Panel Data Models with Heterogeneity and Endogeneity

1 Panel data Models with Heterogeneity and EndogeneityJeff WooldridgeMichigan State UniversityProgramme Evaluation for Policy AnalysisInstitute for Fiscal StudiesJune 20121. Introduction2. General Setup and Quantities of Interest3. Assumptions with Neglected Heterogeneity4. Models with Heterogeneity and Endogeneity5. Estimating Some Popular When Panel data Models contain unobserved Heterogeneity andomitted time-varying variables, control function methods can be used toaccount for both problems. Under fairly week assumptions can obtain consistent, asymptoticallynormal estimators of average structural functions provided suitableinstruments are available. Other issues with Panel data : How to treat dynamics? Models withlagged dependent variables are hard to estimate when Heterogeneity andother sources of Endogeneity are Approaches to handling unobserved Heterogeneity :1. Treat as parameters to estimate. Can work well with largeTbut withsmallTcan have incidental parameters problem.

2 Bias adjustments areavailable for parameters and average partial effects. Usually weakdependence or even independence is assumed across the Remove Heterogeneity to obtain an estimating equation. Works forsimple linear Models and a few nonlinear Models (via conditional MLEor a quasi-MLE variant). Cannot be done in general. Also, may not beable to identify interesting partial Correlated Random Effects: Mundlak/Chamberlain. Requires somerestrictions on distribution of Heterogeneity , although these can benonparametric. Applies generally, does not impose restrictions ondependence over time, allows estimation of average partial effects. Canbe easily combined with CF methods for Endogeneity . Can try to establish bounds rather than estimate parameters or , Fern ndez-Val, Hahn, and Newey (2009) is a Setup and Quantities of Interest Static, unobserved effects probit model for Panel data with an omittedtime-varying variablerit:P yit 1|xit,ci,rit xit ci rit ,t 1.

3 ,T. (1)What are the quantities of interest for most purposes?(i) The element of ,the j. These give the directions of the partialeffects of the covariates on the response probability. For any twocontinuous covariates, the ratio of coefficients, j/ h, is identical to theratio of partial effects (and the ratio does not depend on the covariatesor unobserved Heterogeneity ,ci).5(ii) The magnitudes of the partial effects. These depend not only on thevalue of the covariates, sayxt, but also on the value of the unobservedheterogeneity. In the continuous covariate case, P yt 1|xt,c,rt xtj j xt c rt . (2) Questions: (a) Assuming we can estimate , what should we do aboutthe unobservables c,rt ? (b) If we can only estimate up-to-scale, canwe still learn something useful about magnitudes of partial effects? (c)What kinds of assumptions do we need to estimate partial effects?6 Let xit,yit :t 1,..,T be a random draw from the cross we are interested inE yit|xit,ci,rit mt xit,ci,rit.

4 (3)cican be a vector of unobserved Heterogeneity ,rita vector of omittedtime-varying variables. Partial effects: ifxtjis continuous, then j xt,c,rt mt xt,c,rt xtj, (4)or discrete How do we account for unobserved ci,rit ? If we know enoughabout the distribution of ci,rit we can insert meaningful values for c,rt . For example, if c E ci , rt E rit then we can computethe partial effect at the average (PEA),PEAj xt j xt, c, rt . (5)Of course, we need to estimate the functionmtand c, rt . If we canestimate the distribution of ci,rit , or features in addition to its mean,we can insert different quantiles, or a certain number of standarddeviations from the Alternatively, we can obtain the average partial effect (APE) (orpopulation average effect) by averaging across the distribution ofci:APE xt E ci,rit j xt,ci,rit . (6)The difference between (5) and (6) can be nontrivial. In some leadingcases, (6) is identified while (5) is not.

5 (6) is closely related to thenotion of the average structural function (ASF) (Blundell and Powell(2003)). The ASF is defined asASFt xt E ci,rit mt xt,ci,rit . (7) Passing the derivative through the expectation in (7) gives the with Neglected HeterogeneityExogeneity of Covariates Cannot get by with just specifying a model for the contemporaneousconditional distribution,D yit|xit,ci . The most useful definition of strict exogeneity for nonlinear paneldata Models isD yit|xi1,..,xiT,ci D yit|xit,ci . (8)Chamberlain (1984) labeled (8)strict exogeneity conditional on theunobserved effectsci. Conditional mean version:E yit|xi1,..,xiT,ci E yit|xit,ci . (9)10 Thesequential exogeneityassumption isD yit|xi1,..,xit,ci D yit|xit,ci . (10)Much more difficult to allow sequential exogeneity in in nonlinearmodels. (Most progress has been made for lagged dependent variablesor specific functional forms, such as exponential.) Neither strict nor sequential exogeneity allows for contemporaneousendogeneity of one or more elements ofxit, where, say,xitjis correlatedwith unobserved, time-varying unobservables that Independence In linear Models , serial dependence of idiosyncratic shocks is easilydealt with , either by cluster robust inference or Generalized LeastSquares extensions of Fixed Effects and First Differencing.

6 Withstrictly exogenous covariates, serial correlation never results ininconsistent estimation, even if improperly modeled. The situation isdifferent with most nonlinear Models estimated by MLE. Conditional independence(CI) (under strict exogeneity):D yi1,..,yiT|xi,ci t 1TD yit|xit,ci . (11)12 In a parametric context, the CI assumption reduces our task tospecifying a model forD yit|xit,ci , and then determining how to treatthe unobserved Heterogeneity ,ci. In random effects and correlated random frameworks (next section),CI plays a critical role in being able to estimate the structural parameters and the parameters in the distribution ofci(and therefore, inestimating PEAs). In a broad class of popular Models , CI plays noessential role in estimating about the Unobserved HeterogeneityRandom Effects Generally stated, the key RE assumption isD ci|xi1,..,xiT D ci . (12)Under (12), the APEs are actually nonparametrically identified fromE yit|xit xt.

7 (13) In some leading cases (RE probit and RE Tobit with heterogeneitynormally distributed), if we want PEs for different values ofc, we mustassume more: strict exogeneity, conditional independence, and (12) with a parametric distribution forD ci .14 Correlated Random EffectsA CRE framework allows dependence betweenciandxi, but restrictedin some way. In a parametric setting, we specify a distribution forD ci|xi1,..,xiT , as in Chamberlain (1980,1982), and much worksince. Distributional assumptions that lead to simple estimation homoskedastic normal with a linear conditional mean can Possible to drop parametric assumptions and just assumeD ci|xi D ci|x i , (14)without restrictingD ci|x i . Altonji and Matzkin (2005, Econometrica). Other functions of xit:t 1,..,T are APEs are identified very generally. For example, under (14), aconsistent estimate of the average structural function isASF xt N 1 i 1 Nqt xt,x i , (15)whereqt xit,x i E yit|xit,x i.

8 Need a random sample x i:i 1,..,N for the averaging out Effects The label fixed effects is used differently by different view:ci,i 1,..,Nare parameters to be estimated. Usually leadsto an incidental parameters problem. Second meaning of fixed effects :D ci|xi is unrestricted and welook for objective functions that do not depend oncibut still identifythe population parameters. Leads to conditional MLE if we can find sufficient statistics sisuch thatD yi1,..,yiT|xi,ci,si D yi1,..,yiT|xi,si . (16) Conditional Independence is usually maintained. Key point: PEAs and APEs are generally with Heterogeneity and Endogeneity Letyit1be a scalar response,yit2a vector of endogenous variables,zit1exogenous variables, and we haveE yit1|yit2,zit1,ci1,rit1 mt1 yit2,zit1,ci1,rit1 (17) yit2is allowed to be correlated withrit1(aswellaswithci1). The vector of exogenous variables zit:t 1,..,T withzit1 zitare strictly exogenous in the sense thatE yit|yit2,zi,ci1,rit1 E yit|yit2,zit1,ci1,rit1 D rit1|zi,ci1 D rit1 (18) (19)19 Sometimes we can eliminateciand obtain an equation that can beestimated by IV (linear, exponential).

9 Generally not possible. Now a CRE approach involves modelingD ci1|zi . Generally, we need to model howyit2is related torit1. Control Function methods are convenient for allowing both. Supposeyit2is a scalar andyit2 mit2 zit,z i, 2 vit2E vit2|zi 0D rit1|vit2,zi D rit1|vit2 (20)20 with suitable time-variation in the instruments, the assumptions in(20) allow identification of the ASF if we assume a model forD ci1|zi,vit2 Generally, we can estimateE yit1|yit2,zi,vit2 E yit1|yit2,zit1,z i,vit2 gt1 yit2,zit1,z i,vit2 (21)21 The ASF is now obtained by averaging out z i,vit2 :ASF yt2,zt1 E z i,vit2 gt1 yt2,zt1,z i,vit2 Most of this can be fully nonparametric (Altonji and Matzkin, 2005;Blundell and Powell, 2003) although some restriction is needed onD ci1|zi,vit2 , such asD ci1|zi,vit2 D ci1|z i,vit2 WithTsufficiently large we can add other features of zit:t 1,..,T toz Some Popular ModelsLinear model with Endogeneity Simplest model isyit1 1yit2 zit1 1 ci1 uit1 xit1 1 ci1 uit1E uit1|zi,ci1 0 (22) The fixed effects 2 SLS estimator is common.

10 Deviate variables fromtime averages to removeci1then apply IV: it1 x it1 1 it1z it zit z i23 Easy to make inference robust to serial correlation andheteroskedasticity in uit1 . ( Cluster-robust inference. ) Test for (strict) exogeneity of yit2 :(i) Estimate the reduced form ofyit2by usual fixed effects:yit2 zit 1 ci2 uit2 Get the FE residuals, it2 it2 z it 1. Estimate the augment equationyit1 1yit2 zit1 1 1 it2 ci1 errorit (23)by FE and use a cluster-robust test ofH0: 1 The random effects IV approach assumesci1is uncorrelated withzi,and nominally imposes serial independence on uit1 . Simple way to test the null whether REIV is sufficient. (RobustHausman test comparing REIV and FEIV.)Estimateyit1 1 xit1 1 z i 1 ai1 uit1 (24)by REIV, using instruments 1,zit,z i . The estimator of 1is the FEIV estimator. TestH0: 1 0, preferably using a fully robust test. A rejection isevidence that the IVs are correlated withci, and should use Other than the rank condition, the key condition for FEIV to beconsistent is that the instruments, zit , are strictly exogenous withrespect to uit.