An Extended Class of Instrumental Variables for the Estimation of Causal E¤ects

We examine how structural systems can yield observed variables instrumental in identifying and estimating causal e¤ects. We provide an exhaustive characterization of how structures determine exogeneity and exclusion restrictions that yield moment conditions supporting identi…cation. This provides a comprehensive framework for constructing instruments and control variables. We introduce notions of conditioning and conditional extended instrumental variables (XIVs). These permit identi…cation but need not be traditional instruments as they may be endogenous. We distinguish between observed XIVs and proxies for unobserved XIVs. We emphasize the importance of su¢ ciently specifying causal relations governing the unobservables. JEL Classi…cation Numbers: C10, C13, C21, C31, C51. Keywords: causality, conditional exogeneity, endogeneity, exogeneity, identi…cation, instrumental variables, structural systems. Karim Chalak is Assistant Professor of Economics, Department of Economics, Boston College, 140 Commonwealth Ave., Chestnut Hill, MA 02467 (email: [email protected]); and Halbert White is Chancellor’s Associates Distinguished Professor of Economics, Department of Economics 0508, UCSD, La Jolla, CA 92093 (email: [email protected]). yThe authors thank Kate Antonovics, Julian Betts, Gary Chamberlain, Graham Elliott, Marjorie Flavin, Clive Granger, Jinyong Hahn, James Heckman, Keisuke Hirano, Guido Imbens, Ste¤en Lauritzen, Arthur Lewbel, Mark Machina, Chuck Manski, Salvatore Modica, Judea Pearl, Dimitris Politis, Ross Starr, Elie Tamer, Ruth Williams, and the participants of the 4th annual Advances in Econometrics conference. All errors and omissions are the authors’responsibility.