Reparametrizacion De Modelos Lineles

The Reparameterization of Linear Models Subject to Exact Linear Restrictions1
by Joseph G. Hirschberg Department of Economics University of Melbourne Parkville, Victoria Australia And Daniel J. Slottje Department of Economics Southern Methodist University Dallas, Texas 75275-0496

July 1999

Key words Systems of demand equations, Polynomial lags, Spline lags Classification C51, D12, C63Abstract The estimation of regression models subject to exact linear restrictions, is a widely applied technique, however, aside from simple examples, the reparameterization method is rarely employed except in the case of polynomial lags. 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 theequivalent reparameterized model. In many cases the reparameterization method is computationally more efficient especially when estimation involves an iterative method. The general relationship that converts the two forms of the restricted model is derived. Examples involving systems of demand equations, polynomial lagged equations, and Splines are given in which the transformation from one formto the other are demonstrated. In addition, we demonstrate how a Wald test of the restrictions can be constructed using an augmented version of the reparameterized model. A computer program example is presented to demonstrate the equivalence.

1

We wish to acknowledge our colleagues Thomas B. Fomby and Jenny Lye for their helpful suggestions and comments on this paper, the usual caveat holds.1. Introduction A discussion of the estimation of a regression model subject to exact linear restrictions can be found in most econometrics textbooks (such as Griffiths et al 1993, Greene 1997, and Johnston and DiNardo 1997). The basic exposition on how one solves a set of "normal equations" with the restrictions added has not changed much since Tintner’s (1952) text. However, as Mantell(1973) has shown, it is possible to reconfigure the restrictions to form the reparameterization. More recently Davidson and MacKinnon (1993) (section 1.3) repeated this earlier contribution in the econometrics literature. These authors all propose that the alternative reparameterized model exists, however they assume that a rearrangement of restrictions can be found without any general proposal on howto perform the reparameterization needed. One might suspect that since the techniques of constrained optimization are so familiar to economists from their study of microeconomic theory, that they rightly perceive the minimization of the squared error subject to a linear restriction as a very useful pedagogical analogy to the similar problem in economic theory. This similarity may aid in theinterpretation of resulting Lagrange multipliers in the testing of a set of linear restrictions. The reparameterization of the original regression equation is often referred to informally or only for a specific example, however this approach is usually dropped in the discussion of the general case. In this paper we demonstrate the general relationship between the widely used restricted least squaresestimator generally presented in textbooks and an equivalent estimator found by reparameterization. In many cases the reparameterized model proves to be more insightful and more efficient. It can also be more accurately computed in many instances. A linear regression model subject to a set of linear restrictions is written in the form (1)

Y = Xb + e , s. t . Rb = r ,



where YTx1 thevector of the observations on the dependent variable, XTxk is the matrix of H[SODQDWRU\ YDULDEOHV
kx1

is the set of restricted parameters (note that in this exposition the

unrestricted parameters will be denoted differently), Rmxk is the matrix of m linear combinations of the restricted parameter set, rmx1 is the vector of constraints to which we equate the linear combinations, m < k and the...