Specification testing for conditional moment restrictions under local identification failure

In this paper, we study the asymptotic behavior of specification tests in conditional moment restriction models under first-order local identification failure with dependent data. More specifically, we obtain conditions under which the conventional specification test for conditional moment restrictions retains its standard normal limit when first-order local identification fails but global identification is still attainable. In the process, we derive some novel intermediate results that include extending the first- and second-order local identification framework to models defined by conditional moment restrictions, establishing the rate of convergence of the GMM estimator and characterizing the asymptotic representation for degenerate U-statistics under strong mixing dependence. Importantly, the specification test is robust to first-order local identification failure regardless of the number of directions in which the Jacobian of the conditional moment restrictions is degenerate and remains valid even if the model is first-order identified.