Knowledge that Transforms

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.

[A recent paper by D. J. Roberts and A. Postlewaite [1] shows a possible justification for the assumption that participants in a market behave as price-takers when the number of participants becomes large. The present note compares this justification with other approaches to the same question and argues that the various approaches are complementary rather than alternative since, in order to demonstrate the viability of a system with price-taking behavior, all types of feasible deviations from such behavior must be explored. The note introduces a taxonomy for such deviations which may be useful in comparing various approaches and contributions found in the literature.]

[This paper describes a market in which firms vary their quantities of production according to a new adjustment process. Each firm bases its new production entirely upon a knowledge of its own previous productions and profits. It has no knowledge of the payoff functions of the market. Numerical analysis of the process indicates an approach to equilibrium for all initial states. The set of allowed limit points is rigorously characterized, and determined explicitly in the case of two firms. Some exact solutions are found. The process can be regarded as a way of playing a continuous game with a minimum of information.]

This paper examines the durability of capital goods produced under different market structures when tax considerations are included. Since investment tax credit and depreciation allowances are realized by the owner of the durable good, the durability of products produced by an industry which sells its output differs from that of an industry which rents. For each of these two commercial forms we consider both monopolistic and competitive market structure. Potential gains from different forms of regulation are discussed.

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.