We examine the question of whether Leontief's [15] conditions for separability and aggregation need be approximately satisfied if only approximate aggregates are to be constructed. It is found that they must be approximately satisfied unless the functions involved exhibit increasingly irregular behavior as the approximation involved in the aggregate gets better. The irregularities involved are related to the violation of Lipschitz Conditions. The stringency of the Leontief conditions for aggregation is therefore not easily evaded.
This paper compares the simple Markov process commonly used in migration studies with an economic model of migration where interregional wage differences are the equilibrating variables. Using the economic model, it appears unlikely that regional exit and entry rates will remain stable as the population is redistributed. As a result, both theory and empirical interstate migration evidence suggest that Markov migration projections will usually understate the population changes required before stochastic equilibrium is reached. IN RECENT YEARS the social sciences, and particularly economics, have experienced
Abstract : For the solution of convex quadratic programming problem, a number of efficient methods have been developed. The most well-known methods are the Simplex method for quadratic programming, discovered by Dantzig and, together with the closely related dual method, further developed by van de Panne and Whinston, and methods developed by Beale, Houthakker and Wolfe. The authors have shown that the methods by Beale and Houthakker can be considered as variants of the Simplex method for quadratic programming or are closely related to it. Compared with the Simplex tableaux used in linear programming, quadratic programming tableaux have a larger size. A tableau for a linear programming problem with n variables and m constraints had (m + l) (n + l) nontrivial elements, while a Simplex tableau for a quadratic programming problem with the same number of variables and constraints has (m + n + l) elements. In the Simplex method for quadratic programming, a considerable number of tableaux will be in standard form, which means that the tableau can be divided in symmetric and skew-symmetric parts, so that the number of elements to be computed and stored is reduced by nearly one half. However, nonstandard tableaux do not have these symmetry properties, so that all elements of these tableaux must be computed. This paper gives a reformulation of the Simplex method in which all tableaux are in standard form, so that use can be made of the symmetry properties in every tableau. The actual number of nontrivial elements in a quadratic Simplex tableau is therefore decreased by a factor of 2. This symmetric formulation has other advantages as well. (Author)
The range of admissible price and income elasticity estimates is materially reduced in this investigation of a variety of procedures, including new methods for analyzing deflation bias. Mostly small effects are found for choice of index formula, base year, logarithmic versus linear form, definition of food consumption (e.g., physical versus expenditure measure), procedures related to time, and aggregation. However, the supply elasticity of food is found to be identifiable and to have a value greater than many have believed, providing a basis for deciding whether to favor demand estimates based on consumption-dependent or pricedependent regressions. The most frequently used deflators are inconsistent with the SlutskySchultz relation. Estimates are developed of three types of bias due to improper deflation (weight of food in deflator, correlation of deflator with residual, and formula nonconformities).
In their paper on the three stage least squares method of estimation of a simultaneous equation system, Zellner and Theil [5] make the interesting observation that the large sample efficiency of the estimates of the parameters in the group of over-identified equations is unaffected if the three stage method is applied to this subsystem alone, ignoring all the exactly identified equations. It is shown in this paper that the estimates themselvesnot just their large sample efficiency-are unaffected if one follows the above mentioned simplified procedure. This result is established for the general case of a simultaneous equation system subject to linear homogeneous a priori restrictions, whereas many other known properties of the three stage least squares method have been demonstrated only for the special case of simple restrictions on the structural coefficients. An error in the expression giving the estimates for the exactly identified equations in Zellner-Theil's paper is also corrected. The results of this paper have obvious significance for the problem of developing an efficient computer program for finding the three stage least squares estimates. ZELLNER AND THEIL [5] have proposed a three stage least squares method of estimation of the parameters of a simultaneous equation system as an alternative to the full information maximum likelihood method. The three stage least squares method has been shown to possess a number of attractive properties. Rothenberg and Leenders [2] have shown that when the variance-covariance matrix Z of the disturbance terms is unrestricted, the estimates have the same asymptotic variancecovariance matrix as the full information maximum likelihood method. Under the same condition of unrestricted Z matrix, Sargan [3] has shown that the difference between the estimates by the three stage least squares method and the estimates by the full information maximum likelihood method is of stochastic order 1/T, where T is the sample size. Though the three stage least squares method is much simpler computationally than the full information maximum likelihood method, it still involves the inversion of a moment matrix of a very large order and the difficulties of computing the inverse matrix accurately, especially when it may be near singular due to the presence of intercorrelation among the explanatory