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Re: st: Cross-Sectional Time Series


From   anirban basu <abasu@midway.uchicago.edu>
To   Mark Schaffer <M.E.Schaffer@hw.ac.uk>
Subject   Re: st: Cross-Sectional Time Series
Date   Tue, 25 Jun 2002 17:36:48 -0500 (CDT)


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. 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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