Some higher-order theory for a consistent non-parametric model specification test

Abstract

We provide higher-order approximations for a smoothing-based model specification test. We derive the asymptotic cumulants and give the formal Edgeworth distributional approximation valid to a third order. We also prove the validity of the expansion in a special case where the error in the Edgeworth expansion is of order n − ε 2 for some 0<ε 2< 1 3 . The proof is based on new results for degenerate weighted U-statistics. There is a trade-off between size distortion and local power, so that large bandwidths are good for power and bad for size distortion, and vice-versa. One finding of practical importance is that this trade-off is not affected by the dimensionality of the regressors. This is because there are no smoothing bias terms under the null hypothesis and one can simply take larger bandwidth when dimensions are large. We provide an application to computing size-corrected critical values whose null rejection frequency improves on the normal critical values. Our simulations confirm the efficacy of this method in moderate sized samples.