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 :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.

2 But there are differences for Models nonlinear inendogenous variables even if they are Linear in Parameters . 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. 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.

3 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. (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.

4 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. 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.)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).

5 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. 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.

6 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. 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.

7 For example, supposey2is a binary the CF approach involves estimatingE y1|z,y2 z1 1 1y2 E u1|z,y2 ,andsowemustbeabletoestimateE u1|z,y2 .Ify2 1 z 2 e2 0 , u1,e2 is independentofz,E u1|e2 1e2,ande2~Normal 0, 1 , thenE u1|z,y2 E E u1|z,e2 |z,y2 1E v2|z,y2 1 y2 z 2 1 y2 z 2 ,where / is the inverse Mills ratio (IMR). A simple two-step estimator is toobtain the probit estimator 2and then to add the generalized residual, gri2 yi2 zi 2 1 yi2 zi 2 as a regressor:yi1onzi1,yi2,gri2,i 1,.., of the CF estimators hinges on the model forD y2|z being correctly specified,along with linearity inE u1|v2 (and some sort of independence withz). Of course, if we justapply 2 SLS directly to ( ), it makes no distinction among discrete, continuous, or somemixture fory2. 2 SLS is consistent ifL y2|z z 2actually depends onz2and ( ) holds. So,while estimating ( ) using CF methods wheny2is binary is somewhat popular (Stata s treatreg even has the option of full MLE, where u1,e2 is bivariate normal), one shouldremember that it is less robust than standard IV might one use the binary nature ofy2in IV estimation?

8 AssumeE u1|z 0 and,nominally, assume a probit model forD y2|z . Obtain the fitted probabilities, zi 2 ,fromthefirst stage probit, and then use these as IVs foryi2. This method is fully robust tomisspecification of the probit model ; the standard errors need not be adjusted for the first-stageprobit (asymptotically); and it is the efficient IV estimator ifP y2 1|z z 2 andVar u1|z 12. But it is probably less efficient than the CF estimator if the additionalassumptions needed for CF consistency hold. (Note: Using zi 2 as an IV foryi2is not thesame as using zi 2 as a regressor in place ofyi2.)To summarize: except in the case wherey2appears linearly and a Linear reduced form isestimated fory2, the CF approach imposes extra assumptions not imposed by IV , in more complicated Models , it is hard to beat the CF Random Coefficient Models4 Imbens/Wooldridge, Lecture Notes 6, Summer 07 Control function methods can be used for random coefficient Models that is, modelswhere unobserved heterogeneity interacts with endogenous explanatory variables.

9 However, insome cases, standard IV methods are more robust. To illustrate, we modify equation ( ) asy1 1 z1 1 a1y2 u1, ( )wherez1is 1 L1,y2is the endogenous explanatory variable, anda1, the coefficient ony2 an unobserved random variable. [It is now convenient to set apart the intercept.] We couldreplace 1with a random vector, sayd1, and this would not affect our analysis of the IVestimator (but would slightly alter the Control function estimator). Following Heckman andVytlacil (1998), we refer to ( ) as acorrelated random coefficient(CRC) is convenient to writea1 1 v1where 1 E a1 is the object of interest. We canrewrite the equation asy1 1 z1 1 1y2 v1y2 u1 1 z1 1 1y2 e1, ( )wheree1 v1y2 u1. Equation ( ) shows explicitly a constant coefficient ony2(which wehope to estimate) but also an interaction between the observed heterogeneity,v1, , ( ) is a population model . For a random draw, we would writeyi1 1 zi1 1 1yi2 vi1yi2 ui1, which makes it clear that 1and 1are Parameters toestimate andvi1is specific to discussed in Wooldridge (1997, 2003), the potential problem with applying instrumentalvariables (2 SLS) to ( ) is that the error termv1y2 u1is not necessarily uncorrelated withthe instrumentsz, even if we make the assumptionsE u1|z E v1|z 0, ( )which we maintain from here on.

10 Generally, the termv1y2can cause problems for IVestimation, but it is important to be clear about the nature of the problem. If we are allowingy2to be correlated withu1then we also want to allowy2andv1to be correlated. In other words,E v1y2 Cov v1,y2 1 0. But a nonzero unconditional covariance isnota problemwith applying IV to ( ): it simply implies that the composite error term,e1,has(unconditional) mean 1rather than a zero. As we know, a nonzero mean fore1means that theorginal intercept, 1, would be inconsistenly estimated, but this is rarely a , we can allow Cov v1,y2 , the unconditional covariance, to be unrestricted. Butthe usual IV estimator is generally inconsistent if E v1y2|z depends onz. (There are still cases,which we will cover in Part IV, where the IV estimator is consistent.). Note that, because5 Imbens/Wooldridge, Lecture Notes 6, Summer 07E v1|z 0, E v1y2|z Cov v1,y2|z . Therefore, as shown in Wooldridge (2003), asufficient condition for the IV estimator applied to ( ) to be consistent for 1and 1isCov v1,y2|z Cov v1,y2.