposed by Shao and Zhang (2014). It considers block-wise serial dependence of all lags, therefore, is consistent against general block-wise nonparametric Pitman’s local alternatives at the parametric rate n−1/2, where n is the sample size, and free of a user-chosen parameter. In order to cope with the highdimensionality in the sense that the dimension of time series is comparable to or even greater than the sample size, it is pivotal to employ a bias-reduced estimator for each individual MDD in the test statistic. Monte Carlo simulations reveal that the bias-reduced statistic generally performs better than its competitors substantially. Moreover, it is robust to heteroskedasticity of unknown forms and heavy-tails in the data generating processes. We apply our approach to test the efficient market hypothesis on the US stock market, using data sets on the monthly and daily data of portfolios sorted by industry. Our test results provide strong evidence against the efficient market hypothesis with respect to the US stock market at monthly frequency