A Test of U-type for Goodness-of-fit in Regression Models Through Martingale Difference Divergence

Abstract

Based on the martingale difference divergence, a recently proposed metric for quantifying conditional mean dependence, we introduce a consistent test of U-type for the goodness-of-fit of linear models under conditional mean restriction. Methodologically, our test allows heteroscedastic regression models without imposing any condition on the distribution of the error, utilizes effectively important information contained in the distance of the vector of covariates, has a simple form, is easy to implement, and is free of the subjective choice of parameters. Theoretically, our mathematical analysis is of own interest since it does not take advantage of the empirical process theory and provides some insights on the asymptotic behavior of U-statistic in the framework of model diagnostics. The asymptotic null distribution of the proposed test statistic is derived and its asymptotic power behavior against fixed alternatives and local alternatives converging to the null at the parametric rate is also presented. In particular, we show that its asymptotic null distribution is very different from that obtained for the true error and their differences are interestingly related to the form expression for the estimated parameter vector embodied in regression function and a martingale difference divergence matrix. Since the asymptotic null distribution of the test statistic depends on data generating process, we propose a wild bootstrap scheme to approximate its null distribution. The consistency of the bootstrap scheme is justified. Numerical studies are undertaken to show the good performance of the new test.