A kernel-based measure for conditional mean dependence

A novel metric, called kernel-based conditional mean dependence (KCMD), is proposed to measure and test the departure from conditional mean independence between a response variable Y and a predictor variable X, based on the reproducing kernel embedding and the Hilbert-Schmidt norm of a tensor operator. The KCMD has several appealing merits. It equals zero if and only if the conditional mean of Y given X is independent of X, i.e. E(Y|X)=E(Y) almost surely, provided that the employed kernel is characteristic; it can be used to detect all kinds of conditional mean dependence with an appropriate choice of kernel; it has a simple expectation form and allows an unbiased empirical estimator. A class of test statistics based on the estimated KCMD is constructed, and a wild bootstrap test procedure to the conditional mean independence is presented. The limit distributions of the test statistics and the bootstrapped statistics under null hypothesis, fixed alternative hypothesis and local alternative hypothesis are given respectively, and a data-driven procedure to choose a suitable kernel is suggested. Simulation studies indicate that the tests based on the KCMD have close powers to the tests based on martingale difference divergence in monotone dependence, but excel in the cases of nonlinear relationships or the moment restriction on X is violated. Two real data examples are presented for the illustration of the proposed method.