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Re: st: Cross-Sectional Time Series


From   John Neumann <neumannj@bu.edu>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Cross-Sectional Time Series
Date   Thu, 27 Jun 2002 12:29:33 -0400

Since I started the trouble :), I must enthusiastically
make it a trio!  Thanks all!

John

Mark Schaffer wrote:

> 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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