Revisiting Instrumental Variables and the Classic Control Function Approach, with Implications for Parametric and Non-Parametric Regressions

Kyoo il Kim, Amil Petrin

NBER Working Paper No. 16679
Issued in January 2011
NBER Program(s):   TWP

It is well-known from projection theory that two-stage least squares (2SLS) and the classic control function (CF) estimator in the linear simultaneous equations models are numerically equivalent. Yet the classic CF approach assumes that the regression error in the outcome equation is mean independent of the instruments conditional on the CF control while 2SLS does not. We resolve this puzzle by showing that the classic CF approach omits a generalized control function that may depend on the instruments and control. This term is (asymptotically) uncorrelated with the endogenous regressors given the control under the unconditional moment restrictions of 2SLS. We also show that imposing the 2SLS unconditional moment restrictions in the classic CF setup allows the mean of the error to depend on the instruments and control. In contrast to the linear setting, the non-linear and non-parametric control function setting of Newey, Powell, and Vella (1999) (NPVCF) is no longer consistent if the classic CF condition is violated. This dependence can occur in many economic settings including returns to education, production functions, and demand or supply with non-separable reduced forms for equilibrium prices. We use our results to develop an estimator for this setting that is consistent when the structural error may depend on the instruments given the CF control. Our approach achieves identification by augmenting the NPVCF setting with conditional moment restrictions. Our estimator is a multi-step least squares estimator and thus maintains the simplicity of the NPVCF estimator. Our monte carlos are motivated by our economic examples and they show that our new estimator performs well while the classical CF estimator and the non-parametric analog of NPVCF can be biased in non-linear or nonparametric settings when the conditional mean of the error depends on the instruments given the control.

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This paper was revised on December 5, 2011

Acknowledgments

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