Control Function Instrumental Variable Estimation of Nonlinear Causal Effect Models

The instrumental variable method consistently estimates the effect of a treatment when there is unmeasured confounding and a valid instrumental variable. A valid instrumental variable is a variable that is independent of unmeasured confounders and affects the treatment but does not have a direct effect on the outcome beyond its effect on the treatment. Two commonly used estimators for using an instrumental variable to estimate a treatment effect are the two stage least squares estimator and the control function estimator. For linear causal effect models, these two estimators are equivalent, but for nonlinear causal effect models, the estimators are different. We provide a systematic comparison of these two estimators for nonlinear causal effect models and develop an approach to combing the two estimators that generally performs better than either one alone. We show that the control function estimator is a two stage least squares estimator with an augmented set of instrumental variables. If these augmented instrumental variables are valid, then the control function estimator can be much more efficient than usual two stage least squares without the augmented instrumental variables while if the augmented instrumental variables are not valid, then the control function estimator may be inconsistent while the usual two stage least squares remains consistent. We apply the Hausman test to test whether the augmented instrumental variables are valid and construct a pretest estimator based on this test. The pretest estimator is shown to work well in a simulation study. An application to the effect of exposure to violence on time preference is considered.