Testing conditional moment restrictions

Abstract

Let (x,z) be a pair of observable random vectors. We construct a new "smoothed" empirical likelihood-based test for the hypothesis $\E\{ g(z,\break \theta)|x \} = 0$ w.p.1, where g is a vector of known functions and $\theta$ an unknown finite-dimensional parameter. We show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives. Furthermore, the test is shown to possess an optimality property in large samples. Simulation evidence suggests that it also behaves well in small samples.