The specification for the one-way random-effects model is
Let ),
, and
, with
and
. Define
. Also define
and
as a vector of 1s whose length is
.
In the one-way model, estimation proceeds in a two-step fashion. First, you obtain estimates of the variance of the and
. There are multiple ways to derive these estimates; PROC HPPANEL provides four options. For more information, see the section
One-Way Random-Effects Model in SAS/ETS 13.2 User's Guide.
After the variance components are calculated from any method, the next task is to estimate the regression model of interest.
For each individual, you form a weight (),
where is the
th cross section’s time observations.
Taking the , you form the partial deviations,
where and
are cross section means of the dependent variable and independent variables (including the constant if any), respectively.
The random-effects is then the result of simple OLS on the transformed data.