n effective treatment of adding-up restrictions in the inference of a general linear model - Yongge Tian (AIMS Mathematics, 2023)
This article offers a general procedure of carrying out estimation and inference under a linear statistical model $ {\bf y} = {\bf X} \pmb{β} + \pmb{ε} $ with an adding-up restriction $ {\bf A} {\bf y} = {\bf b} $ to the observed random vector $ {\bf y} $. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $ \pmb{β} $ and a matrix equality for the covariance matrix of the error term $ \pmb{ε} $, which can help in combining the two model equations in certain consistent form. We then give the derivations and presentations of analytic expressions of the ordinary least-squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of parametric vector $ {\bf K} \pmb{β} $ using various analytical algebraic operations of the given vectors and matrices in the model.