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Unformatted text preview: Imbens, Lecture Notes 21, ARE213 Spring 06 1 ARE213 Econometrics Spring 2006 UC Berkeley Department of Agricultural and Resource Economics Panel Data II: Fixed Effects In this lecture we consider the same setup, with a linear model: Y it = X it + c i + it , with c i an unobserved individualspecific, timeinvariant component. However, compared to the random effects discussion we relax the assumption that c i is independent of the observed covariates X it . We continue to maintain the exogeneity assumption on the residuals: E [ it  X i 1 , . . . , X iT , c i ] = 0 , and in fact for inference we make the stronger assumption E [ i i  c i , X i ] = 2 I T . We consider a couple of estimators. The first is based on simply adding a Ndimensional vector of timeinvariant covariates Z , with its j th element for unit i in period equal to Z it,j : Z it,j = 1 { i = j } . Then if we define c to be the vector with typical element c i , we can write Y it = X it + Z i c + it . The first estimator is just the least squares estimator for this regression function: min c, i,t ( Y it X it  Z i c ) 2 . Imbens, Lecture Notes 21, ARE213 Spring 06 2 The estimators for both and c are unbiased. However, the estimators for c are not con sistent. As we get more and more observations, we do not get more information about c i ....
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 Spring '06
 IMBENS

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