Testing, Estimation and Higher Order Expansions in GMM with Semi-Weak Instruments

In this paper we analyze GMM with semi-weak instruments. This case includes the standard GMM and the nearly-weak GMM. In the nearly-weak GMM the correlation between the instruments and the first order conditions decline at a slower rate than root T. We find an important difference between the semi-weak case and the weak case. Inference with point estimates are possible with Wald, Likelihood ratio and LM tests in GMM with semi-weak instruments. This is important from an applied perspective since tests on the weak case do depnd on the true value and can only test simple null. Even though we may have all nearly-weak instruments in GMM it is still possible to test various hypotheses of interest. We also find a difference between the two subcategories in the semi-weak case. We derive higher order expansions for test statistics in the semi-weak case, and we show that with declining quality of instrumnents finite sample behavior of these tests get worser, so standard GMM finite sample behavior is always better than nearly-weak GMM. Unlike standard GMM, in the nearly-weak GMM we cannot eliminate the second order terms from these test statistics expansions.