Model Checks in a Kernel Ridge Regression Framework

We propose new reproducing kernel-based tests for model checking in conditional moment restriction models. By regressing estimated residuals on kernel functions via kernel ridge regression (KRR), we obtain a coefficient function in a reproducing kernel Hilbert space (RKHS) that is zero if and only if the model is correctly specified. We introduce two classes of test statistics: (i) projection-based tests, using RKHS inner products to capture global deviations, and (ii) random location tests, evaluating the KRR estimator at randomly chosen covariate points to detect local departures. The tests are consistent against fixed alternatives and sensitive to local alternatives at the $n^{-1/2}$ rate. When nuisance parameters are estimated, Neyman orthogonality projections ensure valid inference without repeated estimation in bootstrap samples. The random location tests are interpretable and can visualize model misspecification. Simulations show strong power and size control, especially in higher dimensions, outperforming existing methods.