# Random Effects

The random effects estimator treats the individual differences as being randomly assigned to the individuals. Rather than estimate them as parameters as we did in the fixed effects model, here they are incorporated into the model’s error, which in a panel will have a specific structure. The ви term in equation 15.3 is modeled:

віі = ві + Ui (15.5)

where the Ui are random individual differences that are the same in each time period.

Vit = ві + в2Х2 it + взхзи + (eit + Ui)
= ві + в2Х2 it + взХз it + Vit

where the combined error is

vit — ui + eit

the key property of the new error term is that it is homoskedastic

o — var (vit) — var (ui + eit) — oU + ol

and serially correlated. For individual i, that covariance among his errors is

cov (Vit, Vis) — oU

for i — s. The covariance between any two individuals is zero. One of the key advantages of the random effects model is that parameters on time invariant regressors can be estimated. That means that coefficients on black and educ can be estimated. Not so with fixed effects.

The parameter estimates are actually obtained through feasible generalized least squares. Equa­tion 15.8 contains two parameters that describe the variances and covariances in the model. These are estimated and used to perform FGLS. The process is described in some detail in POE4 and will not be discussed in much detail here. However, when gretl estimates the model as specified, it refers to the results as ‘GLS’.

The transformation that is used on the variables of the model is sometimes referred to as quasi-demeaning. It is based on the computation of

This parameter 9 is estimated from the data and the transformation are

Vit — Vit – 9vi, xit — 1 – 9, x*2it — xnt – 9хц, xit — X3it – 9хзі (15.10)

The bars over the variables indicate means for the ith individual taken over the available time periods. Gretl estimates 9 and the variances. In the wage equation the estimate of o^, oU and 9 are, respectively:

’Within’ variance = 0.0380681 ‘Between’ variance = 0.115887 theta used for quasi-demeaning = 0.743683

These match the ones in POE4 exactly.

If the random individual effects are correlated with regressors, then the random effects estimator will not be consistent. A statistical test of this proposition should be done whenever this estimator is used in order to reduce the chance of model misspecification.

To estimate the parameters of this model in gretl is easy. Simply specify the model you want to estimate and choose the random effects option.

1 open "@gretldirdatapoenls_panel. gdt"

2 setobs id year —panel-vars

3 list x1 = educ exper exper2 tenure tenure2 union black south

4 panel lwage x1 —random-effects