## Control Function Instrumental Variable Estimation of Nonlinear Causal Effect Models

*Zijian Guo, Dylan S. Small*; 17(100):1−35, 2016.

### Abstract

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.

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