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st: RE: RE: Re: adjusted r square


From   "Schaffer, Mark E" <M.E.Schaffer@hw.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: RE: Re: adjusted r square
Date   Wed, 21 Feb 2007 00:00:14 -0000

Returning to Ranjita's original question, I just noticed something odd
about the adjusted R-square reported by xtreg,fe.

Background: to get the "right" error variance for a fixed effects
regression, a degrees-of-freedom adjustment is needed that incorporates
the number of fixed effects.  Whereas the usual formula applied to an FE
regression would be SSR/(NT-K), the right formula is SSR/(N(T-1)-K).
See e.g. Wooldridge's 2002 textbook, p. 271.  The idea is that the FEs
are incidental parameters that eat up degrees of freedom and this needs
to be recognized.

The standard defnition of the adjusted R-sq is 1-(1-R2)((N-1)/(N-K)).

xtreg,fe seems to be using 1-(1-R2)((NT-1)/(NT-K)).

But arguably it should follow the same logic as that behind the formula
for the "right" error variance and use 1-(1-R2)((N(T-1)-1)/(N(T-1)-K)).
Or should it?

--Mark

> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu 
> [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Nick Cox
> Sent: 20 February 2007 22:59
> To: statalist@hsphsun2.harvard.edu
> Subject: st: RE: Re: adjusted r square
> 
> Uli raises some good questions, which have often been 
> answered in the literature -- but the answers disagree. 
> 
> Some of the concerns here are pointed up by considering 
> perfect models that interpolate the data. The idea of a 1:1 
> map appears as a literary conceit in Lewis Carroll, Jorge 
> Luis Borges, Umberto Eco, etc. It underlines that the perfect 
> model is just as difficult to understand as the original 
> data. Worse, it is less likely than a simpler model to 
> transfer well to other datasets. A overfitted model pays too 
> much attention to quirks of the dataset by "capitalising on chance". 
> 
> What is immediate is that modelling is a trade-off problem 
> between goodness of fit and parsimony or simplicity. However, 
> people can not agree on how to quantify the trade-off. It is 
> far from self-evident even that the number of adjustable 
> parameters is the best metric for complexity. 
> Adjusting R-square I think goes back further than other 
> criteria such as AIC and BIC, but each criterion proposed has 
> its few years of fame before another becomes more 
> fashionable. I regularly read advice such as that AIC is 
> widely agreed to give the wrong answer, to which the reaction 
> has to be, How do they know? 
> 
> Nick 
> n.j.cox@durham.ac.uk 
> 
> Ulrich Kohler
>  
> > However, as an aside: I do not find the arguments for the 
> > adjusted R2 very
> > convincing. It is sometimes said that you have to be punished 
> > for including
> > additional variables in a model. But why? Because the R2 
> > increases? Why do I
> > need to be punished for this? It is just a simple fact that I 
> > can explain
> > more variance with an additional variable. Punishment and 
> > especially the
> > amount of punishment is pure metaphysics. The set of control 
> > variables should
> > be compiled on theoretical reasons alone. If your model 
> contains some
> > variables that should be excluded on theortical reasons, 
> > exclude them (or you
> > will get punished by your reviewer). Likewise, if your model 
> > does not include
> > a variable in the model that should be there on theortical 
> > reasons, include
> > it (or your reviewer will punish you as well).
> > 
> > Needless to say that it might happen that your reviewer might 
> > punish you for not using the adjusted R2. ;-)
> 
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