  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.
