Directional Nonparametric Least Absolute Deviations Method for Estimating the Boundary of a Convex Set

Transcription

1 Drectonal Nonparametrc Least Absolute Devatons Method for Estmatng the Boundary of a Convex Set Tmo Kuosmanen Sebastán Lozano Kansantalousteteen pävät Jyväskylä

2 Background Kuosmanen, T. (2008): Representaton Theorem for Convex Nonparametrc Least Squares, The Econometrc Journal (to appear) least squares subject to shape constrants multple regressors mn ( y f ( x )) f s. t. n = 1 f s ncreasng/decreasng f s concave/convex 2

3 Smulated example y observatons true f g representor of CNLS log-lnear OLS curve x

4 Smulated example y observatons true f g representor of CNLS log-lnear OLS curve x

5 Ths paper Extend nonparametrc estmaton subject to shape constrants to Least absolute devatons (LAD) Drectonal dsturbance term drectonal dstance functon

6 Why least absolute devatons (LAD)? Compared to the least squares estmator, the LAD estmator s more robust to atypcal observatons (outlers) n small and moderate szed samples. Problem can be lnearzed Can be solved by lnear programmng There exsts an equvalent dual problem Game theoretc nterpreraton

7 Settng Consder a non-empty, closed, convex, monotonc m set Θ R The boundary of Θ s { z z z z } B ( Θ ) = Θ > Θ B( Θ) z 2 Θ z 1

8 Drectonal dstance functon Drectonal dstance functon Luenberger (1992) J. Math Econ. Chambers et al. (1996) J. Econ. Theor., Chambers et al. (1998) JOTA { } DD( z, g) = sup δ z + δ g Θ, g > 0 B( Θ) z 2 z g z 1

9 Drectonal dstance functon DD s a complete characterzaton of set Θ and ts boundary DD( z, g) = sup { δ z + δ g Θ } { m z (, ) 0} R DD z g Θ = { m z } R DD z g B( Θ ) = (, ) = 0 B( Θ) z 2 z g z 1

10 Data generatng process Random sample of vectors z, = 1,,n, drawn from B(Θ) Dsturbance term δ n drecton g, such that the observed vectors z satsfy z + δg B( Θ) δ s a random varable wth a mean zero and an unknown symmetrc dstrbuton.

11 Problem Can we estmate boundary B(Θ) n a nonparametrc fashon, relyng on the postulated propertes of monotoncty and convexty? Ths s a relevant ssue e.g. n the context of productve effcency analyss Note: There s no natural dependent varable. Choosng any of the m dmensons of vectors z to serve as the dependent varable of the regresson analyss would be arbtrary. Errors n all varables

12 Nonparametrc least absolute devatons (NLAD) estmator mn d, T s. t. z D = + d g n = 1 d B( T ) T s convex and monotonc

13 Representaton theorem mn d, T s. t. z Infnte dmensonal problem D = n = 1 + d g B( T ) d T s convex and monotonc Fnte dmensonal LP problem mn D + = ( d + d ) d, d, ω s. t. = ω ( z z ) ( d d ) ( d d ) h, ω g = 1 d + n h h h ω 0 +, d 0

14 Representaton theorem mn d, T s. t. z Infnte dmensonal problem D = n = 1 + d g B( T ) d T s convex and monotonc Fnte dmensonal LP problem mn D + = ( d + d ) d, d, ω s. t. = ω ( z z ) ( d d ) ( d d ) h, ω g = 1 d + n h h h ω 0 +, d 0 D * = D *

15 Illustraton B( Θ) z 2 z ω g z 1

16 Dualty Prmal problem Dual problem max λ, τ s. t. n = 1 1 λ λ 1 0 h, z λ + τ g λ z λ h h h h h h h h h h τ mn D + = ( d + d ) d, d, ω s. t. = ω ( z z ) ( d d ) ( d d ) h, ω g = 1 d + n h h h ω 0 +, d 0

17 NLAD vs. DEA NLAD problem DEA problem max λ, τ s. t. n = 1 1 λ λ 1 0 h, z λ + τ g λ z λ h h h h h h h h h h τ max λ, τ s. t. n = 1 z + τ g λ z h h h h λ h λ z h h τ = 1 0 h,

18 Game theoretc nterpretaton Performance evaluaton game 1 Prncpal, n Agents Vector Z represents observed performance of Agent n terms of m dfferent crtera Pure strateges: Prncpal selects one of m crtera for performance evaluaton; can dffer from one agent to another Agent selects one of n agents as the benchmark Mxed strateges allowed

19 Payoffs: Game theoretc nterpretaton Agent : δ = ( z + δ ) z hk h k h s the benchmark agent, k s the crteron Prncpal: n C c δ = 1

20 Nash equlbrum The NLAD model characterzes the mxed-strategy Nash equlbrum of the game. dstance measures + d = d d, = 1,..., n, represent the net payoff receved by Agent, multpler weghts ω (mn problem) represent the optmal mxed strategy of the Prncpal, ntensty weghts λ h (max problem) represent the optmal mxed strategy of Agent n the equlbrum.

21 Applcaton to Olympc games Performance of countres n Athens 2004 Olympc games Crtera: Populaton GDP # Gold medals # Slver medals # Bronze medals

22 Applcaton to Olympc games Note: total number of medals s fxed Ignored n DEA assessments NLAD model redstrbutes the medals accordng to resources Drecton vector g = (0,0,1,1,1) Weght constrants ω Gold ω Slver ω Bronze

23 Top 10 countres Russa Australa Cuba Ukrane South Korea Chna Romana Hungary Netherlands Greece NLAD resdual DEA effcency

24 Bottom 10 countres Portugal Belgum Canada Venezuela Ngera Colomba Brazl Indonesa Mexco Inda NLAD resdual DEA effcency

25 Conclusons A new approach for estmatng the boundary of a convex set Combne deas from three dfferent felds of lterature: Convex nonparametrc least squares Least absolute devatons (LAD) Drectonal dstance functon Compellng nterpretaton as the Nash equlbrum of a Prncpal Agent game Applcable to performance evaluaton and productvty & effcency analyss