[This paper presents conditions on the coefficients of Nth order linear constant coefficient functional equations in generalized differences, necessary and sufficient for asymptotic stability. These conditions are analogous to the Schur-Cohn conditions for difference equations in dated form, and to the Routh-Hurwitz conditions for differential equations.]
The statistical properties of the certainty equivalence control rule and of the least squares estimates generated by this rule are examined experimentally in a linear model with two unknown parameters. It is found that the least squares certainty equivalence rule converges to its true value with probability one and is asymptotically efficient, having an asymptotic distribution with a variance as small as any other strongly consistent rule. However, while a linear combination of the parameter estimates is consistent, the evidence does not confirm that the individual estimates themselves are consistent. If these converge to their true values at all, they do so very slowly (on the order of (log t)').
[A process which combines a planning procedure for the allocation of final products and a multilateral nonrecontracting trading process for allocating primary and intermediate goods is defined and shown to satisfy Malinvaud's criteria for evaluating planning procedures. Central processing costs are lower than in the Malinvaud procedure since the central planner only collects information on final products.]
[When the policymaker's utility function is asymmetric, structural shifts in uncertainty have quite different implications for optimal policymaker behavior under uncertainty depending on the source of those shifts. Changes in uncertainty about exogenous variables do not lead to changes in the policymaker's bias. However, this is not necessarily the case when there is a change in uncertainty about policy response parameters. By contrast, symmetric policymaker utility functions do not imply these distinctions.]
This paper reports on an experimental study of expectation formation-and revision in a time series context. In an adaptive expectations framework, it is shown that the speed of adjustment seems to fall in turning point periods. Expectations are considered as probability density functions, and a scoring system is devised and employed that gives subjects an incentive to report a measure of the dispersion of these functions. This measure, which is inversely related to the confidence with which expectations are held, seems to be inversely related to past forecasting performance. THIS PAPER REPORTS an empirical exploration of the way individuals form and hold expectations about future values of time series variables. In the application of economic models in which expectations about the future play a major role in determining behavior, these expectations are rarely directly observable, and the econometrician is generally forced to assume that a technical rule generates expectations as a simple function only of past observations. One way to see what sort of technical rules make sense in such applications might be to attempt to use this indirect approach to discriminate among possible functional forms. Usually, however, this is computationally burdensome and not terribly revealing. Another approach, currently receiving attention, involves direct analysis of realworld expectations data.2 A third approach, and the ope followed here, is to create and analyze an experimental situation in which the rule followed must be technical because no information other than the past history of the time series in question is available. The main reason for the attractiveness of the experimental approach here, however, lies in the two aspects of expectation formation with which this study is principally concerned. The first of these concerns the influence of turning points in a time series context. The basic hypothesis is due to F. M. Fisher [9, p. 48]:
Two probabilistic theories of choice behavior (the model of independent random utility and the model of elimination by aspects) imply a testable property, the multiplicative inequality, according to which the probability of selecting an alternative x from an offered set A u B is at least as large as the product of the probabilities of selecting x from A and from B. WHEN FACED WITH a choice among complex alternatives (e.g., commodity bundles, investment plans, job offers) people often exhibit inconsistency. That is, they do not always select the same alternative under seemingly identical conditions. In order to accommodate this fact and obtain an adequate conception of choice behavior, psychologists and economists (e.g., Thurstone [18 and 19], GeorgescuRoegen [8], Luce [12], and Marschak [14]) developed models of choice in which the traditional concept of preference is replaced by the notion of choice probability. These models were investigated by many authors (e.g., Davidson and Marschak [5], Debreu [6], Chipman [4], Luce and Suppes [13], and Tversky [20]) from both mathematical and experimental standpoints, and they have also been applied to various aspects of economic theory such as equilibrium analysis (Hildenbrand [10] and Bhattacharya and Majumdar [2]) and consumer behavior (GeorgescuRoegen [9], Quandt [17], Mossin [16], and McFadden and Richter [15]). Two general forms of probabilistic choice models, called random utility and constant utility, were investigated (see Luce and Suppes [13]). In the random utility form, the subjective values undergo random fluctuations, and the alternative with the highest momentary value is selected. In the constant utility form, choice probability is expressed as a function of some (constant) scale values. Thus, the two forms differ regarding the locus of the random element in the choice process. The random utility form attributes uncertainty to the determination of subjective value, whereas the constant utility form attributes uncertainty to the application of the decision rule. (Some choice models, however, can be expressed in either form.) This paper investigates one prominent example of each form: the independent random utility model and the model of elimination by aspects. It shows that both models satisfy two testable properties which provide (fairly tight) upper and lower bounds for all choice probabilities. To formulate the results, we introduce the following definitions:
A. D. Owen, A Proof that Both the Bias and the Mean Square Error of the Two-Stage Least Squares Estimator are Monotonically Non-Increasing Functions of Sample Size, Econometrica, Vol. 44, No. 2 (Mar., 1976), pp. 409-411