Want coefficient estimates of time-invariant covariates in your panel-data model? Fit a random-effects model. Want to allow for correlation between covariates and unobserved panel-level effects? Fit a fixed-effects model. Want both? With xtreg, cre, you can now fit a correlated random-effects model.

Let's see it work

We study the effect on wages of time-varying variables, such as age or tenure. At the same time, we are interested in the effects of time-invariant variables, such as race. An FE model will omit any variable that remains constant across time and thus cannot fully answer our research question. An RE model may yield inconsistent estimates because of the possible correlation between individual time-invariant heterogeneity and the regressors age and tenure.

We can use a CRE model to circumvent both problems. Let's see it in action.

. webuse nlswork
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. xtreg ln_wage tenure age i.race, cre vce(cluster idcode)
note: 2.race omitted from xt_means because of collinearity.
note: 3.race omitted from xt_means because of collinearity.

Correlated random-effects regression             Number of obs     =     28,101
Group variable: idcode                           Number of groups  =      4,699

R-squared:                                       Obs per group:
     Within  = 0.1296                                          min =          1
     Between = 0.2346                                          avg =        6.0
     Overall = 0.1890                                          max =         15

                                                 Wald chi2(4)      =    1685.18
corr(xit_vars*b, xt_means*γ) = 0.5474            Prob > chi2       =     0.0000

                              (Std. err. adjusted for 4,699 clusters in idcode)


             
 
               Robust                                           
     ln_wage 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]
 
 
 
xit_vars      
 
                                                                
      tenure 
 
   .0211313   .0012113    17.44   0.000     .0187572    .0235055
         age 
 
   .0121949   .0007414    16.45   0.000     .0107417     .013648
             
 
                                                                
        race 
 
                                                                
      Black  
 
  -.1312068   .0117856   -11.13   0.000    -.1543061   -.1081075
      Other  
 
   .1059379   .0593177     1.79   0.074    -.0103225    .2221984
             
 
                                                                
       _cons 
 
     1.2159   .0306965    39.61   0.000     1.155736    1.276064
 
 
 
xt_means     
 
                                                                
      tenure 
 
   .0376991    .002281    16.53   0.000     .0332283    .0421698
         age 
 
  -.0011984   .0013313    -0.90   0.368    -.0038077    .0014109
             
 
                                                                
        race 
 
                                                                
      Black  
 
          0  (omitted)                                          
      Other  
 
          0  (omitted)                                          
 
 
 
     sigma_u 
 
  .33334407                                                     
     sigma_e 
 
  .29808194                                                     
         rho 
 
  .55567161   (fraction of variance due to u_i)                 

Mundlak test (xt_means = 0): chi2(2) = 331.5144             Prob > chi2 = 0.0000

xtreg, cre reports coefficients for the variables in the model (xit_vars) and for their respective panel means (xt_means). In CRE models, panel means are added to the regression to control for potential endogeneity and to correct bias. This procedure gives us the same coefficients we obtain from the corresponding FE model for the time-varying regressors:

. xtreg ln_wage tenure age, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =     28,101
Group variable: idcode                          Number of groups  =      4,699

R-squared:                                      Obs per group:
     Within  = 0.1296                                         min =          1
     Between = 0.1916                                         avg =        6.0
     Overall = 0.1456                                         max =         15

                                                F(2, 4698)        =     766.79
corr(u_i, Xb) = 0.1302                          Prob > F          =     0.0000

                             (Std. err. adjusted for 4,699 clusters in idcode)


             
 
               Robust                            Equal-tailed   
     ln_wage 
 
 Coefficient  std. err.      t    P>|t|     [95% conf. interval]
 
 
 
      tenure 
 
   .0211313   .0012112    17.45   0.000     .0187568    .0235059
         age 
 
   .0121949   .0007414    16.45   0.000     .0107414    .0136483
       _cons 
 
   1.256467   .0194187    64.70   0.000     1.218397    1.294537
 
 
 
     sigma_u 
 
  .39034493                                                     
     sigma_e 
 
  .29808194                                                     
         rho 
 
  .63165531   (fraction of variance due to u_i)                 

We get the benefits of an FE model but do not lose information about time-invariant features of our model.

xtreg, cre performs a Mundlak specification test to help you choose between an RE and FE or CRE model. Unlike a Hausman test, this test is fully robust and remains valid even when a robust vce(), such as vce(cluster idcode), is specified. From the earlier output of xtreg, cre, the Mundlak test, reported beneath the coefficient table, provides strong evidence in favor of the CRE model.