On the equivalence of the weighted least squares and the generalised least squares estimators

This paper is concerned with the equivalence of the weighted least squares estimators (WLSE) and the generalised least squares estimators (GLSE). Necessary and su±cient conditions for the WLSE to be best linear unbiased esti- mators are derived, generalising Anderson (1948, 1971) theorem on the equivalence between the ordinary least squares estimators and the GLSE. Procedures for ob- taining the optimal kernel for a given covariance structure are also described, where optimality is to be intended in the Gauss-Markov sense. The results are illustrated in the context of local polynomial regression methods for the estimation of the underlying trend of a time series.