Data-Driven Tuning in Kernel Specification Tests: Selection Asymptotics, Validity, and Power

We develop model checks for conditional moment restrictions whose tuning parameters, the kernel bandwidth and the regularization level, are selected by cross-validation. The tests regress the estimated residuals on the covariates via kernel ridge regression (KRR), yielding a coefficient function in a reproducing kernel Hilbert space (RKHS) that vanishes under the null and is nonzero for any misspecification not annihilated by the Neyman projection. Our central contribution is a complete asymptotic theory of the data-driven tuning parameter: its limit law under the null, under fixed alternatives, and under local alternatives, and bootstrap size control under sample splitting with no convergence of the selector required. An exact energy identity shows that cross-validation maximizes the signal captured by the test (its noncentrality) and, absent signal, shrinks its noise floor. The tests are consistent against fixed alternatives, achieve $n^{-1/2}$-rate sensitivity to local alternatives, and use Neyman-orthogonal projections so the model is never re-estimated within bootstrap resamples. Simulations show size close to nominal without user-chosen tuning, power comparable to the kernel conditional moment (KCM) test at its best kernel parameter, and robustness where fixed kernel parameters fail in higher dimensions. Three empirical applications illustrate the diagnostic role of the tests; we recommend a $15$--$20\%$ selection share for the sample-splitting protocol.