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st: Fixed effects model with time dummies and other variations


From   "Pi, Ron" <Ron.Pi@jud.ca.gov>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Fixed effects model with time dummies and other variations
Date   Tue, 10 Aug 2004 11:10:29 -0700

Dear all,

I have a panel data set consisting of 58 counties in California and 19 years from 1980 to 1998. I am interested in using the various xt commands to estimate the impact of population growth on the level of auto tort filings in the courts, with automobile accidents (measured as # of persons injured) and other variables as controls. My question is which one of the many models provides the most reliable (or reasonable) estimate. Shown at the bottom are results from 4 models: (1) fixed effects (xtreg, fe); (2) fixed effects with AR(1) (xtregar, fe); (3) xtgls with AR1 and heteroskedastic; and lastly as an experiment, (4) fixed effects with time dummies.

The coef. for population in the first model is .13; it dropped to .04 (along with a significant drop in t-score) in the xtregar model. In the xtgls model the coef. increased to .49, which is quite close to the result from the last model with fixed effects and time dummies (.45). 

I'm inclined to accept the results from the xtgls estimate and xtreg with time dummies as more reasonable in capturing the underlying relationship between population growth and tort filings. As a background, over the 19-year time frame auto tort filings in California rose initially for a few years and have declined since late 1980s, whereas population growth has shown a steady upward trend throughout the entire period. It appears a fixed- effects model (either xtreg or xtregar) would be inappropriate for fitting a linear population trend to a curvilinear filings trend. Theoretically I consider the xtgls model and xtreg fixed effects with time dummies as kind of veering away from the time-series within estimate and thus the results less subject to the year-to-year fluctuations. Am I correct in my interpretation? Any comments will be greatly appreciated.

Thanks,
Ron
------------------------------------------------------------------------------------
Ron Pi
Senior Research Analyst
Office of Court Research
Judicial Council of California - Administrative Office of the Courts
455 Golden Gate Avenue
San Francisco, CA 94102-3688
415-865-7652, Fax 415-865-4330, ron.pi@jud.ca.gov 

www.courtinfo.ca.gov <http://www.courtinfo.ca.gov/ 

"Serving the courts for the benefit of all Californians"
------------------------------------------------------------------------------------


(1)
. xtreg lg_auto lg_pop lg_persinj, fe

Fixed-effects (within) regression               Number of obs      =      1102
Group variable (i): cnty_id                     Number of groups   =        58

R-sq:  within  = 0.1669                         Obs per group: min =        19
       between = 0.9775                                        avg =      19.0
       overall = 0.9537                                        max =        19

                                                F(2,1042)          =    104.36
corr(u_i, Xb)  = 0.2423                         Prob > F           =    0.0000

------------------------------------------------------------------------------
     lg_auto |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lg_pop |   .1301814   .0691205     1.88   0.060    -.0054498    .2658126
  lg_persinj |   1.070011   .0780321    13.71   0.000     .9168932    1.223129
       _cons |  -4.280112   .9003019    -4.75   0.000    -6.046724   -2.513501
-------------+----------------------------------------------------------------
     sigma_u |  .31447197
     sigma_e |  .32886923
         rho |  .47763231   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(57, 1042) =    15.86            Prob > F = 0.0000

(2)
. xtregar lg_auto lg_pop lg_persinj, fe

FE (within) regression with AR(1) disturbances  Number of obs      =      1044
Group variable (i): cnty_id                     Number of groups   =        58

R-sq:  within  = 0.0607                         Obs per group: min =        18
       between = 0.9771                                        avg =      18.0
       overall = 0.9537                                        max =        18

                                                F(2,984)           =     31.81
corr(u_i, Xb)  = 0.9297                         Prob > F           =    0.0000

------------------------------------------------------------------------------
     lg_auto |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lg_pop |   .0422914   .1162172     0.36   0.716    -.1857707    .2703535
  lg_persinj |   .7162647   .0906067     7.91   0.000     .5384602    .8940692
       _cons |  -.6454783   .8010065    -0.81   0.421    -2.217356    .9263991
-------------+----------------------------------------------------------------
      rho_ar |   .4537742
     sigma_u |  .85230185
     sigma_e |  .30011025
     rho_fov |  .88969031   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(57,984) =     5.74              Prob > F = 0.0000

(3)
. xtgls lg_auto lg_pop lg_persinj, pan(correlate hetero) corr(ar1)

Cross-sectional time-series FGLS regression

Coefficients:  generalized least squares
Panels:        heteroskedastic with cross-sectional correlation
Correlation:   common AR(1) coefficient for all panels  (0.6745)

Estimated covariances      =      1711          Number of obs      =      1102
Estimated autocorrelations =         1          Number of groups   =        58
Estimated coefficients     =         3          Time periods       =        19
                                                Wald chi2(2)       =  26360.56
Log likelihood             =         .          Prob > chi2        =    0.0000

------------------------------------------------------------------------------
     lg_auto |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lg_pop |   .4957223   .0509799     9.72   0.000     .3958035     .595641
  lg_persinj |    .707074   .0547812    12.91   0.000     .5997049    .8144432
       _cons |  -5.911482   .2030015   -29.12   0.000    -6.309357   -5.513606
------------------------------------------------------------------------------

(4)
. xi: xtreg lg_auto lg_pop lg_persinj i.year, fe
i.year            _Iyear_1980-2000    (naturally coded; _Iyear_1980 omitted)

Fixed-effects (within) regression               Number of obs      =      1102
Group variable (i): cnty_id                     Number of groups   =        58

R-sq:  within  = 0.2870                         Obs per group: min =        19
       between = 0.9783                                        avg =      19.0
       overall = 0.9578                                        max =        19

                                                F(20,1024)         =     20.61
corr(u_i, Xb)  = 0.6570                         Prob > F           =    0.0000

------------------------------------------------------------------------------
     lg_auto |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lg_pop |   .4521733   .1189174     3.80   0.000     .2188237    .6855228
  lg_persinj |   .5979279   .0857454     6.97   0.000     .4296712    .7661847
 _Iyear_1981 |  -.0087351   .0570119    -0.15   0.878    -.1206086    .1031384
 _Iyear_1982 |  -.0120469   .0576697    -0.21   0.835    -.1252111    .1011173
 _Iyear_1983 |  -.0932685   .0574431    -1.62   0.105    -.2059881     .019451
 _Iyear_1984 |   .0697582   .0576145     1.21   0.226    -.0432977    .1828141

...additional years deleted here...

       _cons |  -4.653511   1.289873    -3.61   0.000    -7.184607   -2.122415
-------------+----------------------------------------------------------------
     sigma_u |  .39874337
     sigma_e |  .30689493
         rho |  .62799507   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(57, 1024) =    18.02            Prob > F = 0.0000


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