Boundedly pivotal structural change tests in continuous updating GMM with strong, weak identification and completely unidentified cases

This paper develops structural change tests in the continuous updating GMM framework that are robust to weak identification. We propose likelihood ratio-like, Anderson–Rubin [1949. Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics 20, 46–63], and Kleinbergen [2005. Testing parameters in GMM without assuming that they are identified. Manuscript. Brown University] types of tests. Since the limits of the test statistics are not nuisance parameter free, bounds for the limit of the test statistics are derived. The bounds are nuisance parameter free and robust to identification problems. Simulations show that the Anderson–Rubin (1949) and Kleinbergen [2005. Testing parameters in GMM without assuming that they are identified. Manuscript. Brown University] type of tests have very good small sample properties. In the case of weak instruments, the sup LM test of Andrews [1993a. Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856] rejects the true null of parameter stability more than the nominal level, and, also has low power in the weak instrument setup.