Knowledge that Transforms

Recent formulations of social decisions problems focus on rules by which a group of individuals (the society) arrives at a choice among available alternatives. Such a rule, which we shall call the outcome function (Gibbard's "game form, " see [ 3 J), specifies the

[This paper considers procedures for testing for autocorrelation when there are missing observations on both the dependent and explanatory variables. These procedures include Durbin-Watson type tests given the vector of residuals, tests based on a set of uncorrelated residuals, and large sample likelihood ratio and Wald tests.]

In Pesaran [9], the test developed by Cox for comparing separate families of hypo-theses was applied to the choice between two non-nested linear single-equation econometric models. In this paper, the analysis is extended to cover multivariate nonlinear models whenever full information maximum likelihood estimation is possible. This allows formal comparisons not only of competing explanatory variables but also of alternative functional forms. The largest part of the paper derives the results and shows that they are recognizable as generalizations of the single-equation case. It is also shown that the calculation of the test statistic involves very little computation beyond that necessary to estimate the models in the first place. The paper concludes with a practical application of the test to the analysis of the U.S. consumption function and it is demon-strated that formal tests can give quite different results to conventional informal selection procedures. Indeed, in the case examined, five alternative hypotheses, some of which appear to perform quite satisfactorily, can all be rejected using the test. 1.

THE CHOW TEST is a widely used procedure for testing for the equality of sets of coefficients in two linear regression models. However, when the number of observations in one of the models is less than the number of regression coefficients, the Chow test is incapable of testing the hypothesis of equality against that of inequality. It can never be concluded from the Chow test itself that the two sets are equal, although at times it may be possible to conclude that they are unequal. This point is implicit in [1], but has not been specifically discussed heretofore. The indeterminacy of the Chow test results from the insufficient number of observations. The two linear regression models, each of which is assumed to satisfy the conditions of the standard normal linear regression model, can be written as

[A generalization of the St. Petersburg paradox has led Menger to observe that utility functions must be bounded to insure existence of expected utility when probability distributions are unrestricted. It is clear that the admissible set of utility functions can be expanded as restrictions are imposed on the distribution functions under consideration. This paper provides a schema for determining the admissible utility functions for each probability distribution set defined by a minimum order requirement on the moments of the distribution.]

It is known that the price mechanism whereby the rate of change of a price is proportional to the excess demand of the corresponding commodity need not converge to a competitive equilibrium for a pure exchange economy with more than two commodities. On the other hand, there exist convergent price mechanisms, similar to the Newton iterative process, where the rate of change of the prices is determined by the excess demand and the marginal excess demands of all the commodities. This is a considerable informational requirement. It is shown that this requirement cannot be substantially reduced for any convergent price mechanisms, that is for price mechanisms expressed in terms of a difference or differential equation where the solutions converge to a competitive equilibrium.

[A gap is filled in a proof of a central limit theorem, for regression in a time series context, which has been used in some econometric theory.]