Optimal Minimax Rates of Specification Testing with Data-driven Bandwidth

Abstract

This study investigates optimal minimax rates of specification testing for linear and non-linear instrumental variable regression models. The rate implies that the uniform power of tests reduces when the dimension of instruments is large. The test constructed by non-parametric kernel techniques can be rate optimal when bandwidths satisfy two order conditions that depend on the dimensions of instruments and the smoothness of alternatives. Since bandwidths are often chosen in a data-dependent way in empirical studies, the rate optimality of the test with data-driven bandwidths are investigated. Bandwidths selected by the least squares cross-validation can satisfy conditions for the rate optimality.