Entropy-Based Moment Selection in the Presence of Weak Identification

Hall et al. (2007 Hall , A. R. , Inoue , A. , Jana , K. , Shin , C. (2007). Information in generalized method of moments estimation and entropy based moment selection. Journal of Econometrics 138:488–512.Crossref] , Google Scholar]) propose a method for moment selection based on an information criterion that is a function of the entropy of the limiting distribution of the Generalized Method of Moments (GMM) estimator. They establish the consistency of the method subject to certain conditions that include the identification of the parameter vector by at least one of the moment conditions being considered. In this article, we examine the limiting behavior of this moment selection method when the parameter vector is weakly identified by all the moment conditions being considered. It is shown that the selected moment condition is random and hence not consistent in any meaningful sense. As a result, we propose a two-step procedure for moment selection in which identification is first tested using a statistic proposed by Stock and Yogo (2003 Stock , J. H. , Yogo , M. ( 2003 ). Testing for weak instruments in linear IV regression . Discussion paper, Kennedy School of Government, Harvard University, Cambridge, MA . Google Scholar]) and then only if this statistic indicates identification does the researcher proceed to the second step in which the aforementioned information criterion is used to select moments. The properties of this two-step procedure are contrasted with those of strategies based on either using all available moments or using the information criterion without the identification pre-test. The performances of these strategies are compared via an evaluation of the finite sample behavior of various methods for inference about the parameter vector. The inference methods considered are based on the Wald statistic, Anderson and Rubin's (1949 Anderson , T. W. , Rubin , H. ( 1949 ). Estimation of the parameters of a single equation in a complete system of stochastic equations . Annals of Mathematical Statistics 20 : 46 – 63 .Crossref] , Google Scholar]) statistic, Kleibergen (2002 Kleibergen , F. ( 2002 ). Pivotal statistics for testing structural parameters in instrumenatl variables regression . Econometrica 70 : 1781 – 1803 .Crossref], Web of Science ®] , Google Scholar]) K statistic, and combinations thereof in which the choice is based on the outcome of the test for weak identification.