1 Quick Overview of the Basic Model - The …

1 Imbens/Wooldridge, Lecture Notes 2, Summer 07 What s New in Econometrics?NBER,Summer 2007 Lecture 2,Monday,July 30th, amLinear Panel data ModelsThese notes cover some recent topics in linear panel data models. They begin with a modern treatment of the Basic linear Model , and then consider some embellishments, such asrandom slopes and time-varying factor loads. In addition, fully robust tests for correlatedrandom effects, lack of strict exogeneity, and contemporaneous endogeneity are 4 considers estimation of models without strictly exogenous regressors, and Section 5presents a unified framework for analyzing pseudo panels (constructed from repeated crosssections). Overview of the Basic ModelMost of these notes are concerned with an unobserved effects Model defined for a largepopulation.

2 Therefore, we assume random sampling in the cross section dimension. Unlessstated otherwise, the asymptotic results are for a fixed number of time periods,T, with thenumber of cross section observations,N, getting some of what we do, it is critical to distinguish the underlying population Model ofinterest and the sampling scheme that generates data that we can use to estimate the populationparameters. The standard Model can be written, for a genericiin the population, asyit t xit ci uit,t 1,..,T, ( )where tis a separate time period intercept (almost always a good idea),xitis a 1 Kvector ofexplanatory variables,ciis the time-constant unobserved effect, and the uit:t 1,..,T areidiosyncratic errors. Thanks to Mundlak (1978) and Chamberlain (1982), we view theciasrandom draws along with the observed variables.

3 Then, one of the key issues is whetherciiscorrelated with elements probably makes more sense to drop theisubscript in ( ), which would emphasize thatthe equation holds for an entire population. But ( ) is useful to emphasizing which factorschange only acrosst, which change only change acrossi, and which change issometimes convenient to subsume the time dummies out correlation (for now) betweenuitandxit, a sensible assumption iscontemporaneous exogeneity conditional on ci:E uit|xit,ci 0,t 1,..,T. ( )This equation really defines in the sense that under ( ) and ( ),1 Imbens/Wooldridge, Lecture Notes 2, Summer 07E yit|xit,ci t xit ci, ( )so the jare partial effects holding fixed the unobserved heterogeneity (and covariates otherthanxtj).

4 As is now well known, is not identified only under ( ). Of course, if we addedCov xit,ci 0for anyt, then is identified and can be consistently estimated by a crosssection regression using periodt. But usually the whole point is to allow the unobserved effectto be correlated with can allow general correlation if we add the assumption ofstrict exogeneity conditionalon ci:E uit|xi1,xi2,..,xiT,ci 0,t 1,..,T, ( )which can be expressed asE yit|xi1,..,xiT,ci E yit|xit,ci t xit ci. ( )If the elements of xit:t 1,..,T have suitable time variation, can be consistentlyestimated by fixed effects (FE) or first differencing (FD), or generalized least squares (GLS) orgeneralized method of moments (GMM) versions of them. If the simpler methods are used, andeven if GLS is used, standard inference can and should be made fully robust toheteroksedasticity and serial dependence that could depend on the regressors (or not).

5 Theseare the now well-known cluster standard errors. With largeNand smallT, there is littleexcuse not to compute them.(Note: Some call ( ) or ( ) strong exogeneity. But in the Engle, Hendry, and Richard(1983) work, strong exogeneity incorporates assumptions on parameters in differentconditional distributions being variation free, and that is not needed here.)The strict exogeneity assumption is always violated ifxitcontains lagged dependentvariables, but it can be violated in other cases wherexi,t 1is correlated withuit a feedbackeffect. An assumption more natural than strict exogeneity issequential exogeneity conditionon ci:E uit|xi1,xi2,..,xit,ci 0,t 1,..,T ( )orE yit|xi1,..,xit,ci E yit|xit,ci t xit ci. ( )This allows for lagged dependent variables (in which case it implies that the dynamics in the2 Imbens/Wooldridge, Lecture Notes 2, Summer 07mean have been completely specified) and, generally, is more natural when we take the viewthat xit might react to shocks that affectyit.

6 Generally, is identified under sequentialexogeneity. First differencing and using lags ofxitas instruments, or forward filtering, can beused in simple IV procedures or GMM procedures. (More later.)If we are willing to assumeciandxiare uncorrelated, then many more possibilities arise(including, of course, identifying coefficients on time-constant explanatory variables). Themost convenient way of stating the random effects (RE) assumption isE ci|xi E ci , ( )although using the linear projection in place ofE ci|xi suffices for consistency (but usualinference would not generally be valid). Under ( ), we can used pooled OLS or any GLSprocedure, including the usual RE estimator. Fully robust inference is available and shouldgenerally be used. (Note: The usual RE variance matrix, which depends only on c2and u2,need not be correctly specified!)

7 It still makes sense to use it in estimation but make inferencerobust.)It is useful to define twocorrelated random effectsassumptions:L ci|xi xi , ( )which actually is not an assumption but a definition. For nonlinear models, we will have toactually make assumptions aboutD ci|xi , the conditional distribution. Methods based on ( )are often said to implement theChamberlain device, after Chamberlain (1982).Mundlak (1978) used a restricted version, and used a conditional expectation:E ci|xi x i , ( )wherex i T 1 t 1 Txit. This formulation conserves on degrees of freedom, and extensions areuseful for nonlinear we writeci xi aiorci x i aiand plug into the original equation, forexampleyit t xit x i ai uit ( )(absorbing into the time intercepts), then we are tempted to use pooled OLS, or REestimation becauseE ai uit|xi 0.

8 Either of these leads to the FE estimator of , and to asimple test ofH0: 0. Later, when we discuss control function methods, it will be handy torun regressions directly that include the time averages. (Somewhat surprisingly, obtain the3 Imbens/Wooldridge, Lecture Notes 2, Summer 07same algebraic equivalence using Chamberlain s devise. The pooled OLS estimator of is stillthe FE estimator, even though the tmight change substantially acrosst.)Some of us have been pushing for several years the notion that specification tests should bemade robust to assumptions that are not directly being tested. (Technically, they should berobust to assumptions that they have no asymptotic power for detecting violations of.) Muchprogress has been made, but one still sees Hausman statistics computed that maintain a full setof assumptions under the null.

9 Take comparing random effects to fixed effects. The keyassumption is ( ). whetherVar vi|xi has the random effects structure, wherevit ci uit,should not be a critical issue. It makes no sense to report a fully robust variance matrix for FEand RE but then to compute a Hausman test that maintains the full set of RE assumptions. (Inaddition to ( ) and ( ), these areVar ui|xi,ci u2 ITandVar ci|xi Var ci .) Theregression-based Hausman test from ( ) is very handy for obtaining a fully robust specifically, suppose the Model contains a full set of year intercepts as well astime-constant and time-varying explanatory variables:yit gt zi wit ci , it is clear that, because we cannot estimate by FE, it is not part of the Hausman testcomparing RE and FE. What is less clear, but also true, is that the coefficients on the timedummies, , cannot be included, either.

10 (RE and FE estimation only with aggregate timeeffects are identical.) In fact, we can only compare theM 1 estimates of ,say FEand include FEand REwe introduce a nonsingularity in the asymptotic variance matrix. Theregression based test, from the pooled regressionyitongt,zi,wit,w i,t 1,..,T;i 1,..,Nmakes this clear (and that the areMrestrictions to test). (Mundlak (1978) suggested this testand Arellano (1993) described the robust version.). Unfortunately, the usual form of theHausman test does not, and, for example, Stata gets it wrong and tries to include the yeardummies in the test (in addition to being nonrobust). The most important problem is thatunwarranted degrees of freedom are added to the chi-square distribution, often many extra df,which can produce seriously Insights Into Old EstimatorsIn the past several years, the properties of traditional estimators used for linear models,particularly fixed effects and its instrumental variable counterparts, have been studied under4 Imbens/Wooldridge, Lecture Notes 2, Summer 07weaker assumptions.