Identifying Multiple Marginal Effects With A Single Instrument

This paper proposes a new strategy for the identification of the marginal effects of an endogenous multivalued variable (which can be continuous, or a vector) in a model with an Instrumental Variable (IV) of lower dimension, which may even be a single binary variable, and multiple controls. Despite the failure of the classical order condition, we show that identification may be achieved by exploiting heterogeneity of the “first stage” in the controls through a new rank condition that we term covariance completeness. The identification strategy justifies the use of interactions between instruments and controls as additional exogenous variables and can be straightforwardly implemented by parametric, semiparametric, and nonparametric two-stage least squares estimators, following the same generic algorithm. Monte Carlo simulations show that the estimators have excellent performance in moderate sample sizes. Finally, we apply our methods to the problem of estimating the effect of air quality on house prices, based on Chay and Greenstone (2005, Journal of Political Economy 113, 376–424). All methods are implemented in a companion Stata software package.