Knowledge that Transforms

[The chief difficulty in applying Aitken estimators to large linear systems stems from the dimensionality of the inverse. Here the conjugate gradient algorithm is applied to the problem, leading to substantial savings in storage and some savings in time. Given that the information matrix is not computed, an inference procedure is developed which involves two easily computed statistics which straddle the conventionally estimated standard errors. The paper is intended to open the way for the application of more efficient estimation procedures to large econometric systems.]

[In Kaneko [6] we considered the relationship between the ratio equilibria and the core of the voting game G(N, W,r), in which a fixed ratio is given. In this paper we present a new voting game G(N, W) in which no fixed ratio is given, and consider the relationship between the ratio equilibria and the core of G(N, W). We prove that the core of G(N, W) coincides with the ratio equilibria.]

[Consider a small country with a strictly convex production possibility set, where non-traded goods as well as traded goods are produced. Suppose that tariffs are the only causes of the distortions in this economy. In the present paper, we will prove that in this economy reduction of the highest tariff rate to the level of the second highest rate will improve welfare if (i) inferior goods do not exist, (ii) the good on which the highest tariff rate is imposed is substitutive to all the other goods both in consumption and in production, and (iii) the non-traded goods are substitutive to all the other goods both in consumption and in production.]

The Shapley value is shown to be avon Neumann-Morgenstern utility function. The concept of strategic risk is introduced, and it is shown that the Shapley value of agame equals its utility if and only if the underlying preferences are neutral to both ordinary and strategic risk.

Edgeworth series expansions are obtained of the finite sample distributions of the least squares estimator and the associated t ratio test statistic in the context of a first-order noncircular stochastic difference equation. General formulae are given for these expansions up to 0(Th1) where T is the sample size and explicit representations of these in terms of the true parameters are derived up to 0(12). Some numerical comparisons of the approximations and the exact distributions are made in the case of the least squares estimator.

A firm minimizes cost or maximizes profit subject to the constraint implied by a production function. Demand equations for inputs, formulated in terms of changes over time, are described in two steps. The first is the total input decision, which describes the Divisia input volume index in terms of the change in output (for cost minimization) or price changes (for profit maximization). The second is the input allocation decision, which describes the changes in the demand for the individual inputs in terms of the Divisia input volume index and the input price changes. It is shown that the input allocation decision allows a simple transformation so that (1) the change in the demand for each transformed input is independent of the changes in the relative prices of all others and (2) the log-change in output is the sum of certain components, each representing the contribution of one transformed input, in such a way that the interaction of these inputs is confined to terms of the third order of smallness.

["If one is willing to interpret Q̃ [the Goldberger, Nagar, Odeh reduced form coefficient covariance estimate] as a covariance matrix of the random parameter π around the constant extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi$ extless/tex-math extgreater, rather than as a covariance matrix of the random estimates extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi$ extless/tex-math extgreater, then using extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi$ extless/tex-math extgreater for extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi$ extless/tex-math extgreater and Q̃ for Q extlesstex-math extgreater$[ extbackslashtilde\ extbackslashpi$ extless/tex-math extgreater and Q are the mean and covariance matrix of the random parameter π] will provide an approximate solution to the evaluation of expectations required in our optimal control problem" [3, p. 641], italics added).]