T. W. Anderson, Takamitsu Sawa, Distributions of Estimates of Coefficients of a Single Equation in a Simultaneous System and Their Asymptotic Expansions, Econometrica, Vol. 44, No. 6 (Nov., 1976), p. 1329
[One should be very careful in using the "non-informative" priors suggested in the Bayesian econometric literature for the covariance matrix of residuals in simultaneous equations models. To highlight the inadequacies of the prior, this paper shows that the prior leads to sharp posterior distributions even in under identified models. Similar problems also arise with the 2SLS method, but one can apply tests for underidentification. Something similar has to be done in the Bayesian context.]
PROBLEMS OF COMPARING or choosing among models of a stochastic process are frequently encountered in empirical research. In many such situations, conventional statistical procedures offer little guidance since they assume that the model is given. If the alternative models can be nested in a more general model, standard estimation and testing procedures can be employed. Often, however, such general models are not readily available and other considerations may dictate against their use. Recently, there has been considerable progress in the development of methods for comparing alternative non-rested models. A review of this work both Bayesian and non-Bayesian, is given in Gaver and Geisel [1]. Discussions of the Bayesian approach to the comparison of linear regression models are given in Zellner [3, Ch. 10] and Lempers [2], among others. In this paper we consider Bayesian comparison of linear models in which the disturbances have non-scalar covariance matrices. General posterior odds expressions are given and specialized to the case of first order autoregressive disturbances. We also consider a specification error problem in this context; that is, we examine the effect of ignoring the non-scalar covariance structure on the posterior odds ratio. For the first order autoregressive disturbance case we give an approximate expression indicating the magnitude of the error involved in computing the posterior odds ignoring the serial correlation. The accuracy of this approximation is investigated via a small sampling experiment. We use the following notatiori: Let the ith model, Mi (i = 1, 2,. .., N), be y = Xi/3i + yi where y is a T x 1 vector of observations on the random (dependent) variable of interest, Xi is a T x ki matrix of observations on the explanatory variables of Mi (Xi is assumed to be non-stochastic with rank ki), p3i is a ki x 1 vector of unknown parameters of Mi, and ui is a T x 1 vector of disturbances of Mi (yi is assumed to have a normal distribution with E(yi) = 0, and E(yiyii) = U22i where 2Ji is an unknown T x T positive definite symmetric matrix with trace (i) = T).2 Probability functions for the models are denoted by P( ), densities for parameters by 7r( ), and densities for observations by p( ).
[Ostroy and Starr [3] have recently shown how money (a good which may be traded although it satisfies no excess supply/demand) allows decentralized achievement of competitive equilibrium allocations via a round of bilateral trades. Following Jevons [1], monetary exchange is here defined as any trade which decreases the utility of any participant. The consequences of this alternative definition for the Ostroy/Starr thesis are investigated and generalized to cover the case where exchange takes place multilaterally (instead of just bilaterally).]
Recent papers by Houthakker [3 and 4], Samuelson [8 and 9], Sir John Hicks [2], and others deal with the question of the existence of a nontrivial preference ordering which exhibits the mathematical properties in terms of both the direct and indirect utility functions. It is shown that homogeneity and separability are compatible with both the direct and indirect utility functions, but that direct and indirect additivity is consistent with only limited classes of utility functions. Samuelson has raised the question of whether there exists a nontrivial self-dual preference ordering which requires more stringent conditions than homogeneity and separability. By a self-dual preference is meant a preference ordering such that the direct utility function is identical with the corresponding indirect utility function. The purpose of this paper is to present a complete solution to the problem of self-duality. First, we elaborate on the'necessary and sufficient conditions for an exactly or strongly self-dual utility function (in Samuelson's sense). Then, using some well-known concepts of the continuous group theory of transformations, we study the case of weakly self-dual preference orderings and give a precise formulation of the concept of the same mathematical form. Some special classes of self-dual preferences are subjected to detailed analysis. RECENT PAPERS BY Houthakker [3 and 4], Samuelson [8 and 9], Sir John Hicks [2], Pollak [6], and Lau [5] deal with the question of the existence of a nontrivial preference ordering which exhibits the mathematical properties in terms of both the direct and indirect utility functions. Some of the mathematical properties investigated in these works are homogeneity, separability, and additivity. It is shown that homogeneity and separability are compatible with both the direct and indirect utility functions, but that direct and indirect additivity is consistent with only limited classes of utility functions ([2, 8, and 9]). Samuelson has raised the question of whether there exists a nontrivial self-dual preference ordering which may require more stringent conditions than homogeneity and separability [8]. By a self-dual preference is meant a preference ordering such that the direct utility function is identical with the corresponding indirect utility function [8].2 Although partial answers given by Houthakker [4], Pollak [6], and Russell [7] are very illuminating, the solution is far from complete. The purpose of this paper is to present a more complete solution to the problem of self-duality. First, we elaborate on the necessary and sufficient conditions for an exactly self-dual utility function (in Samuelson's sense). Then, using some well-known concepts in the theory of the continuous family of transformations, we study the case of weakly self-dual preference orderings and give a precise formulation of the concept of the same mathematical form. Some special classes of self-dual preference orderings which are convenient for empirical estimation are subjected to detailed analysis.
[The parameters of dynamic simultaneous equation models are often estimated using methods which are appropriate only when the errors of the equations are serially independent. The purpose of this paper is to propose a large sample test for serial correlation to replace the invalid Durbin-Watson test. The test requires only simple calculations and can be easily added to standard two-stage least squares/instrumental variables programs. The treatment of serial correlation is discussed. An example is given to illustrate the test procedure.]
This paper concerns utility functions for money. A measure of risk aversion in the small, the risk premium or insurance premium for an arbitrary risk, and a natural concept of decreasing risk aversion are discussed and related to one another. Risks are also considered as a proportion of total assets.
[A representative consumer exists if market behavior corresponds to a representative income or utility level which is a function of the income distribution. Necessary and sufficient conditions are given on micro behavior and macro behavior (whether maximizing or not) for a representative consumer to exist. Nonlinear Engel curves and taste differences are permitted. If the representative income level is restricted to be mean income, we obtain the traditional linear Engel curves solution. A striking result on economy of information in the representation of a social welfare function is given.]
The impatience implications of continuous time utility indicators are interesting to the extent that they differ from the discrete time results. The class of tFaditional integral utility indicators are considered and impatience implications are shown to depend on the dif- ferent convergence implications of the continuous time case. The stronger separability assumptions of continuous time utility indicators allow a weakening of compactness assumptions often required to demonstrate impatience. presence of impatience. Specific separability assumptions were invoked by Koopmans (8) and Koopmans, Diamond, and Williamson (9) in order to demonstrate the presence of impatience in problems involving choice over an infinite program horizon. From a paper by Diamond (4) one,can infer much of the relationship between separability assumptions and impatience implications. Diamond employed several intertemporal non-complementary assumptions to demonstrate eventual impatience for a case in which the consumption space was not compact in the topology of the norm. The use of non-complementary axioms seems justifijable as their economic implications are straightforward while those of compactness assumptions are not immediately obvious.2 Moreover the natural extension of Diamond's first axiom to all time periods yields a condition equivalent to the independence assumption employed by Debreu (3) in representing preferences by an additive function. Consequently, this paper analyzes separable utility indicators directly for impatience implications; the analysis considers the continuous time case as it subsumes the discrete time analog. However the discrete time case will be discussed in order to facilitate analogy construction.