Robust Stochastic Frontier Analysis: a Minimum Density Power Divergence Approach

Transcription

1 Robust Stochastic Frontier Analysis: a Minimum Density Power Divergence Approach Federico Belotti Giuseppe Ilardi CEIS, University of Rome Tor Vergata Bank of Italy Workshop on the Econometrics and Statistics of Efficiency Analysis: Recent Developments and Perspectives Lecce, June 21, 2015

2 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Motivation Maximum Likelihood (ML) remains the most widely used approach to estimate SF models parametrically. Computational simplicity and asymptotic efficiency but very poor robustness properties. Only Nonparametric literature on both deterministic and stochastic frontier models estimation deals with the robustness issue (See Wilson, 1993; Cazals et al., 2002; Florens and Simar, 2014, among others). Belotti, Ilardi Robust Stochastic Frontier Analysis

3 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Fragile behavior of ML in presence of extreme values Belotti, Ilardi Robust Stochastic Frontier Analysis

4 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Aim of the paper We propose to exploit a density-based minimum distance approach, the so-called Minimum Density Power Divergence (MDPD, Basu et al., 1998) to robustify the parametric estimation of Stochastic Frontier (SF) models. We investigate the small samples properties of MDPD focusing on the normal-half normal SF model. We also study the finite sample properties of a simple Hausman-like test for the presence of outliers. Belotti, Ilardi Robust Stochastic Frontier Analysis

5 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks The Density Power Divergences Consider a parametric family of models F θ, indexed by the unknown parameter θ Θ R k, possessing densities f θ ; let G be the class of all distributions G having densities g. The family of divergences by Basu et al. (1998), as a function of α, is d α(g, f ) = [ f 1+α θ (y) (1 + 1 α )g(y)f θ α (y) + 1 ] α g 1+α (y) dy (1) When α = 0, the integral in (1) is undefined d 0(g, f ) = lim α 0 d α(g, f ) = is the Kullback-Leibler (KL) divergence. g(y)log [ ] g(y) dy (2) f (y) Belotti, Ilardi Robust Stochastic Frontier Analysis

6 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Link between ML and MDPD estimators When α = 0, define T 0(G) the minimum KL divergence functional at G and assume that it exists and is unique. Then, given a random sample y 1,..., y N from G, T 0(G N ) maximizes logfθ (y)dg N (y), and is therefore the ML estimate of θ, where G N is the empirical distribution function. In the same way, define T α(g) the minimum density power divergence functional at G. Then, given the data, for each α ˆθ MDPD = arg min θ Θ is the MDPD estimator of θ. f 1+α θ (y)dy (1 + 1 α )n 1 N i=1 f α θ (y i) (3) Belotti, Ilardi Robust Stochastic Frontier Analysis

7 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks MDPD robusteness (α = 1) Belotti, Ilardi Robust Stochastic Frontier Analysis

8 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Properties of the MDPD estimators The parameter α controls the trade-off between robustness and efficiency. Asymptotic properties follows immediately from existing theory since the MDPD family of estimators are M-estimators (Huber, 1981), i.e. solve an equation of the form N ψ(yi, θ) with i=1 ψ(y i, θ) = u θ (y)fθ α (y) u θ (z)f 1+α θ (y)dy (4) and where u θ (y) = logf θ (y)/ θ is the maximum likelihood score function. Under mild regularity conditions, there exists ˆθ MDPD such that, as N 1 ˆθ MDPD is consistent for θ, and 2 N 1 2 (ˆθ MDPD θ) N (0, A 1 BA 1 ). Belotti, Ilardi Robust Stochastic Frontier Analysis

9 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Why the Density Power Divergences approach? Other divergences, like the Hellinger divergence (Beran, 1977), show strong robustness retaining first-order efficiency, but they force to use some form of non-parametric smoother to produce an estimate of the true density g. Among the huge class of density-based minimum distance methods, it has been shown that MDPD has strong robustness properties with a negligible loss in terms of asymptotic efficiency relative to ML (Basu et al., 1998; Lee and Sriram, 2013; Lee and Song, 2013; Kang and Lee, 2014). Can be easily applied to a wide range of SF models. Belotti, Ilardi Robust Stochastic Frontier Analysis

10 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks The SF model We consider the Aigner et al. (1977) model y i = x iβ + v i u i (5) where v i N (0, ψ 2 ) and u i N + (0, σ 2 ). Thus, ε = v i u i SN(0, σ, λ), with λ = σ/ψ and θ = (β, σ, λ). In this case the MDPD estimator is easily obtained by ˆθ MDPD = arg min θ Θ f (ε) 1+α dε (1 + 1 α )n 1 where the first integral has been numerically approximated using Gauss-Hermite quadrature. N f (ε i) α (6) i=1 Belotti, Ilardi Robust Stochastic Frontier Analysis

11 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Monte Carlo Design β = 0.5, σ = 0.5, ψ = 0.25 φ = 0, 0.01, 0.02, 0.03, 0.04, 0.05 Measurement error in y y i = x iβ + v i u i v i = φv 2i + (1 φ)v 1i v 1i N (0, ψ) v 2i N (0, 4ψ) Heterogeneous technology y i = φy 2i + (1 φ)y 1i y 1i = x iβ + v i u i y 2i = x i(4β) + v i u i Belotti, Ilardi Robust Stochastic Frontier Analysis

12 Results: φ = 0 Table : MSE N=500 α = 0 α = 0.25 α = 0.5 α = 1 β σ ψ E(u ε) N=1000 α = 0 α = 0.25 α = 0.5 α = 1 β σ ψ E(u ε)

13 Results: measurement error in y - β

14 Results: measurement error in y - σ

15 Results: measurement error in y - ψ

16 Results: measurement error in y - û

17 Results: heterogeneous technology - β

18 Results: heterogeneous technology - σ

19 Results: heterogeneous technology - ψ

20 Results: heterogeneous technology - û

21 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Testing for the presence of outliers Denote with ˆθ 1 the ML estimator and with ˆθ 2 the MDPD estimator with α = 1. Because of asymptotic normality of our estimators, an asymptotic test may be based on the test statistic ξ = ˆ ^V ˆ, where ˆ = ˆθ 1 ˆθ 2, ^V = D ^WD, D = [I k, I k ], and [ ] [S ^W = H 1 SH 1 H 1 = S 1 S 0 H 1 1 S ] [ ] 2 H 1 S 2 2 S 1 S 2 S H 1. 2 Under H 0 nξ d χ 2 k, where k = rank(v). Belotti, Ilardi Robust Stochastic Frontier Analysis

22 Small samples properties of the test

23 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks Concluding remarks and future directions MDPD appears to be quite promising in robustifying the SF model. This approach may be used to test for the presence of outliers. 1) Panel data extension; 2) Use MDPD in a semiparametric SF model. Belotti, Ilardi Robust Stochastic Frontier Analysis

24 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks References I Aigner, D., Lovell, C., and Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6(1): Basu, A., Harris, I. R., Hjort, H. L., and Jones, M. C. (1998). Robust and efficient estimation by minimizing a density power divergence. Biometrika, 85: Beran, R. (1977). Minimum hellingher distance estimation for parametric models. The Annals of Statistics, 5: Cazals, C., Florens, J.-P., and Simar, L. (2002). Nonparametric frontier estimation: a robust approach. Journal of Econometrics, 106(1):1 25. Florens, J.-P. and Simar, L. (2014). Nonparametric robust stochastic frontier analysis: a tikhonov regularization approach. ISBA Discussion Paper, 23. Huber, P. (1981). Robust Statistics. New York: Wiley. Kang, J. and Lee, S. (2014). Minimum density power divergence estimator for poisson autoregressive models. Computational Statistics and Data Analysis, 80(0): Belotti, Ilardi Robust Stochastic Frontier Analysis

25 Aim of the paper The Minimum Density Power Divergence framework Monte Carlo evidence Concluding remarks References II Lee, J. and Sriram, T. N. (2013). On the performance of l 2 e estimation in modelling heterogeneous count responses with extreme values. Journal of Statistical Computation and Simulation, pages Lee, S. and Song, J. (2013). Minimum density power divergence estimator for diffusion processes. Annals of the Institute of Statistical Mathematics, 65(2): Wilson, P. W. (1993). Detecting outliers in deterministic nonparametric frontier models with multiple outputs. Journal of Business and Economic Statistics, 11(3):pp Belotti, Ilardi Robust Stochastic Frontier Analysis

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