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The instrumental variable method consistently estimates the effect of atreatment when there is unmeasured confounding and a valid instrumentalvariable. A valid instrumental variable is a variable that is independent ofunmeasured confounders and affects the treatment but does not have a directeffect on the outcome beyond its effect on the treatment. Two commonly usedestimators for using an instrumental variable to estimate a treatment effectare the two stage least squares estimator and the control function estimator.For linear causal effect models, these two estimators are equivalent, but fornonlinear causal effect models, the estimators are different. We provide asystematic comparison of these two estimators for nonlinear causal effectmodels and develop an approach to combing the two estimators that generallyperforms better than either one alone. We show that the control functionestimator is a two stage least squares estimator with an augmented set ofinstrumental variables. If these augmented instrumental variables are valid,then the control function estimator can be much more efficient than usual twostage least squares without the augmented instrumental variables while if theaugmented instrumental variables are not valid, then the control functionestimator may be inconsistent while the usual two stage least squares remainsconsistent. We apply the Hausman test to test whether the augmentedinstrumental variables are valid and construct a pretest estimator based onthis 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 isconsidered.

Control Function Instrumental Variable Estimation of Nonlinear Causal Effect Models