Specification Testing for Conditional Moment Restrictions Under Local Identification Failure

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In this paper, we study the asymptotic behaviour of the specification in conditional moment restrictions model under first-order local identification failure with dependent data. More specifically, we obtain conditions under which the conventional specification test for conditional moment restrictions remains valid when first-order local identification fails but global identification is still attainable. In the process, we obtain some novel intermediate results that include extending the first- and second-order local identification framework to models defined by conditional moment restrictions, characterizing the rate of convergence of the GMM estimator and the limiting representation for degenerate U-statistics under strong mixing dependence. Simulation and empirical results illustrate the properties and the practical relevance of the proposed testing framework.