Mis-classified, Binary, Endogenous Regressors: Identification and Inference
This paper studies identification and inference for the effect of a mis-classified, binary, endogenous regressor when a discrete-valued instrumental variable is available. We begin by showing that the only existing point identification result for this model is incorrect. We go on to derive the sharp identified set under mean independence assumptions for the instrument and measurement error, and that these fail to point identify the effect of interest. This motivates us to consider alternative and slightly stronger assumptions: we show that adding second and third moment independence assumptions suffices to identify the model. We then turn our attention to inference. We show that both our model, and related models from the literature that assume regressor exogeneity, suffer from weak identification when the effect of interest is small. To address this difficulty, we exploit the inequality restrictions that emerge from our derivation of the sharp identified set under mean independence only. These restrictions remain informative irrespective of the strength of identification. Combining these with the moment equalities that emerge from our identification result, we propose a robust inference procedure using tools from the moment inequality literature. Our method performs well in simulations.
We thank Daron Acemoglu, Manuel Arellano, Kristy Buzard, Xu Cheng, Bernardo da Silveira, Bo Honoré, Arthur Lewbel, Chuck Manski, Sophocles Mavroeidis, Francesca Molinari, Yuya Takahashi, and seminar participants at Cambridge, CEMFI, Chicago Booth, Manchester, Northwestern, Oxford, Penn State, Princeton, UCL, the 2016 Greater New York Area Econometrics Colloquium, Camp Econometrics IX, and the 2017 North American Summer Meeting of the Econometric Society for valuable comments and suggestions. This document supersedes an earlier version entitled “On Mis-measured Binary Regressors: New Results and Some Comments on the Literature.” The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.