[A logically consistent specification of the adaptive expectations hypothesis in continuous time is derived from an underlying discrete time model. We distinguish between (i) the time interval between predictions and (ii) the time horizon over which predictions are made. Taking limits of the expectation equation as these time intervals approach zero, we derive a mixed difference-differential equation and a mixed total-partial differential equation, respectively, describing actual changes. When these are combined with other equations in an economic model, the expectation mechanism provides a simultaneous determination of both expected and actual dynamics. New results are obtained in three separate applications: (i) the short-run stability of multi-asset markets, (ii) a heterogeneous capital goods model, and (iii) a Phillips curve model of wage-price inflation.]
[The paper derives a function that describes the size distribution of incomes. The two functions most often used are the Pareto and the lognormal. The Pareto function fits the data fairly well towards the higher levels but the fit is poor towards the lower income levels. The lognormal fits the lower income levels better but its fit towards the upper end is far from satisfactory. There have been other distributions suggested by Champernowne, Rutherford, and others, but even these do not result in any considerable improvement. The present paper derives a distribution that is a generalization of the Pareto distribution and the Weibull distribution used in analyses of equipment failures. The distribution fits actual data remarkably well compared with the Pareto and the lognormal.]
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.]