Knowledge that Transforms

This paper attempts to determine conditions under which distributed lag analysis is appropriate. Results indicate that lag functions are stable and linear under fairly general (but constant) objective criteria and decision constraints as long as the underlying economic environment is characterized by a stationary Gauss-Markov process, and observed environmental variables are Gaussian perturbations of that process. The results appear particularly useful for specifying the lag distribution inherent in subjective parameters in specific decision problem contexts. 1. INTRODUC'TION IN AN EARLIER PAPER, Taylor [11] dealt with the problem of determining lag distributions on the basis of optimization assumptions in a dynamic model of uncertainty. Taylor's conclusions, however, are somewhat disturbing in that they indicate only a narrow class of decision problems with uncertainty can be studied in a distributed lag framework. Taylor addresses only an exemplary model of inventory control and by assuming that (i) the firm's cost function can be expressed as a sum of strictly quadratic and linear terms, (ii) all production constraints are linear equalities, and (iii) observed demands are perturbations of unobserved components which follow a Gauss-Markov process, he is able to show that a distributed lag exists in the decision rule. By making yet additional assumptions, he is also able to demonstrate stability of the lag distribution. The purpose of this paper is to generalize the class of decision-makers' objective criteria and the constraint set description under which distributed lag analysis can find some theoretical justification. Although Taylor relies heavily on Kalman filtering theory to derive his results, the equivalent Bayesian approach is used explicitly in this paper for purposes of completeness and continuity.2 Results show that the lag function is stable and linear under more general conditions than Taylor's, although the lag function may enter the resulting econometric model nonlinearly. If econometric investigators are willing to consider nonlinear functions of lag distributions (or approximation of nonlinear functions by linear functions), the results should provide a basis for distributed lag analysis in a much broader class of problems than do Taylor's previous results. Furthermore, it is found that econometric equations which include the distributed lag linearly may exist outside of the set of cases considered by Taylor. The following section begins by specifying the general decision theoretic framework in which we shall operate throughout the paper. The existence of lag functions is made evident. In Section 3, Taylor's environmental system is

[A new formulation of the linear quadratic control, LQC, problem with known coefficients, called the LAG, is presented. The LAG generally does not generate a recursive Ricatti system. For models with long lags an important issue is whatformulation leads to an efficient algorithm both with respect to storage and speed. At present the most efficient known formulation is the minimum state variable representation, MSV. The LAG requires much less storage than the MSV as the LAG does not require conversion to state space representation. For short time horizons the LAG is computationally faster than the MSV. As the time horizon increases, the efficiency of the LAG relative to the MSV declines. Numerical comparisons of the Theil, Chow, MSV, and LAG formulations are shown.]

abstract: the conventional approach to the design of stochastic systems operates, either directly or indirectly, by minimising the variance of the model. this procedure can be regarded as a natural extension of the stability analysis of a deterministic system, according to which the degree of stability is inversely related to the absolute value of its largest characteristic root. very often, however, the policymaker is not indifferent to the frequency composition of economic fluctuations. he may, for example, have a marked dislike for short-term fluctuations. we formalize this notion by setting up a spectral utility function as a criterion for steady-state optimisation, and show that the results from such an optimisation may conflict with those yielded by the conventional approach. large characteristic roots may not necessarily be bad! the scheme of the paper is as follows. because the ideas involved may be unfamiliar we shall spend some time on a rather intuitive motivation for what follows. this is done in section i. in section ii the notion of a spectral utility function is introduced and its evaluation discussed. we then return to the example of section i to give it a more precise treatment. section iii contains extensions, principally to the multivariate case.;

The first part of this article integrates the concept of (relative) risk aversion with respect to income (r) with the static analysis of demand for many commodities. Alternative representations of preferences and demand functions, using duality, give rise to many alternative representations and interpretations of r, and to theorems regarding attitudes towards risk in bundles of quantities and in prices. In the second part, a previous analysis by Deschamps is corrected and completed by specifying the general form of preferences and demands such that r is a function of the utility level only, independent of relative prices. Finally, preferences and demand functions associated with constant r (previously analyzed by Stiglitz and Deschamps) are specified more explicitly and completely. A general conclusion emerging is that demand behavior under certainty can hardly throw any light on the nature of attitudes towards risk. THE CONCEPT OF THE relative risk aversion function as a unit-free measure of individual aversion to income risk under expected utility maximization, was defined by Arrow [1] and Pratt [15], and has proved useful in various applications. Stiglitz [21] studied relations between an individual's aversion to income risk, and his indirect utility and demand functions for many commodities, obtained under certainty in competitive markets. In particular, he analyzed the cases of risk indifference and constant relative risk aversion (r). Deschamps [4] extended this analysis to study implications of alternative assumptions: (i) That absolute risk aversion R = rly is independent of prices, nominal or relative, for given income. This is equivalent, however, to constant r (which is the case analyzed by Stiglitz) for both cases. (ii) That R or r are constant on each indifference surface. This is shown to be impossible for R, but meaningful and interesting for r = r(u). Unfortunately, however, Deschamps could not show the utility and demand functions for this case, and conducted an indirect analysis based on the second-order differential equation implied, failing to note that any such r(u) is compatible with homothetic preferences. In addition, his analysis contains errors (e.g., the case r = 1 constant) and may be subject to misleading interpretations. The purpose of this article is twofold: (i) To complete the analysis of risk aversion with many commodities, by using various alternative formulations of the relative risk aversion function to study general relations between income risk aversion and attitudes towards risk with respect to quantities (e.g., when both relative prices and income are subject to risk), or with respect to prices. (ii) To complete the analysis of Deschamps by showing the general forms of utility and demand functions when r = r(u), and to correct some errors of analysis and interpretation.

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).

[A bargaining situation is described by a set of alternative which are feasible to n individuals when they do cooperate, and an alternative which comes about when they do not cooperate. The paper addresses the question of which cooperative outcome will be chosen. A Nash-type approach is used to prove that, under plausible axioms describing the underlying bargaining process, the individuals must be doing interpersonal comparison of utility. The model and the solution overcome some difficulties recently described by Nydegger and Owen.]

[This paper investigates an operational inconsistency between the constrained priority rationing scheme introduced by Manove in 1973 and the criterion used to determine an "optimal" priority matrix. A truncation of the optimality criterion is suggested in order to resolve this inconsistency and also reduce a potential source of bias against final demand.]

[In recent years, stability analysis has been extended in two directions, which are useful for economic applications. The first direction concerns differential equations with discontinuous right-hand sides. The second direction concerns difference equations with multivalued right-hand sides. The present paper reviews some of these contributions (in Section 5, 6, and the Appendix), brings out their similarities, and illustrates their applications to economic problems. The illustration concerns an economy with both private and public goods (Section 2). It is shown (Section 3) how an efficient allocation for that economy can be reached through a globally stable process, combining a price-guided market allocation of private goods and a quantitative planning procedure for public goods. A discrete version of the planning procedure, using an internally defined variable speed of adjustment, is also shown (Section 4) to be quasi-stable.]

This paper deals with the theoretical development of some aspects of the trend removal problem. The objective is to show the difference between the two most popular trend removal methods: first differences and linear least squares regression. On the one hand, we show that if first differences are used to eliminate a linear trend, the series of residuals would be stationary but would not be white noises as they contain a first lag autocorrelation of -0.50. Furthermore, the spectral density function (SDF) of these residuals relative to that of a white noise series would be exaggerated at the high frequency portion and attenuated at the low frequency portion. On the other hand, we show that the regression residuals from the linear detrending of a random walk series would contain large positive autocorrelations in the first few lags. Relative to that of white noises, the SDF of the regression residuals would be exaggerated at the low frequency portion and attenuated at the high frequency portion.