## Parks Method (Autoregressive Model)

Parks (1967) considered the first-order autoregressive model
in which the random errors
*u*_{it} , *i* = 1, 2, ... , *N*,
*t* = 1, 2, ... , *T*, 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 can be
removed (asymptotically) by the usual transformation of taking
weighted differences.
That is, for *i* = 1,2, ... ,*N*,

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 to
apply ordinary least squares to the preceding
transformed model, obtaining

from which the consistent estimator of _{ij} is calculated:

where

EGLS then proceeds in the usual manner,

where is the derived consistent estimator of
**V**. For computational purposes, it should be pointed out that
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).
Parks demonstrates that his estimator is consistent
and asymptotically, normally distributed with

*Standard Corrections*

For the PARKS option, the first-order autocorrelation coefficient
must be estimated for each cross section. Let be the
*N**1 vector
of true parameters and
*R* = (*r*_{1}, ... ,*r*_{N})'
be the corresponding vector of estimates.
Then, to ensure that only range-preserving estimates are used in
PROC TSCSREG, the following modification for R is made:

where

and

Whenever this correction is made, a warning message is printed.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.