Using the Eaton and Lipsey mode, one shows that a hierarchical system of central places is socially optimal: firms having less frequent purchases are clustered with firms having more frequent purchases in any configuration minimizing total transport and production costs
EMPIRICAL STUDIES OF TWO SIDED MATCHING have so far concentrated on markets in which certain kinds of market failures were addressed by resorting to centralized, deterministic matching procedures. Loosely speaking, the results of these studies are that those centralized procedures which achieved stable outcomes resolved the market failures, while those markets organized through procedures that yielded unstable outcomes continued to fail.2 So the market failures seem to be associated with instability of the outcomes. But many entry-level labor markets and other two-sided matching situations don't employ centralized matching procedures, and yet aren't observed to experience such failures. So we can conjecture that at least some of these markets may reach stable outcomes by means of decentralized decision making. And decentralized decision making in complex environments presumably introduces some randomness into what matchings are achieved. However, as far as we are aware, no nondeterministic models leading to stable outcomes have yet been studied. The present paper demonstrates that, starting from an arbitrary matching, the process of allowing randomly chosen blocking pairs to match will converge to a stable matching with probability one. (This resolves an open question raised by Knuth (1976), who showed that such a process may cycle.) Furthermore, every stable matching can arise
In the general structural equation model only the direction of the vector of coefficients of the endogenous variables is determined.The traditional normalization rule defines the coefficients that are of interest but should not be embodied in the estimation procedure: we show that the properties of the traditionally defined ordinary least squares and two stage least squares estimators are distorted by their dependence on the normalization rule.Symmetrically normalized analogues of these estimators are defined and are shown to have essentially similar properties to those of the limited information maximum likelihood estimator.
IN NONLINEAR MODELS the power function is often approximated by asymptotic methods. The most common approach is to consider the asymptotic local power function. The local power function is monotonic and it has essentially the same shape as the power function in the classical normal linear regression model. However, the accuracy of the approximation can be poor at nonlocal alternatives. This note examines the exact powers of the Wald test in the case of a one parameter nonlinear regression model with normal errors. The model is based on the exponential response function f( x, O) = exp( Ox). The results show that the exact power function of the Wald statistic can be nonmonotonic. For selected designs the exact powers of the Wald test first increase and then eventually decline as the distance between the hypothesized and the true values of the parameter increases. The exponential structure appears in many nonlinear models; see Gallant (1975, 1987) and Bates and Watts (1988). This suggests that nonmonotonicity of the Wald test is a feature of a wide class of nonlinear models. Indeed, Nelson and Savin (1988) show that it arises in standard logit, probit, and Tobit models as well. The focus here on the nonlinear regression model is for expository convenience. While the existence of nonmonotonic power is not new, the surprising results are that this phenomenon occurs in very simple nonlinear models and that it can be quite severe. In such cases the asymptotic local power approximation provides a very poor guide to the performance of alternative tests.
Following Jureckova (1981) we introduce a finite-sample measure of performance of regression estimators based on tail behavior.The least squares estimator is studied in detail, and we find that it may achieve good tail performance under strictly Gaussian conditions.However, the tail performance of the least-squares estimator is found to be extremely poor in the case of heavy-tailed error distributions or when leverage points are present.Further analysis of the least-squares estimator with light-tailed errors indicates the strong influence of the design matrix in determining tail performance.Turning to the tail behavior of various robust estimators of the parameters of the linear model, we focus on tail performance under heavy (algebraic) tailed errors.The /^estimator is seen to be a leading case: we find a simple characterization of its tail behavior in terms of the design configuration and show that a broad class of M-estimators have the same performance.Perhaps most significantly, it is shown that our finite-sample measure of tail performance is, for heavy tailed error distributions, essentially the same as the finite sample concept of breakdown point introduced by Donoho and Huber(1983).This finding provides an important probabilistic interpretation of the breakdown point and clarifies the role of tail behavior as a quantitative measure of robustness.This link is further explored for high-breakdown regression estimators including Rousseeuw's (1982) least-median-of-squares estimator.
This paper considers estimation and hypothesis testing in linear time series when some or all of the variables have (possibly multiple) unit roots. The motivating example is a vector autoregression with some unit roots in the companion matrix, which might include polynomials in time as regressors. Parameters that can be written as coefficients on mean zero, nonintegrated regressors have jointly normal asymptotic distribution, converging at the rate of T(superscript "one-half") In general, the other coefficients (including the coefficient on polynomials in time), and associated t and F test statistics, have nonstandard asymptotic distributions. Copyright 1990 by The Econometric Society.
We provide a theorem giving a set of conditions under which public observation of an aggregate of individual expectations leads to consensus. The key concept unkerlying the result is that of common knowledge, as formalized by Aumann (1976)
but any convex combination of pairs of optimal strategies such that p(2, 2) = 0 satisfies p(1, 1) > 2 (with the obvious notation p(i, j) for the induced probability of row i and column j). However, the following is easily checked. Let I and J be the sets of pure strategies of player 1 and player 2 respectively in a zero-sum game G with value v. Then p = [p(i, i)I(, J) IXJ is a correlated equilibrium distribution for G if and only if for every E J such that p(jo) > 0, the conditional probability of player 2 over player l's actions given jo' [p (iIjo)]I , is an optimal strategy for player 1, yielding exactly v against jo and similarly for [p(jlio)]jEj, io E , p(io) > 0. Hence as conjectured by R. Aumann, if a pure strategy pair occurs with positive probability in a correlated equilibrium, then it occurs with positive probability in a pair of optimal strategies. Also, if one of the players has a unique optimal strategy, then every correlated equilibrium distribution concentrates on a pair of optimal strategies.
The authors extend E. Maskin's results on Nash implementation. First, they establish a condition that is both necessary and sufficient for Nash implementability if there are three or more agents (the case covered by Maskin's sufficiency result). Second--and more important--they examine the two-agent case (for which there existed no general sufficiency results). The two-agent model is the leading case for applications to contracting and bargaining. For this case, too, they establish a condition that is both necessary and sufficient. The authors use their theorems to derive simpler sufficiency conditions that are applicable in a wide variety of economic environments. Copyright 1990 by The Econometric Society.