Knowledge that Transforms

This paper describes a semiparametric estimator for binary response models in which there may be arbitrary heteroskedasticity of unknown form. The estimator is obtained by maximizing a smoothed version of the objective function of C. Manski's maximum score estimator. The smoothing procedure is similar to that used in kernel nonparametric density estimation. The resulting estimator's rate of convergence in probability is the fastest possible under the assumptions that are made. The centered, normalized estimator is asymptotically normally distributed. Methods are given for consistently estimating the parameters of the limiting distribution and for selecting the bandwidth required by the smoothing procedure. Copyright 1992 by The Econometric Society.

THERE HAS BEEN MUCH INTEREST over the years in estimating discrete choice models. However, with the exception of multinomial logit models, agents have been restricted to few choices so that multivariate integrals could be feasibly integrated numerically.2 Pakes and Pollard (1989) and McFadden (1989) have independently developed the method of simulated moments (MSM) to deal with estimating a wide class of models of which high order discrete choice models are a subset. One of the problems a researcher must handle when using MSM for a discrete choice model is how to smooth the discrete simulated random variables. This is necessary to keep small the number of draws required to simulate derivatives and to simulate variation in the data. Geweke (1989) and McFadden (1989) have suggested importance sampling methods (and other methods) to smooth the simulated variables for general error structures. In this paper, I present a factor analytic smoothing method that can be applied to probit problems. Although it is not as general as the methods Geweke and McFadden suggest, it is easy to use and has clear intuition. Furthermore, simulated probabilities will be unbiased, will be bounded between zero and unity, and will have smaller variances than unsmoothed probabilities always and smaller variances than importance sampling estimates for a large class of probabilities. The second section of this paper develops notation, defines the problem, and presents the smoothing method and an algorithm to employ it. The last section presents Monte Carlo comparisons of the smoothing method to the importance sampling method.

The authors consider the problem of cost sharing in the case of a fixed group of agents sharing a one input, one output technology with decreasing returns. They introduce and analyze the serial cost sharing method. Among agents endowed with convex and monotonic preferences, serial cost sharing is dominance solvable and its unique Nash equilibrium is also robust to coalitional deviations. The authors show that no other smooth cost sharing mechanism yields a unique Nash equilibrium at all preference profiles. Copyright 1992 by The Econometric Society.

IN A STIMULATING RECENT PAPER, Abreu and Matsushima (1992) (hereafter A-M) show how a class of social choice functions can be virtually implemented in iteratively undominated strategies. This work has several important features: the mechanisms used are finite and not too difficult to understand, and so less objectionable in this regard than many in the literature; the class of social choice functions implemented is large; and the solution concept-Nash equilibrium determined uniquely by iterative elimination of strongly dominated strategies-is relatively uncontroversial. The point of this note is to argue that the mechanisms used by A-M unfortunately tend to generate games in which the iterative removal of strongly dominated strategies sometimes is indeed (or ought to be) controversial.2 We proceed by first examining an example of a related but simpler implementation problem in which the argument is easily exposed, then indicating how the argument applies generally in the A-M setup. Consider the following much-discussed two-player coordination game:

One elicitation diagnostic identifies a family of prior distributions that are so diffuse that they are practically equivalent to the completely diffuse prior. Another elicitation diagnostic identifies a family of prior distributions that concentrate enough mass in the neighborhood of zero that they are practically equivalent to the dogmatic prior that sets a parameter exactly equal to zero. If either question thus posed can be answered in the affirmative then there is no need to go to the expense of a more accurate elicitation of the prior distribution. Copyright 1992 by The Econometric Society.

The feasibility of Bertrand undercutting with nonuniform prices is established and properties are derived for Bertrand equilibrium in nonuniform price strategies. With free entry, equilibrium entails zero-profit minimum average cost production. If there is more than one producing firm, all prices collapse to a minimum average cost uniform price. An existence condition is compared to conditions from uniform price theory. Without free entry equilibrium, prices may not collapse to a uniform price. Positive profit may occur but all firms earn equal profit and incur equal marginal cost, while consumers pay average outlay no greater than marginal cost. Copyright 1992 by The Econometric Society.

This paper supposes an individual cares about his/her own wealth not only directly but also via the relative standing that this wealth induces. The implications for risk-taking are investigated in particular. Such a model provides a natural explanation of the "concave-convex-concave" utility described by M. Friedman and L. Savage (1948). However, there are a number of key differences between the present model and any model based on own wealth alone. For example, an equilibrium wealth distribution here may have a middle class. Further, the status interaction involves an externality and an equilibrium wealth distribution may be Pareto inefficient. Copyright 1992 by The Econometric Society.

The purpose of this note is to point out that the bimodality is a consequence of the special case considered by N-S. The exact finite sample distribution of the IV estimator has been derived in the economic literature earlier, but its tabulation or plotting involves some assumptions about the nuisance parameters. In this note we plot the densities of the exact finite sample distribution of the IV estimator for several parameter values and show that the bimodality feature noted by N-S is a consequence of a singular covariance matrix and not of a poor instrument.

This paper is concerned with a problem of implementation of a given social choice correspondence. The authors introduces an essential monotonicity condition and show that any implementable social choice correspondence satisfies this condition. Conversely, in a case of three or more participants, any essentially monotone social choice correspondence is implementable. In a case of two participants, the essential monotonicity condition must be completed by a requirement that the social choice correspondence is close to an individually rational correspondence. Copyright 1992 by The Econometric Society.