On the existence of equilibria in economies with an infinite dimensional commodity space

We prove an infinite dimensional extension of the Gale–Nikaido–Debreu lemma which includes all necessary limiting processes and allows a proof of the existence of equilibria under standard assumptions in an economy with infinitely many commodities which exactly parallels the proof of Debreu (1959) for the finite dimensional case.     

Equilibria in markets with a Riesz space of commodities

We present a new proof of the existence of competitive equilibrium for an economy with an infinite dimensional space of commodities.     

Unit root log periodogram regression

Log periodogram (LP) regression is shown to be consistent and to have a mixed normal limit distribution when the memory parameter d=1$d=1$. Gaussian errors are not required. The proof relies on a new result showing that asymptotically infinite collections of discrete Fourier transforms (dft's) of a short memory process at the fundamental frequencies in the vicinity of the origin can be treated as asymptotically independent normal variates, provided one does not include too many dft's in the collection.     

Nonlinear supporting prices : The superadditive case

Assumptions on allocations and preferences sufficient to allow superadditive price support are considered for an exchange economy with a finite number of traders when the commodity space is an ordered topological vector space. The main requirements are a uniformmonotonicity assumption on preferences and that no permutation of the allocation among agents yields a Pareto improvement. No convexity assumption on preferences is used, the positive orthant need not have interior, while the proof is constructive. Applications to finite and infinite dimensional commodity spaces are discussed.     

A regularization approach to the many instruments problem

This paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite. In order to improve the small sample properties of standard instrumental variable (IV) estimators, we propose three modified IV estimators based on three different ways of inverting the covariance matrix of the instruments. These inverses involve a regularization or smoothing parameter. It should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation. We show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound. Higher-order analysis of the MSE reveals that the bias of the modified estimators does not depend on the number of instruments. Finally, we suggest a data-driven method for selecting the regularization parameter. Interestingly, our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument.     

Some new characterization of rational expectation equilibria in economies with asymmetric information

We compare the family of rational expectation equilibria with the ex-post Aubin core allocations in the framework of an economy with uncertainty and asymmetric information. We do it in a very general setting: an arbitrary space of agents and an infinite dimensional commodity space. We also present a characterization of the ex–post Aubin core by allocations belonging to the core of the associated complete information economies, and we use a suitable definition of Edgeworth equilibria to characterize the family of rational expectation equilibria. Moreover, we study some generalizations of the ex-post Aubin core obtained by making some restriction on the families of blocking coalitions.     

Functional partially linear quantile regression model

This paper considers estimation of a functional partially quantile regression model whose parameters include the infinite dimensional function as well as the slope parameters. We show asymptotical normality of the estimator of the finite dimensional parameter, and derive the rate of convergence of the estimator of the infinite dimensional slope function. In addition, we show the rate of the mean squared prediction error for the proposed estimator. A simulation study is provided to illustrate the numerical performance of the resulting estimators.     

An Introduction to Best Empirical Models when the Parameter Space is Infinite Dimensional*

Ploberger and Phillips (Econometrica, Vol. 71, pp. 627–673, 2003) proved a result that provides a bound on how close a fitted empirical model can get to the true model when the model is represented by a parameterized probability measure on a finite dimensional parameter space. The present note extends that result to cases where the parameter space is infinite dimensional. The results have implications for model choice in infinite dimensional problems and highlight some of the difficulties, including technical difficulties, presented by models of infinite dimension. Some implications for forecasting are considered and some applications are given, including the empirically relevant case of vector autoregression (VAR) models of infinite order.     

Existence of maximal elements and equilibria in linear topological spaces

We present some mathematical theorems which are used to generalize previous results on the existence of maximal elements and of equilibrium. Our main theorem in this paper is a new existence proof for an equilibrium in an abstract economy, which is closely related to a previous result of Borglin–Keiding, and Shafer–Sonneschein, but allows for an infinite number of commodities and a countably infinite number of agents.     

On games with incomplete information and the Dvoretsky–Wald–Wolfowitz theorem with countable partitions

It has remained an open question as to whether the results of Milgrom–Weber [Milgrom, P.R., Weber, R.J., 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10, 619–632] are valid for action sets with a countably infinite number of elements without additional assumptions on the abstract measure space of information. In this paper, we give an affirmative answer to this question as a consequence of an extension of a theorem of Dvoretzky, Wald and Wolfowitz (henceforth DWW) due to Edwards [Edwards, D.A., 1987. On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29, 205–220]. We also present a direct elementary proof of the DWW theorem and its extension, one that may have an independent interest.     

Subsampling Inference with K Populations and a Non‐standard Behrens–Fisher Problem

We revisit the methodology and historical development of subsampling, and then explore in detail its use in hypothesis testing, an area which has received surprisingly modest attention. In particular, the general set‐up of a possibly high‐dimensional parameter with data from K populations is explored. The role of centring the subsampling distribution is highlighted, and it is shown that hypothesis testing with a data‐centred subsampling distribution is more powerful. In addition we demonstrate subsampling’s ability to handle a non‐standard Behrens–Fisher problem, i.e., a comparison of the means of two or more populations which may possess not only different and possibly infinite variances, but may also possess different distributions. However, our formulation is general, permitting even functional data and/or statistics. Finally, we provide theory for K ‐ sample U ‐ statistics that helps establish the asymptotic validity of subsampling confidence intervals and tests in this very general setting.     

The number of equilibria of smooth infinite economies

We construct an index theorem for smooth infinite economies that shows that generically the number of equilibria is odd. As a corollary, this gives a new proof of existence and gives conditions that guarantee global uniqueness of equilibria.     

Asymptotic expansion of S-estimators of location and covariance

By means of a straightforward application of empirical process theory, we show that S-estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent vector and matrix valued random elements respectively. This provides an alternative proof of asymptotic normality of S-estimators and clearly explains the limiting covariance structure. It also leads to a relatively simple proof of asymptotic normality of the length of the shortest α-fraction.     

The equilibrium set of infinite dimensional Walrasian economies and the natural projection

The natural projection plays a fundamental role to understand the behavior of the Walrasian economies. In this paper, we extend this method to analyze the behavior of infinite dimensional economies. We introduce the definition of the social equilibrium set, and we show that there exists a bijection between this set and the Walrasian equilibrium set of an infinite dimensional economy. In order to describe the main topological characteristics of both sets, we analyze the main differential characteristics of the excess utility function and then, we extend the method of the natural projection as suggested by Y. Balasko.     

On the asymptotic Fisher information in order statistics

Metrika - We extend the result of Efron and Johnstone (1990), who expressed the Fisher information in terms of the hazard function, to express the Fisher information in order statistics as an...     

The solution of the infinite horizon tracking problem for discrete time systems possessing an exogenous component

In this paper we derive an algorithm that yields, for a discrete-time system, a control minimizing a quadratic cost functional. The system considered is linear and possesses an exogenous component. The cost functional is a quadratic tracking equation over an infinite time horizon with positive semi-definite weighting matrices such that a weighted sum of these matrices is positive definite. The infinite planning horizon Minimum Variance cost criterion and the Linear Quadratic regulator are special cases. For stabilizable systems we give a characterization of the asymptotically admissible reference trajectories.     

Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences

We prove the existence of a competitive equilibrium in a production economy with infinitely many commodities and a measure space of agents whose preferences are price dependent. We employ a saturated measure space for the set of agents and apply recent results for an infinite dimensional separable Banach space such as Lyapunov’s convexity theorem and an exact Fatou’s lemma to obtain the result.     

Testing conditional independence via empirical likelihood

We construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter. One class is based on the CDF; another is based on smoother functions. We show that the test statistics are asymptotically normal under the null hypothesis and a sequence of Pitman local alternatives. We also show that the tests possess an asymptotic optimality property in terms of average power. Simulations suggest that the tests are well behaved in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.     

Fisher information in order statistics and their concomitants in bivariate censored samples

We evaluate the Fisher information (FI) contained in a collection of order statistics and their concomitants from a bivariate random sample. Special attention is given to Type II censored samples. We present a general decomposition result and recurrence relations that are useful in finding the FI in all types of censored samples. We also obtain some asymptotic results for the FI. For the bivariate normal parent, we obtain explicit and asymptotic expressions for the elements of the FI matrix for Type II censored samples. We discuss implications of our findings on inference on the bivariate normal parameters, especially on the correlation. The first author’s research was supported in part by National Institutes of Health, USA, Grant # M01 RR00034 and the second author’s research was supported by a training grant from the Egyptian government     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.