Control Function Methods Lecture

Control Function Methods Lecture - Imbens/Wooldridge...

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Imbens/Wooldridge, Lecture Notes 6, Summer ’07 What s New in Econometrics ? NBER , Summer 2007 Lecture 6 , Tuesday , July 31st , 9 . 00 - 10 . 30 am Control Function and Related Methods These notes review the control function approach to handling endogeneity in models linear in parameters, and draws comparisons with standard methods such as 2SLS. Certain nonlinear models with endogenous explanatory variables are most easily estimated using the CF method, and the recent focus on average marginal effects suggests some simple, flexible strategies. Recent advances in semiparametric and nonparametric control function method are covered, and an example for how one can apply CF methods to nonlinear panel data models is provided. 1 . Linear - in - Parameters Models : IV versus Control Functions Most models that are linear in parameters are estimated using standard IV methods – either two stage least squares (2SLS) or generalized method of moments (GMM). An alternative, the control function (CF) approach, relies on the same kinds of identification conditions. In the standard case where a endogenous explanatory variables appear linearly, the CF approach leads to the usual 2SLS estimator. But there are differences for models nonlinear in endogenous variables even if they are linear in parameters. And, for models nonlinear in parameters, the CF approach offers some distinct advantages. Let y 1 denote the response variable, y 2 the endogenous explanatory variable (a scalar for simplicity), and z the 1 L vector of exogenous variables (which includes unity as its first element). Consider the model y 1 z 1 1 1 y 2 u 1 (1.1) where z 1 is a 1 L 1 strict subvector of z that also includes a constant. The sense in which z is exogenous is given by the L orthogonality (zero covariance) conditions E z u 1 0 . (1.2) Of course, this is the same exogeneity condition we use for consistency of the 2SLS estimator, and we can consistently estimate 1 and 1 by 2SLS under (1.2) and the rank condition, Assumption 2SLS.2. Just as with 2SLS, the reduced form of y 2 – that is, the linear projection of y 2 onto the exogenous variables – plays a critical role. Write the reduced form with an error term as y 2 z 2 v 2 E z v 2 0 (1.3) (1.4) where 2 is L 1. Endogeneity of y 2 arises if and only if u 1 is correlated with v 2 . Write the 1
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Imbens/Wooldridge, Lecture Notes 6, Summer ’07 linear projection of u 1 on v 2 ,inerrorform ,as u 1 1 v 2 e 1 , (1.5) where 1 E v 2 u 1 /E v 2 2 is the population regression coefficient. By definition, E v 2 e 1 0, and E z e 1 0 because u 1 and v 2 are both uncorrelated with z .
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Control Function Methods Lecture - Imbens/Wooldridge...

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