An effective treatment of adding-up restrictions in the inference of a general linear model

This article offers a general procedure of carrying out estimation and inference under a linear statistical model $\mathbf{y}=\mathbf{X}\beta \beta +\epsilon \epsilon$ with an adding-up restriction $\mathbf{A}\mathbf{y}=\mathbf{b}$ to the observed random vector $\mathbf{y}$. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $\beta \beta$ and a matrix equality for the covariance matrix of the error term $\epsilon \epsilon$, 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 $\mathbf{K}\beta \beta$ using various analytical algebraic operations of the given vectors and matrices in the model.