Parametric and nonparametric regression in the presence of endogenous control variables

The aim of this paper is to convey to a wider audience of applied statisticians that nonparametric (matching) estimation methods can be a very convenient tool to overcome problems with endogenous control variables. In empirical research one is often interested in the causal effect of a variable X on some outcome variable Y. With observational data, i.e. in the absence of random assignment, the correlation between X and Y generally does not reflect the treatment effect but is confounded by differences in observed and unobserved characteristics. Econometricians often use two different approaches to overcome this problem of confounding by other characteristics. First, controlling for observed characteristics, often referred to as selection on observables, or instrumental variables regression, usually with additional control variables. Instrumental variables estimation is probably the most important estimator in applied work. In many applications, these control variables are themselves correlated with the error term, making ordinary least squares and two-stage least squares inconsistent. The usual solution is to search for additional instrumental variables for these endogenous control variables, which is often difficult. We argue that nonparametric methods help to reduce the number of instruments needed. In fact, we need only one instrument whereas with conventional approaches one may need two, three or even more instruments for consistency. Nonparametric matching estimators permit root n consistent estimation without the need for (additional) instrumental variables and permit arbitrary functional forms and treatment effect heterogeneity.