Identifying Multiple Marginal Effects with a Single Binary Instrument or by Regression Discontinuity

This paper proposes a new strategy for the identification of all the marginal effects of an endogenous multi-valued variable (which can be continuous, or a vector) in a regression model with one binary instrumental variable. The unobservables must be separable from the endogenous variable of interest in the model. Identification is achieved by exploiting heterogeneity of the "first stage" in covariates. The covariates themselves may be endogenous, and their endogeneity does not need to be modeled. With some modifications, the identification strategy is extended to the Regression Discontinuity Design (RDD) with multi-valued endogenous variables, thereby showing that adding covariates in RDD may improve identification. This paper also provides parametric, semiparametric and nonparametric estimators based on the identification strategy, discusses their asymptotic properties, and shows that the estimators have satisfactory performance in moderate samples sizes. All the proposed estimators can be implemented as Two-Stage Least Squares (TSLS). Finally, we apply our methods to the problem of estimating the effect of air quality on house prices, based on Chay and Greenstone (2005)