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Re: st: Cross-Sectional Time Series


From   Mark Schaffer <M.E.Schaffer@hw.ac.uk>
To   statalist@hsphsun2.harvard.edu, anirban basu <abasu@midway.uchicago.edu>
Subject   Re: st: Cross-Sectional Time Series
Date   Wed, 26 Jun 2002 09:56:52 +0100 (BST)

Anirban,

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

> 
> 
> On Tue, 25 Jun 2002, Mark Schaffer wrote:
> 
> > 
> > Not quite sure what you mean, so apologies if I'm off
> target.  The 
> > coefficient estimates with -regress- won't be the same as
> with
> > -xtreg, fe- (unless the former is estimating the same model
> by 
> > explicitly including the fixed effects as dummy vars).  Both
> sets of 
> > coefficients will be different again from those produced by
> > -xtreg, re-.
> > 
> > --Mark
> 
> 
> You may be right about this. However, I did get the same coeff
> from
> -regress-, xtreg, fe and xtreg, re using a simulated dataset
> with
> exchangeable corr and no dummy vars.

I reproduced your example.  The coefficients are virtually, but not 
exactly, the same - they differ after about 12 decimal places.
I think what's happening is that because of the way you set up your 
artificial dataset, all 3 estimators are giving you unbiased and 
consistent estimates of an assumed linear relationship, and all 3 get 
very close.  (The true relationship isn't linear but that doesn't 
matter here.)

--Mark


> ALso, I got different
> coeffs by
> running an exponential corr model as expected.. Here are the
> details. I
> may have missed something here..Thanks for your input,
> 
> Anirban
> 
> 
> . 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)
> (obs 1000)
> 
> . gen id=_n
> 
> . reshape long y , i(id) j(time)
> (note: j = 1 2 3 4)
> 
> Data                               wide   ->   long
> --------------------------------------------------------------------
---------
> Number of obs.                     1000   ->    4000
> Number of variables                   5   ->       3
> j variable (4 values)                     ->   time
> xij variables:
>                            y1 y2 ... y4   ->   y
> --------------------------------------------------------------------
---------
> 
> . 
> . reg y time, cluster(id)
> 
> Regression with robust standard errors                 Number
> of obs =4000
>                                                        F( 
> 1,999) =42518.72
>                                                        Prob >
> F      =0.0000
>                                                       
> R-squared     =0.7897
> Number of clusters (id) = 1000                         Root
> MSE      =1.0946
> 
> --------------------------------------------------------------------
----------
>              |               Robust
>            y |      Coef.   Std. Err.      t    P>|t|     [95%
> Conf. Interval]
> -------------+------------------------------------------------------
----------
>         time |   1.896646   .0091981   206.20   0.000    
> 1.878596 1.914696
>        _cons |  -.9866896   .0358667   -27.51   0.000   
> -1.057072 -.916307
> --------------------------------------------------------------------
----------
> 
> . 
> . tsset id time
>        panel variable:  id, 1 to 1000
>         time variable:  time, 1 to 4
> . iis id
> . tis time
> 
> . xtreg y time, fe
> 
> Fixed-effects (within) regression               Number of obs 
>     =4000
> Group variable (i) : id                         Number of
> groups   =1000
> 
> R-sq:  within  = 0.9027                         Obs per group:
> min =4
>        between = 0.0000                                       
> avg =4.0
>        overall = 0.7897                                       
> max =4
> 
>                                                 F(1,2999)     
>     =27835.94
> corr(u_i, Xb)  = -0.0000                        Prob > F      
>     =0.0000
> 
> --------------------------------------------------------------------
----------
>            y |      Coef.   Std. Err.      t    P>|t|     [95%
> Conf. Interval]
> -------------+------------------------------------------------------
----------
>         time |   1.896646    .011368   166.84   0.000    
> 1.874356 1.918936
>        _cons |  -.9866896   .0311325   -31.69   0.000   
> -1.047733 -.9256464
> -------------+------------------------------------------------------
----------
>      sigma_u |  .84490704
>      sigma_e |  .80383774
>          rho |  .52489398   (fraction of variance due to u_i)
> --------------------------------------------------------------------
----------
> F test that all u_i=0:     F(999, 2999) =     4.42          
> Prob > F =0.0000
> 
> . xtreg y time, re
> 
> Random-effects GLS regression                   Number of obs 
>     =4000
> Group variable (i) : id                         Number of
> groups   =1000
> 
> R-sq:  within  = 0.9027                         Obs per group:
> min =4
>        between = 0.0000                                       
> avg =4.0
>        overall = 0.7897                                       
> max =4
> 
> Random effects u_i ~ Gaussian                   Wald chi2(1)  
>     =27835.94
> corr(u_i, X)       = 0 (assumed)                Prob > chi2   
>     =0.0000
> 
> --------------------------------------------------------------------
----------
>            y |      Coef.   Std. Err.      z    P>|z|     [95%
> Conf. Interval]
> -------------+------------------------------------------------------
----------
>         time |   1.896646    .011368   166.84   0.000    
> 1.874365 1.918927
>        _cons |  -.9866896   .0390072   -25.30   0.000   
> -1.063142 -.9102369
> -------------+------------------------------------------------------
----------
>      sigma_u |  .74318848
>      sigma_e |  .80383774
>          rho |  .46085639   (fraction of variance due to u_i)
> --------------------------------------------------------------------
----------
> 
> . prais y time
> 
> Number of gaps in sample:  999   (gap count includes panel
> changes)
> (note: computations for rho restarted at each gap)
> 
> Iteration 0:  rho = 0.0000
> Iteration 1:  rho = 0.4034
> Iteration 2:  rho = 0.4136
> Iteration 3:  rho = 0.4140
> Iteration 4:  rho = 0.4140
> Iteration 5:  rho = 0.4140
> 
> Prais-Winsten AR(1) regression -- iterated estimates
> 
>       Source |       SS       df       MS              Number
> of obs =4000
> -------------+------------------------------           F(  1, 
> 3998) =8914.61
>        Model |  8913.21448     1  8913.21448           Prob >
> F      =0.0000
>     Residual |  3997.37575  3998  .999843859          
> R-squared     =0.6904
> -------------+------------------------------           Adj
> R-squared =0.6903
>        Total |  12910.5902  3999  3.22845467           Root
> MSE      =.99992
> 
> --------------------------------------------------------------------
----------
>            y |      Coef.   Std. Err.      t    P>|t|    
> [95%Conf. Interval]
> -------------+------------------------------------------------------
----------
>         time |   1.953367   .0157109   124.33   0.000    
> 1.922565 1.984169
>        _cons |  -1.061051   .0456138   -23.26   0.000    
> -1.15048 -.9716229
> -------------+------------------------------------------------------
----------
>          rho |   .4140074
> --------------------------------------------------------------------
----------
> Durbin-Watson statistic (original)    0.925711
> Durbin-Watson statistic (transformed) 1.672284
> 
> 
> 
> 
> 
> > 
> > > 
> > > 
> > > Anirban
> > > 
> > > ______________________________________
> > > ANIRBAN BASU
> > > Doctoral Student
> > > Harris School of Public Policy Studies
> > > University of Chicago
> > > (312) 563 0907 (H)
> > >
> ________________________________________________________________
> > > 
> > > 
> > > On Tue, 25 Jun 2002, Mark Schaffer wrote:
> > > 
> > > > Hi everybody.
> > > > 
> > > > Just a couple of clarifying details on -cluster- vs.
> -xtreg-
> > > and 
> > > > Anirban's response to John.
> > > > 
> > > > The -cluster- option for -regress- doesn't really impose
> a
> > > particular 
> > > > within-cluster correlation structure on the data.  If I
> > > understand it 
> > > > correctly, what -cluster- does instead is loosen the
> usual
> > > assumption 
> > > > of independence of observations to independence of
> clusters.
> > >  The 
> > > > correlation between observations within clusters can be
> > > arbitrary.  
> > > > The way this works is basically by treating all the
> > > observations in a 
> > > > cluster as a kind of "super-observation" and then
> applying
> > > the robust 
> > > > ("sandwich") formula to these super-observations in
> order to
> > > 
> > > > calculate the standard errors of the coefficients
> produced
> > > by -
> > > > regress-.  See the manual entry for -regress-, p. 87.
> > > > 
> > > > The estimated coefficients (the betas) produced by
> -regress-
> > > are the 
> > > > same whether or not the -cluster- option is used; the
> only
> > > thing that 
> > > > is different is the standard errors.
> > > > 
> > > > With fixed effects, you _do_ impose a particular
> correlation
> > > 
> > > > structure, namely all the observations within a cluster
> > > share U(k) in 
> > > > Anirban's notation.  If you use -xtreg- with -fe- to
> > > estimate, Stata 
> > > > does not, however, use a first-difference estimator - it
> > > uses a fixed 
> > > > effects estimator.  In other words, it doesn't
> > > first-difference to 
> > > > get rid of the fixed effects, it uses the mean-deviation
> 
> > > > transformation to get rid of them.
> > > > 
> > > > Hope this helps.
> > > > 
> > > > --Mark
> > > > 
> > > > Quoting anirban basu <abasu@midway.uchicago.edu>:
> > > > 
> > > > > Hi John,
> > > > > 
> > > > > 
> > > > > With reg command and cluster option, one basically
> imposes
> > > an
> > > > > exchangeable
> > > > > correlation structure on the data. i.e assume corr
> (y(i),
> > > > > y(j)) = rho,
> > > > > where i ne j and  i,j are any two observation from the
> > > same
> > > > > cluster. Rho
> > > > > is constant for every pair of observation within a
> > > cluster.
> > > > > So, one can
> > > > > visuaize it in terms of a random effects model where :
> > > > > 
> > > > > Y(k) = Xb + U(k) + e, where k represents clusters and
> U(k)
> > > is
> > > > > a
> > > > > cluster-specific random effect that is common to all
> > > > > observation in that
> > > > > cluster. However, -reg- does not give estimates of
> this
> > > random
> > > > > effect. It
> > > > > just estimates -betas- assuming this structure.
> > > > > 
> > > > > However, this estimation is correct only if U(k) are
> > > > > uncorrelated with
> > > > > Xs. i.e. the unobserved characteristics of a cluster
> over
> > > time
> > > > > is
> > > > > uncorrelated with the X over time. If not then fixed
> > > effects
> > > > > is useful.
> > > > > 
> > > > > 
> > > > > With fixed effects, one evades the correlation problem
> by
> > > > > taking
> > > > > differences. i.e for any cluster k:
> > > > > 
> > > > > Y(ik) - Y(1k) = [X(ik) - X(1k)]b + [e(ik) - e(1k)]
> > > > > 
> > > > > Note that by taking the difference, the unobserved
> U(k) is
> > > > > eliminated.
> > > > > However, fixed effects assume the U(k) is fixed over
> time
> > > for
> > > > > any cluster
> > > > > k. i.e. the unobserved characteristics of a cluster is
> not
> > > > > changing over
> > > > > time. Also, since we are taking a difference, fixed
> > > effects
> > > > > model cannot
> > > > > estimate the betas for baseline covariates since they
> > > cancel
> > > > > out in the
> > > > > difference.
> > > > > 
> > > > > Hope this helps,
> > > > > 
> > > > > Anirban
> > > > > 
> > > > > 
> > > > > 
> > > > > ______________________________________
> > > > > ANIRBAN BASU
> > > > > Doctoral Student
> > > > > Harris School of Public Policy Studies
> > > > > University of Chicago
> > > > > (312) 563 0907 (H)
> > > > >
> > >
> ________________________________________________________________
> > > > > 
> > > > > 
> > > > > On Tue, 25 Jun 2002, John Neumann wrote:
> > > > > 
> > > > > > Hello all,
> > > > > > 
> > > > > > Since I frequently see panel data questions flying
> > > around
> > > > > the
> > > > > > list, I'm thinking that some of you can provide me
> with
> > > a
> > > > > > very succinct answer to the following question, and
> in
> > > so
> > > > > > doing clarify conceptually for me the data-related
> > > issue:
> > > > > > 
> > > > > > I have data on investment products, by year.  Not
> all
> > > > > > products have data in each year.  The dependent
> > > > > > variable is scaled in such a way as to make time
> series
> > > > > > variation in its levels of no concern.  Here's the
> > > question:
> > > > > > 
> > > > > > What is the difference between using the reg
> command,
> > > > > > with the robust and cluster option, vs. the xtreg
> > > command
> > > > > > fixed effects model?  The cluster variable using reg
> > > would
> > > > > > naturally be the i( ) parameter for xtreg ...
> > > > > > 
> > > > > > Thanks!
> > > > > > 
> > > > > > John Neumann
> > > > > > Boston University
> > 

Prof. Mark Schaffer
Director, CERT
Department of Economics, School of Management
Heriot-Watt University, Edinburgh EH14 4AS
tel +44-131-451-3494 / fax +44-131-451-3008
email: m.e.schaffer@hw.ac.uk
web: http://www.som.hw.ac.uk/ecomes
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