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The PANEL Procedure

Parks Method (Autoregressive Model)

Parks (1967) considered the first-order autoregressive model in which the random errors , , and have the structure




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




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 follows:




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).

Parks demonstrates that this 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 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:






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

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