Identifying Multiple Marginal Effects With a Single Instrument

Abstract

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.

Regression Discontinuity Designs With a Continuous Treatment

“…14 By this procedure, insignificant estimates (at the 5% significance level) of 1 O q.u/ along with some significant but small estimates will be trimmed out and the asymptotic behavior of our estimator is not affected. 13 If one wishes to focus on quantiles such that j1q.u/j > c for some small c > 0, then one can define the trimming parameter to be c n D c C n , where n is defined the same way. 14 Consider the bandwidth sequences h D cn a and Q h D cn b for some constants 0 < a; b < 1 and c > 0.…”

“…3 Our paper is also related to the IV literature using first-stage heterogeneity for identification. See, for recent examples, Brinch et al, (2017) and Caetano and Escanciano (2017). These existing studies consider treatment response heterogeneity in covariates.…”

Regression Discontinuity Designs with a Continuous Treatment

“…14 By this procedure, insignificant estimates (at the 5% significance level) of 1 O q.u/ along with some significant but small estimates will be trimmed out and the asymptotic behavior of our estimator is not affected. 13 If one wishes to focus on quantiles such that j1q.u/j > c for some small c > 0, then one can define the trimming parameter to be c n D c C n , where n is defined the same way. 14 Consider the bandwidth sequences h D cn a and Q h D cn b for some constants 0 < a; b < 1 and c > 0.…”

Partial Identification of Nonseparable Models Using Binary Instruments

“…D' Haultfoeuille and Février (2015) consider the case in which the instrumental variable takes more than two values, thus showing point identification can be achieved using group and dynamical systems theories even when F X|Z (x|z) and F X|Z (x|z ′ ) have no intersection. Caetano and Escanciano (2020) provide alternative results for the identification of nonseparable models with continuous endogenous variables and binary instruments. They use the observed covariates to identify the structural function.…”