Further Results on the Consistent Directions of Least Squares Estimators

Abstract

Wu (1980) defined the consistent directions of the least squares estimator in a linear model as the linear combinations of parameter estimates that are asymptotically consistent. For the polynomial regression model, a characterization of the space of consistent directions $S$ was obtained in terms of the convergence rates of the corresponding design sequence to its limit points. By employing a more general and yet simpler approach, we obtain here a similar result for any regression model with one independent variable and smooth regression function. When $f_i(x)$ is an extended Tchebycheff system, the above characterization is further refined and the consistency region $C$ is shown to be either the set of limit points of the design sequence or the whole real line.