Journal Article The Demand Curves from a Quadratic Utility Indicator Get access L. L. Wegge L. L. Wegge University of California, Davis Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 35, Issue 2, April 1968, Pages 209–224, https://doi.org/10.2307/2296549 Published: 01 April 1968

An over-identified model could be defined as an exactly identified model that is subject to over-identifying restrictions. One could therefore define a constrained indirect least squares estimator for systems of equations similar to generalized least squares estimators under constraints for single equations. The estimator differs from three stage least squares in using the indirect least squares estimated covariance instead of the two stage least squares estimated covariance. With linear constraints, the estimator is linear. Under the overall null hypothesis with all constraints obtaining, the constrained indirect least squares estimator has the same asymptotic properties as the full infornhtation maximum likelihood estimator. The main advantage of the estimator lies in its easy adaptability to the multiple comparisonist's preferred testing procedure given the exactly identified model as maintained hypothesis. In this paper we stay with the likelihood principle and the corresponding preliminary Wald-type multiple X tests. 1. PROPERTIES OF SEQUENTIALLY CONSTRAINED MAXIMUM LIKELIHOOD ESTIMATORS BELOW WE DEFINE a family of estimators obtained by adding one or a group of constraints after another. To verify the properties of these estimators, we first compare the covariances in the asymptotic distribution of maximum likelihood estimators of models that differ in the number of prior constraints on the structural parameter. References are [1, 13, and 14], but we state the comparisons in a form that shows more of the details. Let f( ; xt, 0) be the density of the endogenous variables yt E R G conditional on the exogenous variables x, E R K and the reduced form parameter 0 E R m. For a sequence (yt), t = 1, . n of n independently selected endogenous variables,

In this paper a family of functional iterations is introduced. One member of this family is the Newton-Raphson method and another member, obtained from a generalization of Steffensen's method to a system of equations, has been considered in [7]. The general member of the family is derived from a regulafalsi construction, due to Gauss, for a particular choice of points in the iteration. From the computational point of view, all the members of the family of iterations, except the Newton-Raphson method, have the property that the partial derivatives of the system of equations are used almost never if a computing device with unlimited precision is utilized. Further, the asymptotic speed of convergence for any member is at least of order two. In view of the difficulties of obtaining the functional form of the second order partials of the likelihood function for general linear and nonlinear simultaneous systems, the method proposed here may be recommended in the computation of full information maximum likelih Dod estimates. Even if the partials of the system of equations are easily calculated, then some member of the family may still lead to convergence if the Newton-Raphson method does not. Practically speaking, the proposed method can be used to determine an approximate solution and this approximate solution will be closer to the solution if the precision of the computations is higher.