Partial identification using random set theory

This paper illustrates how the use of random set theory can benefit partial identification analysis. We revisit the origins of Manski’s work in partial identification (e.g., [Manski, 1989] and [Manski, 1990]) focusing our discussion on identification of probability distributions and conditional expectations in the presence of selectively observed data, statistical independence and mean independence assumptions, and shape restrictions. We show that the use of the Choquet capacity functional and the Aumann expectation of a properly defined random set can simplify and extend previous results in the literature. We pay special attention to explaining how the relevant random set needs to be constructed, depending on the econometric framework at hand. We also discuss limitations in the applicability of specific tools of random set theory to partial identification analysis.     

Random scheduling with deadlines under dichotomous preferences

A group of individuals share a deterministic server which is capable of serving one job per unit of time. Every individual has a job and a cut off time slot (deadline) where service beyond this slot is as worthless as not getting any service at all. Individuals are indifferent between slots which are not beyond their deadlines (compatible slots). A schedule (possibly random) assigns the set of slots to individuals by respecting their deadlines. We only consider the class of problems where for every set of relevant slots (compatible with at least one individual) there are at least as many individuals who have a compatible slot in that set: we ignore the case of underdemand. For this class, we characterize the random scheduling rule which attaches uniform probability to every efficient deterministic schedule (efficient uniform rule) by Pareto efficiency, equal treatment of equals, and probabilistic consistency (Chambers, 2004). We also show that a weaker version of the probabilistic consistency axiom is enough to achieve our result. Finally we show that efficient uniform rule is strategyproof.     

Weighted majoritarian rules for the location of multiple public facilities

We consider collective decision problems given by a profile of single-peaked preferences defined over the real line and a set of pure public facilities to be located on the line. In this context, Bochet and Gordon (2012) provide a large class of priority rules based on efficiency, object-population monotonicity and sovereignty. Each such rule is described by a fixed priority ordering among interest groups. We show that any priority rule which treats agents symmetrically — anonymity — respects some form of coherence across collective decision problems — reinforcement — and only depends on peak information — peak-only — is a weighted majoritarian rule. Each such rule defines priorities based on the relative size of the interest groups and specific weights attached to locations. We give an explicit account of the richness of this class of rules.     

How risky is a random process?

The riskiness of random processes is compared by (a) employing a decision theoretic equivalence between processes and lotteries on path-spaces to identify the riskiness of the former with that of the latter, and (b) using the theory of comparative riskiness of lotteries over vector spaces to compare the riskiness of lotteries on a given path-space. We derive the equivalence used in step (a) and contribute a new criterion to the theory applied in step (b). The validity of the new criterion, which applies second order stochastic dominance to utility distributions, is established by showing its equivalence to the benchmark decision theoretic criterion when comparing the riskiness of lotteries over any vector space. We demonstrate the theory’s tractability via diverse economic applications.     

Random assignments of bundles

We propose a model studying the random assignments of bundles with no free disposal. The key difference between our model and the one where objects are allocated (see Bogomolnaia and Moulin (2001)) is one of feasibility. The implications of this difference are significant. Firstly, the characterization of sd-efficient random assignments is more complex. Secondly, we are able to identify a preference restriction, called essential monotonicity, under which the random serial dictatorship rule (extended to the setting with bundles) is equivalent to the probabilistic serial rule (extended to the setting with bundles). This equivalence implies the existence of a rule on this restricted domain satisfying sd-efficiency, sd-strategy-proofness, and equal treatment of equals. Moreover, this rule only selects random assignments which can be decomposed as convex combinations of deterministic assignments.     

The design of feedback rules in linear stochastic rational expectations models

This paper is concerned with optimal and sub-optimal feedback rules for linear stochastic continuous time models with rational expectations. We consider four types of feedback rules: (1) the optimal but time-inconsistent rule which is available if the controller is able to commit himself or herself; (2) quasi-optimal and time-inconsistent rules of the form w = Dy where w is the vector of instruments, y the state vector and D a matrix of constants possibly with constraints; (3) the optimal time-consistent rule which is also linear in y; (4) ‘over-stable’ rules which have ‘too many’ stable roots. We show that rules of type (1) can be expressed and implemented as a form of integral control, all except type (2) satisfy certainty equivalence and that rules of type (4) will always be inferior to the optimal rule (1). These results are demonstrated in two illustrative examples.     

Project financing,entrepreneurial activity,and investment in the presence of asymmetric information

We analyze a two-period signaling model in which a representative entrepreneur in a regional economy has a project that generates a random cash flow and that requires investment that the entrepreneur raises from a competitive market. The project's type is known to the entrepreneur but not to the investors. Further, the entrepreneur is restricted to issuing debt only or equity only. We first show that there is no separating perfect Bayesian equilibrium (PBE) contract involving the issuance of equity only, that there exists a pooling PBE contract involving the issuance of equity only, and that a debt contract is preferred to an equity contract by our entrepreneur. Next, we suppose that the entrepreneur incurs a non-pecuniary cost of financial distress F > 0 whenever he is unable to make a repayment at time t = 1. We provide conditions on F under which a pooling PBE contract with debt exists and a separating PBE contract with debt and equity exists. Finally, we examine whether a high type entrepreneur will prefer a setting with a cost of financial distress (F > 0) or a setting in which there is no such cost (F = 0).     

The structure of strategy-proof random social choice functions over product domains and lexicographically separable preferences

We characterize the class of dominant-strategy incentive-compatible (or strategy-proof) random social choice functions in the standard multi-dimensional voting model where voter preferences over the various dimensions (or components) are lexicographically separable. We show that these social choice functions (which we call generalized random dictatorships) are induced by probability distributions on voter sequences of length equal to the number of components. They induce a fixed probability distribution on the product set of voter peaks. The marginal probability distribution over every component is a random dictatorship. Our results generalize the classic random dictatorship result in Gibbard (1977) and the decomposability results for strategy-proof deterministic social choice functions for multi-dimensional models with separable preferences obtained in LeBreton and Sen (1999).     

Bounds for the mean residual life function of a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X _t  = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.     

Increasing sales by introducing non‐salable items

Rationality implies that adding ‘irrelevant’ and, in particular, inferior alternatives to the opportunity set cannot increase the choice probability of some other alternative. In this study, we propose a novel approach that can rationalize an intended addition of such alternatives because it strictly increases the choice probability of some existing alternative. The driving force behind the existence and extent of such an increase is the random nature of individual preferences, that implies intransitivity, and the random nature of the applied choice procedures. We study the case of a firm interested in increasing the sales of some of its existing products by introducing a new and inferior (non‐salable) product. Our main results focus on the feasibility and potential advantage of a successful such strategy. We first establish necessary and sufficient conditions for an increase in the sale probability and then derive the maximal possible absolute and relative increase in this probability, when the firm has extremely limited information on the characteristics of the consumers. We then derive analogous results, assuming that the existing line of products consists of just two items and that the firm has accurate information on the consumers' stochastic preferences over the existing products. These later results are illustrated using some experimental evidence. The applicability of the approach is finally briefly discussed in the context of branding policy. Copyright © 2006 John Wiley & Sons, Ltd.     

On stable rules for selecting committees

A voting rule is said to be stable if it always elects a fixed-size subset of candidates such that there is no outside candidate who is majority preferred to any candidate in this set whenever such a set exists. Such a set is called a Weak Condorcet Committee (WCC). Four stable rules have been proposed in the literature. In this paper, we propose two new stable rules. Since nothing is known about the properties of the stable rules, we evaluate all the identified stable rules on the basis of some appealing properties of voting rules. We show that they all satisfy the Pareto criterion and they are not monotonic. More, we show that every stable rule fails the reinforcement requirement.     

Ex-post incentive compatible and individually rational assignments in housing markets with interdependent values

This paper presents a model for housing markets with interdependent values. We introduce private information on the quality of a house (i.e., high or low), which is known only to the initial owner. Interdependency means that the ex-post preference of an agent depends on the private information of the other agents with regard to the quality of houses. We prove that on a domain satisfying a richness condition, the no-trade rule is the only rule that satisfies ex-post incentive compatibility and ex-post individual rationality.     

An optional scrambled randomized response technique for practical surveys

Eichhorn and Hayre (J Stat Plan Inference 7:307–316, 1983) introduced the scrambled response technique to gather information on sensitive quantitative variables. Singh and Joarder (Metron 15:151–157, 1997), Gupta et al. (J Stat Plan Inference 100:239–247, 2002) and Bar-Lev et al. (Metrika 60:255–260, 2004) permitted the respondents either to report their true values on the sensitive quantitative variable or the scrambled response and developed the optional randomized response (ORR) technique based on simple random sample with replacement (SRSWR). While developing the ORR procedure, these authors made the assumption that the probability of disclosing the true response or the randomized response (RR) is the same for all the individuals in a population. This is not a very realistic assumption as in practical survey situations the probability of reporting the true value or the RR generally varies from unit to unit. Moreover, if one generalizes the ORR method as developed by these authors relaxing the ‘constant probability’ assumption, the variance of an unbiased estimator for the population total or mean can not be estimated as this involves the unknown parameter, ‘the probability of revealing the true response’. Here we propose a modified ORR procedure for stratified unequal probability sampling after relaxing the assumption of ‘constant probability’ of providing the true response. It is also demonstrated with a numerical exercise that our procedure produces better estimator for a population total than that provided by the method suggested by the earlier authors.     

Condorcet meets Bentham

We show that if the statistical distribution of utility functions in a population satisfies a certain condition, then a Condorcet winner will not only exist, but will also maximize the utilitarian social welfare function. We also show that, if people’s utility functions are generated according to certain plausible random processes, then in a large population, this condition will be satisfied with very high probability. Thus, in a large population, the utilitarian outcome will be selected by any Condorcet consistent voting rule.     

Uncertain tax rules and futures hedging

This paper considers the optimal futures hedging decision under uncertain tax treatment. If the Corn Products (CP) rule applies, gains or losses from futures trading can offset business gains or losses. However, under the Arkansas Best (AB) doctrine, offsetting is not allowed. We show that the risk neutral firm will not trade futures contracts if the probability the CP rule prevails is small. When the probability is sufficiently large, the firm will assume an underhedge. A risk averse firm is likely to trade, even if the AB rule prevails. As long as the CP ruling is not a sure thing, the firm will engage in underhedge. The effects of average business profits, the volatility of business profits, and risk aversion on the optimal futures position are provided. Copyright © 1999 John Wiley & Sons, Ltd.     

Some remarks on the construction of independent identically distributed random variables,Markov chains and martingales

In this note, we make some remarks on the construction of sequences of independent identically distributed random variables and of Markov chains concretely on a probability space (Ω,A,P). We also show that there are non non-trivial martingales of exponential type.     

Exchange-rate forecasts with simultaneous nearest-neighbour methods: evidence from the EMS

In this paper we extend nearest-neighbour predictors to allow for information content in a wider set of simultaneous time series. We apply these simultaneous nearest-neighbour (SNN) predictors to nine EMS currencies, using daily data for the 1st January 1978–31st December 1994 period. When forecasting performance is measured by Theil's U statistic, the (nonlinear) SNN predictors perform marginally better than both a random walk and the traditional (linear) ARIMA predictors. Furthermore, the SNN predictors outperform the random walk and the ARIMA models when producing directional forecasts.When formally testing for forecast accuracy, in most of the cases the SNN predictor outperforms the random walk at the 1% significance level, while outperforming the ARIMA model in three of the nine cases. On the other hand, our results suggest that the probability of correctly predicting the sign of change is higher for the SNN predictions than the ARIMA case.     

On the natural selection rule in general linear models

We consider a problem of selecting the best treatment in a general linear model. We look at the properties of the natural selection rule. It is shown that the natural selection rule is minimax under to “0–1” loss function and it is a Bayes rule under a monotone permutation invariant loss function with respect to a permutation invariant prior for every variance balanced design. Some other condition on the design matrix is given so that a Bayes rule with respect to a normal prior will be of simple structure.     

On the properties of the coefficient of determination in regression models with infinite variance variables

We examine the asymptotic properties of the coefficient of determination, R²$R^2$, in models with α-stable$\alpha \text{-stable}$   random variables. If the regressor and error term share the same index of stability α<2$\alpha <2$, we show that the R²$R^2$  statistic does not converge to a constant but has a nondegenerate distribution on the entire [0,1]$[0,1]$ interval. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable, and we show that the density function is unbounded at 0 and 1. If the indices of stability of the regressor and error term are unequal, we show that the coefficient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability, irrespective of the value of the slope coefficient. In an empirical application, we revisit the Fama and MacBeth (1973) two-stage regression and demonstrate that in the infinite-variance case the R²$R^2$  statistic of the second-stage regression converges to 0 in probability even if the slope coefficient is nonzero. We deduce that a small value of the R²$R^2$  statistic should not, in itself, be used to reject the usefulness of a regression model.