Instrumental Variables Methods with Heterogeneity and Mismeasured Instruments

We study the consequences of using an error-laden proxy W for instruments Z on the interpretation of Wald, local instrumental variables (LIV), and instrumental variables (IV) estimands in ordered discrete choice structural systems with heterogeneity. A proxy W need only satisfy an exclusion restriction and that the treatment and outcome are mean independent from W given Z. Unlike Z, W need not satisfy monotonicity and exogeneity. For example, W could code Z with error, missing observations, or coarsely. We show that Wald, LIV, and IV estimands using W identify weighted averages of local or marginal treatment e¤ects. We study necessary and su¢ cient conditions for nonnegative weights. Further, we study a condition under which the Wald or LIV estimand usingW identi…es the same local or marginal e¤ects that would have been recovered had Z been observed. For example, this holds for binary Z, therefore the Wald estimand using W identi…es the same “average causal response,” or local average treatment e¤ect for binary treatment, that would have been recovered using Z. Also, under this condition, LIV using W can be used to identify average treatment e¤ects for e.g. the population, treated, and untreated. Keywords: causality, instrumental variables, local average treatment e¤ect, marginal treatment e¤ect, measurement error, monotonicity, proxy, quadrant dependence, Wald method. JEL Classi…cation Codes: C31, C35, C36. Assistant Professor, Department of Economics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467 (email: [email protected]). yAn earlier version of this paper was circulated under the title “Identi…cation of Local Treatment Effects Using a Proxy for an Instrument.”The author thanks the participants in the California Econometrics Conference 2010, the 2011 North American Winter Meeting of the Econometric Society, the Boston College labor lunch seminar, and the econometrics seminars at UCSD, Brown, UC Riverside, Ohio State University, Yale, UCL, Oxford, UBC, Northwestern, Harvard/MIT, UC Davis, and Stanford as well as Gary Chamberlain, Xavier d’Haultfoeuille, Scott Fulford, Peter Gottschalk, Guido Imbens, Stefan Hoderlein, Ivana Komunjer, Tobias Klein, Arthur Lewbel, Ke-Li Xu, Mathis Wagner, and Halbert White for helpful comments and suggestions. Any errors are the author’s responsibility.