Testing conditional moment restrictions

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, θ)|x} = 0 w.p.1, where g is a vector of known functions and θ 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.

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