X2 X3

0.9980 0.9743"

1 0.9664

0.9664 1

Note that in (C.10.20) we have bordered the correlation matrix by the variables of the model so that we can readily identify which variables are involved in the computation of the correlation coefficient. Thus, the coefficient 0.9980 in the first row of matrix (C.10.20) tells us that it is the correlation coefficient between Y and X2 (that is, r12). From the zero-order correlations given in the correlation matrix (C.10.20) one can easily derive the first-order correlation coefficients. (See exercise C.7.)

C.11 GENERALIZED LEAST SQUARES (GLS)

On several occasions we have mentioned that OLS is a special case of GLS. To see this, return to Eq. (C.2.2). To take into account heteroscedastic variances [the elements on the main diagonal of (C.2.2)] and autocorrelations in the error terms [the elements off the main diagonal of (C.2.2)], assume that

where V is a known n x n matrix. Therefore, if our model is:

y = X p + u where E(u) = 0 and var-cov(u) = a2V. In case a2 is unknown, which is typically the case, V then represents the assumed structure of variances and covariances among the random errors ut.

948 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL

Under the stated condition of the variance-covariance of the error terms, it can be shown that pgls - (X'V—1X)—1X'V—1y (C.11.2)

pgls is known as the generalized least-squares (GLS) estimator of p.

It can also be shown that var-cov(pgls) - o2(X'V—1X)—1 (C.11.3)

It can be proved that pgls is the best linear unbiased estimator of p.

If it is assumed that the variance of each error term is the same constant a2 and the error terms are mutually uncorrelated, then the V matrix reduces to the identity matrix, as shown in (C.2.3). If the error terms are mutually uncorrelated but they have different (i.e., heteroscedastic) variances, then the V matrix will be diagonal with the unequal variances along the main diagonal. Of course, if there is heteroscedasticity as well as autocorrelation, then the V matrix will have entries on the main diagonal as well as on the off diagonal.

The real problem in practice is that we do not know a2 as well as the true variances and covariances (i.e., the structure of the V matrix). As a solution, we can use the method of estimated (or feasible) generalized least squares (EGLS). Here we first estimate our model by OLS disregarding the problems of heteroscedasticity and/or autocorrelation. We obtain the residuals from this model and form the (estimated) variance-covariance matrix of the error term by replacing the entries in the expression just before (C.2.2) by the estimated u, namely, u. It can be shown that EGLS estimators are consistent estimators of GLS. Symbolically, pegls - (X'V—11X)—1KX'V-11y) (C.11.4)

where V is an estimate of V.

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