Panel data

For panel data, the usual set up is:

$$y_{it}=\alpha +X_{it}\beta +v_i+{ϵ}_{it}$$

Fixed effect

A fixed effect model can be done with OLS on

$$(y_{it}-{\overline{y}}_i+\overline{\overline{y}})=\alpha +(X_{it}-\overline{X}+\overline{\overline{X}})\beta +({ϵ}_{it}-{\overline{ϵ}}_i+\overline{v})+\overline{\overline{ϵ}}$$ This is basically OLS on de-meaned $Y$ and $X$.

Random effect

The random effect looks at this at different angle: it treated $v_i+{ϵ}_{it}$ as the error term. There are two components of the error term. Suppose they are estimated as ${\hat{\sigma }}_e^2$ (idiosyncratic component), and ${\hat{\sigma }}_u^2$ (individual component). Then we can do GLS transformation:

$z_{it}^{\ast }=z_{it}-{\hat{\theta }}_i{\overline{z}}_i$

and ${\hat{\theta }}_i=1-\sqrt{\frac{{\hat{\sigma }}_e^2}{T_i{\hat{\sigma }}_u^2+{\hat{\sigma }}_e^2}}$

where $T_i$ is the number of observations for individual $i$.

Given estimates of ${\hat{\sigma }}_e^2$ and ${\hat{\sigma }}_u^2$, we can run OLS on transformed variables (including $y$ and all $X$’s). We can iterate the process.

Correlated random effect

Correlated random effect (CRE) can be done by running a random effect model of $y_{it}$ on $X_{it}$, ${\overline{X}}_i$, $z_i$ and a constant.

The shortcoming of RE model is that it has to assume $v_i$ and $X_{it}$ are uncorrelated to have a consistent estimator. If that is not true (most eocnomists don’t think it is), then we have an inconsistent estimator. Meanwhile, FE estimator is consistent, because $v_i$ is not in the error term, it gets cancelled out.

The disadvantage of FE model is that it cannot include any time-invariant covariate, such as race, gender, etc.

The benefit of CRE is that it can include $z_i$’s which are time-invariant, while remain consistent. In fact, Mundlak and Wooldridge pointed out that CRE estimates on $X_{it}$ are the same as FE estimates.

There is also a Mundlak test to choose between RE or CRE.

Example

Stata 19 implemented CRE in the xtreg command, with “cre” option. The following example is from Stata’s website, using the nlswork dataset.

webuse nlswork

xtreg ln_wage tenure age i.race, cre vce(cluster idcode)

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

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

To compare with a fixed effect model:

webuse nlswork

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

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

note: 2.race omitted because of collinearity.
note: 3.race omitted because of collinearity.

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
     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
             |
        race |
      Black  |          0  (omitted)
      Other  |          0  (omitted)
             |
       _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 see the coefficient estimates are the same for “tenure” and “age”, but CRE model allows you to estimate the effect of “race”.

We can also manually do it by using a RE model on $X$, $\overline{X}$ and $z$:

webuse nlswork
egen age_mean = mean(age), by(idcode)
egen tenure_mean = mean(tenure), by(idcode)
xtreg ln_wage tenure tenure_mean age age_mean i.race, vce(cluster idcode)

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

(1 missing value generated)

(12 missing values generated)


Random-effects GLS 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(6)      =    2688.49
corr(u_i, X) = 0 (assumed)                      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]
-------------+----------------------------------------------------------------
      tenure |   .0211328    .001211    17.45   0.000     .0187592    .0235064
 tenure_mean |   .0376962   .0022835    16.51   0.000     .0332207    .0421718
         age |   .0121935   .0007412    16.45   0.000     .0107409    .0136462
    age_mean |  -.0011922   .0013353    -0.89   0.372    -.0038094     .001425
             |
        race |
      Black  |  -.1312062    .011785   -11.13   0.000    -.1543044    -.108108
      Other  |   .1059734   .0593175     1.79   0.074    -.0102868    .2222337
             |
       _cons |   1.215738   .0307699    39.51   0.000      1.15543    1.276046
-------------+----------------------------------------------------------------
     sigma_u |  .33338665
     sigma_e |  .29808194
         rho |  .55573468   (fraction of variance due to u_i)
------------------------------------------------------------------------------

This is what stata’s “cre” option is doing behind the scene.