Knowledge that Transforms

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.

The authors compare two methods for a monopolist to sell information to traders in a financial market. In a direct sale, information buyers observe versions of the seller's signal while in an indirect sale the seller sells shares in a portfolio based on his private information. It is shown that, when traders are identical and pricing is linear, there is a trade-off between optimal surplus extraction that is possible under direct sale and more effective control of the usage of information that is possible under indirect sale. The optimal selling method depends on how much information is revealed by equilibrium prices. Copyright 1990 by The Econometric Society.

A method for computing (approximate) cost-of-living index numbers for arbitrary reference income levels is presented. It combines the use of fairly disaggregated data with a relatively modest use of econometric methods. The basis ingredients are (1) a series of chained Tornqvist price index numbers relating to per capital total expenditure, (2) price index numbers for the commodities distinguished, and (3) the income parameters of a differential demand system. This method attains second order precision and is equivalent to a full econometric method based on a translog utility function. The method is demonstrated on Netherlands data for 1952-81. Copyright 1990 by The Econometric Society.

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According to this paper, the analysis of the Nash bargaining problem with risky outcomes must include the cases where the disagreement outcome itself is risky as well. Thus, one generalizes the Roth and Rothblum results to models with risky disagreement outcome as well. It turns out that their result will depend, in some cases, upon the «degree of change» in risk aversion. Particularly, when the potential agreement, has an outcome which makes player 2 worse off compared to the disagreement, and if the disagreement does not dominate that outcome (i.e., its support contains a worse outcome), then when player 2 becomes «sufficiently» more risk averse player 1 becomes better off. Thus one obotains that increase in risk aversion may be disadvantageous to player 2 even if the risky disagreement is preferred to some outcome of the (risky) agreement