## The One-Way Random Effects Model

The specification for the one-way random
effects model is

Let **Z**_{0} = *diag*(**j**_{Ti}), , and
**Q**_{0} = *diag*(**E**_{Ti}), with and
.
Define
and .

The fixed effects estimator of is still unbiased
under the random effects assumptions, so you need to calculate only
the estimate of .

In the balanced data case, the estimation method
for the variance components is the fitting constants method
as applied to the one way model; refer to Baltagi and Chang (1994).
Fuller and Battese (1974) apply this method to the two-way model.

Let

The estimator of the error variance is given by

and the estimator of the cross-sectional variance component is
given by

The estimation of the one-way unbalanced data model is performed
using a specialization (Baltagi and Chang 1994) of the approach
used by Wansbeek and Kapteyn (1989)
for unbalanced two-way models.

The estimation of the variance
components is performed by using a
quadratic unbiased estimation (QUE) method. This involves focusing
on quadratic forms of the centered residuals, equating
their expected values to the realized quadratic forms, and solving
for the variance components.

Let

where the residuals are given by
if there is an intercept and by
if there is not.

Consider the expected values

and
are
obtained by equating the quadratic forms to their expected values.
The estimated generalized least squares procedure substitutes
the QUE estimates into the covariance
matrix of *u*_{it}, which is given by

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