Testing parameters in GMM without assuming that they are identifiied

We propose a Generalized Method of Moments (GMM) Lagrange multiplier statistic, i.e. the K-statistic, that uses the Jacobian at the evaluated parameter value instead of the expected Jacobian. To obtain its limit behavior, we use a novel assumption that brings GMM closer to maximum likelihood and which is easily satisfied. The usual asymptotic x2 distribution of the K-statistic then holds under a wider set of circumstances, like weak and many instrument asymptotics and combinations thereof, than the standard full rank case for the Jacobian. The behavior of the K-statistic can be spurious around inflexion points and the maximum of the objective function since the moment conditions are then not satisfied. Combinations of the K-statistic with statistics that test the validity of the moment equations overcome the spurious behavior. We conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series.