Four equivalent definitions of reparametrizations and restrictions in linear models

Abstract

Reparametrizations and restrictions are transformations of linear models. Four equivalent definitions of these transformations are proposed in this article. Each of these definitions is based on one of the following concepts: (1) estimation sets, (2) column spaces, (3) residual vectors, and (4) degrees of freedom. These four definitions include as special cases all other definitions of reparametrizations and restrictions found in the literature. Unlike those used by other authors, the definitions in this article allow the original and transformed models to have different response vectors. This approach is useful in tests of nonhomogeneous hypotheses and other applications. However, a transformation of the original model into a reparametrized or restricted model with different response may produce counterintuitive effects on the coefficient of determination and other statistics.

Keywords:

• linear model
• regression
• affine transformation
• determination coefficient
• residual vector
• hypothesis test