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Re: st: Cross-Sectional Time Series


From   Mark Schaffer <M.E.Schaffer@hw.ac.uk>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Cross-Sectional Time Series
Date   Thu, 27 Jun 2002 17:15:25 +0100 (BST)

Seconded.  That was very instructive.  Thanks!

--Mark

Quoting anirban basu <abasu@midway.uchicago.edu>:

> Hi David and Vince,
> 
> Thanks for your insights and helpful comments. This was a good
> learning
> experience..
> 
> Anirban
> 
> ______________________________________
> ANIRBAN BASU
> Doctoral Student
> Harris School of Public Policy Studies
> University of Chicago
> (312) 563 0907 (H)
> ________________________________________________________________
> 
> 
> On Wed, 26 Jun 2002, Vince Wiggins, StataCorp wrote:
> 
> > I have one additional comment in the continuing thread
> comparing the results
> > of -regress-, -xtreg, fe-, and -xtreg , re-.  
> > 
> > While I agree with the comparisons between the models
> presented by Mark
> > Schaffer <M.E.Schaffer@hw.ac.uk> and David Drukker
> <ddrukker@stata.com>, there
> > is a more mundane reason why the example presented by
> Anirban Basu
> > <abasu@midway.uchicago.edu> elicits virtually identical
> estimates from
> > -regress-, -xtreg, fe-, and -xtreg, re-.  The short answer
> is they have to be
> > identical, at least to machine precision of the
> computations.
> > 
> > Anirban Basu asks us to generate data in the following
> manner,
> > 
> >     . mat C= (1, 0.6, 0.6, 0.6 \  0.6, 1, 0.6, 0.6 \ 0.6,
> 0.6, 1, 0.6 \  /*
> >  	*/ 0.6, 0.6, 0.6, 1)
> >     . drawnorm y1 y2 y3 y4, n(1000) means(1 3 4 7) corr(C)
> >     . gen id=_n
> >     . reshape long y , i(id) j(time)
> > 
> > Anirban is using -drawnorm- to create 4 correlated variables
> and then
> > -reshape- to turn these into a panel data with 4 values for
> a single y.  This
> > is a fine way to create data with a random effect.  Here are
> the first three
> > panels:
> > 
> > . list in 1/12
> > 
> >             id       time          y
> >   1.         1          1  -.0939699
> >   2.         1          2   2.265574
> >   3.         1          3   2.323656
> >   4.         1          4   6.053069
> >   5.         2          1   1.367081
> >   6.         2          2   3.062155
> >   7.         2          3   4.830178
> >   8.         2          4   7.105754
> >   9.         3          1   1.145398
> >  10.         3          2   4.087784
> >  11.         3          3    3.99791
> >  12.         3          4   6.942679
> > 
> > 
> > Anirban, the asks us to try the OLS, fixed-effects, and
> random-effects
> > estimators on this data by typing,
> > 
> >      . regress y time
> > 
> >      . xtreg   y time , i(id) fe 
> > and,
> >      . xtreg   y time , i(id) re 
> > 
> > What is unusual about this model is that we are including
> -time- as a
> > regressor.  Note that we have perfectly balanced panels of 4
> observations
> > each, and that the variable -time- exactly repeats itself --
> counting 1, 2, 3,
> > 4 in each panel.
> > 
> > What does this mean for the fixed-effects (FE)
> transformation?  The FE
> > transformation just subtracts the panel mean for each
> variable (dependent and
> > independent) from each value.  The panel mean for time is
> 2.5 in every panel.
> > This means the the FE transformation just subtracts a
> constant value from
> > -time-.  Subtracting a constant from a regressor does not
> have any effect on
> > its estimated coefficient.  
> > 
> > But wait, we also subtracted the panel means from the
> dependent variable y and
> > those means were not the same for each panel.  As it turns
> out, when panels
> > are balanced, the FE transformation of any variable produces
> a variable that
> > has a regression coefficient of exactly 1 when regressed
> against the
> > untransformed variable.  Thus, the relationship with a
> variable that has not
> > been transformed (like -time-, that had only a constant
> subtracted) remains
> > exactly the same.
> > 
> > So, with only a single independent variable that repeats
> exactly in each
> > balanced panel, OLS and fixed-effects regression will
> produce the same
> > estimate of the coefficient on the regressor (within machine
> tolerance of the
> > different computations performed).
> > 
> > Side-note:  While I was aware of the behaviour of variables
> that repeat within
> > panel for balanced panels, I hadn't previously considered
> why the FE
> > transformation of the dependent variable has no effect.  A
> little scribbling
> > on the white board from Bobby Gutierrez
> <rgutierrez@stata.com> shows that when
> > the FE transformation is expressed in matrix form it is
> idempotent for balanced
> > panels.  That causes the transformation to essentially fall
> out of regression
> > of y on y-transformed leaving a coefficient of 1.
> > 
> > What about the random-effects (RE) estimator?  The GLS
> random-effects
> > estimator is just a matrix-weighted combination of the FE
> estimator and the
> > between-effects (BE) estimator.  The BE estimator is a
> regression of the
> > panel-level mean of each variable (again, dependent and
> independent).  As we
> > saw above, the panel-level mean for -time- is a constant 2.5
> in every panel
> > and thus is collinear with the constant.  This means that
> the between
> > estimator cannot estimate B_time and provides no additional
> information for
> > this coefficient.  It has no contribution to the RE
> estimator.  So, the RE
> > estimator must be identical to the FE estimator in a model
> with a single
> > covariate that repeats exactly within each balanced panel.
> > 

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