Fixed effects vs Correlated Random Effects coefficients

[Clara Martinez Toledano]

I am running a fixed effects regression and a correlated random effects regression (following Mundlak, 1978). The theory says that the time-varying coefficients of both regressions should be the same across the two specifications (in my case, difftax dummysigned and specvd), but I get slightly different coefficients. Do you know why this can be the case? Thank you in advance. See my output attached.

[Carlo Lazzaro]

Clara:
you're dealing with a bit unbalanced panel dataset and this nuisance explains the slight differences you experienced in the time-vaying coefficients that are common to both your specifications.
That said, given the coefficients of the panel-specific means for the time-invariant coefficients reported in the hybrid model outcome table (that you should -test-, though) , I'm under the impression that -fe- is not the way to go and -re- should be preferred.

[Clara Martinez Toledano]

Many thanks, Carlo. This was helpful.

[Enrique Pinzon (StataCorp)]

Dear Clara,

When comparing correlated random effects and fixed effects coefficients, it is important to construct the panel-level means using the same estimation sample that -xtreg, fe- uses. I would suggest you do that and report back. Below is an example with an unbalanced and with one of the covariates having missing values in the estimation sample. You get exactly the same slope coefficient estimates for both fitted models.

Here is the code

webuse nlswork, clear 
// Panel is unbalanced
xtset idcode year
// Fit fixed effects model with some observations missing in wks_ue
xtreg ln_wage c.wks_ue##i.south, fe 
// Store my estimates to display using -etable- later
estimates store fe
// Getting the estimation sample for panel means
generate sample = e(sample)
// Getting a variable list to do Mundlak
fvexpand c.wks_ue##i.south if sample 
local varlist "`r(varlist)'"
//Generating Mundlak regressors
local k: list sizeof varlist
forvalues i=1/`k' {
    local x: word `i' of `varlist'
    quietly bysort idcode: egen double mx`i' = mean(`x') if sample
}
// Fitting correlated-random-effects model
xtreg ln_wage `varlist' mx*
estimates store mundlak
etable, estimates(fe mundlak) cstat(_r_b, nformat(%9.6f))

here is the output

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

. // Panel is unbalanced
. xtset idcode year

Panel variable: idcode (unbalanced)
 Time variable: year, 68 to 88, but with gaps
         Delta: 1 unit

. // Fit fixed effects model with some observations missing in wks_ue
. xtreg ln_wage c.wks_ue##i.south, fe 

Fixed-effects (within) regression               Number of obs     =     22,823
Group variable: idcode                          Number of groups  =      4,645

R-squared:                                      Obs per group:
     Within  = 0.0029                                         min =          1
     Between = 0.0593                                         avg =        4.9
     Overall = 0.0504                                         max =         14

                                                F(3,18175)        =      17.48
corr(u_i, Xb) = 0.1963                          Prob > F          =     0.0000

--------------------------------------------------------------------------------
       ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
---------------+----------------------------------------------------------------
        wks_ue |  -.0024934   .0004682    -5.33   0.000    -.0034111   -.0015756
       1.south |  -.0665757   .0139186    -4.78   0.000    -.0938575   -.0392939
               |
south#c.wks_ue |
            1  |   .0014072   .0007115     1.98   0.048     .0000126    .0028017
               |
         _cons |    1.66299   .0061003   272.61   0.000     1.651032    1.674947
---------------+----------------------------------------------------------------
       sigma_u |  .42311267
       sigma_e |  .32369196
           rho |  .63080977   (fraction of variance due to u_i)
--------------------------------------------------------------------------------
F test that all u_i=0: F(4644, 18175) = 5.88                 Prob > F = 0.0000

. // Store my estimates to display using -etable- later
. estimates store fe

. // Getting the estimation sample for panel means
. generate sample = e(sample)

. // Getting a variable list to do Mundlak
. fvexpand c.wks_ue##i.south if sample 

. local varlist "`r(varlist)'"

. //Generating Mundlak regressors
. local k: list sizeof varlist

. forvalues i=1/`k' {
  2.         local x: word `i' of `varlist'
  3.         quietly bysort idcode: egen double mx`i' = mean(`x') if sample
  4. }

. // Fitting correlated-random-effects model
. xtreg ln_wage `varlist' mx*
note: mx2 omitted because of collinearity.
note: mx4 omitted because of collinearity.

Random-effects GLS regression                   Number of obs     =     22,823
Group variable: idcode                          Number of groups  =      4,645

R-squared:                                      Obs per group:
     Within  = 0.0029                                         min =          1
     Between = 0.0719                                         avg =        4.9
     Overall = 0.0702                                         max =         14

                                                Wald chi2(6)      =     437.99
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

--------------------------------------------------------------------------------
       ln_wage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
---------------+----------------------------------------------------------------
        wks_ue |  -.0024934   .0004656    -5.35   0.000     -.003406   -.0015808
       1.south |  -.0665757    .013841    -4.81   0.000    -.0937037   -.0394478
               |
south#c.wks_ue |
            1  |   .0014072   .0007075     1.99   0.047     .0000205    .0027938
               |
           mx1 |  -.0122406   .0016735    -7.31   0.000    -.0155206   -.0089607
           mx2 |          0  (omitted)
           mx3 |  -.1121623   .0199403    -5.62   0.000    -.1512446   -.0730799
           mx4 |          0  (omitted)
           mx5 |  -.0037978   .0025897    -1.47   0.143    -.0088735    .0012779
         _cons |   1.746899    .008868   196.99   0.000     1.729518     1.76428
---------------+----------------------------------------------------------------
       sigma_u |  .36636858
       sigma_e |  .32369196
           rho |   .5616091   (fraction of variance due to u_i)
--------------------------------------------------------------------------------

. estimates store mundlak

. etable, estimates(fe mundlak) cstat(_r_b, nformat(%9.6f))

-----------------------------------------------------------
                                         ln_wage   ln_wage 
-----------------------------------------------------------
Weeks unemployed last year              -0.002493 -0.002493
1 if south                                                 
  1                                     -0.066576 -0.066576
1 if south # Weeks unemployed last year                    
  1                                      0.001407  0.001407
mx1                                               -0.012241
mx3                                               -0.112162
mx5                                               -0.003798
Intercept                                1.662990  1.746899
Number of observations                      22823     22823
-----------------------------------------------------------

[Joro Kolev]

The reason for slightly different results in unbalanced panels is what Enrique explained: If one wants to get identical results, the panel averages have to be calculated over the same sample the regression is run. Eric de Souza taught me this no so long ago on this thread here
https://www.statalist.org/forums/for...an-test-result

You might find useful this thread too
https://www.statalist.org/forums/for...dlak-procedure

[John Moran]

Speaking further about fixed and correlated random effects models, the paper by Schunk & Perales (SJ 17) "Within- and between-cluster effects in generalized linear mixed models: A discussion of approaches and the xthybrid command" gives a more than useful discussion of the whole scenario (IMHO)

john moran