Efficient Inference with Poor Instruments: a General Framework

We consider a general framework where weaker patterns of identifcation may arise: typically, the data generating process is allowed to depend on the sample size. However, contrary to what is usually done in the literature on weak identification, we do not give up the efficiency goal of statistical inference: even fragile information should be processed optimally for the purpose of both efficient estimation and powerful testing. Our main contribution is actually to consider that several patterns of identification may arise simultaneously. This heterogeneity of identification schemes paves the way for the device of optimal strategies for inferential use of information of poor quality. More precisely, we focus on a case where asymptotic efficiency of estimators is well-defined through the variance of asymptotically normal distributions. Standard efficient estimation procedures still hold, albeit with rates of convergence slower than usual. We stress that these are feasible without requiring the prior knowledge of the identification schemes.