Weighted linear regression models with fixed weights and spherical disturbances

Abstract

In linear regression models weights are usually employed within the framework of generalized least squares (GLS) to deal with heteroscedastic errors. In this paper some aspects of estimation and inference are addressed when the weights are used to give the observations different fixed weights in the estimation process of the parameters while the assumption of spherical (i.e. independent and identically normal distributed) disturbances is maintained. Here, this model is referred to as weighted linear regression (WLR) model. Applications of WLR-type models are discounted least squares, a standard procedure in time series forecasting, geographically weighted regression and local regression. For WLR-models the covariance matrix of the estimated coefficients is substantially different in comparison to the GLS-case in terms of the structure of the unscaled covariance matrix and in terms of the error variance estimator. The expressions valid for the GLS-case are not appropriate in the framework considered here and in fact their application is strongly misleading. The results of a simulation study suggest that in most instances the common distributions can be used as approximations for inferential purposes within the WLR-framework. The results derived in this paper will not only provide new inferential procedures for the mentioned applications but may be also beneficial in other applications as well. The usefulness of the approach is demonstrated by a real data example.