A Gaussian process approach to model checks

Abstract

This paper proposes a Gaussian process (GP) approach for testing conditional moment restrictions. Tests are based on squared Neyman orthogonal function-parametric processes integrated with respect to a GP distribution. This methodology leads to a general unified framework of kernel-based tests having the following properties: (i) bootstrap tests are easy to implement in the presence of nuisance parameters (they are simple quadratic forms, and there is no need to reestimate the nuisance parameters in each bootstrap replication); and (ii) the new tests are valid under general conditions, including higher-order conditional moments of unknown form, regularized estimators (e.g., Lasso) or parameters at the boundary of the parameter space. Novel applications include distance kernel tests for zero conditional treatment effects. The paper introduces Neyman orthogonal kernels, a new asymptotic theory and a detailed local power analysis. Monte Carlo experiments and a real data application illustrate the sensitivity of tests to the dimension of covariates and to the mean and covariance kernel of the GP.