Local disaggregation of negative demand and excess demand functions

The literature on the characterization of aggregate excess and market demand has generated three types of results: global, local, or ‘at a point’. In this note, we study the relationship between the last two approaches. We prove that within the class of functions satisfying standard conditions and whose Jacobian matrix is negative semi-definite, only n/2+1$n/2+1$ agents are needed for the ‘at’ decomposition. We ask whether, within the same class, the ‘around’ decomposition also requires only n/2+1$n/2+1$ agents.     

High dimensional covariance matrix estimation using a factor model

High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality p$p$ tends to ∞$\infty$ as the sample size n$n$ increases. Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model is employed to reduce dimensionality and to estimate the covariance matrix. The factors are observable and the number of factors K$K$ is allowed to grow with p$p$. We investigate the impact of p$p$ and K$K$ on the performance of the model-based covariance matrix estimator. Under mild assumptions, we have established convergence rates and asymptotic normality of the model-based estimator. Its performance is compared with that of the sample covariance matrix. We identify situations under which the factor approach increases performance substantially or marginally. The impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied. The asymptotic results are supported by a thorough simulation study.     

The first-order approach when the cost of effort is money

We provide sufficient conditions for the first-order approach in the principal-agent problem when the agent’s utility has the nonseparable form u(y−c(a))$u(y-c\left(a\right))$ where y$y$ is the contractual payoff and c(a)$c\left(a\right)$ is the money cost of effort. We first consider a decision-maker facing prospects which cost c(a)$c\left(a\right)$ and with distributions of returns y$y$ that depend on a$a$. The decision problem is shown to be concave if the primitive of the cdf of returns is jointly convex in a$a$ and y$y$, a condition we call Concavity of the Cumulative Quantile (CCQ) and which is satisfied by many common distributions. Next we apply CCQ to the distribution of outcomes (or their likelihood-ratio transforms) in the principal-agent problem and derive restrictions on the utility function that validate the first-order approach. We also discuss another condition, log-convexity of the distribution, and show that it allows binding limited liability constraints, which CCQ does not.     

Risk aversion for variational and multiple-prior preferences

The objective of this paper is to identify variational preferences and multiple-prior (maxmin) expected utility functions that exhibit aversion to risk under some probability measure from among the priors. Risk aversion has profound implications on agents’ choices and on market prices and allocations. Our approach to risk aversion relies on the theory of mean-independent risk of Werner (2009). We identify necessary and sufficient conditions for risk aversion of convex variational preferences and concave multiple-prior expected utilities. The conditions are stability of the cost function and of the set of probability priors, respectively, with respect to a probability measure. The two stability properties are new concepts. We show that cost functions defined by the relative entropy distance or other divergence distances have that property. Set of priors defined as cores of convex distortions of probability measures or neighborhoods in divergence distances have that property, too.     

Calibrating the wealth effects of decoupled payments: Does decreasing absolute risk aversion matter?

Arrow’s hypotheses regarding the relationship between wealth and risk aversion measures have formed the basis for a large body of empirical research and theory. For example, many have suggested that decoupled farm subsidy payments may increase production as they decrease farmers’ risk aversion. This paper develops a new calibration technique designed to measure the minimum change in concavity of a utility of wealth function necessary to describe a particular change in production behavior for some discrete change in wealth. I conclude that measurable changes in production levels should not be produced by changing levels of risk aversion except when wealth changes are a substantial portion of wealth. This tool draws into question the usefulness of Arrow’s hypotheses in many current applications.     

Core and competitive equilibria of a coalitional exchange economy with infinite time horizon

The core and competitive equilibria of a large exchange economy on the commodity space ?^∞${?}^{\infty }$ will be discussed. We define the economy as a measure on the space of consumers’ characteristics following Hart and Kohlberg (1974), and prove the existence of competitive equilibria and their equivalence with the core without assuming the convexity of preferences.     

Tests of risk premia in linear factor models

We show that statistical inference on the risk premia in linear factor models that is based on the Fama–MacBeth (FM) and generalized least squares (GLS) two-pass risk premia estimators is misleading when the β$\beta$’s are small and/or the number of assets is large. We propose novel statistics, that are based on the maximum likelihood estimator of Gibbons [Gibbons, M., 1982. Multivariate tests of financial models: A new approach. Journal of Financial Economics 10, 3–27], which remain trustworthy in these cases. The inadequacy of the FM and GLS two-pass t$t$/Wald statistics is highlighted in a power and size comparison using quarterly portfolio returns from Lettau and Ludvigson [Lettau, M., Ludvigson, S., 2001. Resurrecting the (C)CAPM: A cross-sectional test when risk premia are time-varying. Journal of Political Economy 109, 1238–1287]. The power and size comparison shows that the FM and GLS two-pass t$t$/Wald statistics can be severely size distorted. The 95% confidence sets for the risk premia in the above-cited work that result from the novel statistics differ substantially from those that result from the FM and GLS two-pass t$t$-statistics. They show support for the human capital asset pricing model although the 95% confidence set for the risk premia on labor income growth is unbounded. The 95% confidence sets show no support for the (scaled) consumption asset pricing model, since the 95% confidence set of the risk premia on the scaled consumption growth consists of the whole real line, but do not reject it either.     

A note on fourth-order risk aversion

Keenan et al. (J Risk Uncertain 24:264–277, 2002) introduced a measure of downside risk aversion (third-order risk aversion), and in Theorem 1, they showed four equivalent definitions of increased downside risk aversion. This result is thought as a higher-order extension of Theorem 3 in Diamond et al. (J Econ Theory 8:337–360, 1974). We consider fourth-order risk aversion and show four equivalent definitions of increased fourth-order risk aversion. Our result is thought as a higher-order extension of Theorem 1 in Keenan et al. (2002).     

Developments in non-expected utility theories: an empirical study of risk aversion

This paper investigates different developments in non-expected utility theories. Our focus is to study the agent’s attitude towards risk in a context of monetary gambles. Based on simulated data of the “Deal or No Deal” TV game show, we first compare the performance of the expected utility model versus a loss-aversion model. We find that the loss-aversion model has a better performance compared to the expected utility model. We then study the attitude towards risk according to two parameters: the relative risk aversion coefficient defined over the value function and the probability weighting coefficient proposed by the Cumulative Prospect Theory. We find evidence for probability weighting being undertaken by contestants reflecting less risk aversion over large stakes. We also explore the performance of two models of rank-dependant utility: the Quiggin (1982) and the power probability weighting models. We find that the probability weighting coefficient is still significant for both models. Finally, we integrate initial wealth into the contestants’ preferences function and we show that the initial wealth level affects the estimates of risk attitudes.     

Asymptotics of the principal components estimator of large factor models with weakly influential factors

This paper introduces a drifting-parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components (PC) estimator in situations when the factors’ explanatory power does not strongly dominate the explanatory power of the cross-sectionally and temporally correlated idiosyncratic terms. Under our asymptotics, the PC estimator is inconsistent. We find explicit formulae for the amount of the inconsistency, and propose an estimator of the number of factors for which the PC estimator works reasonably well. For the special case when the idiosyncratic terms are cross-sectionally but not temporally correlated (or vice versa), we show that the coefficients in the OLS regressions of the PC estimates of factors (loadings) on the true factors (true loadings) are asymptotically normal, and find explicit formulae for the corresponding asymptotic covariance matrix. We explain how to estimate the parameters of the derived asymptotic distributions. Our Monte Carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n$n$ and T$T$. We apply our theoretical results to test a hypothesis about the factor content of the US stock return data.     

Informativeness of experiments for meu

The celebrated Blackwell’s theorem demonstrates the equivalence of a notion of statistical informativeness and economic valuableness for the class of preferences that are represented by the subjective expected utility. This note shows that this equivalence holds for a larger class of preferences, namely maxmin$maxmin$ expected utility.     

Edgeworth expansions and normalizing transforms for inequality measures

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second-order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second-order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n⁻¹)$O\left(n^{-1}\right)$. This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n^−3/2)$O\left(n^{-3/2}\right)$. We illustrate our procedure with an application to regional inequality measurement in Côte d’Ivoire.     

Relevant coherent measures of risk

We introduce and study the _f0$f_0$-relevance property of a coherent measure of risk on a positions vector space with vector ordering. We show that it is equivalent to a special no arbitrage condition on bounded positions spaces. Continuity from below leads to representations of _f0$f_0$-relevant coherent measures of risk based on equivalent functionals in Banach subspaces of the order dual. We define and describe _f0$f_0$-martingales in a lattice, and present a solution to the hedging price problem: the asset price process is an order convergent _f0$f_0$-martingale. Under the _f0$f_0$-relevance hypothesis we study the relationship between worst conditional mean and value at risk.