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Re: st: Cross-Sectional Time Series


From   anirban basu <abasu@midway.uchicago.edu>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Cross-Sectional Time Series
Date   Wed, 26 Jun 2002 16:45:30 -0500 (CDT)

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.
> 
> 
> -- Vince
>    vwiggins@stata.com
> 
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