Characterizing the Consistent Directions of Least Squares Estimates

Abstract

Given a sequence of $p \times 1$ vectors $\mathbf{v} = \{\nu_i\}^\infty_{i=1}$ such that $M_n = \Sigma^n_{i=1} \nu_i\nu'_i$ is positive definite for some $n$, the linear space $\{u: u'M^{-1}_n u \rightarrow \infty\}$ is characterized in terms of the limiting properties of $\mathbf{v}$. This characterization result is applied to give a necessary and sufficient condition for the asymptotic consistency of any best linear unbiased estimator in terms of the limiting properties of the design sequence. For the polynomial regression model, it can be further related to the geometry of the polynomial system.