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 The TSCSREG Procedure

## Parks Method (Autoregressive Model)

Parks (1967) considered the first-order autoregressive model in which the random errors uit , 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 = (r1, ... ,rN)' 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.

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