Stochastic Nonparametric Envelopment of Data (StoNED) in the Case of Panel Data: Timo Kuosmanen

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1 Stochastic Nonparametric Envelopment of Data (StoNED) in the Case of Panel Data: Timo Kuosmanen NAPW 008 June 5-7, New York City NYU Stern School of Business

2 Motivation The field of productive efficiency analysis has been divided between two dominating paradigms: DEA is a deterministic, mathematical programming approach to productive efficiency analysis SFA is a probabilistic, parametric, regression-based approach to frontier estimation

3 Motivation Bridging the gap between DEA and SFA is increasingly recognized as one of the key methodological challenges to this field Banker and Maindiratta (199) JPA; Park and Simar (1994) JASA; Fan, Li, Weersink (1996) JBES; Kneip and Simar (1996) JPA; Park, Sickles and Simar (1998, 003, 006) J. Ectr.; Post, Cherchye and Kuosmanen (00) OR; Hall and Simar (00) JASA; Griffin and Steel (004) J. Ectr.; Kuosmanen, Post and Scholtes (007) J. Ectr.; Kumbhakar, Park, Simar and Tsionas (007) J. Ectr., etc. etc.

4 The key idea DEA can be interpreted as nonparametric least squares regression subject to monotonicity and concavity of the frontier and a sign-constraint on residuals Kuosmanen (008): Representation Theorem for Convex Nonparametric Least Squares, Econometric Journal, forthcoming. Kuosmanen and Johnson (Session TB, Thursday)

5 The key idea DEA can be interpreted as nonparametric least squares regression subject to a sign-constraint on residuals DEA is a nonparametric counterpart to the parametric programming (PP) approach by Aigner & Chu (1968) AER Kuosmanen and Johnson (Session TB, Thursday)

6 The key idea DEA can be interpreted as nonparametric least squares regression subject to a sign-constraint on residuals DEA is a nonparametric counterpart to the parametric programming (PP) approach by Aigner & Chu (1968) AER StoNED is a stochastic counterpart to DEA in the same way as SFA is a stochastic counterpart to PP

7 Earlier work on StoNED StoNED model for the cross-sectional setting: Nonparametric DEA-style production frontier Stochastic SFA-style noise and inefficiency components Encompasses both DEA and SFA as its special cases Kuosmanen T. (006): Stochastic Nonparametric Envelopment of Data: Combining Virtues of SFA and DEA in a Unified Framework. MTT Discussion Paper 3/006. Kuosmanen T, and Kortelainen M. (007): Stochastic Nonparametric Envelopment of Data: Cross-sectional Frontier Estimation Subject to Shape Constraints, University of Joensuu, Economics Discussion Paper 46.

8 Cross-sectional StoNED model y = f ( x ) u + v, i i i i Function f is an increasing and concave frontier production function with an unknown functional form (similar to DEA) u σ is the inefficiency term v i i. i. d (0, u ) (0, v ) σ is the noise term i i. i. d N N

9 Estimating the StoNED model In two steps Step 1: Estimate E(y i x i ) = f(x i ) -µ by convex nonparametric least squares (CNLS) Step : Given residuals from Step 1, apply the method of moments or maximum pseudolikelihood techniques to estimate parameters σ u andσ v. Conditional expected inefficiency E(u i ε i ) can be calculated using the result by Jondrow et al. (198)

10 Cross section vs. panel data In the cross-sectional setting, it is impossible to distinguish inefficiency from noise without imposing some distributional assumptions, E.g., u i ~ N(0,σ u ), v i ~N(0,σ v ). If we observe the same firm over many periods, the noise component can be averaged out => fully nonparametric estimation subject to noise is possible, assuming just monotonicity and concavity of the frontier f

11 Earlier SFA and DEA approaches SFA models routinely utilize the panel data features Random effects: Lee and Tyler (1978) J. Econometrics Fixed effects: Schmidt and Sickles (1984) JBES In DEA, possibilities of the panel data rarely utilized Apply DEA separately to each time period y, x Apply DEA to the averaged data : Ruggiero (004) JORS Semi- and nonparametric panel data models Kernel regression: Kneip and Simar (1996) JPA; Park et al. (1998, 003, 006) J. Econometrics; etc. i i

12 StoNED model for panel data y it = f(x it ) u i + v it y it = output of firm i in period t x it = input vector of firm i in period t u i = time invariant inefficiency of firm i v it = random disturbance, firm i in period t f belongs to the set of a continuous, monotonic, and concave production functions (F )

13 Estimation In a single step Apply Convex Nonparametric Least Squares (CNLS) regression adapted to the panel setting: CNLS in single-input case: Hildreth (1954), JASA, CNLS in multi-input case: Kuosmanen (008), Ectr. J.

14 Estimation The firm-specific time-invariant inefficiency term can be estimated by using fixed or random effects Fixed effects inefficiency term can be correlated with the inputs cannot distinguish inefficiency from other time-invariant firmspecific factors Random effects inefficiency term must be uncorrelated with the inputs allows time-invariant firm-specific factors to enter the model as inputs (or as other explanatory variables) fewer unknowns

15 CNLS with fixed effects min f, u, v s. t. T t= 1 i= 1 y = f ( x ) u + v f F v it n it it i it

16 CNLS with fixed effects Infinite dimensional problem Quadratic programming problem min f, u, v s. t. T t= 1 i= 1 y = f ( x ) u + v i; t f F v it n it it i it min α, β, u, v s. t. y = α + β x u + v i; t α T t = 1 i = 1 + β x α + β x i, j; t, s β 0 n v it it it it it i it it it it it js js it i; t

17 Representation Theorem Infinite dimensional problem Quadratic programming problem min f, u, v s. t. T t= 1 i= 1 y = f ( x ) u + v i; t f F v it n it it i it = min α, β, u, v s. t. y = α + β x u + v i; t α T t = 1 i = 1 + β x α + β x i, j; t, s β 0 n v it it it it it i it it it it it js js it i; t The proof is analogous to that of Theorem.1 in Kuosmanen (008) Ectr. J.

18 CNLS regression observations CNLS curve

19 CNLS regression observations CNLS curve

20 Estimating inefficiency terms u Inefficiencies cannot be identified directly CNLS estimates of u i can be negative COLS correction applied to the fixed effects Schmidt and Sickles (1984) JBES { } uˆ = u min u i i h h

21 Estimating production function f CNLS estimator of the production function f is a piecewise linear function αˆ { } 11 + βˆ 11x min uh, h αˆ + { } 1 βˆ 1x min uh, fˆ( x ) = min h... αˆ + ˆ min { } nt βnt x uh, h whereα^, β^ are the estimated CNLS coefficients. Function f^ is one of the optimal solutions to the original, infinite dimensional CNLS problem

22 Random effects estimation Rewrite the StoNED model as y it = g(x it ) + ε it g(x it ) = f(x it ) E(u i ) is the average production function ε it = E(u i ) u i + v it is the normalized composite error with zero mean Estimate g by Feasible Generalized CNLS (adapt the Aitken estimator to CNLS) Infer inefficiecy based on the within group residuals of the FGNLS regression.

23 y Example f(x) = ln(x)+1, 5 firms, 0 periods scatter of {x i, f(x i )-u i } 4 3 Firm 1 Firm 1 Firm 3 Firm 4 Firm x

24 Example true f and scatter of {x i, y i } 4 3 Firm 1 Firm Firm 3 1 Firm 4 Firm 5 f(x )

25 Example Estimated StoNED frontier 4 3 Firm 1 Firm Firm 3 1 Firm 4 Firm 5 f(x ) 0 StoNED frontier

26 Example StoNED frontier vs. SFA frontier (CD, FE) 4 3 Firm 1 Firm Firm 3 Firm 4 1 Firm 5 f(x ) StoNED frontier 0 SFA frontier

27 y Example scatters {x i, f(x i )-u i } and {x i, f^(x i )-u i^} Firm 1 Firm Firm 3 Firm 4 Firm 5 StoNED 1 StoNED StoNED 3 StoNED 4 StoNED x

28 Extensions Technical progress (or regress): f(x it,t) Efficiency changes: u it Heteroskedasticity and autocorrelation: FGLS Heterogeneity: true fixed effects modeling

29 Conclusions StoNED melds together - Nonparametric frontier of DEA (f(x)) - Stochastic composite error of SFA (ε i = v i u i ) Fully nonparametric StoNED approach developed for the panel data setting Estimation by CNLS regression augmented with fixed or random effects treatment Combining the virtues of SFA and DEA is possible -> New opportunities as well as challenges

30 Thank you for your attention! Questions and comments are welcome: Visit the StoNED homepage for more information, news, working papers, computer codes: