Hsiao’s model (1974, 1975) is obtained as a special case of the model of the preceding subsection by assuming 2,, and 2* are diagonal and putting 2€ =
Hsiao (1975) proposed the following method of estimating 2,,, 2Л, and a1: For simplicity assume that X does not contain a constant term. A simple modification of the subsequent discussion necessary for the case in which X contains the constant term is given in the appendix of Hsiao (1975). Consider the time series equation for the /th individual:
Уі = ВД + Л) + X? A + €/. (6.7.8)
If we treat Ці as if it were a vector of unknown constants (which is permissible so far as the estimation of 2д and a2 is concerned), model (6.7.8) is the heteroscedastic regression model considered in Section 6.5.4. Hsiao suggested estimating 2* and a2 either by the Hildreth-Houck estimator (6.5.24) or their alternative estimator described in note 7 (this chapter). In this way we obtain N independent estimates of 2Д and a2. Hsiao suggested averaging these N estimates. (Of course, these estimates can be more efficiently combined, but that may not be worth the effort.) By applying one of the Hildreth-Houck estimators to the cross-section equations for T time periods, 2A can be similarly estimated. (In the process we get another estimate of a2.)
Hsiao also discussed three methods of estimating fl. The first method is FGLS using the estimates of the variances described in the preceding paragraph. The second method is an analog of the transformation estimator defined for ECM in Section 6.6.1. It is defined as the LS estimator of fi obtained from (6.7.2) treating ц and A as if they were unknown constants. The third method is MLE derived under the normality ofy, by which fi, 2Д, and a2
are obtained simultaneously. Hsiao applied to his model the method of scoring that Anderson (1969) derived for a very general random coefficients model, where the covariance matrix of the error term can be expressed as a linear combination of known matrices with unknown weights. (Note that Anderson’s model is so general that it encompasses all the models considered in this chapter.) Hsiao essentially proved that the three estimators have the same asymptotic distribution, although his proof is somewhat marred by his assumption that these estimators are of the probabilistic order of (NT)~1/2.