Locally Robust Kernel Specification Tests for Conditional Moment Restrictions

We develop kernel-based specification tests for semiparametric conditional moment models with high-dimensional nuisance parameters, extending existing conditional moment tests---which typically require asymptotically linear nuisance estimators---to accommodate modern machine-learning methods.

The proposed locally robust kernel tests combine Neyman-orthogonal moments, cross-fitting, and reproducing kernel Hilbert space methods, yielding inference that is first-order insensitive to nuisance estimation error.

We establish oracle equivalence between the feasible and infeasible test processes under local alternatives and weak nuisance-rate conditions, and characterize the resulting local power.

A fast multiplier bootstrap avoids nuisance re-estimation.

Applications include specification testing in high-dimensional linear and logistic regression, significance testing with machine-learning regressions, and tests of constant conditional treatment effects.

Monte Carlo simulations and an application to the National Supported Work program illustrate the finite-sample performance of the proposed tests.