Abolfazl Keshvari

Stochastic non-convex envelopment of data: Applying isotonic regression to frontier estimation

Isotonic nonparametric least squares (INLS) is a regression method for estimating a monotonic function by fitting a step function to data. In the literature of frontier estimation, the free disposal hull (FDH) method is similarly based on the minimal assumption of monotonicity. In this paper, we link these two separately developed nonparametric methods by showing that FDH is a sign-constrained variant of INLS. We also discuss the connections to related methods such as data envelopment analysis (DEA) and convex nonparametric least squares (CNLS). Further, we examine alternative ways of applying isotonic regression to frontier estimation, analogous to corrected and modified ordinary least squares (COLS/MOLS) methods known in the parametric stream of frontier literature. We find that INLS is a useful extension to the toolbox of frontier estimation both in the deterministic and stochastic settings. In the absence of noise, the corrected INLS (CINLS) has a higher discriminating power than FDH. In the case of noisy data, we propose to apply the method of non-convex stochastic envelopment of data (non-convex StoNED), which disentangles inefficiency from noise based on the skewness of the INLS residuals. The proposed methods are illustrated by means of simulated examples.

AN EXTENDED NUMERATION METHOD FOR SOLVING FREE DISPOSAL HULL MODELS IN DEA

Production Possibility Set (PPS) based on Free Disposal Hull assumption describes the minimum PPS for evaluating efficiency of DMUs and presents one reference for each unit. Tulkens (Journal of Productivity Analysis, 4(1), 183–210) proposed a mathematical program and a procedure for solving FDH model that can be used for only VRS technology. In this paper, we extend the method for solving all four standard technologies (VRS, CRS, NDRS and NIRS) by a numeration algorithm without using LP or MILP regular solving methods.

A penalized method for multivariate concave least squares with application to productivity analysis

We propose a penalized method for the least squares estimator of a multivariate concave regression function. This estimator is formulated as a quadratic programing (QP) problem with O(n2) constraints, where n is the number of observations. Computing such an estimator is a very time-consuming task, and the computational burden rises dramatically as the number of observations increases. By introducing a quadratic penalty function, we reformulate the concave least squares estimator as a QP with only non-negativity constraints. This reformulation can be adapted for estimating variants of shape restricted least squares, i.e. the monotonic-concave/convex least squares. The experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem. The Matlab and R codes for implementing the penalized problems are provided in the paper.

Discrete and Integer Valued Inputs and Outputs in Data Envelopment Analysis

Standard axioms of free disposability, convexity and constant returns to scale employed in Data Envelopment Analysis (DEA) implicitly assume continuous, real-valued inputs and outputs. However, the implicit assumption of continuous data will never hold with exact precision in real world data. To address the discrete nature of data explicitly, various formulations of Integer DEA (IDEA) have been suggested. Unfortunately, the axiomatic foundations and the correct mathematical formulation of IDEA technology has caused considerable confusion in the literature. This chapter has three objectives. First, we re-examine the axiomatic foundations of IDEA, demonstrating that some IDEA formulations proposed in the literature fail to satisfy the axioms of free disposability of continuous inputs and outputs, and natural disposability of discrete inputs and outputs. Second, we critically examine alternative efficiency metrics available for IDEA. We complement the IDEA formulations for the radial input measure with the radial output measure and the directional distance function. We then critically discuss the additive efficiency metrics, demonstrating that the optimal slacks are not necessarily unique. Third, we consider estimation of the IDEA technology under stochastic noise, modeling inefficiency and noise as Poisson distributed random variables.

Segmented concave least squares: A nonparametric piecewise linear regression

Abstract In this paper, segmented concave least squares (SCLS) is introduced. SCLS is a nonparametric piecewise linear regression problem in which the estimated function is (monotonic) concave and the number of linear segments (k) is pre-specified. Ordinary least squares (k = 1) and concave least squares (k = n, the number of observations) are two extreme cases of this problem. An application of SCLS is to estimate a hedonic function. Using this method, observations are categorized into k groups and a piecewise linear hedonic function is estimated such that there is one linear segment for every group. The estimated hedonic function holds the principle of diminishing marginal utility. In this paper, SCLS is used to categorize hotels in Finland into three groups. A trade-off between the number of groups and the goodness of fit measure is used to determine the number of groups. Based on the similarities of the pricing methods, hotels in the sample are endogenously classified and the shadow prices for each group are calculated. The results reveal that the hotels do not value hotel attributes similarly and there are significant differences among groups. Hedonic pricing model via SCLS provides a novel categorization of hotels that cannot be obtained by using ordinary least squares.

Solving cardinality constrained mean-variance portfolio problems via MILP

Controlling the number of active assets (cardinality of the portfolio) in a mean-variance portfolio problem is practically important but computationally demanding. Such task is ordinarily a mixed integer quadratic programming (MIQP) problem. We propose a novel approach to reformulate the problem as a mixed integer linear programming (MILP) problem for which computer codes are readily available. For numerical tests, we find cardinality constrained minimum variance portfolios of stocks in S&P500. A significant gain in robustness and computational effort by our MILP approach relative to MIQP is reported. Similarly, our MILP approach also competes favorably against cardinality constrained portfolio optimization with risk measures CVaR and MASD. For illustrations, we depict portfolios in a portfolio map where cardinality provides a third criterion in addition to risk and return. Fast solution allows an interactive search for a desired portfolio.