PANEL Procedure
Parks Method for Autoregressive Models (PARKS Option)
Parks (1967) considered the first-order autoregressive model in which the random errors 
, 
, and 
have the structure
where
The model assumed is first-order autoregressive with contemporaneous correlation between cross sections. In this model, the covariance matrix for the vector of random errors u can be expressed as
where
The matrix V is estimated by a two-stage procedure, and 
is then estimated by generalized least squares.
The first step in estimating V involves the use of ordinary least squares to estimate and obtain the fitted residuals, as follows:
A consistent estimator of the first-order autoregressive parameter is then obtained in the usual manner, as follows:
Finally, the autoregressive characteristic of the data is removed (asymptotically) by the usual transformation of taking weighted differences. That is, for 
,
which is written
Notice that the transformed model has not lost any observations (Seely and Zyskind 1971).
The second step in estimating the covariance matrix V is applying ordinary least squares to the preceding transformed model, obtaining
from which the consistent estimator of 
is calculated as
where
Estimated generalized least squares (EGLS) then proceeds in the usual manner,
where 
is the derived consistent estimator of V. For computational purposes, 
is obtained directly from the transformed model,
where 
.
The preceding procedure is equivalent to Zellner’s two-stage methodology applied to the transformed model (Zellner 1962).
The variance estimate is
Standard Corrections
For the PARKS option, the first-order autocorrelation coefficient must be estimated for each cross section. Let 
be the 
vector of true parameters and 
be the corresponding vector of estimates. Then, to ensure that only range-preserving estimates are used in PROC PANEL, the following modification for R is made:
where
and
Whenever this correction is made, a warning message is printed.