Anne Vanhems

Statistical inference for DEA estimators of directional distances

In productivity and efficiency analysis, the technical efficiency of a production unit is measured through its distance to the efficient frontier of the production set. The most familiar non-parametric methods use Farrell–Debreu, Shephard, or hyperbolic radial measures. These approaches require that inputs and outputs be non-negative, which can be problematic when using financial data. Recently, Chambers et al. (1998) have introduced directional distance functions which can be viewed as additive (rather than multiplicative) measures efficiency. Directional distance functions are not restricted to non-negative input and output quantities; in addition, the traditional input and output-oriented measures are nested as special cases of directional distance functions. Consequently, directional distances provide greater flexibility. However, until now, only free disposal hull (FDH) estimators of directional distances (and their conditional and robust extensions) have known statistical properties (Simar and Vanhems, 2012). This paper develops the statistical properties of directional d estimators, which are especially useful when the production set is assumed convex. We first establish that the directional Data Envelopment Analysis (DEA) estimators share the known properties of the traditional radial DEA estimators. We then use these properties to develop consistent bootstrap procedures for statistical inference about directional distance, estimation of confidence intervals, and bias correction. The methods are illustrated in some empirical examples.

Non‐Parametric Estimation of Exact Consumer Surplus with Endogeneity in Price

This paper analyses a structural microeconomic relation describing the exact consumer surplus in a non-parametric setting with endogenous prices. The exact consumer surplus can be characterized as the solution of a differential equation involving the observed demand function. The strategy put forward in this paper involves two steps: first, estimate the demand function with endogeneity using non-parametric IV, second, plug this estimator into the differential equation to estimate the exact consumer surplus. The rate of convergence for this estimator is derived and is shown to be faster than the rate for the underlying non-parametric IV regression estimator. Solving the differential equation smooths the demand estimator and leads to a faster rate of convergence. The implementation of the methodology is illustrated through a simulation study. Copyright (C) 2010 The Author(s). The Econometrics Journal (C) 2010 Royal Economic Society

Nonparametric Study Of Solutions Of Differential Equations

The solution of differential equations lies at the heart of many problems in structural economics. In econometrics the general nonparametric analysis of consumer welfare is historically the most obvious application, but there are also many applications in finance and other fields. This work considers the general nonparametric form for these problems and identification conditions. It derives a kernel-based estimator and shows consistency and asymptotic normality.In particular, the link with inverse problems allows us to define it in terms of a well-posed inverse problem and to stress the regularity properties of the estimated solution.I thank Jean-Pierre Florens, Richard Blundell, Eric Renault, and Christine Thomas-Agnan for stimulating conversations, suggestions, and advice. I am very grateful to Oliver Linton and two anonymous referees for most helpful comments.

Nonparametric instrumental regression with non-convex constraints

This paper considers the nonparametric regression model with an additive error that is dependent on the explanatory variables. As is common in empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. A classical example in microeconomics considers the consumer demand function as a function of the price of goods and the income, both variables often considered as endogenous. In this framework, the economic theory also imposes shape restrictions on the demand function, such as integrability conditions. Motivated by this illustration in microeconomics, we study an estimator of a nonparametric constrained regression function using instrumental variables by means of Tikhonov regularization. We derive rates of convergence for the regularized model both in a deterministic and stochastic setting under the assumption that the true regression function satisfies a projected source condition including, because of the non-convexity of the imposed constraints, an additional smallness condition.

Portfolio selection in a multi-moment setting: A simple Monte-Carlo-FDH algorithm

Abstract This paper proposes a non-parametric efficiency measurement approach for the static portfolio selection problem in a general inputs–outputs space, where inputs can include variance and kurtosis and outputs can include mean and skewness. Our work is in the vein of Briec, Kerstens, and Jokung (2007) and Jurzenko, Maillet, and Merlin (2006) who develop a directional distance (shortage function) approach to evaluate the performance of portfolios in Mean–Variance–Skewness and in Mean–Variance–Skewness–Kurtosis spaces. Our approach use the Free Disposal Hull (FDH) estimator to derive an algorithm avoiding the heavy and non-robust numerical optimization approaches suggested so far. This new approach is much faster, more robust to reach the optimum and more flexible since it can be extended to more general situations. We illustrate the algorithm with a data set on the French CAC 40 already used in the literature, to compare our method with the numerical optimization approaches. It appears that our approach is much more performant than the numerical optimization techniques, both in terms of computing time, but also in terms of robustness, avoiding the pitfall of local numerical optima.

Nonparametric Hedge Funds and Replication Indices Performance Analysis: A Robust Directional Application

Since the pioneering work by Treynor, Sharpe, and Jensen, many performance measures have been introduced and empirically applied for evaluating the performance of hedge funds (HF) and HF replication indices. Recently, production frontier methods have been used in this field (e.g., Gregoriou, Sedzro, and Zhu, 2005), since they do not require the specification of a benchmark (such as in standard multifactor models) and they do not assume any statistical properties of fund returns (e.g., normality assumption). In addition, they also have the considerable advantage of being multi-dimensional.