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).

2 Overview of the Basic ModelMost of these notes are concerned with an unobserved effects Model defined for a largepopulation. 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.

3 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. 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.

4 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).

5 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.

6 ,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). Theseare the now well-known cluster standard errors. With largeNand smallT, there is littleexcuse not to compute them.

7 (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.

8 ,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. Generally, is identified under sequentialexogeneity. First differencing and using lags ofxitas instruments, or forward filtering, can beused in simple IV procedures or GMM procedures.

9 (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.

10 (Note: The usual RE variance matrix, which depends only on c2and u2,need not be correctly specified! 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).