Re: st: Fixed effects decomposition (was xthtaylor)

[Mario F Rueda Narv�ez <mfrueda@uma.es>]

> 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 Narv�ez
Universidad de M�laga.

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

*
*   For searches and help try:
*   http://www.stata.com/support/faqs/res/findit.html
*   http://www.stata.com/support/statalist/faq
*   http://www.ats.ucla.edu/stat/stata/