Introducing Environmental Variables in Nonparametric Frontier Models: a Probabilistic Approach

Transcription

1 Journal of Productvty Analyss, 24, , Sprnger Scence+Busness Meda, Inc. Manufactured n The Netherlands. Introducng Envronmental Varables n Nonparametrc Fronter Models: a Probablstc Approach CINZIA DARAIO IIT-CNR and Scuola Superore S. Anna, Italy LÉOPOLD SIMAR Insttut de Statstque, Unversté Catholque de Louvan, Belgum cnza@sssup.t; cnza.darao@t.cnr.t smar@stat.ucl.ac.be Abstract Ths paper proposes a general formulaton of a nonparametrc fronter model ntroducng external envronmental factors that mght nfluence the producton process but are nether nputs nor outputs under the control of the producer. A representaton s proposed n terms of a probablstc model whch defnes the data generatng process. Our approach extends the basc deas from Cazals et al. (2002) to the full multvarate case. We ntroduce the concepts of condtonal effcency measure and of condtonal effcency measure of order-m. Afterwards we suggest a practcal way for computng the nonparametrc estmators. Fnally, a smple methodology to nvestgate the nfluence of these external factors on the producton process s proposed. Numercal llustratons through some smulated examples and through a real data set on Mutual Funds show the usefulness of the approach. JEL Classfcaton: C13, C14, D20 Keywords: producton functon, fronter, nonparametrc estmaton, envronmental factors, robust estmaton 1. Introducton Most of the economc theory on effcency analyss dates back to Koopmans (1951) and Debreu (1951) on actvty analyss. We mght consder a producton technology where the actvty of the producton unts s characterzed by a set of nputs x R p + used to produce a set of outputs y Rq +. In ths framework the producton set s the set of techncally feasble combnatons of (x, y). It s defned as ={(x, y) R p+q + x can produce y}. (1.1) Correspondng author.

2 94 DARAIO AND SIMAR Assumptons are usually done on ths set, such as free dsposablty of nputs and outputs, meanng that f (x, y), then (x,y ), as soon as 1 x x and y y. Often convexty of s also assumed, and so on (see e.g. Shephard, 1970, for a modern formulaton of the problem). As far as effcency s of concern, the boundares of are of nterest. For nstance, f we are lookng n the nput drecton, the Farrell Debreu measure of nput-orented effcency score for a unt operatng at the level (x, y) s usually defned as: θ(x,y)= nf{θ (θx,y) }. (1.2) If (x, y) s nsde, θ(x,y) 1 s the proportonate reducton of nputs a unt workng at the level (x, y) should perform to acheve effcency. The correspondng radal effcent fronter n the nput space, for unts producng a level y of outputs, s defned by ponts wth effcency scores equal to 1. Ths fronter s then descrbed as the set (x (y), y), where x (y)=θ(x,y)x s the radal projecton of (x, y) on the fronter, n the nput drecton (orthogonal to the vector y). If we are lookng n the output drecton, the Farrell Debreu measure of outputorented effcency score for a unt operatng at the level (x, y) s smlarly defned as: λ(x, y) = sup{λ (x, λy) }. (1.3) Here λ(x, y) 1 represent the proportonate ncrease of outputs the unt operatng at level (x, y) should attan to be consdered as beng effcent. The effcent fronter corresponds to those ponts where λ(x, y) = 1. In emprcal studes, the set s unknown and so are the effcency scores. The econometrc problem s therefore to estmate these quanttes from a random sample of producton unts X ={(X,Y ) = 1,...,n}. Snce the poneerng work of Farrell (1957), the lterature has developed a lot of dfferent approaches to acheve ths goal. The nonparametrc models are partcularly appealng snce they don t rely on restrctve hypothess on the data generatng process (DGP). The most popular approaches are based on envelopment estmators n the sprt of Farrell approach. Deprns et al. (1984) have proposed the Free Dsposal Hull (FDH) of the set of the observatons to estmate : { } FDH = (x, y) R p+q + y Y,x X, = 1,...,n. (1.4) The convex hull of FDH provdes the data envelopment analyss (DEA) estmator of, popularzed as lnear programmng estmator by Charnes et al. (1978): { n n DEA = (x, y) R p+q + y γ Y ; x γ X for (γ 1,...,γ n ) such that =1 =1 } n γ = 1 ; γ 0,= 1,...,n, (1.5) =1

3 INTRODUCING ENVIRONMENTAL VARIABLES 95 t s the smallest free dsposal convex set coverng all the data. The correspondng estmators of the effcency scores are then obtaned by pluggng n the equatons (1.2) and (1.3) above n place of the unknown. Today, statstcal nference based on DEA/FDH type of estmators s avalable ether by usng asymptotc results (Knep et al., 1998 and Park et al., 2000) or by usng the bootstrap, see Smar and Wlson (2000) for a recent survey of the avalable results. In summary, f the true attanable set s free dsposal, then FDH s a consstent estmator of, but DEA s not. If s free dsposal and convex then both estmators are consstent, but the DEA estmator takes advantage of the convexty assumpton and acheves a slghtly faster rate of convergence. Durng the last decades, the lterature on effcency estmaton has been extended to explore the reasons of dfferent level of effcences across producton unts. The dea was to relate effcency measures to some external or envronmental factors whch mght nfluence the producton process but that are not under the control of the producers. The evaluaton of the nfluence of envronmental factors on the effcency of producers s ndeed a relevant ssue related to the explanatons of effcency, the dentfcaton of economc condtons that create neffcency, and fnally to the mprovement of manageral performance. When categorcal factors are consdered (lke the form of ownershp,... ), we are n the presence of dfferent groups of producers; n ths stuaton, testng ssues for comparng group effcency scores can be proposed usng approprate bootstrap algorthms (n the sprt of Smar and Wlson, 2002). When these external factors z R r are contnuous manly two approaches have been proposed n lterature but both are flawed by restrctve pror assumptons on the DGP and/or on the role of these external factors on the producton process. The frst famly of models s based on a one-stage approach (see e.g., Banker and Morey, 1986; Fare, et al., 1989; Fare et al., 1994, p ), where these factors z are consdered as free dsposal nputs and /or outputs whch contrbute to defne the attanable set R p + Rq + Rr, but whch are not actve n the optmzaton process defnng the effcency scores. For nstance, the analog of (1.2), would be: θ(x,y z) = nf{θ (θx,y,z) }, (1.6) and the estmator of s defned as above by addng the varables z n defnng the FDH and /or the DEA envelopng set, wth a varable z beng consdered as an nput f t s conducve (favorable, advantageous, benefcal) to effcency and as an output f t s detrmental (damagng, unfavorable) to effcency. The drawback of ths approach s twofold: frst we have to know a pror what s the role of z on the producton process, and second we assume the free dsposablty (and eventually convexty, f DEA s used) of the correspondng attanable extended set. The second famly of models s based on a two-stage approach. Here the estmated effcency scores are regressed, n an approprated lmted dependent varable parametrc regresson model (lke truncated normal regresson models) on the envronmental factors z. Some models n ths famly propose also three-stage and fourstage analyss as extenson of the two-stage approach (for more detals see Fred

4 96 DARAIO AND SIMAR et al., 1999; Fred et al., 2002). As ponted out by Smar and Wlson (2003), most of these models are flawed by the fact that usual nference on the obtaned estmates of the regresson coeffcent s not avalable. Smar and Wlson (2003) gve a lst of references where ths approach has been used and propose a bootstrap algorthm to obtan more accurate nference. However, also ths bootstrap-based approach, even when corrected, has two nconvenences. Frst, t reles on a separablty condton between the nput output space and the space of values for z: the extended attanable set s the cartesan product R r and so the value of z does not nfluence the poston of the fronter of the attanable set. Second, the regresson n the second stage reles on some parametrc assumptons (lke lnear model and truncated normal error term). In ths paper, we propose a more general full nonparametrc approach whch overcomes most of the drawbacks mentoned above. It reles on a probablstc defnton of the fronter and of the effcency whch s equvalent to the defnton proposed above but allows an easy ntroducton of envronmental factors. The basc deas where proposed n Cazals et al. (2002) (from hereafter CFS). Here, we extend to a more general multvarate setup and we propose a practcal methodology to evaluate the estmators. We wll defne condtonal effcent fronter and also condtonal order-m fronter and ther correspondng nonparametrc estmators. In partcular, order-m fronter estmators are known as beng more robust to outlers and/or extreme values than the full fronter estmates. We also suggest an easy procedure for evaluatng the mpact of these envronmental factors on the producton process. The paper s organzed as follows. The next secton ntroduces the multvarate probablstc model for defnng the DGP of a producton process. Ths secton ncludes also the defnton of the full fronter and of the order-m fronter. Secton 3 shows how ths framework can easly be adapted to the ntroducton of envronmental factors. Secton 4 addresses some practcal computatonal ssues and Secton 5 llustrates the methodology by usng some smulated data sets and a real data set on mutual funds. Secton 6 concludes. 2. Producton Fronters: a Probablstc Formulaton The producton process s here descrbed by the jont probablty measure of (X, Y ) on R p + Rq +. The support of (X, Y ) s the attanable set. In terms of the jont probablty measure of (X, Y ), the Farrell-Debreu nput effcency defned n (1.2) can also be characterzed, under free dsposablty, as: θ(x,y)= nf{θ F X (θx y)>0}, (2.1) where F X (x y)= Prob(X x Y y). A nonparametrc estmator of θ(x,y) can be provded by pluggng the emprcal verson of F X (x y) n (2.1) gven by n=1 1I (X x, Y y) F X,n (x y)= n=1, (2.2) 1I (Y y)

5 INTRODUCING ENVIRONMENTAL VARIABLES 97 where 1I ( ) s the ndcator functon. Then, the estmator of the nput effcency score for a gven pont (x, y) s the soluton of ˆθ n (x, y) = nf{θ F X,n (θx y)>0}. (2.3) Now, as ponted n CFS, ths concdes to the FDH estmator of θ(x,y) gven by { } ( j X ) ˆθ n (x, y) = nf{θ (θ x,y) FDH }= mn max Y y j=1,...,p x j, (2.4) where a j denotes the jth component of a vector a. We know that under the free dsposal assumpton, ths s a consstent estmator of θ(x,y) but wth a poor rate of convergence n 1/(p+q) : ths s the curse of dmensonalty shared by most nonparametrc estmators (see Park, Smar and Wener, 2000 for more propertes of ˆθ n (x, y)). The FDH estmator FDH s very senstve to extreme ponts, snce as an estmator of the full-fronter, t envelops all the cloud of ponts X. Therefore, CFS propose to estmate an order-m fronter, whch corresponds to another defnton of the benchmark aganst whch unts wll be compared. The dea can be summarzed as follows (we extend somewhat the presentaton of CFS, ntroducng here the concept of order-m effcency). For a gven level of outputs y n the nteror of the support of Y, consder now m..d. random varables X,= 1,...,m generated by the condtonal p-varate dstrbuton functon F X (x y) and defne the set: m (y) ={(x, y ) R p+q + x X,y y, = 1,...,m}. (2.5) Then, for any x, we may defne θ m (x, y) = nf{θ (θx,y) m (y)}. (2.6) Note that θ m (x, y) may be computed by the followng formula: { } ( j X ) θ m (x, y) = mn max =1,...,m j=1,...,p x j. (2.7) θ m (x, y) s a random varable snce the X are random varables generated by F X (x y). Now, adaptng the Defnton 5.1 n CFS for the expected order-m fronter, we can defne the expected order-m nput effcency measure, or n shorter, the order-m nput effcency measure as follows: Defnton 2.1. For any x R p +, the (expected) order-m nput effcency measure denoted by θ m (x, y) s defned for all y n the nteror of the support of Y as: θ m (x, y) = E( θ m (x, y) Y y), (2.8) where we assume the exstence of the expectaton.

6 98 DARAIO AND SIMAR So, n place of lookng for the lower boundary of the support of F X (x y), as was typcally the case for the full-fronter and for the effcency score θ(x,y), the order-m effcency score can be vewed as the expectaton of the mnmal nput effcency score of the unt (x, y), when compared to m unts randomly drawn from the populaton of unts producng more outputs than the level y. Ths s certanly a less extreme benchmark for the unt (x, y) than the absolute mnmal achevable level of nputs: t s compared to a set of m peers producng more than ts level y and we take as benchmark, the expectaton of the mnmal achevable nput n place of the absolute mnmal achevable nput. Note that the order-m effcency score s not bounded by 1: a value of θ m (x, y) greater than one ndcates that the unt operatng at the level (x, y) s more effcent than the average of m peers randomly drawn from the populaton of unts producng more output than y. Then for any x R p +, the expected mnmum level of nputs of order-m s defned as xm (y) = θ m(x,y)x whch can be compared wth the full-fronter x (y) = θ(x,y)x. The order-m nput effcency score shares the followng propertes: Theorem 2.1. For any x R p + and for all y n the nteror of the support of Y,f θ m (x, y) exsts, we have: θ m (x, y) = 0 = θ(x,y)+ (1 F X (ux y)) m du (2.9) θ(x,y) (1 F X (ux y)) m du, (2.10) lm m θ m(x, y) = θ(x,y). (2.11) The theorem s a drect consequence of Theorem 5.1 and Theorem 5.2 of CFS. A nonparametrc estmator of θ m (x, y) s straghtforward: we replace the true F X ( y) by ts emprcal verson, F X,n ( y). Wehave ˆθ m,n (x, y) = Ê( θ m (x, y) Y y) = 0 = ˆθ n (x, y) + (1 F X,n (ux y)) m du, (2.12) ˆθ n (x,y) (1 F X,n (ux y)) m du (2.13) Ths leads to an estmator of the fronter, whch for fnte m, does not envelop all the observed data ponts and so, s less senstve to extreme ponts and /or to outlers. As shown by (2.13), as m ncreases and for fxed n, ˆθ m,n (x, y) ˆθ n (x, y). Smar (2003) proposes a sem-automatc procedure to flag potental outlers by nvestgatng the convergence of ˆθ m,n (x, y) to ˆθ n (x, y) as m ncreases: f ˆθ m,n (x, y) s stll larger than 1 even for large values of m, then (x, y) could be an extreme ponts of the cloud X. CFS analyze the asymptotc propertes of the proposed nonparametrc estmators. In partcular, they show the n-consstency of ˆθ m,n (x, y) to θ m (x, y) for m

7 INTRODUCING ENVIRONMENTAL VARIABLES 99 fxed, as n. Note that we avod the curse of dmensonalty for the nonparametrc estmator of the order-m effcency. We now brefly sketch the man dfferences for the output orented case. The Farrell-Debreu output effcency score can be characterzed as λ(x, y) = sup{λ S Y (λy x)>0}, (2.14) where S Y (y x)= Prob(Y y X x). A nonparametrc estmator of λ(x, y) s provded by the emprcal verson of S Y (y x): n=1 1I (X x,y y) Ŝ Y,n (y x)= n=1. (2.15) 1I (X x) Then, the estmator of the output effcency score for a gven pont (x, y) s the soluton of ˆλ n (x, y) = sup{λ Ŝ Y,n (λy x)>0}, (2.16) whch concdes to the FDH estmator: ˆλ n (x, y) = sup{λ (x, λy) FDH }= max X x { mn j=1,..., q ( )} j Y y j, (2.17) For a gven level of nputs x n the nteror of the support of X, consder m..d. random varables Y,= 1,...,m generated by the condtonal q-varate dstrbuton functon F Y (y x)= Prob(Y y X x) and defne the set: m (x) ={(x,y) R p+q + x x,y y, = 1,...,m}. (2.18) Then, for any y, we may defne λ m (x, y) = sup{λ (x, λy) m (x)} (2.19) { ( )} j Y = max mn =1,...,m j=1,..., q y j. (2.20) The order-m output effcency measure s defned as follows. Defnton 2.2. For any y R q +, the (expected) order-m output effcency measure denoted by λ m (x, y) s defned for all x n the nteror of the support of X as: λ m (x, y) = E( λ m (x, y) X x), (2.21) where we assume the exstence of the expectaton. As above, we obtan

8 100 DARAIO AND SIMAR Theorem 2.2. For any y R q + λ m (x, y) exsts, we have: and for all x n the nteror of the support of X, f λ m (x, y) = 0 = λ(x, y) [ 1 (1 SY (uy x)) m] du (2.22) λ(x,y) 0 (1 S Y (uy x)) m du, (2.23) lm m λ m(x, y) = λ(x, y). (2.24) As above, the theorem derves mmedately from Theorem A.1 and Theorem A.2 of CFS, whch are the adapted versons of ther Theorems 5.1 and 5.2 to the outputorented case. A nonparametrc estmator of λ m (x, y) s gven by: ˆλ m (x, y) = 0 = ˆλ n (x, y) [ 1 (1 ŜY,n (uy x)) m] du (2.25) ˆλ n (x,y) 0 (1 Ŝ Y,n (uy x)) m du. (2.26) 3. Introducng Envronmental Factors The analyss of the precedng secton can easly be extended to the case where addtonal nformaton s provded by other varables Z R r, exogenous to the producton process tself, but whch may explan part of t. The basc dea for ntroducng ths addtonal nformaton n the model s to condton the producton process to a gven value of Z = z. CFS propose the dea for order-m fronters and for the unvarate case (one nput for the nput orented case or one output for the output orented case). We propose below a more general presentaton nspred from Sectons 5.1 and 5.2 of CFS, allowng to handle the mult-nput (mult-output) and the full fronter cases. To save place, we descrbe the basc deas n the nput orented framework. Practcal computatonal ssues are addressed n Secton 4. The jont dstrbuton on (X, Y ) condtonal on Z =z defnes the producton process f Z = z. In partcular the effcency measure defned above n (2.1) has to be adapted to the condton Z = z as follows: θ(x,y z) = nf{θ F X (θx y,z)>0}, (3.1) where F X (x y,z)= Prob(X x Y y,z= z). A nonparametrc estmator of the condtonal full-fronter effcency θ(x,y z) s gven by pluggng a nonparametrc estmator of F X (x y,z). Ths requres some smoothng technques n z. At ths purpose we use a kernel estmator of F X (x y,z)

9 INTRODUCING ENVIRONMENTAL VARIABLES 101 defned as: F X,n (x y,z)= n=1 1I (x x,y y)k((z z )/h n ) n=1, (3.2) 1I (y y)k((z z )/h n ) where K( ) s the kernel and h n s the bandwdth of approprate sze (we dscuss practcal bandwdth selecton ssues n the next secton). Hence, we obtan the condtonal FDH effcency measure as follows: ˆθ n (x, y z) = nf{θ F X,n (θx y,z)>0}. (3.3) Note that the asymptotc propertes of ˆθ n (x, y z) have not yet been derved n the lterature, but we mght expect that the rate of convergence of the usual FDH estmator wll deterorate wth the dmenson of Z, due to the smoothng n gettng F X,n (x y,z). The condtonal order-m nput effcency measure s ntroduced accordngly. For a gven level of outputs y n the nteror of the support of Y, consder the m..d. random varables X,= 1,...,m generated by the condtonal p-varate dstrbuton functon F X (x y,z) and defne the set: z m (y) ={(x, y ) R p+q + x X,y y, = 1,...,m}. (3.4) Note that ths set depends on the value of z snce the X are generated through F X (x y,z). Then, for any x, we may defne θ m z (x, y) = nf{θ (θx,y) z m (y)}. (3.5) Note that θ m z (x, y) may be computed by the followng formula: { θ z m (x, y) = mn =1,...,m max j=1,...,p } ( j X ) x j. (3.6) Now we can defne the condtonal order-m nput effcency measure by followng the dea of Defnton 2.1. Defnton 3.1. For any x R p +, the condtonal order-m nput effcency measure gven that Z = z, denoted by θ m (x, y z) s defned for all y n the nteror of the support of Y as: θ m (x, y z) = E( θ m z (x, y) Y y,z= z), (3.7) where we assume the exstence of the expectaton. Therefore, for any x R p +, the expected mnmum level of nputs of order m, gven that Z =z, s defned as x m (y z)=θ m(x, y z) x. As above we have mmedately the followng theorem.

10 102 DARAIO AND SIMAR Theorem 3.1. For any x R p + θ m (x, y z) exsts, we have: and for all y n the nteror of the support of Y,f θ m (x, y z) = 0 (1 F X (ux y,z)) m du, (3.8) lm m θ m(x, y z) = θ(x,y z). (3.9) The proof s obtaned by followng the same arguments for the proofs of Theorem 5.1 and Theorem 5.2 of CFS. A nonparametrc estmator of θ m (x, y z) s provded by pluggng the nonparametrc estmator of F X (x y,z) proposed above n (3.2). As showed n CFS, the resultng estmator of the order-m effcency measure acheves the rate of convergence nh r n, where r = dm(z), so here, due to the smoothng n Z, we cannot avod the curse of dmensonalty n the dmenson of Z. Formally, the estmator s obtaned as follows ˆθ m,n (x, y z) = Ê( θ m z (x, y) y,z)= (1 F X,n (ux y,z)) m du (3.10) 0 where θ z m (x, y) s defned above n (3.6), and the m random varables X are generated accordng to the estmated F X,n (x y,z). For a gven kernel and a gven bandwdth, the unvarate ntegral n (3.10) can be evaluated for any pont (x, y) and for any level of the envronmental factors Z =z, by usng an approprate numercal method. Note that here agan, for a fxed value of n we have lm m ˆθ m,n (x, y z)= ˆθ n (x, y z). The dervatons of the formulae for the defnton and the estmaton of the condtonal output effcency scores (full-fronter and order-m) are obtaned n a smlar way by replacng n Secton 2, S Y (λy x) by S Y (λy x,z) and Ŝ Y,n (λy x) by Ŝ Y,n (λy x,z). 4. Practcal Computatons Agan, the presentaton here s lmted to the nput orented case to save place FDH and Condtonal FDH Effcency Estmates For any gven pont (x, y), the FDH estmator ˆθ n (x, y) s very easy and fast to compute. The operatonal formula comes from (2.4): ˆθ n (x, y) = nf{θ F X,n (θx y)>0}= mn Y y { max j=1,...,p ( )} j X x j. (4.1)

11 INTRODUCING ENVIRONMENTAL VARIABLES 103 It s easy to show that for any (symmetrc) kernel wth compact support (K(u) = 0 f u > 1, as for the unform, trangle, epanechnkov or quartc kernels), the condtonal FDH effcency estmator s gven by: ˆθ n (x, y z) = nf{θ F X,n (θx y,z)>0}= mn { Y y, Z z h} { max j=1,...,p ( )} j X x j, (4.2) where h s the chosen bandwdth. It s nterestng to note that our plug-n estmates F X,n (θx y,z) > 0 s such that for kernels wth unbounded support, lke the gaussan kernel, ˆθ n (x, y z) ˆθ n (x, y): the estmate of the full-fronter effcency s unable to detect any nfluence of the envronmental factors. Therefore, n ths framework of condtonal boundary estmaton, kernels wth compact support have to be used Order-m and Condtonal Order-m Effcences For the order-m effcency ˆθ m,n (x, y) and ˆθ m,n (x, y z), the unvarate ntegrals (2.13) and (3.10) could be evaluated by numercal methods 2, even when p 1. The algorthms are very fast: the computaton of such ntegrals for one pont, s of the order of a hundredth of a second on a old Pentum III, 450 MHz machne. However numercal ntegraton can be avoded by an easy Monte Carlo algorthm (proposed n CFS for the order-m fronter), that we descrbe below, as fast for small values of m such as m = 10, but much slower when m ncreases: 1. Foragveny, draw a sample of sze m wth replacement among those X such that Y y and denote ths sample by (X 1,b,...,X m,b ); ( 2. Compute θ m b j (x, y) = mn X ) =1,...,m {max,b j=1,...,p. x j 3. Redo [1 2] for b = 1,...,B, where B s large. 4. Fnally, ˆθ m,n (x, y) B 1 Bb=1 θ m b (x, y). The qualty of the approxmaton can be tuned by ncreasng B but n most applcatons, say B = 200, seems to be a reasonable choce (see Smar, 2003, for a code wrtten n Matlab). Ths Monte Carlo algorthm can be adapted as follows for the condtonal order-m effcency score. Suppose that h s the chosen bandwdth for a partcular kernel K( ): 1. Foragveny, draw a sample of sze m wth replacement, and wth a probablty K((z z )/h)/ n j=1 K((z z j )/h), among those X such that Y y. Denote ths sample by (X 1,b,...,X m,b );

12 104 DARAIO AND SIMAR { ( )} 2. Compute θ m b,z j X,b (x, y) = mn =1,...,m max j=1,...,p. x j 3. Redo [1 2] for b = 1,...,B, where B s large. 4. Fnally, ˆθ m,n (x.y z) B 1 Bb=1 θ m b,z (x, y) 4.3. Bandwdth Selecton: A Smple Data-Drven Method It s well known that the choce of the bandwdth s mportant n nonparametrc smoothng. We propose n ths paper a very smple and easy to compute rule based on a k-nearest Neghbor (k-nn) method. The dea s that the smoothng n computng our Z-condtonal effcency estmators (3.3) and (3.10), comes from the smoothng n the estmaton of the condtonal dstrbuton functon F X,n (x y,z) (see equaton (3.2)). Ths s due to the contnuty of the varable Z. Hence, we suggest n a frst step to select a bandwdth h whch optmzes n a certan sense the estmaton of the densty of Z. We propose to use the lkelhood cross valdaton crteron (see Slverman, 1986 for detals), usng a k-nn method: ths allows to obtan bandwdths whch are localzed, nsurng we have always the same number of observatons Z n the local neghbor of the pont of nterest z when estmatng the densty of Z. So, for a grd of values of k, we evaluate the leave-one-out kernel densty estmate of Z, f ˆ ( ) k (Z ) for = 1,...,n and fnd the value of k whch maxmzes the functon: CV (k) = n 1 n =1 ( ) log f ˆ ( ) k (Z ), where f ˆ ( ) k (Z ) = 1 (n 1)h Z n j=1,j ( Zj Z K h Z ), and h Z s the local bandwdth chosen such that there exst k ponts Z j verfyng Z j Z h Z. Afterwards, n a second step, n order to compute F X,n (x y,z), wehavetotake nto account for the dmensonalty of x and y, and the sparsty of ponts n larger dmensonal spaces. Consequently, we expand the local bandwdth h Z by a factor 1 + n 1/(p+q), ncreasng wth (p + q) but decreasng wth n. In the smulatons below, we wll see that the estmaton of the effcency scores s not too senstve to small varaton of h around the value gven by our smple data-drven method.

13 INTRODUCING ENVIRONMENTAL VARIABLES Stressng the Influence of Z on the Producton Process The comparson of ˆθ n (x, y z) wth ˆθ n (x, y) s certanly of nterest for analyzng the global nfluence of Z on the producton process. When Z s unvarate, a scatter plot of the ratos 3 ˆθ n (x, y z)/ ˆθ n (x, y) aganst Z and ts smoothed nonparametrc regresson lne would be helpful to descrbe the nfluence of Z on effcency. If ths regresson s ncreasng, t ndcates that Z s detrmental (unfavorable) to effcency and when ths regresson s decreasng, t specfes a Z factor conducve (favorable) to effcency. We recall ndeed that here we are n an nput orented framework. In the frst case (unfavorable Z) the envronmental varable acts lke an extra undesred output to be produced askng for the use of more nputs n producton actvty, hence Z has a negatve effect on the producton process. In ths case ˆθ n (x, y z), the effcency computed takng Z nto account, wll be much larger than the uncondtonal effcency ˆθ n (x, y) for large values of Z then for small value of Z. Consequently, the ratos ˆθ n (x, y z)/ ˆθ n (x, y) wll ncrease, on average, wth Z. In the second case (favorable Z), the envronmental varable plays a role of a substtutve nput n the producton process, gvng the opportunty to save nputs n the actvty of producton; n ths case, Z has a postve effect on the producton process. It follows that the condtonal effcency ˆθ n (x, y z) wll be much larger than ˆθ n (x, y) for small values of Z (less substtutve nputs) than for large values of Z. Therefore, the ratos ˆθ n (x, y z)/ ˆθ n (x, y) wll, on average, decrease when Z ncreases. Snce we know that full-fronter estmates, and the derved estmated effcency scores, are very senstve to outlers and extreme values, we do also the same analyss for the more robust order-m effcency scores. Thus we present also the nonparametrc smoothed regresson of the ratos ˆθ m,n (x, y z)/ ˆθ m,n (x, y) on Z. Ths could be done for some values of m, knowng that when m ncreases, ths converges to the precedng case (full-fronter). As ponted n CFS, m can also be vewed as a trmmng parameter and several values of m could be used to provde a senstvty analyss. Ths allows to detect potental outlers,.e., ponts such that ther order-m effcency scores are stll larger than 1, even when m ncreases, see Smar (2003). Mutats mutands, the same could be done n the output orented case, wth smlar conclusons to detect the nfluence of Z on effcency. In ths case, the nfluence of Z goes n the opposte drecton: an ncreasng regresson corresponds to favorable envronmental factor and a decreasng regresson ndcates an unfavorable factor. In an output orented framework, a favorable Z means that the envronmental varable operates as a sort of extra nput freely avalable: for ths reason the envronment s favorable to the producton process. Consequently, the value of ˆλ n (x, y z) wll be much smaller (greater effcency) than ˆλ n (x, y) for small values of Z than for large values of Z: the ratos ˆλ n (x, y z)/ˆλ n (x, y) wll ncrease wth Z, on average. In the case of unfavorable Z, the envronmental varable works as a compulsory or unavodable output to be produced to face the negatve envronmental

14 106 DARAIO AND SIMAR condton. Z n a certan sense penalzes the producton of the outputs of nterest. In ths stuaton, ˆλ n (x, y z) wll be much smaller than ˆλ n (x, y) for large values of Z. As a result, the regresson lne of ˆλ n (x, y z)/ˆλ n (x, y) over Z wll be decreasng. Of course, we do not propose any nference here, but only an easy and useful descrptve dagnostc tool. 5. Emprcal Illustratons 5.1. Classroom Smulated Data Sets We begn wth some very smple smulatons where all the unts produce the same quantty of output by usng a sngle nput X. Now suppose that Z s unfavorable to the producton process (suppose each unt has to produce 1 lter of ce from water at 20 C: X s the requred energy and Z s the envronmental temperature: f Z s large, the process, even effcent wll requre more nput). We smulated a sample sze n=100 from Z Unform(1, 10) and compare 3 dfferent scenaros for generatng X. Case 1. X = Z 3/2 + ε, here Z s unfavorable to X for all values of ts range and ε s the random true neffcency ε Exp(3),.e. an exponental r.v. wth mean 3. Case 2. X = 5 3/2 + ε, here Z s ndependent of X and ε s as above. Case 3. X = 5 3/2 1I (Z 5) + Z 3/2 1I (Z > 5) + ε,.e., the unfavorable effect of Z on X starts only after the value of Z larger than 5, wth the same neffcency term ε. The three scenaros correspond to 3 qute dfferent stuatons. For Case 1, a monotone ncreasng fronter for X, as a functon of Z, for Case 2, no nfluence of Z, ths s to check f our procedure does not ntroduce spurous nfluence and Case 3 s a mxed case, no nfluence of Z for small value of Z, then a monotone ncreasng fronter. We computed the FDH, condtonal to Z-FDH, the order-m and condtonal Z-order-m nput effcency scores of all the 100 unts. For the llustraton, we have chosen here m = 25. For larger values of m the results converge very quckly to the full-fronter results. We present the results for a trangle kernel (we obtan very smlar results wth other kernels wth compact support). The three followng pctures (Fgures 1 3) llustrate how the nonparametrc regresson of the ratos between the condtonal and the uncondtonal effcency measures on Z s able to capture the real effect of Z on the producton process. We recover exactly what we expected through the three dfferent smulaton scenaros. So t seems that our estmaton procedure works pretty well. Table 1 gves the average values of the four dfferent nput effcency measures for the three cases. Agan we obtan the expected results under the three dfferent scenaros. For nstance, n case 2, the true mean effcency score s about

15 INTRODUCING ENVIRONMENTAL VARIABLES 107 eff(x,y z)/eff(x,y) Effect of Z on Full fronter values of Z 14 Effect of Z on Order m fronter eff m (x,y z)/eff m (x,y) values of Z Fgure 1. Classroom example, case 1, unfavorable effect of Z on producton effcency (nput orented framework). Scatterplot and smoothed regresson of ˆθ n (x, y z)/ ˆθ n (x, y) on Z (top) and of ˆθ m,n (x, y z)/ ˆθ m,n (x, y) on Z, wth m = 25 (bottom). Here k-nn= /2 /(5 3/2 + 3) Note that the full detaled table of results for the 100 unts (not reproduced here to save place) provdes two nterestng nformaton: for each unt (X,Y ), the number of domnatng unts N,.e., the number of ponts j such that X j X and Y j Y. The same s done for the Z-condtonal measure where N z s the number of ponts domnatng (X,Y ), wth n addton Z j Z h Z. The summary Table 1 gves the average values of N and N z over the n = 100 observatons. Table 1. Average effcency scores over the 100 observatons, for the classroom example. N s the average number of observatons domnatng (x, y) and N z the average number of domnatng ponts gven Z =z. h s the average of the selected local bandwdths (wth k-nn=19). Case N ˆθ n (x, y) ˆθ n,m (x, y) h N z ˆθ n (x, y z) ˆθ n,m (x, y z)

16 108 DARAIO AND SIMAR 3 Effect of Z on Full fronter eff(x,y z)/eff(x,y) values of Z 3 Effect of Z on Order m fronter eff m (x,y z)/eff m (x,y) values of Z Fgure 2. Classroom example, case 2, no effect of Z on producton effcency (nput orented framework). Scatterplot and smoothed regresson of ˆθ n (x, y z)/ ˆθ n (x, y) on Z (top) and of ˆθ m,n (x, y z)/ ˆθ m,n (x, y) on Z, wth m = 25 (bottom). Here k-nn=19. Senstvty to the bandwdth selecton Table 1 gves the results wth a bandwdth determned by our data-drven method. We redd the computatons for other choce of the bandwdth, takng h = h 0.8 and for h = h 1.2. The pctures correspondng to Fgures 1 3wth these other values of h, have the same shape and we don t see any substantal dfferences (they are not reproduced to save space). As expected, the ndvdual Z-condtonal effcency scores wll ncrease when h decreases and the contrary when h ncreases. But the orders of magntude of the Z-condtonal effcency scores reman the same. Table 2 show the correspondng values for the means of ˆθ n (x, y z) and of ˆθ n,m (x, y z), to be compared wth those obtaned n Table 1. The results are globally rather stable to the choce of the bandwdth Multvarate Smulated Data Sets In ths smulated example, we smulate a mult-nput (p = 2) and mult-output (q = 2) data set. We follow the deas proposed by Park et al. (2000) and by Smar

17 INTRODUCING ENVIRONMENTAL VARIABLES Effect of Z on Full fronter eff(x,y z)/eff(x,y) values of Z 3 Effect of Z on Order m fronter eff m (x,y z)/eff m (x,y) values of Z Fgure 3. Classroom example, case 3, unfavorable effect of Z on producton effcency, only after Z>5 (nput orented framework). Scatterplot and smoothed regresson of ˆθ n (x, y z)/ ˆθ n (x, y) on Z (top) and of ˆθ m,n (x, y z)/ ˆθ m,n (x, y) on Z, wth m = 25 (bottom). Here k-nn= 19. Table 2. Senstvty to the choce of the bandwdth, for the classroom example. Case h ˆθ n (x, y z) ˆθ n,m (x, y z) 1 h h h h h h (2003) to smulate the data set and then we ntroduce some dependency to an envronmental factor Z. In ths set-up, the functon descrbng the effcent fronter s gven by: y (2) = (x (1) ) 0.3 (x (2) ) 0.4 y (1) where y (j),(x (j) ), denotes the jth component of y, (of x), for j = 1, 2. We draw X (j) ndependent unforms on (1, 2) and Ỹ (j) ndependent unform on (0.2, 5).

18 110 DARAIO AND SIMAR Then the generated random rays n the output space are characterzed by the slopes S = Ỹ (2) /Ỹ (1). Fnally, the generated random ponts on the fronter are defned by: Y (1),eff = (X(1) ) 0.3 (X (2) ) 0.4 S + 1 Y (2),eff = (X(1) ) 0.3 (X (2) ) 0.4 Y (1),eff. We choose, as above, the effcences generated by exp( U ) where U are drawn from an exponental wth mean µ=1/3. Fnally, n a standard setup (wthout envronmental factors), we defne Y = Y,eff exp( U ). Now we ntroduce the dependency on Z n the latter expresson as follows: Z Unform(1, 4) Case 1. Z s favorable to output producton but dfferently for Y (1) than for Y (2). We defne V = Z and set Y (1) = V 2 Y (1),eff exp( U ) Y (2 ) = V 1/2 Y (2),eff exp( U ). Case 2. Z s ndependent of Y. We defne V =2.5, the mean of Z and use the same latter expressons to generate Y. Agan, Case 2 allows to check f our procedure does not ntroduce spurous nfluence of Z on the Z-condtonal effcency scores. We computed the FDH, condtonal to Z-FDH, the order-m and condtonal Z-order-m output effcency scores of all the unts. We have chosen agan m = 25, for larger values of m, saym 100, the results are very smlar to the full-fronter (FDH) results. We present the results for a trangle kernel: here agan the results are very stable wth respect to other choce of the kernel wth compact support. Fgures 4 and 5 ndcate very clearly the dfferences between the two scenaros even wth a small sample sze of n=100 (remember that we are n a space of dmenson 5). For a larger sample sze (n = 500) the effect of Z on the effcency appears stll more clearly as n Fgure 6. Here also, the dfference between the full fronter and the order-25 fronter s more vsble. Table 3 presents agan a summary of the results, as n the precedng classroom example. We see here, by lookng at the average values of N and N z, that n ths 4 (or 5) dmensonal space, t s better to rely on more observatons than n = 100 to get more sensble results, at least for the full-fronter estmates (ths s the curse of dmensonalty of the FDH and Z-FDH estmators). However, even when n = 100, the Fgures 4 and 5 allows to detect the effect of Z on the producton process. Senstvty to the bandwdth selecton Here agan, we redd the computatons for other choces of the bandwdth h, takng h = h 0.8 and for h = h 1.2. We obtan globally the same conculsons as for

19 INTRODUCING ENVIRONMENTAL VARIABLES Effect of Z on Full fronter eff(x,y z)/eff(x,y) values of Z Effect of Z on Order m fronter eff m (x,y z)/eff m (x,y) values of Z Fgure 4. Multvarate example, case 1, n = 100: postve effect of Z on producton effcency (output orented framework). Scatterplot and smoothed regresson of ˆλ n (x, y z)/ˆλ n (x, y) on Z (top) and of ˆλ m,n (x, y z)/ˆλ m,n (x, y) on Z, wth m = 25 (bottom). Here k NN = 17. Table 3. Multvarate example, Average effcency scores: N s the average number of observatons domnatng (x, y) and N z the average number of domnatng ponts gven Z = z. h s the average of the selected local bandwdths. Case N ˆλ n (x, y) ˆλ n,m (x, y) h N z ˆλ n (x, y z) ˆλ n,m (x, y z) Case 1, n = Case 2, n = Case 1, n = the classroom example. The pctures correspondng to Fgures 4 6 and obtaned wth these new values of h, have the same shape and we don t see any substantal dfferences (they are not reproduced to save space). The orders of magntude of the Z-condtonal effcency scores reman the same. Table 4 show the correspondng values for the means of ˆλ n (x, y z) and of ˆλ n,m (x, y z), to be compared wth those obtaned n Table 3. The results are remarkably stable to the choce of the bandwdth.

20 112 DARAIO AND SIMAR 1.5 Effect of Z on Full fronter eff(x,y z)/eff(x,y) values of Z 1.5 Effect of Z on Order m fronter eff m (x,y z)/eff m (x,y) values of Z Fgure 5. Multvarate example, case 2, n=100: no effect of Z on producton effcency (output orented framework). Scatterplot and smoothed regresson of ˆλ n (x, y z)/ˆλ n (x, y) on Z (top) and of ˆλ m,n (x, y z)/ˆλ m,n (x, y) on Z, wth m = 25 (bottom). Here k-nn=19. Table 4. Senstvty to the choce of the bandwdth, for the multvarate example. Case h ˆλ n (x, y z) ˆλ n,m (x, y z) 1 h h h h h h Mutual Funds Data We llustrate our methodology analyzng US Mutual Funds data. We use a crosssecton data set, collected by the reputed Mornngstar, whch conssts of the US Mutual Funds unverse updated at Among ths unverse we select the Aggressve-Growth (AG) category of Mutual Funds. These are funds that seek rapd growth of captal and that may nvest n emergng market growth companes.

21 INTRODUCING ENVIRONMENTAL VARIABLES Effect of Z on Full fronter eff(x,y z)/eff(x,y) values of Z 1.5 Effect of Z on Order m fronter eff m (x,y z)/eff m (x,y) values of Z Fgure 6. Multvarate example, case 1, n = 500: postve effect of Z on producton effcency (output orented framework). Scatterplot and smoothed regresson of ˆλ n (x, y z)/ˆλ n (x, y) on Z (top) and of ˆλ m,n (x, y z)/ˆλ m,n (x, y) on Z, wth m = 25 (bottom). Here k-nn=40. From a frst data set of 247 observatons, we end up, for ths llustraton, wth a sample of 129 mutual funds, after droppng 103 observatons for mssng values and 15 observatons detected by the Smar s (2003) procedure as beng outlers. The selecton of varables has been done by takng the same varables chosen n earler studes (Murthy et al. 1997; Sengupta, 2000) that used a (determnstc) nonparametrc approach. Followng these prevous studes, we apply an nput orented framework n order to evaluate the performance of mutual funds n terms of ther rsk (as expressed by standard devaton of return) and transacton costs (ncludng expense rato, loads and turnover) management. Murthy et al. (1997) used as nputs: rsk (standard devaton, or volatlty of the return), expense rato (the percentage of fund assets pad for operatng expenses, management fees, admnstratve fees, and all other asset-based costs), loads (percentage for the front-end and back-end sales charges of each fund) and turnover rato (a measure of the fund s tradng actvty). The three latter nputs are consdered as a measure of the transacton costs. The tradtonal output n ths framework s the total return of funds (the annual return at the , expressed n percentage terms). Most returns where negatve n ths perod, hence we shft them to get all postve returns by addng 100.

22 114 DARAIO AND SIMAR Ths does not change our nput orented analyss. Sengupta (2000) uses market rsks of mutual funds (the percentage of fund s movements that can be explaned by movements n ts benchmark ndex) as an nput n hs analyss, underlyng that the effect of market rsks s conducve for mutual funds performance. In our llustraton we use ths varable (market rsks) as envronmental varable, to nvestgate ts effect on our data,.e., f t s detrmental or favorable to the performance of mutual funds n the perod under consderaton. In our llustraton we decded to elmnate one of the nputs prevously consdered, the loads, for the followng reasons: the curse of dmensonalty (6 varables, wth only 129 observatons); loads n the data set s typcally a dscrete varable wth not many dfferent values (round to the percentage), wth a majorty of funds havng loads equal to zero; and fnally, the correlaton of ths varable wth any of the fve others (X, Y and Z) s smaller than 0.07, whch mght ndcate an orthogonal aspect of the actvty. So, we end up wth three nputs, 1 output, 1 envronmental factor and 129 observatons. Table 5 dsplays some summary statstcs of the chosen varables. Table 6 presents the results comng from our Matlab code. In order to save place, we present only 15 funds chosen at random from the full table (presented at length n the Appendx). We have chosen a trangle kernel for the smoothng and the lkelhood cross-valdaton procedure provded k = 21 as the optmal choce for the k-nn method. We select agan the value of m = 25, although we dd the computatons for several values of m. If10 m 50 we obtan very smlar results and when m s larger than, say, 100 we obtan very smlar results as for the full-fronter effcency scores. Lookng at the last row of Table 6, we see that the global effect of the market rsk factor Z on the full effcency measures s an ncrease from to For the order-m fronter we have a smlar mean effect gong from to The effect s more mportant for the full FDH fronter, as expected, snce these measures are more senstve to extreme ponts. We propose here some descrptve comments on the fgures of Table 6: a few funds have a huge ncrease of ther effcences when Z-condtonal measures are consdered (funds lke #1,#2,#3,..., even some lke fund #6 becomes effcent). Some other funds have a very poor performance, even f we take the envronmental factor nto account: these are funds lke #37, #74, #127,.... In practcal applcatons, these funds should deserve more attenton. To have a global dea of the effect of the rsk factor Z on mutual funds performance, we regress nonparametrcally the ratos between the condtonal effcency measures and the uncondtonal effcency measures on Z: we obtan the pcture dsplayed n Fgure 7. Lookng at ths pcture, we can see a global postve effect of the market rsk factor Z on the performance of mutual funds. When lookng at the full condtonal effcency measures (top panel of Fgure 7), ths effect seems to be more mportant when Z 0.5. Note that for low values of Z, the regresson lne n ths case s attracted to low values of solated ponts on the left of the pcture. Ths global effect s confrmed on the bottom panel of the same pcture, where

23 INTRODUCING ENVIRONMENTAL VARIABLES 115 Table 5. Summary statstcs for the n = 129 Aggressve-Growth US Mutual Funds. Average, standard devaton, mnmum, maxmum and nterquartle range. Varable Mean Std Mn Max qr Y = return X (1) = volatlty X (2) = turnover X (3) = exp. rato Z = market rsk the effect seems to start around Z = 0.2. These pctures confrm that wth our data market rsk acts as a substtutve nput n the mutual funds management process. 6. Concluson In ths paper, developng deas proposed by Cazals et al. (2002), we provde a full probablstc formulaton of a nonparametrc fronter model and of a nonparametrc fronter model of order-m. Ths formulaton allows the ntroducton n both models (full fronter and order-m fronter) of envronmental factors whch may nfluence the producton process but that are nether nputs nor outputs under the control of the producer. Table 6. Results from 15 selected funds from the Aggressve-Growth US Mutual Funds. N s the number of observatons domnatng (x, y) and N z the number of domnatng ponts gven Z = z. h s the selected local bandwdth (k-nn=21). Last row s the average over all the 129 observatons. Fund N ˆθ n (x, y) ˆθ n,m (x, y) h N z ˆθ n (x, y z) ˆθ n,m (x, y z) mean

24 116 DARAIO AND SIMAR Effect of Z on Full fronter 2 eff(x,y z)/eff(x,y) values of Z Effect of Z on Order m fronter 2 eff m (x,y z)/eff m (x,y) values of Z Fgure 7. Aggressve-Growth US Mutual Funds. Scatterplot and smoothed regresson of the ratos ˆθ n (x, y z)/ ˆθ n (x, y) on Z (top) and of ˆθ m,n (x, y z)/ ˆθ m,n (x, y) on Z, wth m = 25 (bottom). Here k-nn=21. The presentaton allows general mult-nput/mult-output stuatons and provdes a practcal way for evaluatng the nonparametrc estmators. A data-drven procedure for choosng the bandwdth, based on a k-nearest neghbor method s suggested. Furthermore, we propose a useful graphcal tool for hghlghtng the eventual nfluence of Z on the producton process. Our method wll tell us f the envronmental factor s conducve or detrmental to the producton actvty. The approach s llustrated by some smulated data set and wth a real data set on US mutual funds, where the rsk of the market shows a postve nfluence on the performance (management process) of mutual funds. Some nterestng theoretcal ssues are stll open n ths framework. For nstance, what are the statstcal propertes of the condtonal full fronter effcency estmator? Or, how could we select optmal bandwdth n the estmaton procedure? We propose a very smple sensble technque, based on lkelhood cross-valdaton for the densty of Z, but other crteron could be nvestgated.

25 INTRODUCING ENVIRONMENTAL VARIABLES 117 Appendx. A.1. Appendx A.1. Full Table of Results for the Mutual Funds Example. Unts N ˆθ n (x, y) ˆθ n,m (x, y) h N z ˆθ n (x, y z) ˆθ n,m (x, y z)