Econometrica: Sep 2018, Volume 86, Issue 5
Overidentification in Regular Models
Xiaohong Chen, Andres Santos
In the unconditional moment restriction model of Hansen (1982), specification tests and more efficient estimators are both available whenever the number of moment restrictions exceeds the number of parameters of interest. We show that a similar relationship between potential refutability of a model and existence of more efficient estimators is present in much broader settings. Specifically, a condition we name local overidentification is shown to be equivalent to both the existence of specification tests with nontrivial local power and the existence of more efficient estimators of some “smooth” parameters in general semi/nonparametric models. Under our notion of local overidentification, various locally nontrivial specification tests such as Hausman tests, incremental Sargan tests (or optimally weighted quasi likelihood ratio tests) naturally extend to general semi/nonparametric settings. We further obtain simple characterizations of local overidentification for general models of nonparametric conditional moment restrictions with possibly different conditioning sets. The results are applied to determining when semi/nonparametric models with endogeneity are locally testable, and when nonparametric plug‐in and semiparametric two‐step GMM estimators are semiparametrically efficient. Examples of empirically relevant semi/nonparametric structural models are presented.
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Supplement to "Overidentification in Regular Models"
This Online Supplementary Appendix contains additional results and proofs to support the main text. Appendix D contains the proof of the limiting experiment results in Appendix A and additional lemmas. Appendix E presents the technical lemmas and their proofs that are used in the proofs of Appendix B and Appendix C. Appendix F contains the proofs of the results in Section 4. Appendix G provides sufficient conditions for verifying Assumption 3.1 in the general nonparametric conditional moment restriction models studied in Section 4. Appendix H provides additional discussion of the Examples in Section 4 as well as a final example.