Robust M Tests for Distribution Symmetry with Time Series Data

Abstract

We propose a class of robust M tests for distribution symmetry of time series data. The proposed tests are robust to the estimation effect of replacing the unknown location parameter with its consistent estimator and are constructed by extending the M tests of Kuan and Lee (2006) and Lee (2007) to tests that permit non-differentiable odd functions. As compared with the skewness-coefficient-based test of Bai and Ng (2005, BN), our tests do not require consistent estimation of any nuisance parameters in the asymptotic variance and hence circumvent the problems arising from such consistent estimation. Moreover, with bounded odd functions and some robust location estimators, the proposed tests can even be robust to lack of any higher-order moments. Our simulations also reveal that the proposed tests are properly sized and, with proper odd functions, they are quite powerful for leptokurtic data. Our empirical study on US stock index returns also shows that the BN test produces conflicting results for NYSE. For DJIA and NASDAQ that are highly leptokurtic, distribution asymmetry is captured by the proposed tests but not the BN test.