Stata for FE/RE models

[River Huang]

Dear All, Consider the following standard estimation of FE (fixed-effect) and RE (random-effect) model in (1) and (2):

webuse grunfeld, clear
xtset company year
// (1) fixed-effect
xtreg invest mvalue kstock i.year, fe robust
outreg2 using fere, word dec(4) ctitle(fe) replace
// (2) random-effect
xtreg invest mvalue kstock i.year, robust
outreg2 using fere, word dec(4) ctitle(re) append
// (3) fixed-effect + random-effect?
xtreg invest mvalue kstock i.company i.year, robust
outreg2 using fere, word dec(4) ctitle(fe+re) append

However, I have seen several papers using (3) to estimate the FE model (compared to (1), same estimates but different standard errors). What would be the problem by doing so? Thanks.

[Andrew Musau]

// (3) fixed-effect + random-effect? xtreg invest mvalue kstock i.company i.year, robust outreg2 using fere, word dec(4) ctitle(fe+re) append

#3 is a FE model (there are no random effects, even though technically, you are estimating using the RE estimator). I personally don't think that there is an issue with either estimation #1 or #3. #1 is referred to as the within-estimator whereas #3 is referred to as least squares dummy variables (LSDV). The (slight) differences in the standard errors reflect the estimation method. You can get the same standard errors by adding the undocumented -dfadj- option to xtreg, fe.

webuse grunfeld, clear
xtreg invest mvalue kstock i.year, fe dfadj
xtreg invest mvalue kstock i.company i.year

Res.:

. xtreg invest mvalue kstock i.year, fe dfadj

Fixed-effects (within) regression               Number of obs      =       200
Group variable: company                         Number of groups   =        10

R-sq:  within  = 0.7985                         Obs per group: min =        20
       between = 0.8143                                        avg =      20.0
       overall = 0.8068                                        max =        20

                                                F(21,169)          =     31.90
corr(u_i, Xb)  = -0.3250                        Prob > F           =    0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .1177158   .0137513     8.56   0.000     .0905694    .1448623
      kstock |   .3579163    .022719    15.75   0.000     .3130667    .4027659
             |
        year |
       1936  |  -19.19741   23.67586    -0.81   0.419    -65.93593    27.54112
       1937  |  -40.69001   24.69541    -1.65   0.101    -89.44122    8.061213
       1938  |   -39.2264   23.23594    -1.69   0.093    -85.09647    6.643667
       1939  |  -69.47029   23.65607    -2.94   0.004    -116.1698   -22.77083
       1940  |  -44.23507   23.80979    -1.86   0.065      -91.238     2.76785
       1941  |  -18.80446     23.694    -0.79   0.429     -65.5788    27.96987
       1942  |  -21.13979   23.38163    -0.90   0.367    -67.29748    25.01789
       1943  |  -42.97762   23.55287    -1.82   0.070    -89.47334    3.518104
       1944  |  -43.09876    23.6102    -1.83   0.070    -89.70766    3.510134
       1945  |  -55.68303   23.89561    -2.33   0.021    -102.8554   -8.510689
       1946  |  -31.16928   24.11598    -1.29   0.198    -78.77665    16.43809
       1947  |  -39.39223   23.78368    -1.66   0.100    -86.34361    7.559141
       1948  |  -43.71651   23.96965    -1.82   0.070    -91.03501    3.601991
       1949  |   -73.4951   24.18292    -3.04   0.003    -121.2346   -25.75559
       1950  |  -75.89611   24.34553    -3.12   0.002    -123.9566    -27.8356
       1951  |   -62.4809   24.86425    -2.51   0.013    -111.5654   -13.39637
       1952  |  -64.63233    25.3495    -2.55   0.012    -114.6748   -14.58987
       1953  |  -67.71796   26.61108    -2.54   0.012    -120.2509   -15.18501
       1954  |  -93.52622   27.10786    -3.45   0.001    -147.0399   -40.01257
             |
       _cons |  -32.83631   18.87533    -1.74   0.084     -70.0981    4.425483
-------------+----------------------------------------------------------------
     sigma_u |  91.798268
     sigma_e |  51.724523
         rho |  .75902159   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 169) = 52.36                     Prob > F = 0.0000

. 
. xtreg invest mvalue kstock i.company i.year

Random-effects GLS regression                   Number of obs     =        200
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.7985                                         min =         20
     between = 1.0000                                         avg =       20.0
     overall = 0.9517                                         max =         20

                                                Wald chi2(30)     =    3329.49
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .1177158   .0137513     8.56   0.000     .0907638    .1446679
      kstock |   .3579163    .022719    15.75   0.000     .3133878    .4024447
             |
     company |
          2  |   207.0542   35.17275     5.89   0.000     138.1169    275.9915
          3  |  -135.2308   35.70897    -3.79   0.000    -205.2191   -65.24252
          4  |    95.3538   50.72211     1.88   0.060     -4.05971    194.7673
          5  |  -5.438636   57.83052    -0.09   0.925    -118.7844    107.9071
          6  |   102.8886   54.17388     1.90   0.058    -3.290244    209.0674
          7  |   51.46657   58.17922     0.88   0.376     -62.5626    165.4957
          8  |   67.49048   50.97092     1.32   0.185     -32.4107    167.3917
          9  |   30.21752   55.72307     0.54   0.588    -78.99769    139.4327
         10  |   126.8371   58.52545     2.17   0.030     12.12931    241.5448
             |
        year |
       1936  |  -19.19741   23.67586    -0.81   0.417    -65.60124    27.20643
       1937  |  -40.69001   24.69541    -1.65   0.099    -89.09212    7.712108
       1938  |   -39.2264   23.23594    -1.69   0.091      -84.768    6.315193
       1939  |  -69.47029   23.65607    -2.94   0.003    -115.8353   -23.10524
       1940  |  -44.23507   23.80979    -1.86   0.063    -90.90141    2.431264
       1941  |  -18.80446     23.694    -0.79   0.427    -65.24385    27.63492
       1942  |  -21.13979   23.38163    -0.90   0.366    -66.96694    24.68736
       1943  |  -42.97762   23.55287    -1.82   0.068    -89.14038     3.18515
       1944  |  -43.09876    23.6102    -1.83   0.068     -89.3739    3.176369
       1945  |  -55.68303   23.89561    -2.33   0.020    -102.5176   -8.848489
       1946  |  -31.16928   24.11598    -1.29   0.196    -78.43574    16.09718
       1947  |  -39.39223   23.78368    -1.66   0.098    -86.00739    7.222923
       1948  |  -43.71651   23.96965    -1.82   0.068    -90.69617    3.263145
       1949  |   -73.4951   24.18292    -3.04   0.002    -120.8927   -26.09745
       1950  |  -75.89611   24.34553    -3.12   0.002    -123.6125   -28.17976
       1951  |   -62.4809   24.86425    -2.51   0.012    -111.2139   -13.74786
       1952  |  -64.63233    25.3495    -2.55   0.011    -114.3164   -14.94822
       1953  |  -67.71796   26.61108    -2.54   0.011    -119.8747   -15.56119
       1954  |  -93.52622   27.10786    -3.45   0.001    -146.6567   -40.39578
             |
       _cons |  -86.90019   56.04663    -1.55   0.121    -196.7496    22.94919
-------------+----------------------------------------------------------------
     sigma_u |          0
     sigma_e |  51.724523
         rho |          0   (fraction of variance due to u_i)
------------------------------------------------------------------------------

.

[River Huang]

Dear Andrew, Thanks a lot for this explanation and illustration.