The distributions of the LIML and TSLS estimates of the coefficient of an endogenous variable in a single equation can be approximated by asymptotic expansions. This paper relates the expansions in terms of the noncentrality parameter and the sample size going to infinity, the noncentrality parameter going to infinity with the sample size held fixed, and the standard deviation of the disturbance going to zero (small-o). 1. INTRODUCriON RECENTLY, ASYMPTOTIC EXPANSIONS of the distributions of estimates of coefficients of a single equation in a system of simultaneous equations have been made by Anderson [1], Anderson and Sawa [2], Mariano [6 and 7], and Sargan and Mikhail [11]. The expansions have usually been carried out on the basis that the sample size increases and that the effect of the exogenous variables (the noncentrality parameter) increases along with the sample size. In this paper we consider the case of the covariance matrix of the disturbances known and alternatively the case of the sample size fixed. We relate these three cases to the approach of letting the disturbance decrease (the small-o- approach). The estimates treated are two-stage least squares (TSLS) and limited information maximum likelihood (LIML).

By looking at approximate multivariate risk premiums a matrix measure of multivariate local risk aversion is introduced for a multi-attributed utility function u. This matrix function R(x) = [-uij(x)/ui(x)] generalizes the univariate measure of Pratt [11] and the conditional measure of Keeney [7]. It has particular advantages in assessing the attitude of a decision-maker toward correlated risks, a concern of Richard [13], and is more informative than the scalar measure proposed by Kihlstrom and Mirman [8]. Simple characteristics of the absolute risk aversion matrix R determine whether a utility function is additive or concave. Assumptions of either constancy or proportionality of R are shown to lead to specific restrictions on the form of u which are more stringent than those of Rothblum [15].

Previous attempts to estimate labor supply functions based upon the constrained maximization of a utility function with leisure and income as arguments have assumed that all time not spent at work on the job is leisure time. In this paper we formulate a model of a household in which time is allocated between work on the job, leisure, and housework, where leisure is defined to be net of time spent on housework. The model is estimated under two alternative stochastic specifications and the results compared to those obtained (i) assuming that housework time is exogenous and (ii) assuming that housework time is part of leisure time.

Much of the development of economic theory has been based on assumptions that credit economic agents with perfect discriminating power and perfect consistency in their choice behavior, but recently a range of less restrictive psychological choice theories has been introduced into economic analysis. These theories may be classed into two typesthose that treat preference (and hence choice) as a probabilistic phenomenon, and those that retain the deterministic structure of the traditional analysis but relax in some way the assumption that individual preferences be transitive. in previous work others have examined the implications for economic equilibrium of assuming preferences to be random; our purpose in this paper is to examine the equilibrium implications of a particularly attractive deterministic choice theory, that of semiordered preferences. We first examine an exchange ec such an economy is shown to have an equilibrium, and further, for interior equilibria, it is shown that a range of allocations to consumers is consistent with equilibrium. Finally, we show that in a semiorder market economy the set of price equilibria has a non-empty interior relative to the price simplex.