Knowledge that Transforms

THE SMALL SAMPLE PROPERTIES of the estimators of parameters in a single equation of a system of equations have been investigated in several ways. Some of the many Monte Carlo studies that have been conducted are reviewed in Johnston [8]. Nagar [14] used expansions in powers of terms whose order in probability was T1, where T is the sample size, to obtain approximations of the small-sample behavior of the members of Theil's k-class estimators [18]. Basmann [4, 5], Kabe [9], Richardson [15], Sawa [17], and others, present exact distributions for the two-stage least squares estimators for certain models. Mariano and Sawa [13] give the exact distribution of the limited information estimator for the coefficient in a single equation containing two endogenous variables. The studies of the exact distributions show that the limited information estimator does not possess moments and that the first two moments of the two-stage least squares estimator exist only for certain levels of overidentification. Kadane [11] presents an approximation to the bias and mean square error of the k-class estimator and the limited information estimator in terms of an expansion in o-, where o_2 iS (a multiple of) the variance of the residuals in the equation. Kadane's results for the k-class agree with those of Nagar [14]. Recently asymptotic expansions of tie distribution function of the estimators of a single equation have appeared (e.g., see Anderson [1], and Sargan and Mikhail [16]). We present a modification of the limited information estimator and demonstrate that the modified estimator possesses finite moments and that one member of the class has bias of order T2. The estimator is a member of the k-class estimators originally introduced by Theil [18]. We also introduce a modification of the Nagar [14] fixed k-class estimators to ensure finite moments. The modified estimators have the same limiting distribution as the unmodified estimators, and the approximations presented by Nagar hold for the first two moments of the modified k-class estimator. Restricting the modified k-class estimator and the modified limited information estimator to have the same, but arbitrary, bias we show that to order T2 the modified limited information estimator dominates the k-class 1 Journal Paper number J-8374 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa; Project 2039. This research was partly supported by the United States Bureau of the Census through Joint Statistical Agreements 74-1 and 75-1.

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].

There has recently been a resurgence of interest in specifying more completely the relationship between Walrasian microeconomic models of economic behavior and Keynesian macroeconomic models.On the face of it, these two approaches to economic reality seem very different.The Walrasian model assumes agents engage in maximizing behavior taking as given a common perception of relative prices.The relative prices then adjust to equilibrate the system.The Keynesian model specifies that agents' be- havior obeys certain ad hoc rules relating quantity variables of the sys- tem.These quantities then adjust to equilibrate the system.Keynesian analysis is often thought to concern itself primarily with a case where price signals are fixed or adjust very slowly; Walrasian

This paper reports the testing of hypotheses concerning: (i) whether the household is better viewed as planning over a single-period versus a multiperiod horizon; (ii) whether the household is better viewed as planning in a single-asset or a multiasset framework; (iii) the relative importance of substitution and wealth effects as sources of change in the stock demand for automobiles. The findings are that a multiperiod, multiasset model best describes stock demand, that the separation theorem which implies a zero wealth effect is rejected, and that substitution effects are seven times more important than wealth effects. THE ECONOMIC LITERATURE CONTAINS several empirical studies of household automobile demand [3, 7, 8, and 10] and theoretical models of the household [1, 4, 5, 6, and 14] which are or could be applied to automobile demand. Two aspects of theory which are not fully reflected in the empirical studies are the implications of a multiperiod horizon and the possibility of substitution among assets. Theoretical models [5 and 15] which assume a multiperiod horizon imply that relevant asset prices are user costs and the appropriate constraint is wealth. In contrast, most empirical studies use purchase prices rather than user costs, and income rather than wealth. In addition, theoretical models [4 and 5] permit substitution over a variety of goods, whereas most empirical studies restrict substitutions to automobiles and consumption goods. To the extent that estimated equations are misspecified, the prevailing conclusion that income effects are more important than substitution effects may be due to left-out-variable bias. This paper investigates each of these three issues-the length of the horizon, the range of substitutions, and the relative importance of substitution and wealth effects-by estimating over the same set of data a variety of alternative equations which reflect different assumptions about the horizon and range of substitutions. Initially, a multiperiod, multiasset model of the household consumption-saving decision is stated and used to derive the appropriate arguments for the broadest estimating equation. A linear approximation of this equation is estimated using quarterly United States data covering the years 1952-1972. Then this estimate is compared to competing equations derived under restrictions on the multiperiod, multiasset model. Specifically, demand equations derived under multiperiod, single-asset, single-period, single-asset, and single-period, multiasset assumptions are estimated and compared to the broadest multiperiod, multiasset equation. In addition, versions of the restricted equations which have appeared in the literature are estimated and compared. The findings are: (i) a multiperiod, multiasset equation best describes automobile stock demand, (ii) estimates of substitution and wealth effects are quite sensitive to specification bias, and (iii) substitution effects are seven times more important than wealth effects in the dominant equation.

Ti-H CONCERN of this paper is the estimation of the parameters of a probabilistic choice model when choices rather than decision makers are sampled. Existing estimation methods presuppose an exogeneous sampling process, that is one in which a sequence of decision makers are drawn and their choice behaviors observed. In contrast, in choice based sampling processes, a sequence of chosen alternatives are drawn and the characteristics of the decision makers selecting those alternatives are observed. The problem of estimating a choice model from a choice based sample has suibstantive interest because data collection costs for such processes are often considerably smaller than for exogeneous sampling. Particular instances of this differential occur in the analysis of transportation behavior. For example, in studying choice of mode for work trips, it is often less expensive to survey transit users at the station and auto users at the parking lot than to interview commuters at their homes. Similarly, in examining choice of destination for shopping trips, surveys conducted at various shopping centers offer significant cost savings relative to home interviews.2 While interest in transportation applications provided the original motivation for our work, it has become apparent that choice based sampling processes can be cost effective in the analysis of numerous decision problems. In particular, wherever decision makers are physically clustered according to the alternatives they select, choice based sampling processes can achieve economies of scale not available with exogeneous sampling. Some non-transportation decision problems in which decision makers do cluster as described include the schooling decisions of students, the job decisions of workers, the medical care decisions of patients and the residential location decisions of households. Realization of the sampling cost benefits of choice based samples presupposes of course that the parameters of the underlying choice model can logically be inferred from such samples and that a tractable estimator with desirable statistical properties can be found. We shall, in this paper, confirm the logical supposition, develop a suitable estimator, and characterize the behavior of existing, exogeneous sampling, estimators in the context of choice based samples. An outline of the presentation and summary of major results follows.

[This paper describes a decentralized planning procedure which converges to a global optimum--as seen by a central planning board--whether or not the production possibility sets of the firms are convex. All information is exchanged in the form of quantities: the planning board proposes quotas and the firms respond with feasible production programs.]

[Research on the spatial firm has shown that spatial results often differ from non-spatial. This paper considers the relationship between the price of a spatial monopoly and the price of a spatially competitive firm. The conditions under which the competitive price will exceed the monopoly price are outlined.]

[The paper analyzes borrowing and lending on uncertain future income, with a positive probability of bankruptcy. Creditor and debtor play a strategic game, in which it is shown that optimal creditor behavior is not generally well defined. The model suggests that under uncertainty the availability of credit may be restricted below that which would be predicted by classical microeconomic theory.]