Stata FAQ
What is the relationship between xtreg-re, xtreg-fe and xtreg-be?

xtreg with its various options performs regression analysis on panel datasets. In this FAQ we will try to explain the differences between xtreg-re and xtreg-fe with an example that is taken from analysis of variance. The example (below) has 32 observations taken on eight subjects, that is, each subject is observed four times. The eight subjects are evenly divided into two groups of four.

The design is a mixed model with both within-subject and between-subject factors. The within-subject factor (b) has four levels and the between-subject factor (a) has two levels. To keep the analysis simple we will not consider the a*b interaction.

----------------------------------          
        s |   b1    b2    b3    b4
----------+-----------------------
a1        |
        1 |    3     4     7     7
        2 |    6     5     8     8
        3 |    3     4     7     9
        4 |    3     3     6     8
----------+-----------------------
a2        |
        5 |    1     2     5    10
        6 |    2     3     6    10
        7 |    2     4     5     9
        8 |    2     3     6    11
----------------------------------

We will begin by looking at the within-subject factor using xtreg-fe. The fe option stands for fixed-effects which is really the same thing as within-subjects. Notice that there are coefficients only for the within-subjects (fixed-effects) variables. Following the xtreg we will use the test command to obtain the three degree of freedom test of the levels of b.

use http://www.ats.ucla.edu/stat/stata/faq/spf24

 xi: xtreg y i.a i.b, i(s) fe
 
i.a               _Ia_1-2             (naturally coded; _Ia_1 omitted)
i.b               _Ib_1-4             (naturally coded; _Ib_1 omitted)

Fixed-effects (within) regression               Number of obs      =        32
Group variable (i): s                           Number of groups   =         8

R-sq:  within  = 0.8722                         Obs per group: min =         4
       between =      .                                        avg =       4.0
       overall = 0.8259                                        max =         4

                                                F(3,21)            =     47.77
corr(u_i, Xb)  = 0.0000                         Prob > F           =    0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _Ia_2 |  (dropped)
       _Ib_2 |        .75   .5824824     1.29   0.212    -.4613384    1.961338
       _Ib_3 |        3.5   .5824824     6.01   0.000     2.288662    4.711338
       _Ib_4 |       6.25   .5824824    10.73   0.000     5.038662    7.461338
       _cons |       2.75   .4118772     6.68   0.000     1.893454    3.606546
-------------+----------------------------------------------------------------
     sigma_u |   .6681531
     sigma_e |  1.1649647
         rho |  .24752475   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(7, 21) =     0.99               Prob > F = 0.4669

test _Ib_2 _Ib_3 _Ib_4

 ( 1)  _Ib_2 = 0
 ( 2)  _Ib_3 = 0
 ( 3)  _Ib_4 = 0

       F(  3,    21) =   47.77
            Prob > F =    0.0000

Next, we will use the be option to look at the between-subject effect. This time notice that only the coefficient for a is given as it represents the between-subjects effect.

xi: xtreg y i.a i.b, i(s) be

i.a               _Ia_1-2             (naturally coded; _Ia_1 omitted)
i.b               _Ib_1-4             (naturally coded; _Ib_1 omitted)

Between regression (regression on group means)  Number of obs      =        32
Group variable (i): s                           Number of groups   =         8

R-sq:  within  =      .                         Obs per group: min =         4
       between = 0.2500                                        avg =       4.0
       overall = 0.0133                                        max =         4

                                                F(1,6)             =      2.00
sd(u_i + avg(e_i.))=      .625                  Prob > F           =    0.2070

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _Ia_2 |      -.625   .4419417    -1.41   0.207    -1.706392    .4563925
       _Ib_2 |  (dropped)
       _Ib_3 |  (dropped)
       _Ib_4 |  (dropped)
       _cons |     5.6875      .3125    18.20   0.000      4.92284     6.45216
------------------------------------------------------------------------------

test _Ia_2

 ( 1)  _Ia_2 = 0

       F(  1,     6) =    2.00
            Prob > F =    0.2070

Now it is time to get both the within and between with a single xtreg-re command. Notice that there are now estimates for both a and b. Since the xtreg-re test command gives us a chi-square and not an F-ratio, we have to rescale the chi-square by dividing by the degrees of freedom. The coefficients and test for the re model are the same as the coefficients and test from the separate fe and be models.

xi: xtreg y i.a i.b, i(s) re

i.a               _Ia_1-2             (naturally coded; _Ia_1 omitted)
i.b               _Ib_1-4             (naturally coded; _Ib_1 omitted)

Random-effects GLS regression                   Number of obs      =        32
Group variable (i): s                           Number of groups   =         8

R-sq:  within  = 0.8722                         Obs per group: min =         4
       between = 0.2500                                        avg =       4.0
       overall = 0.8392                                        max =         4

Random effects u_i ~ Gaussian                   Wald chi2(4)       =    145.32
corr(u_i, X)       = 0 (assumed)                Prob > chi2        =    0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _Ia_2 |      -.625   .4419417    -1.41   0.157     -1.49119    .2411899
       _Ib_2 |        .75   .5824824     1.29   0.198    -.3916445    1.891644
       _Ib_3 |        3.5   .5824824     6.01   0.000     2.358356    4.641644
       _Ib_4 |       6.25   .5824824    10.73   0.000     5.108356    7.391644
       _cons |     3.0625    .474224     6.46   0.000     2.133038    3.991962
-------------+----------------------------------------------------------------
     sigma_u |  .22658174
     sigma_e |  1.1649647
         rho |  .03645008   (fraction of variance due to u_i)
------------------------------------------------------------------------------

test _Ia_2

 ( 1)  _Ia_2 = 0

           chi2(  1) =    2.00
         Prob > chi2 =    0.1573

test _Ib_2 _Ib_3 _Ib_4

 ( 1)  _Ib_2 = 0
 ( 2)  _Ib_3 = 0
 ( 3)  _Ib_4 = 0

           chi2(  3) =  143.32
         Prob > chi2 =    0.0000
  
/* convert chi-square to F */

display "F = " r(chi2)/r(df)

F = 47.77193

Stata's xtreg random effects model is just a matrix weighted average of the fixed-effects (within) and the between-effects. In our example, because the within- and between-effects are orthogonal, thus the re produces the same results as the individual fe and be. With more general panel datasets the results of the fe and be won't necessarily add up in the same manner.