On Sharp Identification Regions for Regression Under Interval Data

The reliable analysis of interval data (coarsened data) is one of the most promising applications of imprecise probabilities in statistics. If one refrains from making untestable, and often materially unjustified, strong assumptions on the coarsening process, then the empirical distribution of the data is imprecise, and statistical models are, in Manski’s terms, partially identified. We first elaborate some subtle differences between two natural ways of handling interval data in the dependent variable of regression models, distinguishing between two different types of identification regions, called Sharp Marrow Region (SMR) and Sharp Collection Region (SCR) here. Focusing on the case of linear regression analysis, we then derive some fundamental geometrical properties of SMR and SCR, allowing a comparison of the regions and providing some guidelines for their canonical construction. Relying on the algebraic framework of adjunctions of two mappings between partially ordered sets, we characterize SMR as a right adjoint and as the monotone kernel of a criterion function based mapping, while SCR is indeed interpretable as the corresponding monotone hull. Finally we sketch some ideas on a compromise between SMR and SCR based on a set-domained loss function. This paper is an extended version of a shorter paper with the same title, that is conditionally accepted for publication in the Proceedings of the Eighth International Symposium on Imprecise Probability: Theories and Applications. In the present paper we added proofs and the seventh chapter with a small Monte-Carlo-Illustration, that would have made the original paper too long

A Characterization of the Distributions That Imply Existence of Linear Equilibria in the Kyle-Model

The existence of a linear equilibrium in Kyle's model of market making with multiple, symmetrically informed strategic traders is implied for any number of strategic traders if the joint distribution of the underlying exogenous random variables is elliptical. The reverse implication has been shown for the case in which the random variables are independent and have finite second moments. Here we extend this result to the case in which the underlying random variables are not necessarily independent and their joint distribution is determined by its moments

Matching Heterogeneous Agents with a Linear Search Technology

Steady state equilibria in heterogeneous agent matching models with search frictions have been shown to exist in Shimer and Smith (2000) under the assumption of a quadratic search technology. We extend their analysis to the commonly investigated linear search technology.Search, Matching, Steady State Equilibrium

On the Existence of Linear Equilibria in the Rochet-Vila Model of Market Making

This paper derives necessary and sucient conditions for the existence of linear equilibria in the Rochet-Vila model of market making. In contrast to most previous work on the existence of linear equilibria in models of market making, we do not impose independence of the underlying random variables. For distributions that are determined by their moments we show that a linear equilibrium exists if and only if the joint distribution of noise trade and asset payoff is elliptical.Market Microstructure, Market Making, Linear Equilibria

A Characterization of the Distributions That Imply Existence of Linear Equilibria in the Kyle-Model

The existence of a linear equilibrium in Kyle's model of market making with multiple, symmetrically informed strategic traders is implied for any number of strategic traders if the joint distribution of the underlying exogenous random variables is elliptical. The reverse implication has been shown for the case in which the random variables are independent and have finite second moments. Here we extend this result to the case in which the underlying random variables are not necessarily independent and their joint distribution is determined by its moments.Market Microstructure; Kyle Model; Linear

Existence of Linear Equilibria in the Kyle Model with Multiple Informed Traders

We consider Kyle's market order model of insider trading with multiple informed traders and show: if a linear equilibrium exists for two different numbers of informed traders, asset payoff and noise trading are independent and have finite second moments, then these random variables are normally distributed.insider trading; Kyle model; linear equilibrium; normal distribution

Complexity of Nested Circumscription and Nested Abnormality Theories

The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L_{CIRC}, which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains CONP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth International Joint Conference on Artificial Intelligence (IJCAI-01), pages 169--174. Morgan Kaufmann, 200

A Characterization of the Distributions That Imply Existence of Linear Equilbria in the Kyle-Model

The existence of a linear equilibrium in Kyle's model of market making with multiple, symmetrically informed strategic traders is implied for any number of strategic traders if the joint distribution of the underlying exogenous random variables is elliptical. The reverse implication has been shown for the case in which the random variables are independent and have finite second moments. Here we extend this result to the case in which the underlying random variables are not necessarily independent and their joint distribution is determined by its moments.Market Microstructure, Kyle Model, Linear Equilibria, Elliptical Distributions