Alternative Forms for Restricted Regressions

Abstract

The estimation of regression models subject to linear restrictions, is a widely applied technique, however, aside from simple examples, the equivalence between the linear restricted case to the reparameterization or substitution case is rarely employed. We believe this is due to the lack of a general transformation method for changing from the definition of restrictions in terms of the unrestricted parameters to the equivalent reparameterized model and conversely, from the reparameterized model to the equivalent linear restrictions for the unrestricted model. In many cases the reparameterization method is computationally more efficient especially when estimation involves an iterative method. But the linear restriction case allows a simple method for adding and removal of restrictions.In this paper we derive a general relationship that allows the conversion between the two forms of the restricted models. Examples involving systems of demand equations, polynomial lagged equations, and splines are given in which the transformation from one form to the other are demonstrated as well as the combination of both forms of restrictions. In addition, we demonstrate how an alternative Wald test of the restrictions can be constructed using an augmented version of the reparameterized model. 1 We wish to acknowledge Thomas B. Fomby for helpful suggestions and comments on this paper, the usual caveat holds.1

Providing Intuition to the Fieller Method with Two Geometric Representations using STATA and EVIEWS

“…This then implies that 1 A X X = and 2 R . X X + = For further details see Hirschberg, Lye and Slottje (2005) This approach is easily implemented in any computer program that plots the confidence ellipse provided the ellipse has the correct dimensions. Note that the confidence ellipse produced as part of the OLS options in Eviews 5.0 is specified as:…”