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Re: st: Fixed effects decomposition (was xthtaylor)


From   Mario F Rueda NarvŠez <mfrueda@uma.es>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Fixed effects decomposition (was xthtaylor)
Date   Wed, 11 Feb 2004 12:15:38 +0100

My question about the decomposition still stands though, how do I estimate
g in the model below when using fixed effects.
    y(it)=b*x(it) + g*z(i) + a(i) + e(it)
where x(it) are time varying characteristics, z(i) are time invariant
characteristics, and a(i) are the fixed effects.
One answer is that there is no fixed-effect estimator of g. In the fixed-effect (within) model the assumption is that every variable in X and Z is correlated with the (fixed) effect a(i). More formally, fixed-effect estimation is consistent whether those correlations exist or not, while random effects are only consistent (and efficient) if no explanatory variable is correlated with a(i). (I think this is well illustrated in William Gould's faq at http://www.stata.com/support/faqs/stat/xt.html , even if it is not the main subject there). The way the fixed-effect method avoids the problem is by not using the cross-sectional information in the data (differences between two different indidivuals, if your panel consists of individual information at different moments) , but only the variation within individuals. To do so, the variables (as -xtreg, fe- does) are transformed into deviations of individual means. Unfortunately, when you do that transformation to the variables in Z you end up with rows full of 0's and z(i) has to be consequently dropped from the analysis. That is, you can't use "within" variation in the case of time invariant characteristis because there is not such a variation.


My proposal is to estimate fixed effects model, predict a(i), then regress
a(i) on z.  I would bootstrap to get the standard errors right on the
auxilliary regression
     xtreg y x, fe
     predict a, u
     by id : keep if _n==1
     regress a z
Which connects with my second answer. Hausman and Taylor (1981) discuss in some detail the "fixed-effects" estimation of g and the model which would be estimated with the above code. They conclude that regressing the a(i) ressiduals on z(i) will only yield a consistent estimator of g if z(i) and a(i) are uncorrelated, and that is not the assumption under the fixed-effects approach. Besides, if that is the case, you would be better off with a random effects model -xtreg, fe- or, if some characteristics in x(i) are endogenous with respect to a(i), the appropiate -xthtaylor- specification. Further, they suggest a variation of your last regression (regress a z), but using an instrumental variable approach (where some or all the z's are assumed endogenous and at least that many x's have to be assume exogenous). a would be regressed on z, with the individual means of the exogenous x's as instruments. If, in your code, x is uncorrelated with a but you suspect z is, it would be someting like this:

xtreg y x, fe
predict a, u
by id: gen count_ = _n *this way you don't have to drop any observations*
egen x_bar = mean(x), by(id)
ivreg a (z = x_bar) if count_ == 1

(Please note that I don't recall exactly if H&T use "a" or other *very* related variable as depvar). This is what they call the "within estimator" of the g parameter(s). However, I think I recall that their use of the "witinh estimator" term is not because of how it is estimated, but because estimating g this way is consistent (if x is really exogenous), but doesn't have to be the best choice in terms of efficiency. If you can find more x(i) exogenous variables that z(i) endogenous ones, then the HT estimator is more efficient. If you have an equal number, -xthtaylor- is the same as above and should get the same or very similar results, and if you don't have enough exogenous x's, estimation of g is not feasible.

That said, I'm pretty sure selecting only one observation out of each individual (when "id" is your i variable) is correct in the case of a balanced panel, but the thing gets trickier with unbalanced panels. I haven't read anything that appeared conclusive on how to handle between models with unbalanced panels and would really thank any references.

Reference:

Jerry A. Hausman; William E. Taylor
Econometrica, Vol. 49, No. 6 (Nov., 1981) , pp. 1377-1398

Hope this helps,

Mario F. Rueda NarvŠez
Universidad de MŠlaga.

PS: The HT estimator (and subsequents proposals) is explained quite nicely in Marno Veerbek's "A Guide to Modern Econometrics", which could be helpful.

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