Re: [Gretl-users] panel model residuals
Am 19.12.2015 um 22:22 schrieb Allin Cottrell:
On Fri, 18 Dec 2015, Allin Cottrell wrote:
> Now let's consider the random-effects model. This can be represented as
>
> y_{it} = \mu + X_{it}\beta + u_i + e_{it}
>
If there's a reasonably uncontroversial way to recover estimates of the
u_i it would be nice to mention that in the doc. A simple (too simple?)
way would be just to take the individual means of the composite
residuals, as in
series ui = pmean($uhat)
However, that does not agree with Stata.
I think it's relatively intuitive that this (pmean) is conceptually what
the fixed effects would be. And given that the fixed effects can absorb
a maximum amount of variation (the incidental parameter problem),
another intuition would say that the random effects therefore have to
different and also less "fitted" to the data.
After a little bumbling around
it became apparent that, given a balanced panel, the Stata ui values
differ from the above by a multiplicative constant, and after a bit more
trial and error it emerged that the constant is
1 - (1 - \theta)^2
where \theta is the GLS coefficient. So (for a balanced panel) we have
ui_stata = (1 - (1 - theta)^2) * pmean($uhat)
Can anyone supply an econometric rationale for that formula?
I think the point is to split up the means such that the estimated
variance of the group effects (which enter in theta to make the GLS
feasible) will be justified by that split. However, I am not sure at all
whether this split would always be unique, i.e. whether another
"allocation" of the means among the two error components (and differing
across groups) would also be possible. In other words, whether the
variance restriction is identifying by itself.
cheers,
sven