Score tests in GMM: Why use implied probabilities?

While simple to implement and thus attractive in practice, the GMM score test of Newey and West (1987) often displays upward size distortion under common scenarios involving skewed moment vectors or models with weak identification. Inference based on the Generalized Empirical Likelihood (GEL) is seen as a general solution to this problem. Kleibergen (2005) and, more generally, Guggenberger and Smith (2005) devised an elegant theory for the GEL score tests. However, strictly speaking, the GEL score tests do not nest the Newey–West score test. Our paper provides a unified framework for score tests in GMM that nests all of the above as special cases and helps us to understand the precise mechanism by which the standard first order asymptotic theory on size and power well approximates the finite sample behavior of some score tests (namely, a subset of the GEL score tests) but not others. Special attention is paid to models with weak identification. We also argue that the apparent computational burden of GEL can be overcome in practice by recognizing the fundamental common role played by the GEL implied probabilities under all special cases of our framework. In particular, we show that all the GEL implied probabilities are asymptotically equivalent at a higher order – both under the null and under appropriate sequences of alternatives – and thus are exchangeable across computationally burdensome (e.g. Empirical Likelihood) and easy (e.g. Euclidean Empirical Likelihood) GEL score tests without affecting the first order asymptotics. Extensive simulation evidence is provided to corroborate our theoretical results. The simulation results also support a simple and yet important insight on the power of the tests: the use of implied probabilities to efficiently estimate the components of the score statistic, namely, the Jacobian and the asymptotic variance of the moment vector, can significantly improve the power of the score test in finite samples.