Estimation of Theoretically Consistent Stochastic Frontier Functions - PowerPoint PPT Presentation

Estimation of Theoretically Consistent Stochastic Frontier Functions in R Arne Henningsen Department of Agricultural Economics University of Kiel, Germany

Outline Theoretically Consistent Stochastic Frontier Functions Arne Henningsen Stochastic Frontier Analysis Introduction Theoretical Consistency Stochastic Frontier Restricted Estimation of Frontier Functions Analysis Theoretical (Empirical Example) Consistency Summary and Outlook Restricted Estimation Empirical Example Summary and Outlook 2 / 12

Stochastic Frontier Analysis Theoretically Consistent Stochastic Frontier Production economics Functions Assumption of traditional empirical analyses: Arne Henningsen all producers always manage to optimize their production process Introduction ⇒ All departures from the optimum Stochastic Frontier = random statistical noise Analysis ⇒ y = f ( x , β ) + v , e.g. with v ∼ N (0 , σ 2 ) Theoretical Consistency Practice: producers do not always succeed in optimizing Restricted their production Estimation Empirical Stochastic Frontier Analysis (SFA) accounts for failures in Example optimization (Meeusen & van den Broeck 1977; Aigner, Summary and Outlook Lovell & Schmidt 1977) 3 / 12

Stochastic Frontier Analysis Theoretically Consistent Stochastic Output (e.g. haircuts) Frontier y = f ( x , β ) e − u e v Functions Arne ln y = ln f ( x , β ) − u + v Henningsen with u ≥ 0 Introduction E [ u ] = f ( z , δ ) Stochastic Frontier Analysis Theoretical Consistency Restricted Estimation Empirical Example 0 y − f ( x , β ) Summary and Outlook Input (e.g. working hours) 4 / 12

Software for Stochastic Frontier Analysis Theoretically Consistent Stochastic Frontier Functions LIMDEP Arne Henningsen STATA FRONTIER (Version 4.1) Introduction Stochastic ⇒ Tim Coelli (CEPA, Univ. of Queensland, Brisbane) Frontier ⇒ freely available for download (including FORTRAN source) Analysis ⇒ but not really free (no license specified) Theoretical Consistency ⇒ command line interface / “instruction file” Restricted ⇒ THE software for SFA for a long time Estimation ⇒ development stopped in 1996 Empirical ⇒ LIMDEP and STATA have more features today, Example but FRONTIER is still widely used Summary and Outlook 5 / 12

Theoretical Consistency Theoretically Consistent Stochastic Frontier Functions Arne Microeconomic theory requires several properties Henningsen of a production function y = f ( x , β ) Introduction Most important: “monotonicity” Stochastic Frontier ⇒ f ( . ) monotonically increasing in inputs Analysis ⇒ all marginal products ∂ f /∂ x i are non-negative Theoretical Consistency Restricted Monotonicity even more important Estimation in Stochastic Frontier Analysis (SFA) Empirical Example Summary and Outlook 6 / 12

Non-monotone Production Frontier Theoretically Consistent Stochastic Output (e.g. haircuts) Frontier Functions Arne Henningsen Introduction Stochastic Frontier Analysis Theoretical Consistency Firm A Firm B Restricted Estimation “inefficient” “efficient” Empirical Example Summary and Outlook Input (e.g. working hours) 7 / 12

Restricted Estimation of Frontier Functions Theoretically Consistent Stochastic Frontier Not available in standard software packages Functions Arne Econometric approaches for restricted estimations Henningsen ⇒ ML estimation with restrictions imposed at the sample Introduction mean (e.g. Bokusheva and Hockmann: Production Risk and Stochastic Technical Inefficiency in Russian Agriculture , ERAE, 2006) Frontier Analysis ⇒ MCMC estimation with restrictions imposed at all data points (O’Donnell & Coelli: A Bayesian Approach to Imposing Theoretical Consistency Curvature on Distance Functions , JE, 2005) Restricted ⇒ Three-Step Estimation with monotonicity imposed at all Estimation data points (Henningsen & Henning: Estimation of Theoretically Empirical Example Consistent Stochastic Frontier Functions with a Simple Three-Step Summary and Procedure , unpublished, 2008) Outlook 8 / 12

Three-Step Estimation Theoretically Consistent based on Koebel, Falk & Laisney: Imposing and Testing Stochastic Frontier Curvature Conditions on a Box-Cox Cost Function , JBES, Functions 2003 Arne Henningsen 1 Unrestricted frontier estimation (FRONTIER, R:micEcon) Introduction ln y = ln f ( x , β ) − u + v , E [ u ] = z ′ δ Stochastic ⇒ unrestricted parameters ˆ β , their covariance matrix ˆ Σ β Frontier Analysis 2 Minimum distance estimation (R:constrOptim | solve.QP | optim) Theoretical β 0 = argmin �� � � �� Consistency ˆ β 0 − ˆ ˆ Σ − 1 ˆ β 0 − ˆ ˆ β β | nlm | Rdonlp2) β Restricted s.t. f ( x , ˆ β 0 ) satisfies theoretical conditions Estimation Empirical β 0 , “frontier” output y max = f ( x , ˆ ⇒ restricted param. ˆ β 0 ) Example Summary and 3 Final frontier estimation (FRONTIER, R:micEcon) Outlook ln y = α 0 + α 1 ln y max − u + v , E [ u ] = z ′ δ 0 ⇒ y max = ˆ α 0 f ( x , ˆ α 1 , E [ e − u ], ˆ β 0 ) ˆ δ 0 9 / 12

Empirical Example Theoretically Consistent rice production in the Philippines Stochastic Frontier Functions translog production function Arne 1 output (rice), 3 inputs (labour, land, fertiliser) Henningsen 2 variables explaining efficiency (education, upland fields) Introduction 43 rice producers, 8 years Stochastic Frontier unrestricted frontier estimation Analysis Theoretical ⇒ monotonicity violated at 39 observation Consistency ⇒ quasiconcavity violated at 4 observation Restricted Estimation minimum distance estimation Empirical ⇒ monotonicity and quasiconcavity fulfilled at all observation Example second frontier estimation Summary and Outlook ⇒ virtually no adjustment: α 0 = 0 . 0005, α 1 = 0 . 9999 ⇒ efficiency estimates ... 10 / 12

Efficiency Estimates Theoretically 1.0 Consistent Stochastic ● ● ● ● ● ● ● technical efficiency calculated from the restricted model ● ● ● ● ● ● Frontier ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Functions ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.8 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Arne ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Henningsen ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● correlation: ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Pearson: 0.996 Introduction ● ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● Spearman: 0.995 ● ● ● Stochastic ● ● ● ● ● ● ● ● ● ● ● ● ● Frontier ● ● ● ● Kendall: 0.954 ● ● ● Analysis ● ● ● ● ● ● ● ● ● 0.4 ● ● Theoretical ● ● ● Consistency ● ● Restricted Estimation 0.2 ● Empirical Example Summary and 0.0 Outlook 0.0 0.2 0.4 0.6 0.8 1.0 technical efficiency calculated from the unrestricted model 11 / 12

Summary and Outlook Theoretically Consistent Summary Stochastic Frontier Functions SFA is an important tool in production/firm analysis Arne Theoretical consistency is important especially for frontier Henningsen functions. Introduction Imposing restrictions by a three-step estimation procedure Stochastic Frontier ⇒ relatively simple compared to other restricted frontier Analysis estimations Theoretical Consistency ⇒ can be done easily in R (using also FRONTIER) Restricted Estimation Outlook Empirical Integrating FRONTIER into an R package Example Summary and Adding further functions for SFA (e.g. MCMC estimation) Outlook Coworkers and contributors are welcome! 12 / 12