Conditional symmetry test based on empirical characteristic function

In this paper, we propose a method for testing the conditional symmetry of multivariate random variables, specifically testing whether or not the conditional distribution of one random variable is symmetric around zero given another. Both the conditional symmetry test for observable data and unobserved error terms in parametric models are considered. The proposed test statistics are built on the weighted $ L_2 $ L2-distance between empirical characteristic functions and require no distributional assumption about the data distribution. The asymptotic properties of the test statistics under the null and alternative hypotheses are investigated. Since the limiting null distributions are intractable, a nonparametric Monte Carlo resampling method is used to determine critical values. Simulation results show the good performance of the proposed method in various scenarios. Two real datasets are analysed to illustrate the practical usefulness of the proposed tests.