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

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

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

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.

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)

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)

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.

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

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)

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

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

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 )

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.

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

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

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

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.

CNLS regression 7 6 5 4 3 observations CNLS curve 1 0 0 1 3 4 5 6 7 8 9 10 11

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

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

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.

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 5 0 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 x

Example true f and scatter of {x i, y i } 4 3 Firm 1 Firm Firm 3 1 Firm 4 Firm 5 f(x ) 0 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19

Example Estimated StoNED frontier 4 3 Firm 1 Firm Firm 3 1 Firm 4 Firm 5 f(x ) 0 StoNED frontier 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19

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 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19

y Example scatters {x i, f(x i )-u i } and {x i, f^(x i )-u i^} 4 3 1 0 Firm 1 Firm Firm 3 Firm 4 Firm 5 StoNED 1 StoNED StoNED 3 StoNED 4 StoNED 5 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 x

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

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

Thank you for your attention! Questions and comments are welcome: E-mail: Timo.Kuosmanen@mtt.fi Visit the StoNED homepage for more information, news, working papers, computer codes: http://www.nomepre.net/stoned