Integrated conditional moment test and beyond: when the number of covariates is divergent

Abstract

Summary The classical integrated conditional moment test is a promising method for model checking and its basic idea has been applied to develop several variants. However, in diverging-dimension scenarios, the integrated conditional moment test may break down and has completely different limiting properties from the fixed-dimension case. Furthermore, the related wild bootstrap approximation can also be invalid. To extend this classical test to diverging dimension settings, we propose a projected adaptive-to-model version of the integrated conditional moment test. We study the asymptotic properties of the new test under both the null and alternative hypotheses to examine if it maintains significance level, and its sensitivity to the global and local alternatives that are distinct from the null at the rate $n^{-1/2}$. The corresponding wild bootstrap approximation can still work for the new test in diverging-dimension scenarios. We also derive the consistency and asymptotically linear representation of the least squares estimator when the parameter diverges at the fastest possible known rate in the literature. Numerical studies show that the new test can greatly enhance the performance of the integrated conditional moment test in high-dimensional cases. We also apply the test to a real dataset for illustration.