Nonparametric Frontier Estimation: recent developments and new challenges

Transcription

1 Nonparametric Frontier Estimation: recent developments and new challenges Toulouse - October 211 Léopold Simar Institut de Statistique Université Catholique de Louvain, Belgium

2 Nonparametric Frontier Estimation: recent developments and new challenges 2 Contents Frontier Models and Efficiency Measures Economy of Production and Farell-Debreu efficiency scores Statistical Paradigm Different models and Different approaches Nonparametric approaches FDH and DEA estimators and Statistical inference Challenges: Drawbacks of FDH/DEA and Solutions Robustness to outliers: Partial-order frontier (order-m and order-α quantile) Economic interpretation of the frontier: Parametric approximations Hetrogeneity: introducing Environmental Factors Introducing noise: Stochastic Nonparametric Frontiers

3 ² Nonparametric Frontier Estimation: recent developments and new challenges 3 I. Frontier Models and Efficiency Measures

4 Nonparametric Frontier Estimation: recent developments and new challenges 4 The Frontier Model -1- Economic Theory Koopmans (1951), Debreu (1951): Activity Analysis x R p + vector of inputs y R q + vector of outputs Production set Ψ of physically attainable points (x,y): Ψ = {(x,y) R p+q + x can produce y}. The input (output) correspondence sets Ψ can be described by its sections: y Ψ, X(y) = {x R p + (x,y) Ψ} x Ψ, Y (x) = {y R q + (x,y) Ψ}. We have (x,y) Ψ, x X(y) y Y (x).

5 ² Nonparametric Frontier Estimation: recent developments and new challenges 5 8 Production set Y(x ) Ψ output: y 4 y X(y ) x input: x 12 Isoquants in input space 12 Isoquants in output space 1 y 2 > y 1 1 x 1 <x input 2 : x 2 6 output 2 : y X(y 2 ) 4 Y(x 2 ) 2 X(y 1 ) 2 Y(x 1 ) input : x output : y 1 1 Top panel: Production set Ψ for p = q = 1. Bottom Panels: Correspondence sets X(y) and Y (x) for p = 2 and q = 2

6 Nonparametric Frontier Estimation: recent developments and new challenges 6 The Frontier Model -2- Usual Assumptions (a.o.): (Shephard, 197) Free Disposability of inputs and outputs (x,y) Ψ, then if x x,y y, (x,y ) Ψ Convexity: if (x 1,y 1 ),(x 2,y 2 ) Ψ, then for all α [,1] we have: (x,y) = α(x 1,y 1 )+(1 α)(x 2,y 2 ) Ψ No Free Lunches: (x,y) / Ψ if x = and y,y =. Farrell-Debreu Efficiency scores radial measures of distance to the boundary of Ψ Input oriented: θ(x,y) = inf{θ (θx,y) Ψ} 1 Output oriented: λ(x,y) = sup{λ (x,λy) Ψ} 1

7 ² Nonparametric Frontier Estimation: recent developments and new challenges 7 8 Production set 7 6 M Ψ 5 output measure output: y R 4 Q input measure P=(x,y) N input: x 12 Isoquants in input space 12 Isoquants in output space Q=(x, y (x) ) input 2 : x 2 6 * P=(x,y ) 1 output 2 : y 2 6 Y(x 2 ) y 4 Q X(y 2 ) y 2 > y 1 x 4 P=(x,y ) 2 2 X(y 1 ) 2 Y(x 1 ) O input : x output : y 1 1 Top panel: θ P = RQ / RP 1 and λ P = NM / NP 1. Bottom panels: θ P = OQ / OP 1 and λ P = OQ / OP 1

8 Nonparametric Frontier Estimation: recent developments and new challenges 8 The Frontier Model -3- Extensions Hyperbolic Distances: adjusts simultaneously input and output levels (Färe et al., 1985, Färe and Grosskopf, 24). γ(x,y Ψ) = sup{γ > (γ 1 x,γy) Ψ}. Directional Distances: Projection of (x, y) onto the technology frontier in a direction d = ( d x,d y ). (Chambers et al., 1998, Färe and Grosskopf, 2). δ(x,y d x,d y,ψ) = sup{δ (x δd x,y + δd y ) Ψ}. Additive: allow negative values of x and/or y. Special cases: If d = ( x,) with x > : δ(x,y d x,d y,ψ) = 1 θ(x,y Ψ) 1 If d = (,y) with y > : δ(x,y d x,d y,ψ) = λ(x,y Ψ) 1 1

9 Nonparametric Frontier Estimation: recent developments and new challenges 9 The Frontier Model -4- Under free disposability, characterization of the technology δ(x,y d x,d y,ψ) if and only if (x,y) Ψ δ(x,y d x,d y,ψ) = if (x,y) is on the frontier. 3 Production set 2.5 Y Output 1 (x 2,y 2 ).5 g Y.5 g x (x 1,y 1 ) X Input Presentation today and below: Radial cases, but can be extended (Wilson, 211, Simar and Vanhems, 21, Simar, Vanhems and Wilson, 211)

10 ² Nonparametric Frontier Estimation: recent developments and new challenges 1 II. The Statistical Paradigm

11 Nonparametric Frontier Estimation: recent developments and new challenges 11 The Statistical Paradigm In practice, Ψ is unknown θ(x, y) and/or λ(x, y) are also unknown. Estimation based on a sample X = {(x i,y i ), i = 1,...,n} 8 Production set 12 Isoquants in input space output: y ?? P Ψ? y input 2 : x ? P 2 2 X(y)? input: x input : x 1 1

12 Nonparametric Frontier Estimation: recent developments and new challenges 12 The Statistical Paradigm -2- Different Approaches Deterministic Frontiers: Prob{(x i,y i ) Ψ} = 1, pour tout i = 1,...,n. No noise on the data, no random shocks... Distance to frontier is pure inefficiency. Drawback: sensitivity to outliers (superefficient units or errors) Stochastic Frontiers Random noise: some observations may / Ψ. Distance to frontier has 2 components (noise and inefficiency) Drawback: identification problems Different Models: for frontier function and for the law of (X,Y), F(x,y) Parametric Models: very restrictive, standard methods (MLE, OLS,...) e.g. SFA Y i = β X i +V i U i, where V i N(,σV 2 ),U i N + (,σu 2 ), indep. Nonparametric Models: very flexible but more difficult and more challenging.

13 Nonparametric Frontier Estimation: recent developments and new challenges 13 Choosing a Model: A Summary Models Parametric P Nonparametric NP Deterministic D Analytical models for frontier No specific model for frontier and for F(x,y) and for F(x,y) Stochastic S Analytical models for frontier No specific model for frontier for F(x, y) including noise and for F(x, y) including noise (Some structure on noise) Remarks: D S and P NP Horizontal and Vertical comparisons are legitimate and may be useful. Semiparametric Models: combine P and N P (see below)

14 Nonparametric Frontier Estimation: recent developments and new challenges 14 Choosing a Model: Inference Inference is: Parametric P Nonparametric NP Deterministic D Very Easy Easy COLS, MOLS, MLE (restrictive) FDH: ˆF n (x,y) F(x,y) Two-stages: P fit of NP DEA: convexify FDH Bootstrap for efficiency scores Bootstrap Stochastic S Easy Complicated MOLS, MLE (restricted models) Identification problems Identification problems (deconvolution problem) (noise vs inefficency) Localizing P and SFDH/SDEA Sensitivity: Bagging Semi-(non)parametric models Bootstrap is needed everywhere!

15 Nonparametric Frontier Estimation: recent developments and new challenges 15 One Example Efficiency Analysis of Air Controlers (Mouchart and Simar, 22). Data are available on the activity of 37 european air controler units in 2, four outputs: total flight hours controlled, number of air movements controlled, number of sectors controlled and sum of sector hours worked. two inputs: the number of air controllers in EFT and the total number of hours worked by air controlers. For the example: aggregated in one output and one input

16 Nonparametric Frontier Estimation: recent developments and new challenges Values of the output Values of the input Values of the output Values of the input The frontier estimates for the Air controlers data: left various deterministic estimates (Solid: Shifted OLS, dash-dotted: MLE-Gamma), right stochastic (N+HN) (solid for MLE).

17 Nonparametric Frontier Estimation: recent developments and new challenges 17 DEA VRS and DEA CRS frontiers: one input one output output factor input factor DEA estimators VRS (solid) and CRS (dash-dot) for Air Controlers data.

18 Nonparametric Frontier Estimation: recent developments and new challenges 18 The Statistical Paradigm -3- Statistical Inference Estimation individual inefficiencies ( rankings ) Confidence intervals for these measures Specification tests Aggregation of inputs and/or outputs Relevance of the chosen variables Hypothesis testing on the shape of the efficient frontier ( technology ) Convexity Returns to scale (increasing/decreasing/constant) Evolution over time Panel data Gain or loss of productivity? Technical progress or gain of efficiency?

19 Nonparametric Frontier Estimation: recent developments and new challenges 19 The Literature Parametric deterministic or stochastic frontier models: hundreds of papers in Econometric literature (Journal of Econometrics,...) Easier but are the parametric assumptions reasonable ones? Nonparametric deterministic frontier models: thousands of papers in hundreds of different journals (Management sciences, OR, Econometrics) Very popular (flexibility) but some drawbacks (see below). Nonparametric stochastic frontier models: very recent, very few applications (theoretical econometric literature) Flexible but so far, hard to use: work in progress... Applications: Banks, Transports (Air, Railways,...), Public Services, Municipalities, Post, School, Education, Research, University, Insurance, Hospitals, Finance, Mutual funds, Industry, Electric plants, Food industry, Agronomy, Macroeconomic, Economy of development, Regional economy,... (Journal of Productivity Analysis)

20 ² Nonparametric Frontier Estimation: recent developments and new challenges 2 III. Nonparametric Approaches

21 Nonparametric Frontier Estimation: recent developments and new challenges 21 Nonparametric Estimators: FDH -1- Envelopment Estimators: estimate Ψ by ˆΨ which envelops at best the cloud of n data points X. Free Disposal Hull: FDH Deprins, Simar, Tulkens (1984) ˆΨ FDH (X) = { (x,y) R p+q + y y i, x x i, (x i,y i ) X } FDH efficiency scores ˆθ(x,y ) = inf{θ (θx,y ) ˆΨ FDH (X)} ˆλ(x,y ) = sup{λ (x,λy ) ˆΨ FDH (X)}. Practical computations: fast and easy (sorting algorithms) The set dominating points: D = {i (x i,y i ) X, x i x, y i y } ( ) ( ) x j i y j i ˆθ(x,y ) = min max ; ˆλ(x i D j=1,...,p x j,y ) = max min i D j=1,...,q y j

22 Nonparametric Frontier Estimation: recent developments and new challenges 22 Nonparametric Estimators: FDH -2-8 FDH estimator 7 6 Ψ FDH 5 (x,y ) output: y 4 3 (x,y ) i i 2 Free Disposability input: x FDH estimator ˆΨ FDH of the production set Ψ: the are the observations.

23 Nonparametric Frontier Estimation: recent developments and new challenges 23 Nonparametric Estimators: DEA -1- Data Envelopment Analysis: DEA If Ψ is convex: Take the convex hull of ˆΨ FDH (Farrell, 1957, Charnes, Cooper and Rhodes, 1978) ˆΨ DEA = {(x,y) IR p+q y such that n γ i y i ;x i=1 n γ i x i for (γ 1,...,γ n ) i=1 n γ i = 1;γ i,i = 1,...,n}. i=1 Estimation of efficiency score ˆθ(x,y) = inf{θ (θx,y) ˆΨ DEA (X)} ˆλ(x,y) = sup{λ (x,λy) ˆΨ DEA (X)} Computation through linear programs. Available free software: FEAR (Wilson, 28)

24 Nonparametric Frontier Estimation: recent developments and new challenges 24 Nonparametric Estimators: DEA -2-8 DEA estimator 7 6 output: y 5 4 P=(x,y ) Ψ DEA 3 2 (x,y ) i i input: x DEA estimator ˆΨ DEA of the production set Ψ: the are the observations.

25 Nonparametric Frontier Estimation: recent developments and new challenges 25 Nonparametric Estimators: DEA Isoquants in input space 1 8 input 2 : x 2 6 P=(x,y) y 4 2 Q=(x (y),y) Q DEA =(x DEA (y),y) X DEA (y) X(y) input : x 1 1 Properties of DEA estimators: Relations between ˆθ DEA (x,y) and θ(x,y)?

26 Nonparametric Frontier Estimation: recent developments and new challenges 26 Statistical Inference: State of the Art -1- Properties: recent survey, Simar and Wilson (28) Consistency and rate of convergence: (ˆθ(x,y) θ(x,y) ) = O p (n τ ), as n? FDH: Korostelev, Simar and Tsybakov (1995a) and Park, Simar and Weiner (2). Rate is n 1/(p+q). Recent Extensions: Daouia, Florens and Simar (21) DEA: Korostelev, Simar and Tsybakov (1995b) and Kneip, Park and Simar (1998). Rate is n 2/(p+q+1). Park, Jeong and Simar (21) (CRS case), rate is n 2/(p+q). Nice! but not very useful for the practitionners. Curse of dimensionality: bad rates if p+q.

27 Nonparametric Frontier Estimation: recent developments and new challenges 27 Statistical Inference: State of the Art -2- Is Inference possible? Asymptotic sampling distribution: n τ (ˆθ(x,y) θ(x,y) ) Q(η), as n? FDH: Park, Simar and Weiner (2), Badin, Simar (29), Daouia, Florens and Simar (21); Q(η) is a Weibull distribution with unknown parameters to be estimated: not easy to handle and need large sample sizes if p+q increases. DEA: Gijbels, Mammen, Park and Simar (1999), Kneip, Simar and Wilson (28), Park, Jeong, Simar (21); Q(η) is a Regular distribution depending on unknown parameters but no closed forms available (untractable for practical purposes) when p or q > 1. No hope? Yes: the bootstrap.

28 Nonparametric Frontier Estimation: recent developments and new challenges 28 The Bootstrap -1- Basic Idea The Real World : The Data Generating Process P (x i,y i ) in X are realizations of iid random variables (X,Y) with probability density function f(x,y) with support Ψ, and Prob ( (X,Y) Ψ ) = 1. ˆΨ(X) is an estimator of Ψ (FDH or DEA) ˆθ(x,y) = inf{θ (θx,y) ˆΨ(X)} is an estimator of θ(x,y) The Bootstrap World : Consider a DGP ˆP, a consistent estimator of P. We can use ˆΨ(X) (FDH or DEA) and some appropriate ˆf(x,y) with support ˆΨ(X), and Prob ( (X,Y) ˆΨ(X) ) = 1. Bootstrap Analogy: Define a new data set X = {(x i,y i ), i = 1,...,n} drawn from ˆP. ˆΨ(X ) is an estimator of ˆΨ(X): here, ˆΨ(X ) is the FDH or DEA set computed with X as reference data set. ˆθ (x,y) = inf{θ (θx,y) ˆΨ(X )} is an estimator of ˆθ(x,y)

29 ² Nonparametric Frontier Estimation: recent developments and new challenges Isoquants in input space 1 * y input 2 : x * Q (x i,y i ) * Q DEA Q * DEA * P=(x,y) * * X* DEA (y) X DEA (y) X(y) input : x 1 1 The Bootstrap idea: the are the original observations (x i,y i ) generated by the unknown, and the * are the pseudo-observations (x i,y i) generated by the known ˆ.

30 Nonparametric Frontier Estimation: recent developments and new challenges 3 The Bootstrap -2- The Key Relation : If the Bootstrap is consistent, for large n, (ˆθ (x,y) ˆθ(x,y)) ˆP (ˆθ(x,y) θ(x,y)) P. The right part is unknown and/or difficult to handle The left part can be approximated by Monte-Carlo simulation methods Inference is now available Bias correction and Standard errors of ˆθ(x,y) are available Confidence intervals for θ(x,y) can be builded How to generate X? Naive bootstrap looks easy: n random drawns of (x i,y i ) from X. But naive bootstrap is inconsistent Simar and Wilson (1998, 1999a, 1999b) The efficient facet, which determines in the original sample X the value of ˆθ, appears too often, and with a fixed probability, in X and this fixed probability does not vanish even when n.

31 Nonparametric Frontier Estimation: recent developments and new challenges 31 The Bootstrap -3- Two Solutions: see Simar and Wilson (1998, 2, 211a), Jeong and Simar (26), Kneip, Simar and Wilson (28) Subsampling: draw from ˆP pseudo-samples of size m = n κ where κ < 1. How to chose m in practice: Simar and Wilson (211a). Smoothing: Use smoothed density estimate ˆf(x,y) and smooth the boundary of ˆΨ when defining ˆP: not easy to implement due to the double smoothing. Simplification: homogeneous bootstrap, Simar and Wilson (1998), similar to homoskedastic assumption in regression. But restrictive... Consistent efficient algorithm in the heterogeneous case: Kneip, Simar and Wilson (211). Testing issues: Returns to scale, Simar and Wilson (22), Comparison of groups of firms, Simar and Zelenyuk (26, 27), Testing significancy of variables and/or aggregation of variables, Simar and Wilson (21), and work in progress (convexity,...). Extensions available: Hyperbolic distances, Wilson (211), Directional distances, Simar and Vanhems (21), Simar, Vanhems and Wilson (211).

32 Nonparametric Frontier Estimation: recent developments and new challenges 32 An Example: Program Follow Through (PFT) Charnes, Cooper, Rhodes (1981): analysis of an experimental education program administered in US schools: data for 49 schools that implemented PFT, and 21 schools that did not, for a total of 7 observations. 5 inputs and 3 outputs x 1 : Education level of the mother (percentage of high school graduates among the mothers), x 2 : Highest occupation of a family member (according a pre-arranged rating scale), x 3 : Parental visit to school index (number of visits to the school) x 4 : Parent counseling index (time spent with child on school related topics) x 5 : Number of teachers of the school. There are three outputs (results to standard tests): y 1 : Total Reading Score (MAT: Metropolitan Achievement Test), y 2 : Total Mathematics Score (MAT) and y 3 : Coopersmith Self-Esteem Inventory (measure of self-esteem). We look for output efficiency of the Schools λ(x,y) using DEA estimators.

33 Nonparametric Frontier Estimation: recent developments and new challenges 33 Units ˆλ(x,y) Units ˆλ(x,y) Questions: What is the real value of λ(x,y) (bias correction, confidence intervals)? Comparaison of the 2 groups of school: Mean of Group A (49 PFT schools): ˆλ A = Mean of Group B (21 Non-PFT schools): ˆλ B = (more efficient?) Is it significant?

34 Nonparametric Frontier Estimation: recent developments and new challenges 34 The Bootstrap Units Eff. Scores Eff. Bias-Corrected Bias Std Lower Bound Upper Bound After bias correction the mean are: Group A (PFT): 1.94 Group B (Non-PFT): 1.74 Formal Test: H : E[λ(X,Y) A] = E[λ(X,Y) B] vs H : E[λ(X,Y) A] > E[λ(X,Y) B] p-value of H =.559: We do not reject H.

35 Nonparametric Frontier Estimation: recent developments and new challenges 35 An Other Example: Role of Innovation on Exports -1- Schubert and Simar (211) analyze the relations between exports and innovation in the sector of Mechanical Engineering in Germany (CIS survey, 27) The economic literature is unclear and divided on the role of innovation Empirical studies, so far, used parametric models with restrictive assumptions and analyze mean behavior of firms (regression) We want to analyze the role of innovation on efficient production plans, not on averages Data: 215 firms 3 inputs: X 1 Expenses for personnel, X 2 Expenses for equipment and materials, X 3 Expenses for innovation 2 outputs: Y 1 Domestic turnover et Y 2 Exports

36 Nonparametric Frontier Estimation: recent developments and new challenges 36 An Other Example: Role of Innovation on Exports -2- NB: here individual efficiency scores are not of interest Results: tests using bootstrap on different models The inputs X 1 and X 2 can be aggregated, without changing the structure of the efficient frontier, but not X 3. The outputs Y 1 and Y 2 can be aggregated Empirical Conclusions: The expenditures in innovation (even measured in monetary terms) are really different than other routine expenses: they influence the shape of the frontier, it is really an input, and not a by-product (as claimed by some economic theory). For efficient firms, there is no clear empirical evidence of a link between the expenses in innovation and the exports (as claimed by some economic theory).

37 ² Nonparametric Frontier Estimation: recent developments and new challenges 37 IV. Challenges

38 Nonparametric Frontier Estimation: recent developments and new challenges 38 Challenges: Drawbacks of DEA/FDH and Solutions Sensitivity to extreme/outliers: robust methods and/or detection of outliers Order-m frontiers: Cazals, Florens and Simar (22), Simar (23), Daouia, Florens and Simar (29). Order-α quantile frontiers: Aragon, Daouia and Thomas (25), Daouia and Simar (25, 27), Daouia, Florens and Simar (29, 21). ² Lack of Economic interpretation: Semiparametric Model, parametric approximations of nonparametric frontiers, Simar (1992), Florens and Simar (25), Daouia, Florens and Simar (28) Heterogeneity: How to explain inefficiency by environmental/external factors? Two-stage methods, Simar and Wilson (27, 211b). Conditional measures of efficiency, Cazals, Florens and Simar (22), Daraio and Simar (25, 26, 27a, 27,b), Jeong, Park and Simar (21), Badin, Daraio and Simar (21, 211). No noise is allowed: deterministic frontiers Prob ( (X,Y) Ψ ) = 1: Nonparametric Stochastic Frontiers?: Simar (27), Kumbhakar, Park, Simar and Tsionas (28), Simar and Zelenyuk, (211), Kneip, Simar and Van Keilegom (21), flexible semiparametric models.

39 ² Nonparametric Frontier Estimation: recent developments and new challenges 39 IV.1 Sensitivity to Outliers

40 Nonparametric Frontier Estimation: recent developments and new challenges 4 Robust Frontier -1 Probabilitic Formulation of DGP The DGP: H(x,y) = Prob(X x,y y), Ψ is the support of H(x,y) Farrell-Debreu Efficiency score (case of input orientation) H(x,y) = Prob(X x Y y) Prob(Y y) = F X Y (x y)s Y (y) θ(x,y ) = inf{θ (θx,y ) Ψ} = inf{θ F X Y (θx y ) > } Nonparametric Estimator: Plug-in the empirical version of H(x,y) ˆH n (x,y) = 1 n n i=1 1I(X i x,y i y), then ˆF X Y,n (x y) = ˆH n (x,y) ˆH n (,y) The FDH estimators: Cazals, Florens and Simar (22) ˆΨ FDH is the support of ˆH n (x,y) Estimation (input) efficiency score: ˆθ(x,y ) = inf{θ ˆF X Y,n (θx y ) > }

41 Nonparametric Frontier Estimation: recent developments and new challenges 41 Robust Frontier -2- Partial order frontiers. Economic interpretation (case of univariate output) Another benchmark frontier less extreme than the full frontier. Order-m: Cazals, Florens, Simar (22) a unit (x,y) is benchmarked against the average maximal output reached by m peers randomly drawn from the population of units using less input than x. As m, order-m frontier converges to the full-frontier. Order-α quantile: Aragon, Daouia, Thomas (25), Daouia and Simar (27) a unit (x,y) is benchmarked against the output level not exceeded by 1(1 α)% of firms in the population of units using less input than x. As α 1, order-α frontier converges to the full-frontier.

42 Nonparametric Frontier Estimation: recent developments and new challenges 42 Robust Frontier -2- Partial order frontiers: Mathematical definition for univariate output Full Frontier Benchmark: φ(x) = inf{y F Y X (y x) 1} and Less Extreme Benchmarks: Order-m frontier: φ m (x) = E [ max(y 1,...,Y m ) X x ] = Order-α quantile frontier: Properties φ α (x) = F 1 Y X (α x) (1 [F Y X (y x)] m )dy = inf{y R + F Y X (y x) α} as m, φ m (x) φ(x) and as α 1, φ α (x) φ(x)

43 Nonparametric Frontier Estimation: recent developments and new challenges 43 Robust Frontier F(y X <= x).6.4 m data points y with X <=x φ m (x).2 φ.8 (x) φ(x) * * * * * * values of y Picture of F Y X (y x) = Prob(Y y X x) Illustration of full and partial frontiers: one output with m = 6 and α =.8

44 Nonparametric Frontier Estimation: recent developments and new challenges 44 Robust Frontier -4- Nonparametric estimators of partial order frontier Plug-in principle ˆφ m,n (x) = (1 [ˆF n,y X (y x)] m )dy ˆφ α,n (x) = inf{y R + ˆF n,y X (y x) α} Properties n-consistency and asymptotic normality: n(ˆφm,n (x) φ m (x)) Convergence to FDH estimator: L N(,σ 2 m(x)) and n(ˆφ α,n (x) φ α (x)) L N(,σ 2 α(x)) as m, ˆφ m,n (x) ˆφ FDH,n (x) and as α 1, ˆφ α,n (x) ˆφ FDH,n (x) Choice of m and α: tune the percentage of points left out estimated partial frontier, see Simar (23), Daraio, Simar (25, 27a).

45 Nonparametric Frontier Estimation: recent developments and new challenges In solid black line, the true frontier y = x.5. In green solid, the FDH frontier estimate, in blue dashed the estimated order-m frontier and in dash-dot red the estimate of the order-α frontier. In black dotted, the shifted OLS estimate and in dash-dot black, the parametric stochastic fit, m = 2 and α =.95.

46 Nonparametric Frontier Estimation: recent developments and new challenges 46 Robust Frontier -5- Robust Nonparametric Estimator of Full-Frontier φ(x), Daouia, Florens, Simar (29, 21) If m = m(n) (and α = α(n)) converges to (to 1) when n, but at a slow rate, we obtain an estimator (after bias correction) that converges to the full frontier with a Normal limiting distribution Easy to build confidence intervals for φ(x) using Normal Tables. For finite n, ˆφ m(n),n (x) and ˆφ α(n),n (x) provide estimators of φ(x) that will not envelop all the data points and so, are more robust to extreme and outliers.

47 Nonparametric Frontier Estimation: recent developments and new challenges 47 Robust Frontier x 14 Data points FDH frontier phi tilde phi m x 14 Data points FDH frontier phi tilde phi m Post Offices in France (from Daouia, Florens, Simar, 29). Left panel: estimation with the 4 extreme points. Right panel: estimation without these 4 points

48 ² Nonparametric Frontier Estimation: recent developments and new challenges 48 IV.2 Lack of Economic Interpretation

49 Nonparametric Frontier Estimation: recent developments and new challenges 49 Parametric Approximation of Deterministic Frontiers -1- Parametric models: easy economic interpretation of the model (returns to scale, elasticities, elasticities of substitution,...) Standard parametric approches: some drawbacks strong restrictive assumptions on the stochastic part of the models sensitive to extreme/outliers most are regression-based models and capture the shape of the cloud of points near its center (not at the efficient boundary) Two stage semiparametric approaches: Simar (1992), Florens, Simar (25), Daouia, Florens, Simar (28) First estimate the efficient frontier using nonparametric or robust nonparametric methods; Then fit, by standard OLS, the approriate parametric model on the obtained nonparametric frontier

50 Nonparametric Frontier Estimation: recent developments and new challenges 5 Parametric Approximation of Deterministic Frontiers -2- More sensible estimator of the parametric frontier model and allows for some noise by tunning the robustness parameter. Asymptotic theory of the resulting estimators (for fix m and fix α): If FDH is used as 1st step: ˆθ n p θ If order-m is used is used as 1st step: n(ˆθ m n θ m ) If order-α is used is used as 1st step: n(ˆθ α n θ α ) L N k (,V m ) L N k (,V α ) where θ, (θ m, θα ), are the pseudo-true values of the parameters of the best approximation of the corresponding frontier φ(x), (φ m (x), φ α (x)). If m(n) and α(n) 1 as n at appropriate rates: ˆθ n a.s. θ ; ˆθm(n) n a.s. θ ; ˆθα(n) n a.s. θ Multivariate case: multi-input/muti-output, see Daraio and Simar (27a)

51 Nonparametric Frontier Estimation: recent developments and new challenges In solid black line, the true frontier y = x.5 homoscedastic inefficiency. In cyan solid, the FDH frontier, in blue dashed the order-m frontier and in dash-dot red the order-α frontier. Here, m = 2 and α = In black dotted, the shifted OLS estimate.

52 Nonparametric Frontier Estimation: recent developments and new challenges Same with 3 outliers included.

53 Nonparametric Frontier Estimation: recent developments and new challenges Same with heteroscedastic inefficiency. In cyan solid, the FDH frontier estimate, in blue dashed the order-m frontier and in dash-dot red the order-α frontier. Here, m = 2 and α = In black dotted, the shifted OLS estimate.

54 Nonparametric Frontier Estimation: recent developments and new challenges Same with 3 outliers included.

55 ² Nonparametric Frontier Estimation: recent developments and new challenges 55 IV.3 Heterogeneity

56 Nonparametric Frontier Estimation: recent developments and new challenges 56 Introducing Environmental Factors -1- Motivation The analysis of productive efficiency should have two components: 1. Estimation of a production frontier (best-practice) which serve as a benchmark against which efficiency of a producer can be measured; 2. Incorporation into the analysis of exogenous variables (Z) which are neither inputs, nor outputs, and so are not under the control of the producer, but which may influence the process. How to explain inefficiencies of firms by these factors? How to introduce heterogeneity in the production process?

57 Nonparametric Frontier Estimation: recent developments and new challenges 57 Introducing Environmental Factors -2- One-stage approaches Banker and Morey (1986) Z is like an input(favorable) or like an output (defavorable) Adapt FDH/DEA Free disposability? Convexity? RTS assumption? Which direction for Z? What if the effect of Z changes? (say, favorable if Z z and then defavorable or neutral for Z > z ) Two-stage approaches Simar and Wilson (27, 211) DEA efficiency scores are regressed on Z (in an appropriate way) Implicit Separability Condition: Z does not influence Ψ Z only affects the probability of being more or less efficient The second stage regression is nonstandard (correlation among efficiency scores, bias,...): inference by bootstrap.

58 Nonparametric Frontier Estimation: recent developments and new challenges 58 Traditional 2-stage approaches First stage get efficiency estimates ˆλ(X i,y i ) (or ˆθ(X i,y i ), ˆγ(X i,y i ),...) with respect to ˆΨ (by DEA or FDH,...) Second stage regression of ˆλ(X i,y i ) on Z. Parametric models (truncated regression, logistic, etc,...) Nonparametric models (truncated, etc,...) Problems: Ψ z = {(x,y) Z = z, x can produce y} Simar and Wilson (27, 211b): If Ψ z = Ψ, what is the Economic meaning of λ(x,y) (and so, of ˆλ(X i,y i ) ), for a unit facing environmental conditions z? Separability issue: condition for giving economic meaning to ˆΨ and ˆλ(x,y). Separability condition: Ψ z = Ψ, for all z. Even if separability holds, Inference in second stage is nonstandard (bootstrap).

59 Nonparametric Frontier Estimation: recent developments and new challenges 59 Separability Condition g(x) = [1 (X 1) 2 ] 1/2 Y = g(x)e (Z 2)2 U Y = g(x)e (Z 2)2 e U 1 1 y*.5 y** z x z x.75 1 Left Panel: Separable, Right Panel: Not Separable

60 Nonparametric Frontier Estimation: recent developments and new challenges 6 Conditional Efficiency -1- Conditional Measures Cazals, Florens, Simar (22), Daraio Simar (25, 27a, 27b), Jeong, Park, Simar (21) The DGP (A Model for the Production process) is now characterized by F(x,y z) = Prob(X x,y y Z = z) or H(x,y z) = Prob(X x,y y Z = z) The attainable set is Ψ z : the support of F(x,y z) Natural and very easy: A firm combines inputs X R p + and outputs Y R q + facing the environmental conditions Z R r No separability conditions No prior information of the role of Z (favorable or not to the process) Note that the separability condition of 2-stages methods relies on: Ψ Ψ z for all z.

61 Nonparametric Frontier Estimation: recent developments and new challenges 61 Conditional Efficiency -2- Conditional efficiency score Same idea as the unconditional measure: λ(x,y z) = sup{λ H XY Z (x,λy z) > } = sup{λ S Y X,Z (λy x,z) > }, where S Y X,Z (y x,z) = H XY Z (x,y z)/h XY Z (x, z) = Prob(Y y X x,z = z). Nonparametric estimator: kernel smoothing on Z (here continuous) ˆH XY,n Z (x,y Z = z) = n i=1 1I(X i x,y i y)k((z i z)/h) n i=1 K((Z i z)/h) ˆS Y X,Z (y x,z) = n i=1 1I(Y i y,x i x)k h (Z i,z) n i=1 1I(X i x)k h (Z i,z)

62 Nonparametric Frontier Estimation: recent developments and new challenges 62 Conditional Efficiency -3- Conditional FDH efficiency estimator: Kernels with compact support, { } Y ˆλ j i FDH (x,y z) = sup{λ ˆS Y X,Z (λy x,z) > } = max min. {i X i x, Z i z h} j=1,...,q y j Conditional FDH attainable set: ˆΨ Z FDH = {(x,y) R p+q + x x i,y y i for i s.t. Z i z h} DEA versions: Convexify the FDH attainable set, see Daraio, Simar (27b) ˆΨ Z DEA = {(x,y) R p+q + x γ i x i, y γ i y i for γ i s.t. {i Z i z h} {i Z i z h} γ i = 1}, ˆλ DEA (x,y z) = sup{λ (x,λy) ˆΨ Z DEA }. {i Z i z h}

63 Nonparametric Frontier Estimation: recent developments and new challenges 63 Conditional Efficiency -4- Properties Optimal bandwidth selection by data-driven methods, Badin, Daraio, Simar (21) Asymptotic properties: similar to FDH/DEA with n replaced by nh r, Jeong, Park, Simar (21) Allow to detect the direction of the influence of Z on efficiency, see Dario, Simar (25, 27a) Inference (confidence intervals) by bootstrap Robust versions (using order-m and order-α) are also available Z can be continuous, categorical or discrete

64 Nonparametric Frontier Estimation: recent developments and new challenges 64 Conditional Efficiency -5- Usefulness Define a pure measure of technical efficiency, Badin, Daraio, Simar (211) Eliminate most of the influence of Z on ˆλ(x,y z) by using a flexible location-scale nonparametric model: ˆλ(x,y z) = μ(z)+σ(z)ε, where μ(z) and σ(z) are unspecified functions ˆε i allows to rank firms facing different operating conditions. N.B.: An other approach: Florens, Simar, Van Keilegom (211). First eliminate influence of Z on inputs X and outputs Y by using two flexible location-scale nonparametric models The residuals are pure inputs and outputs X i and Y i Search for the frontier in these new units, to define pure measure of technical efficiency Full frontier and order-m frontiers

65 Nonparametric Frontier Estimation: recent developments and new challenges 65 Conditional Efficiency, Example -1- A Toy example: No output (Y i 1) and one input (input orientation) Z has no effect on X when Z 5 and then a defavorable effect on X when Z > 5. The input are generated according X i = I(Z i <= 5)+Z 1.5 i 1I(Z i > 5)+U i, where Z i U(1,1), U i Expo(μ = 3) and n = 1.

66 ² Nonparametric Frontier Estimation: recent developments and new challenges Data points for Toy example 2.5 Effect of Z on Full frontier θ(x,y) eff(x,y z)/eff(x,y) Values of Z i θ(x,y z) (x,y,z) derivative of Q z values of Z Effect of Z on Full frontier: derivative Values of X i values of Z Effect of Z on the ratios ˆθ n (x,y z)/ˆθ n (x,y).

67 Nonparametric Frontier Estimation: recent developments and new challenges 67 Conditional Efficiency, Examples -2a- 2 inputs/ 2 outputs : output orientation The efficient frontier is described by: y (2) = 1.845(x (1) ).3 (x (2) ).4 y (1). X (j) i U(1,2) and Y (j) i U(.2,5) for j = 1,2. The output efficient random points on the frontier are where S i = (2) Y i / Y (1) i i,eff = 1.845(X(1) i ).3 (X (2) S i +1 Y (1) i ).4 Y (2) i,eff = 1.845(X(1) i ).3 (X (2) i ).4 Y (1) i,eff. represent the generated random rays in the output space. The efficiencies are simulated according to exp( U i ) The observed output are defined by Y i = Y i,eff exp( U i ) where U i Exp(μ U = 1/2). n = 1.

68 Nonparametric Frontier Estimation: recent developments and new challenges 68 Conditional Efficiency, Examples -2b- Environmental factors Z bivariate We generate two independent uniform variables Z j U(1,4) to build the bivariate variable Z = (Z 1,Z 2 ). The influence of Z on the production process is described by: Y (1) i = (1+2 Z ) Y (1) i,eff exp( U i) Y (2) i = (1+2 Z ) Y (2) i,eff exp( U i). Z 1 pushes the efficient frontier above when far from 2.5, in both directions, with a cubic effect, Z 2 has no effect on the frontier or on the distribution of inefficiencies: Z 2 is irrelevant. Note that there is no interaction between Z 1 and Z 2 (independent) and no interaction between X and Z. Remember: only n = 1 observations, with p = q = r = 2!

69 ² Nonparametric Frontier Estimation: recent developments and new challenges 69 1 eff(x,y z)/eff(x,y) Z Z Smoothed nonparametric surface regression of ˆλ n (x,y z)/ˆλ n (x,y) on Z 1 and Z 2.

70 Nonparametric Frontier Estimation: recent developments and new challenges 7 Effect of Z 1 on Full frontier eff(x,y z)/eff(x,y) eff(x,y z)/eff(x,y) 1.5 values of Z 1 Effect of Z on Full frontier values of Z 2 Simulated example with multivariate Z. Marginal views of the surface regression of ˆλ n (x,y z)/ˆλ n (x,y) on z at the observed points (X i,y i,z i ), viewed as a function of Z 1 (top panel) and as a function of Z 2 (bottom panel).

71 ² Nonparametric Frontier Estimation: recent developments and new challenges 71 IV.4 Introducing Noise

72 Nonparametric Frontier Estimation: recent developments and new challenges 72 Nonparametric Stochastic Frontiers -1- Basic Idea: localize (using kernels) an anchorage parametric model, Kumbhakar, Park, Simar, Tsionas (27) Y i = r(x i )+v i u i u X = x N(,σu 2(x)) and v X = x N(,σ2 v (x)) and u and v being independent conditionally on X. r(x),σu 2(x) and σ2 v (x) are unknown functional parameters Estimation by Local Maximum Likelihhood method: r(x),σ 2 u (x) and σ2 v (x) are approximated by local polynomials (linear or quadratic). Asymptotic properties are available Bandwidths selection by LS cross-validation

73 Nonparametric Frontier Estimation: recent developments and new challenges 73 Nonparametric Stochastic Frontiers -2- Multivariate extension: Simar (27), Simar, Zelenyuk (211) Use (partial-)polar coordinates: (x,y) (ω,η,x), where ω R + is the modulus and η [,π/2] q 1 is the amplitude (angle) of the vector y. The joint density f X,Y (x,y) induces a density on (ω,η,x): f ω,η,x (ω,η,x) = f ω (ω η,x)f η,x (η,x) For a given (x,y) the frontier point y (x,y) = λ(x,y)y has a modulus: ω(y (x,y)) = sup{ω R + f ω (ω η,x) > } Back to a univariate frontier problem! Given (η,x) find ω(y (x,y)).

74 Nonparametric Frontier Estimation: recent developments and new challenges 74 Nonparametric Stochastic Frontiers Isoquant in Output space 1 Q = (x,y (x,y)) = (x,ω(y (x,y)),η) 8 Output 2 : y 2 X o o P= (x,y) = (x,ω,η) η Output 1 : y 1 Polar coordinates in the output space for a particular section Y(x). Output efficiency of P = (x,y) is λ(x,y) = OQ / OP = ω(y (x,y))/ω(x,y) 1.

75 Nonparametric Frontier Estimation: recent developments and new challenges 75 Nonparametric Stochastic Frontiers -4- The Model: The observations are made on noisy data in the output radial-direction The data {(X i,y i ), i = 1,...,n} have polar coordinates (ω i,η i,x i ) ω i = ω(y (X i,y i ))e u i e v i, where u i > is inefficiency and v i is noise (E(v i X i,y i ) = ). ω(y (X i,y i )) is only a function of (η i,x i ). In the log-scale, the model could be written as logω i = r(η i,x i ) u i +v i, with u i > and E(v i η i,x i ) =.

76 Nonparametric Frontier Estimation: recent developments and new challenges 76 Nonparametric Stochastic Frontiers -5- Stochastic Versions of DEA/FDH : Two-stage procedure [1] Whitening the noise : Compute the consistent estimator of the frontier levels ˆr(η i,x i ) for each data points This gives points (X i,yi ) where Yi = exp(ˆr(η i,x i ))Y i /ω i [2] Run a DEA (or FDH) program with reference set (X i,y i ). Summary: Very encouraging results Computationally demanding (cross-validation for bandwidth selection) Below, some bivariate examples (see multivariate examples in Simar and Zelenyuk, 211)

77 Nonparametric Frontier Estimation: recent developments and new challenges Data points Point estimates True frontier DEA with noise DEA no noise Data points Point estimates True frontier DEA with noise DEA no noise Data points Point estimates True frontier DEA with noise DEA no noise Data points Point estimates True frontier DEA with noise DEA no noise Data points Point estimates True frontier DEA with noise DEA no noise Data points Point estimates True frontier DEA with noise DEA no noise (a) n = 1,ρ nts =, (b) n = 13,ρ nts = + 3 outliers, (c) n = 1,ρ nts = 1, (d) n = 2,ρ nts = 1, (e) n = 5,ρ nts = 1, (f) n = 5,ρ nts = 2.

78 Nonparametric Frontier Estimation: recent developments and new challenges Data points Point estimates True frontier FDH with noise FDH no noise Data points Point estimates True frontier FDH with noise FDH no noise Data points Point estimates True frontier FDH with noise FDH no noise Data points Point estimates True frontier FDH with noise FDH no noise Data points Point estimates True frontier FDH with noise FDH no noise Data points Point estimates True frontier FDH with noise FDH no noise (a) n = 2,ρ nts =, (b) n = 23,ρ nts = + 3 outliers, (c) n = 2,ρ nts =.5, (d) n = 2,ρ nts = 1, (e) n = 5,ρ nts = 1, (f) n = 1,ρ nts = 1.

79 Nonparametric Frontier Estimation: recent developments and new challenges 79 Conclusions Nonparametric models NP are Econometric Models Flexible and can be robustified, Inference is available (bootstrap) Noise can be introduced Environmental factors (heterogeneity) can be introduced P and NP are complimentary models NP models can be used to check (test) P models (not the contrary). Parametric approximations of N P models can be useful for economic analysis. Semiparametric models should be developed. Other challenges Panel Data: introduce dynamic behavior of units Theory for functions of DEA/FDH scores: Kneip, Simar and Wilson (211) Useful for justifying testing procedures RTS, Convexity, Simar and Wilson (211a), Separabilty, Daraio, Simar and Wilson (21),...

80 Nonparametric Frontier Estimation: recent developments and new challenges 8 References Basic References Fried, H., Lovell, K. and S. Schmidt (eds) (28),The Measurement of Productive Efficiency, 2nd Edition, Oxford University Press. Kumbhakar, S.C. and C.A.K. Lovell (2), Stochastic Frontier Analysis, Cambridge University Press. Daraio, C. and L. Simar (27a), Advanced Robust and Nonparametric Methods in Efficiency Analysis. Methodology and Applications, Springer, New-York. Simar, L. and P.W. Wilson (28), Statistical Inference in Nonparametric Frontier Models: recent Developments and Perspectives, in The Measurement of Productive Efficiency, 2nd Edition, Harold Fried, C.A.Knox Lovell and Shelton Schmidt, editors, Oxford University Press, 28.

81 Nonparametric Frontier Estimation: recent developments and new challenges 81 References list Aigner, D.J., Lovell, C.A.K. and P. Schmidt (1977), Formulation and estimation of stochastic frontier models, Journal of Econometrics, 6, Aragon, Y. and Daouia, A. and Thomas-Agnan, C. (25), Nonparametric Frontier Estimation: A Conditional Quantile-based Approach, Econometric Theory, 21, Badin, M., Daraio, C. and L. Simar (21), Optimal Bandwidth Selection for Conditional Efficiency Measures: a Data-driven Approach, European Journal of Operational Research, 21, Badin, L., Daraio, C. and L. Simar (211), How to measure the impact of environmental factors in a nonparametric production model? Discussion paper 211/19, Institut de Statistique, UCL. Badin, L. and L. Simar (29), A bias corrected nonparametric envelopment estimator of frontiers, Econometric Theory, 25, 5, Banker, R.D. and R.C. Morey (1986), Efficiency analysis for exogenously fixed inputs and outputs, Operations Research, 34(4), Cazals, C. Florens, J.P. and L. Simar (22), Nonparametric Frontier Estimation: a Robust Approach, in Journal of Econometrics, 16, Chambers, R. G., Y. Chung, and R. Färe (1998), Profit, directional distance functions, and nerlovian efficiency, Journal of Optimization Theory and Applications, 98,

82 Nonparametric Frontier Estimation: recent developments and new challenges 82 Charnes, A., Cooper W.W. and E. Rhodes (1978), Measuring the inefficiency of decision making units, European Journal of Operational Research 2 (6), Daouia, A., J.P. Florens and L. Simar (28), Functional Convergence of Quantile-type Frontiers with Application to Parametric Approximations, Journal of Statistical Planning and Inference, 138, Daouia, A., Florens, J.P. and L. Simar (29), Regularization of Non-parametric Frontier Estimators. Discussion paper 922, Institut de Statistique, UCL. Daouia, A., Florens, J.P. and L. Simar (21), Frontier estimation and extreme values theory. Bernoulli, 16(4), Daouia, A. and L. Simar (25), Robust Nonparametric Estimators of Monotone Boundaries, Journal of Multivariate Analysis, 96, Daouia, A. and L. Simar (27), Nonparametric efficiency analysis: a multivariate conditional quantile approach, Journal of Econometrics, 14, Daraio, C. and L. Simar (25), Introducing environmental variables in nonparametric frontier models: a probabilistic approach, Journal of Productivity Analysis, vol 24, 1, Daraio, C. and L. Simar (26), A Robust Nonparametric Approach to Evaluate and Explain the Performance of Mutual Funds, European Journal of Operational Research, vol 175, 1,

83 Nonparametric Frontier Estimation: recent developments and new challenges 83 Daraio, C. and L. Simar (27a), Advanced Robust and Nonparametric Methods in Efficiency Analysis. Methodology and Applications, Springer, New-York. Daraio, C. and L. Simar (27), Conditional nonparametric frontier models for convex and non convex technologies: a unifying approach, Journal of Productivity Analysis, vol 28, Daraio, C., Simar, L. and P.W. Wilson (21), Testing whether Two-Stage Estimation is Meaningful in Non-Parametric Models of Production, Discussion paper 131, Institut de Statistique, UCL. Debreu, G. (1951), The coefficient of ressource utilization, Econometrica, 19:3, Deprins, D., Simar, L. and H. Tulkens (1984), Measuring labor inefficiency in post offices. In The Performance of Public Enterprises: Concepts and measurements. M. Marchand, P. Pestieau and H. Tulkens (eds.), Amsterdam, North-Holland, Farrell, M.J. (1957), The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A, 12, Färe, R. and S. Grosskopf (2), Theory and application of directional distance functions, Journal of Productivity Analysis 13, Färe, R., and S. Grosskopf (24), New Directions: Efficiency and Productivity, Boston: Kluwer Academic Publishers. Färe, R., S. Grosskopf, and C. A. K. Lovell (1985), The Measurement of Efficiency of Production, Boston: Kluwer-Nijhoff Publishing.

84 Nonparametric Frontier Estimation: recent developments and new challenges 84 Florens, J.P. and L. Simar, (25), Parametric Approximations of Nonparametric Frontier, Journal of Econometrics, vol 124, 1, Fried, H., Lovell, K. and S. Schmidt (eds) (28),The Measurement of Productive Efficiency, 2nd Edition, Oxford University Press. Gijbels, I., E. Mammen, B.U. Park and L. Simar (1999), On Estimation of Monotone and Concave Frontier Functions, Journal of the American Statistical Association, vol 94, 445, Hall, P. and L. Simar (22), Estimating a Changepoint, Boundary or Frontier in the Presence of Observation Error, Journal of the American Statistical Association, 97, Jeong, S.O., B. U. Park and L. Simar (21), Nonparametric conditional efficiency measures: asymptotic properties. Annals of Operations Research, 173, Jeong, S.O. and L. Simar (26), Linearly interpolated FDH efficiency score for nonconvex frontiers, Journal of Multivariate Analysis, 97, Kneip, A., Park, B.U. and Simar, L. (1998). : A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econometric Theory, 14, Kneip, A., Simar, L. and I. Van Keilegom (21), Boundary Estimation in the Presence of Measurement Errors. Discussion paper 146, Institut de Statistique, UCL. Kneip, A, L. Simar and P.W. Wilson (28), Asymptotics and consistent bootstraps for DEA estimators in non-parametric frontier models, Econometric Theory, 24,

85 Nonparametric Frontier Estimation: recent developments and new challenges 85 Kneip, A., Simar, L. and P.W. Wilson (211), A Computational Efficient, Consistent Bootstrap for Inference with Non-parametric DEA Estimators. Discussion paper 93, Institut de Statistique, UCL, in press Computational Economics. Kneip, A., Simar, L. and P.W. Wilson (211), Central Limit Theorems for DEA efficiency scores: when bias can kill the variance. Discussion paper 211/**, Institut de Statistique, UCL. Koopmans, T.C.(1951), An analysis of production as an efficient combination of activities, in Koopmans, T.C. (ed) Activity Analysis of Production and Allocation, Cowles Commision for Research in Economics, Monograph 13, John-Wiley, New-York. Korostelev, A., Simar, L. and A. Tsybakov (1995a), Efficient Estimation of Monotone Boundaries, Annals of Statistics, 23(2), Korostelev, A., Simar, L. and A. Tsybakov (1995b), On Estimation of Monotone and Convex Boundaries, Publications des Instituts de Statistique des Universités de Paris, 1, Kumbhakar, S.C. and C.A.K. Lovell (2), Stochastic Frontier Analysis, Cambridge University Press. Kumbhakar, S.C., Park, B.U., Simar, L. and E.G. Tsionas (27), Nonparametric stochastic frontiers: a local likelihood approach, Journal of Econometrics, 137, 1, Mouchart, M. and L. Simar (22), Efficiency analysis of Air Controlers: first insights, Consulting report 22, Institut de Statistique, Université Catholique de Louvain, Belgium.

86 Nonparametric Frontier Estimation: recent developments and new challenges 86 Park, B.U., Jeong, S.-O. and L. Simar (21), Asymptotic Distribution of Conical-Hull Estimators of Directional Edges. Annals of Statistics, Vol 38, 6, Park, B. Simar, L. and Ch. Weiner (2), The FDH Estimator for Productivity Efficiency Scores: Asymptotic Properties, Econometric Theory, Vol 16, Park, B., L. Simar and V. Zelenyuk (28), Local Likelihood Estimation of Truncated Regression and its Partial Derivatives: Theory and Application, Journal of Econometrics, 146, Ritter, C. and L. Simar (1997), Pitfalls of normal-gamma stochastic frontier models, Journal of Productivity Analysis, 8, Schubert, T. and L. Simar (211), Innovation and export activities in the German mechanical engineering sector: an application of testing restrictions in production analysis. Journal of Productivity Analysis, 36, Shephard, R.W. (197). Theory of Cost and Production Function. Princeton University Press, Princeton, New-Jersey. Simar, L. (1992), Estimating efficiencies from frontier models with panel data: a comparison of parametric, non-parametric and semi-parametric methods with bootstrapping, Journal of Productivity Analysis, 3,