Bounding endogenous regressor coefficients using moment inequalities and generalized instruments

Abstract

The main approach to deal with regressor endogeneity is instrumental variable estimator (IVE), where an instrumental variable (IV) m is required to be uncorrelated to the regression model error term u (COR(m,u)=0) and correlated to the endogenous regressor. If COR(m,u)?0 is likely, then m gets discarded. But even when COR(m,u)?0, often one has a good idea on the sign of COR(m,u). This article shows how to make use of the sign information on COR(m,u) to obtain an one-sided bound on the endogenous regressor coefficient, calling m a generalized instrument or generalized instrumental variable (GIV). If there are two GIV's m1 and m2, then a two-sided bound or an improved one-sided bound can be obtained. Our approach is simple, needing only IVE; no non-parametrics, nor any tuning constants. Specifically, the usual IVE is carried out, and the only necessary modification is that the estimate for the endogenous regressor coefficient is interpreted as a lower/upper bound depending on the prior notion on the sign of COR(m,u) and some estimable moment. A real data application is done to Korean household data with two or more children to illustrate our approach for the issue of child quantityquality trade-off.