A new measure of partial conditional mean independence in Hilbert spaces

A novel metric, called partial martingale difference-angle divergence, is proposed to test and measure partial conditional mean (in)dependence for Hilbert elements. The partial martingale difference-angle divergence has some appealing properties. It is nonnegative and equals zero if and only if the partial conditional mean independence holds; it has a simple expectation form; it does not require the moment condition for the predictor variable. We construct an estimator for partial martingale difference-angle divergence and derive its asymptotic properties. Finite sample simulations show that the proposed test performs well and has strong testing power for nonlinear relationships. Two real data examples are introduced to illustrate the application of the proposed test.