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Alternative forms for restricted regressions

Hirschberg, Joe ; Lye, Jenny ; Slottje, Daniel J. ( 2005-11)

The estimation of regression models subject to linear restrictions, is a widely appliedtechnique, however, aside from simple examples, the equivalence between the linear restrictedcase to the reparameterization or substitution case is rarely employed. We believe this is due tothe lack of a general transformation method for changing from the definition of restrictions interms 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 whenestimation involves an iterative method. But the linear restriction case allows a simple methodfor adding and removal of restrictions.In this paper we derive a general relationship that allows the conversion between the twoforms of the restricted models. Examples involving systems of demand equations, polynomiallagged equations, and splines are given in which the transformation from one form to the otherare demonstrated as well as the combination of both forms of restrictions. In addition, wedemonstrate how an alternative Wald test of the restrictions can be constructed using anaugmented version of the reparameterized model.