Coherent risk measures in general economic models and price bubbles

In this article we study coherent risk measures in general economic models where the set of financial positions is an ordered Banach space E$E$ and the safe asset an order unit _x0$x_0$ of E$E$. First we study some properties of risk measures. We show that the set of normalized (with respect to _x0$x_0$) price systems is weak star compact and by using this result we prove a maximum attainment representation theorem which improves the one of Jaschke and Küchler (2001). Also we study how a risk measure changes under different safe assets and we show a kind of equivalence between these risk measures. In the sequel we study subspaces of E$E$ consisting of financial positions of risk greater or equal to zero and we call these subspaces unsure. We find some criteria and we give examples of these subspaces. In the last section, we combine the unsure subspaces with the theory of price-bubbles of Gilles and LeRoy (1992).     

One-step R-estimation in linear models with stable errors

Classical estimation techniques for linear models either are inconsistent, or perform rather poorly, under α$\alpha$-stable error densities; most of them are not even rate-optimal. In this paper, we propose an original one-step R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators remain root-n$n$ consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While parametric stable-likelihood estimation, due to the absence of a closed form for stable densities, is quite cumbersome, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values (_α0,_b0)$({\alpha }_0,b_0)$ of the tail index α$\alpha$ and skewness parameter b$b$, while preserving root-n$n$ consistency under any (α,b)$(\alpha ,b)$ as well as under usual light-tailed densities. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.     

Treatment effects in sample selection models and their nonparametric estimation

In a sample-selection model with the ‘selection’ variable Q$Q$ and the ‘outcome’ variable Y^∗$Y^{\ast }$, Y^∗$Y^{\ast }$ is observed only when Q=1$Q=1$. For a treatment D$D$ affecting both Q$Q$ and Y^∗$Y^{\ast }$, three effects are of interest: ‘participation  ’ (i.e., the selection) effect of D$D$ on Q$Q$, ‘visible performance  ’ (i.e., the observed outcome) effect of D$D$ on Y≡QY^∗$Y\equiv Q{Y}^{\ast }$, and ‘invisible performance  ’ (i.e., the latent outcome) effect of D$D$ on Y^∗$Y^{\ast }$. This paper shows the conditions under which the three effects are identified, respectively, by the three corresponding mean differences of Q$Q$, Y$Y$, and Y|Q=1$Y|Q=1$ (i.e., Y^∗|Q=1$Y^{\ast }|Q=1$) across the control (D=0$D=0$) and treatment (D=1$D=1$) groups. Our nonparametric estimators for those effects adopt a two-sample framework and have several advantages over the usual matching methods. First, there is no need to select the number of matched observations. Second, the asymptotic distribution is easily obtained. Third, over-sampling the control/treatment group is allowed. Fourth, there is a built-in mechanism that takes into account the ‘non-overlapping support problem’, which the usual matching deals with by choosing a ‘caliper’. Fifth, a sensitivity analysis to gauge the presence of unobserved confounders is available. A simulation study is conducted to compare the proposed methods with matching methods, and a real data illustration is provided.     

Quasi ML estimation of the panel AR(1) model with arbitrary initial conditions

In this paper we show that the Quasi ML estimation method yields consistent Random and Fixed Effects estimators for the autoregression parameter ρ$\rho$ in the panel AR(1) model with arbitrary initial conditions and possibly time-series heteroskedasticity even when the error components are drawn from heterogeneous distributions. We investigate both analytically and by means of Monte Carlo simulations the properties of the QML estimators for ρ$\rho$. The RE(Q)MLE for ρ$\rho$ is asymptotically at least as robust to individual heterogeneity and, when the data are i.i.d. and normal, at least as efficient as the FE(Q)MLE for ρ$\rho$. Furthermore, the QML estimators for ρ$\rho$ only suffer from a ‘weak moment conditions’ problem when ρ$\rho$ is close to one if the cross-sectional average of the variances of the errors is (almost) constant over time, e.g. under time-series homoskedasticity. However, in this case the QML estimators for ρ$\rho$ are still consistent when ρ$\rho$ is local to or equal to one although they converge to a non-normal possibly asymmetric distribution at a rate that is lower than N^1/2$N^{1/2}$ but at least N^1/4$N^{1/4}$. Finally, we study the finite sample properties of two types of estimators for the standard errors of the QML estimators for ρ$\rho$, and the bounds of QML based confidence intervals for ρ$\rho$.     

Panel unit root tests in the presence of a multifactor error structure

This paper extends the cross-sectionally augmented panel unit root test (CIPS$CIPS$) proposed by Pesaran (2007) to the case of a multifactor error structure, and proposes a new panel unit root test based on a simple average of cross-sectionally augmented Sargan–Bhargava statistics (CSB$CSB$). The basic idea is to exploit information regarding the m$m$ unobserved factors that are shared by k$k$ observed time series in addition to the series under consideration. Initially, we develop the tests assuming that m⁰$m^0$, the true number of factors, is known and show that the limit distribution of the tests does not depend on any nuisance parameters, so long as k≥m⁰−1$k\ge m^0-1$. Small sample properties of the tests are investigated by Monte Carlo experiments and are shown to be satisfactory. Particularly, the proposed CIPS$CIPS$ and CSB$CSB$ tests have the correct size for all   combinations of the cross section (N$N$) and time series (T$T$) dimensions considered. The power of both tests rises with N$N$ and T$T$, although the CSB$CSB$ test performs better than the CIPS$CIPS$ test for smaller sample sizes. The various testing procedures are illustrated with empirical applications to real interest rates and real equity prices across countries.     

Semiparametric binary regression models under shape constraints with an application to Indian schooling data

We consider estimation of the regression function in a semiparametric binary regression model defined through an appropriate link function (with emphasis on the logistic link) using likelihood-ratio based inversion. The dichotomous response variable Δ$\Delta$ is influenced by a set of covariates that can be partitioned as (X,Z)$(X,Z)$ where Z$Z$ (real valued) is the covariate of primary interest and X$X$ (vector valued) denotes a set of control variables. For any fixed X$X$, the conditional probability of the event of interest (Δ=1$\Delta =1$) is assumed to be a non-decreasing function of Z$Z$. The effect of the control variables is captured by a regression parameter β$\beta$. We show that the baseline conditional probability function (corresponding to X=0$X=0$) can be estimated by isotonic regression procedures and develop a likelihood ratio based method for constructing asymptotic confidence intervals for the conditional probability function (the regression function) that avoids the need to estimate nuisance parameters. Interestingly enough, the calibration of the likelihood ratio based confidence sets for the regression function no longer involves the usual χ²${\chi }^2$ quantiles, but those of the distribution of a new random variable that can be characterized as a functional of convex minorants of Brownian motion with quadratic drift. Confidence sets for the regression parameter β$\beta$ can however be constructed using asymptotically χ²${\chi }^2$ likelihood ratio statistics. The finite sample performance of the methods are assessed via a simulation study. The techniques of the paper are applied to data sets on primary school attendance among children belonging to different socio-economic groups in rural India.     

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model _Yi=exp(_Xi/2)_ηi$Y_i=exp(X_i/2){\eta }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where only (_Yi)$\left(Y_i\right)$ is observed. The model is rewritten as a particular hidden model: _Zi=_Xi+_εi$Z_i=X_i+{\epsilon }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where (_ξi)$\left({\xi }_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent sequences of i.i.d. noise. Moreover, the sequences (_Xi)$\left(X_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent and the distribution of ε$\epsilon$ is known. Then, our aim is to estimate the functions b$b$ and σ²${\sigma }^2$ when only observations _Z1,…,_Zn$Z_1,\dots ,Z_n$ are available. We propose to estimate bf$bf$ and (b²+σ²)f$(b^2+{\sigma }^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b$b$ and σ²${\sigma }^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.     

Estimating covariation: Epps effect,microstructure noise

This paper is about how to estimate the integrated covariance _〈X,Y〉T${〈X,Y〉}_T$ of two assets over a fixed time horizon [0,T]$[0,T]$, when the observations of X$X$ and Y$Y$ are “contaminated” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previous-tick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a two scales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.     

Essential supremum and essential maximum with respect to random preference relations

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X$X$ endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X$X$ endowed by a random preference relation to the space L⁰(X)$L^0\left(X\right)$ of X$X$-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.     

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.