A data-driven smooth test of symmetry

In this paper we propose a data driven smooth test of symmetry. We first transform the raw data via the probability integral transformation according to a symmetrized empirical distribution, and show that under the null hypothesis of symmetry, the transformed data has a limiting uniform distribution, reducing testing for symmetry to testing for uniformity. Employing Neyman's smooth test of uniformity, we show that only odd-ordered orthogonal moments of the transformed data are required in constructing the test statistic. We present a standardized smooth test that is distribution-free asymptotically and derive the asymptotic behavior of the test and establish its consistency. Extension to dependent data case is discussed. We investigate the finite sample performance of the proposed tests on both homogeneous and mixed distributions (with unobserved heterogeneity). An empirical application on testing symmetry of wage adjustment process, based on heterogeneous wage contracts with different durations, is provided.