Rate Optimal Specification Test When the Number of Instruments is Large

Abstract

We propose a rate optimal specification test for instrumental variable regression models based on the nearest neighbor observation with respect to instruments. The proposed test has uniform power against a set of non-smooth alternatives. The optimal minimax rate is n-1=4 for any dimension of instruments, where n is sample size. This rate coincides with the fastest possible rate achievable by any tests under the local alternative setting when the alternative is constructed by a non-smooth function and/or the dimension of the instrument is large. Since such local alternative belongs to the set of alternatives considered in this study, our test is preferable in a large dimension setting. In the simulation and empirical applications with a large dimension of instruments, we observe that the test works well and the power approaches one reasonably fast as the sample size increase.