1 Linear in Parameters Models IV versus Control …

1 Imbens/Wooldridge, Lecture Notes 6, Summer 07 What s New in Econometrics?NBER,Summer 2007 Lecture 6,Tuesday,July 31st, amControl Function and Related MethodsThese notes review the Control function approach to handling endogeneity in Models linearin Parameters , and draws comparisons with standard methods such as 2 SLS. Certain nonlinearmodels with endogenous explanatory variables are most easily estimated using the CF method,and the recent focus on average marginal effects suggests some simple, flexible 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 Models .

2 IV versus Control FunctionsMost Models that are Linear in Parameters are estimated using standard IV methods eithertwo stage least squares (2 SLS) or generalized method of moments (GMM). An alternative, thecontrol function (CF) approach, relies on the same kinds of identification conditions. In thestandard case where a endogenous explanatory variables appear linearly, the CF approachleads to the usual 2 SLS estimator. But there are differences for Models nonlinear inendogenous variables even if they are Linear in Parameters .

3 And, for Models nonlinear inparameters, the CF approach offers some distinct the response variable,y2the endogenous explanatory variable (a scalar forsimplicity), andzthe 1 Lvector of exogenous variables (which includes unity as its firstelement). Consider the modely1 z1 1 1y2 u1 ( )wherez1is a 1 L1strict subvector ofzthat also includes a constant. The sense in whichzisexogenous is given by theLorthogonality (zero covariance) conditionsE z u1 0. ( )Of course, this is the same exogeneity condition we use for consistency of the 2 SLS estimator,and we can consistently estimate 1and 1by 2 SLS under ( ) and the rank condition,Assumption as with 2 SLS, the reduced form ofy2 that is, the Linear projection ofy2onto theexogenous variables plays a critical role.

4 Write the reduced form with an error term asy2 z 2 v2E z v2 0 ( ) ( )where 2isL 1. Endogeneity ofy2arises if and only ifu1is correlated withv2. Write the1 Imbens/Wooldridge, Lecture Notes 6, Summer 07linear projection ofu1onv2,inerrorform,asu1 1v2 e1, ( )where 1 E v2u1 /E v22 is the population regression coefficient. By definition, E v2e1 0,and E z e1 0becauseu1andv2are both uncorrelated ( ) into equation ( ) givesy1 z1 1 1y2 1v2 e1, ( )where we now viewv2as an explanatory variable in the equation. As just noted,e1,isuncorrelated ,y2is a Linear function ofzandv2,andsoe1is alsouncorrelated uncorrelated withz1,y2,andv2, ( ) suggests a simple procedure forconsistently estimating 1and 1(as well as 1): run the OLS regression ofy1onz1,y2,andv2using a random sample.

5 (Remember, OLS consistently estimates the Parameters in anyequation where the error term is uncorrelated with the right hand side variables.) The onlyproblem with this suggestion is that we do not observev2; it is the error in the reduced formequation fory2. Nevertheless, we can writev2 y2 z 2and, because we collect data ony2andz, we can consistently estimate 2by OLS. Therefore, we can replacev2withv 2, the OLSresiduals from the first-stage regression ofy2onz. Simple substitution givesy1 z1 1 1y2 1v 2 error, ( )where, for eachi,errori ei1 1zi 2 2 , which depends on the sampling error in 2unless 1 0.

6 Standard results on two-step estimation imply the OLS estimators from ( )will be consistent for 1, 1,and OLS estimates from ( ) are Control function estimates. The inclusion of the residualsv 2 controls for the endogeneity ofy2in the original equation (although it does so withsampling error because 2 2).It is a simple exercise in the algebra of least squares to show that the OLS estimates of 1and 1from ( ) areidenticalto the 2 SLS estimates starting from ( ) and usingzas thevector of instruments. (Standard errors from ( ) must adjust for the generated regressor.)

7 It is trivial to use ( ) to testH0: 1 0, as the usualtstatistic is asymptotically validunder homoskedasticity Var u1|z,y2 12underH0 ; or use the heteroskedasticity-robustversion (which doesnotaccount for the first-stage estimation of 2).Now extend the model :2 Imbens/Wooldridge, Lecture Notes 6, Summer 07y1 z1 1 1y2 1y22 u1E u1|z 0. ( ) ( )For simplicity, assume that we have a scalar,z2, that is not also inz1. Then, under ( ) which is stronger than ( ), and is essentially needed to identify nonlinear Models we canuse, say,z22(ifz2is not binary) as an instrument fory22because any function ofz2isuncorrelated withu1.

8 In other words, we can apply the standard IV estimator with explanatoryvariables z1,y2,y22 and instruments z1,z2,z22 ; note that we have two endogenousexplanatory variables, would the CF approach entail in this case? To implement the CF approach in ( ),we obtain the conditional expectation E y1|z,y2 a Linear projection argument no longerworks because of the nonlinearity and that requires an assumption about E u1|z,y2 .Astandard assumption isE u1|z,y2 E u1|z,v2 E u1|v2 1v2, ( )where the first equality follows becausey2andv2are one-to-one functions of each other(givenz) and the second would hold if u1,v2 is independent ofz a nontrivial restriction onthe reduced form error in ( ), not to mention the structural The final assumption islinearity of the conditional expectation E u1|v2 , which is more restrictive than simply defininga Linear projection.

9 Under ( ),E y1|z,y2 z1 1 1y2 1y22 1 y2 z 2 z1 1 1y2 1y22 1v2. ( )Implementing the CF approach means running the OLS regressiony1onz1,y2,y22,v 2,wherev 2still represents the reduced form residuals. The CF estimates arenotthesameasthe2 SLSestimates using any choice of instruments for y2,y22 .The CF approach, while likely more efficient than a direct IV approach, is less robust. Forexample, it is easily seen that ( ) and ( ) imply that E y2|z z 2. A Linear conditionalexpectation fory2is a substantive restriction on the conditional distribution ofy2.

10 Therefore,the CF estimator will be inconsistent in cases where the 2 SLS estimator will be consistent. Onthe other hand, because the CF estimator solves the endogeneity ofy2andy22by adding thescalarv 2to the regression, it will generally be more precise perhaps much more precise than the IV estimator. (I do not know of a systematic analysis comparing the two approaches inmodels such as ( ).)3 Imbens/Wooldridge, Lecture Notes 6, Summer 07 Standard CF approaches impose extra assumptions even in the simple model ( ) if weallowy2to have discreteness in its distribution.