How the sampling variances affect the linear predictor of the Fay-Herriot model

The Fay-Herriot model can be seen as a linear mixed-effects model, with known within-subject variance parameters. These values are given by the sampling variances of the direct estimators of some parameters in the small areas under investigation. The linear predictor of the Fay-Herriot model may be biased. When the linking regression model is not misspecified, bias does not affect the linear predictor with equal sampling variances, because the fixed-effects estimator reduces to the ordinary least squares regression estimator. In most applications, these variances are quite different, and this is a cause of concern in the matter of bias in the likelihood-based estimation procedures. We study how unequal sampling variances may cause bias and worse mean squared error of the linear predictor, also introducing a measure of the efficiency of the predictor itself. Simulations are conducted, in order to evaluate empirically in several scenarios the consequences of the heterogeneity of the sampling variances on the linear predictor, by different shapes of their empirical distribution.