The Asymptotic Properties of GMM and Indirect Inference under Second-order Identi?cation

This paper presents a limiting distribution theory for GMM and Indirect Inference estimators when local identi?cation conditions fail at ?rst-order but hold at second-order. These limit distributions are shown to be non-standard, but we show that they can be easily simulated, making it possible to perform inference about the parameters in this setting. We illustrate our results in the context of a dynamic panel data model in which the parameter of interest is identi?ed locally at second order by non-linear moment restrictions but not at ?rst order at a particular point in the parameter space. Our simulation results indicate that our theory leads to reliable inferences in moderate to large samples in the neighbourhood of this point of ?rst-order identi?cation failure. In contrast, inferences based on standard asymptotic theory (derived under the assumption of ?rst-order local identi?cation) are very misleading in the neighbourhood of the point of ?rst-order local identi?cation failure.