Center for Policy Research Working Paper No. 121

Transcription

1 ISSN: Center for Polcy Research Workng Paper No. 121 ALTERNATIVE TECHNICAL EFFICIENCY MEASURES: SKEW, BIAS, AND SCALE QU FENG AND WILLIAM C. HORRACE Center for Polcy Research Maxwell School of Ctzenshp and Publc Affars Syracuse Unversty 426 Eggers Hall Syracuse, New York (315) Fax (315) e-mal: March 2010 $5.00 Up-to-date nformaton about CPR s research projects and other actvtes s avalable from our World Wde Web ste at www-cpr.maxwell.syr.edu. All recent workng papers and Polcy Brefs can be read and/or prnted from there as well.

2 CENTER FOR POLICY RESEARCH Sprng 2010 Chrstne L. Hmes, Drector Maxwell Professor of Socology Assocate Drectors Margaret Austn Assocate Drector Budget and Admnstraton Douglas Wolf Gerald B. Cramer Professor of Agng Studes Assocate Drector, Agng Studes Program John Ynger Professor of Economcs and Publc Admnstraton Assocate Drector, Metropoltan Studes Program SENIOR RESEARCH ASSOCIATES Bad Baltag... Economcs Robert Bfulco... Publc Admnstraton Leonard Burman.. Publc Admnstraton/Economcs Kalena Cortes Educaton Thomas Dennson... Publc Admnstraton Wllam Duncombe... Publc Admnstraton Gary Engelhardt... Economcs Deborah Freund.. Publc Admnstraton/Economcs Madonna Harrngton Meyer... Socology Wllam C. Horrace... Economcs Duke Kao... Economcs Erc Kngson... Socal Work Sharon Koko..Publc Admnstraton Thomas Knesner... Economcs Jeffrey Kubk... Economcs Andrew London... Socology Len Lopoo... Publc Admnstraton Amy Lutz... Socology Jerry Mner... Economcs Jan Ondrch... Economcs John Palmer... Publc Admnstraton Davd Popp... Publc Admnstraton Chrstopher Rohlfs... Economcs Stuart Rosenthal... Economcs Ross Rubensten... Publc Admnstraton Perry Sngleton Economcs Margaret Usdansky... Socology Mchael Wasylenko... Economcs Jeffrey Wensten Economcs Janet Wlmoth... Socology GRADUATE ASSOCIATES Charles Alamo... Publc Admnstraton Matthew Baer... Publc Admnstraton Mara Brown... Socal Scence Chrstan Buerger... Publc Admnstraton Qanqan Cao... Economcs Il Hwan Chung... Publc Admnstraton Kevn Cook... Publc Admnstraton Alssa Dubnck... Economcs Kate Francs... Publc Admnstraton Andrew Fredson... Economcs Vrglo Galdo... Economcs Pallab Ghosh... Economcs Clorse Harvey... Publc Admnstraton Douglas Honma... Publc Admnstraton Becky Lafrancos... Economcs Hee Seung Lee... Publc Admnstraton Jng L... Economcs Wael Moussa... Economcs Phuong Nguyen... Publc Admnstraton Wendy Parker... Socology Lesle Powell... Publc Admnstraton Kerr Rassan... Publc Admnstraton Amanda Ross... Economcs Ryan Sullvan... Economcs Tre Wentlng... Socology Coady Wng... Publc Admnstraton Ryan Yeung... Publc Admnstraton Can Zhao.....Socal Scence STAFF Kelly Bogart Admnstratve Secretary Martha Bonney Publcatons/Events Coordnator Karen Cmlluca...Offce Coordnator Ktty Nasto.... Admnstratve Secretary Cand Patterson...Computer Consultant Roseann Presutt Admnstratve Secretary Mary Santy.... Admnstratve Secretary

3 Abstract In the fxed-effects stochastc fronter model an effcency measure relatve to the best frm n the sample s unversally employed. Ths paper consders a new measure relatve to the worst frm n the sample. We fnd that estmates of ths measure have smaller bas than those of the tradtonal measure when the sample conssts of many frms near the effcent fronter. Moreover, a two-sded measure relatve to both the best and the worst frms s proposed. Smulatons suggest that the new measures may be preferred dependng on the skewness of the neffcency dstrbuton and the scale of effcency dfferences. Emal: qfeng@ntu.edu.sg, Tel: Dvson of Economcs, School of Humantes and Socal Scences, HSS-04-48, 14 Nanyang Drve, Sngapore Correspondng Author. Emal: whorrace@maxwell.syr.edu, Tel: Fax: Center for Polcy Research, 426 Eggers Hall, Syracuse, NY Key Words: stochastc fronter model, relatve effcency measure, two-sded measure, bas, bootstrap confdence ntervals JEL Classfcaton: C15, C23, D24

4 1 Introducton There are several ways to estmate tme-nvarant techncal effcency n stochastc fronter models for panel data. Compared to maxmum lkelhood or generalzed least-squares estmaton (Battese and Coell, 1988), fxed-effects estmaton (Schmdt and Sckles, 1984) has the advantage of not requrng dstrbutonal assumptons on the error components. Wthout these dstrbutonal assumptons, effcency levels cannot be dentfed drectly. Hence, a measure relatve to the best frm n the sample s unversally employed (Schmdt and Sckles, 1984). In ths case only the effcency dstance to the best frms matters. The worst frm n the sample s gnored. For example, suppose there are 3 frms wth effcency levels 0.30, 0.90 and 0.99 respectvely. It appears that frm 2squteeffcent. If the effcency level of the worst frm mproves to 0.89 due to technologcal change, the dstance between the frm 2 and the best frm s unchanged. However, frm 2 s now almost as neffcent as the worst frm. Ths example shows that usng the worst frm as a reference pont provdes a dfferent perspectve on techncal effcency. Actually, n compettve settngs the worst frm s of partcular mportance because the margnal cost of ths frm may determne prce. 1 Ths paper consders an alternatve effcency measure relatve to the worst frm n the sample and compares ths measure to the tradtonal relatve effcency measure on a varety of metrcs. More generally, t may be nterestng to use both the best frm and the worst frm as reference ponts. Therefore, a two-sded measure relatve to both the best frm and the worst frm n the sample s also proposed. Dfferent from effcency measures relatve to the best or to the worst frms alone, the two-sded measure lnearly scales the effcency level onto the unt nterval wth effcency scores of 0 for the worst frm and 1 for the best frm. Consequently, the dstance of the effcency level between any frms becomes nformatve. Ths paper dscusses fxed-effects estmates of the measure relatve to the worst frm and the 1 Wewouldlketothankarefereeforpontngthsouttous. 1

5 two-sded measure (relatve to the best and the worst). We focus on estmaton bas and nference. The level of the bas of relatve effcency estmates s related to the skewness of the underlyng dstrbuton of techncal (n)effcency. Snce the max operator favors postve nose, the tradtonal estmate has larger bas when there are more effcent frms n the populaton. Qan and Sckles (2008) descrbe ths scenaro as mostly stars, few dogs. When there are "mostly stars" our estmate (relatve to the worst frm) s less based than the tradtonal estmate. However, not surprsngly, the bas results are reversed when there are "mostly dogs." When the dstrbuton of (n)effcency s symmetrc, the bas results of the two estmators are dentcal. These results are borne out n our smulatons based on three parameterzatons of the Beta dstrbuton. The two-sded estmate balances these sources of bas (n a sense) as we shall see n the sequel. Inference on estmated techncal effcency s often mportant, and t proceeds wth constructon of confdence ntervals. When dstrbutonal assumptons are made on the two error components (nose and neffcency), the theory for confdence nterval constructon s straghtforward (Horrace and Schmdt, 1996). The ntervals are vald n both fnte samples and asymptotcally. In the case of fxed-effects estmaton, confdence ntervals for techncal effcency may be based on asymptotc normalty, when the sample sze s large (Horrace and Schmdt, 2000). However, when the tme dmenson of the data s small, the preferred method to construct confdence ntervals wthout dstrbutonal assumptons s to perform the bootstrap. Km, Km and Schmdt (2007) provde a detaled and ntutve survey on constructng varetes of bootstrap confdence ntervals. They argue that the max operator of the tradtonal estmate (relatve to the best frm) produces bas, leadng to low coverage rates when constructng smple bootstrap confdence ntervals. Our proposed estmates suffer from the same source of bas. However, the coverage rates of bootstrap confdence ntervals are a functon of the magntude of the bas whch s related to the skewness of the underlyng dstrbuton of (n)effcency and to the estmate employed (.e., relatve to best, relatve to worst, or two-sded). 2

6 Ultmately, the fxed-effects estmates provde nformaton on the skewness of the underlyng dstrbuton of (n)effcency. Gven ths skewness nformaton there maybe an emprcal trade-off between bas and the effcency measure employed. For example, f the data suggest there are "mostly stars, few dogs", then the tradtonal estmate has large bas and our estmate (relatve to the worst) has small bas. However, a measure relatve to the best frm may be of nterest. In ths case, the emprcst must decde whch s more mportant: bas or the emprcal relevance of the measure. (Ths trade-off also has mplcatons for bootstrap nference.) Of course, f a measure relatve to the worst frm s needed, then there s no trade-off n ths case. Ths trade-off underscores the fact that the proposed estmates are alternatves to the tradtonal estmates and that all three estmates are measurng dfferent quanttes (although they are all normalzatons to the unt nterval, as we shall see). There s no sense n whch the estmates are substtutes; they are complements that smply add to the emprcst s toolbox. The paper s organzed as follows. The next secton dscusses the effcency measure relatve to the worst frm, our proposed estmate of ths measure, and performs smulatons to compare the bas of the estmate to that of the tradton estmate under dfferent (n)effcency dstrbutons. Secton 3, ntroduces a two-sded effcency measure that pegs relatve effcency not only to the most effcent frm n the sample but also to the least effcent frm. A smulaton study of ts bas s also conducted. In secton 4 bootstrap confdence ntervals are dscussed for the dfferent measures, and a smulaton study of coverage rates and nterval wdths s provded n the sprt of Km, Km and Schmdt (2007). Our contrbuton s to demonstrate how these measures and ther estmates perform n fnte samples under dfferent skewnesses of the dstrbuton of techncal neffcency. Secton 5 apples the estmators to a panel of Indonesan rce farms, and the salent features of all three measures are dscussed and compared. The last secton summarzes and concludes. 3

7 2 Relatve Effcency Measures The stochastc fronter model for panel data s: y t = α + x 0 tβ u + v t, =1,,N, t=1,,t. (1) The error term contans two parts: tme-nvarant u 0, a measure of techncal neffcency; and v t d(0,σ 2 v). A large value of u mples that the frm s neffcent. Usually, techncal effcency s defned as r =exp( u ) under a log-lnear specfcaton of the Cobb-Douglas producton functon. 2 Lettng α = α u, slope parameter β can be estmated consstently usng fxed-effects estmaton. Call ths estmate β. b The usual estmate of α s bα = y x 0 b β,wherey and x are wthn-group averages. However, u s undentfed wthout addtonal assumptons. The lterature suggests an effcency measure relatve to the best frm u =max j α j α = u mn j u j and ts estmate bu =max j bα j bα,wherethebα are fxed-effects estmates of α (Schmdt and Sckles, 1984). Correspondngly, when output s n logarthms, relatve techncal effcency s defned as r exp( u ) wth ts fxed-effects estmate br exp( bu ). We call u the max-measure or the tradtonal measure. In the stochastc fronter model (1), the effcent fronter s defned as the best frm usng mn j u j or max j α j. Hence, u measures techncal neffcency as the devaton from ths fronter. Smlarly, we can defne the neffcent fronter as the worst frm usng max j u j or mn j α j, and measure techncal effcency as u = α mn j α j =maxu j u. (2) j Ths s smply the devaton from the neffcent fronter. We call u the mn-measure. Its correspondng techncal effcency score s r =1 exp( u ). Usng exp( x) 1 x for small x, r s an approxmaton of u on the unt nterval. To fx deas, we mage that u has some 2 If Y = e u e v f(x, β), thene u s techncal effcency for y =lny and f(x, β) = k j=1 xβ j j,say.evenfthssnot the producton functon n mnd, emprcsts often use the measure r to normalze u to the unt nterval. 4

8 upper support bound u, so that realzatons of u cannot be too large. Then mn j α j approxmates α u, andu approxmates u u 0, the devaton from the neffcent fronter. The concept of a upper bound for neffcency was recently consdered n Qan and Sckles (2008). Indeed, "usng ths bound as the neffcent fronter, we may defne nverted effcency scores n the same sprt of Inverted DEA descrbed n Entan, Maeda, and Tanaka (2002)." 3 The correspondng fxed-effects estmates of the mn-measure are: û = ˆα mn j ˆα j, (3) ˆr = 1 exp( û ). Usng the same arguments as Schmdt and Sckles (1984), û s consstent for u = u u,as T and N. When the producton functon s Cobb-Douglas and output s n logarthms, the r has a natural nterpretaton: t s the true percentage of the output of frm relatve to the effcent frm for a fxed set of nput, so r s the way we would naturally measure effcency for a Cobb-Douglas producton functon. In ths case the proposed measure, r,doesnothaveths natural nterpretaton, however (as already mentoned) performance relatve to the neffcent frm may be relevant, because the margnal cost of the neffcent frm may equal prce n compettve markets (markets where N s large and u s small). When output s not n logarthms or there s no partcular producton functon n mnd, r s nterpretaton s less clear, and t may be nterpreted as a normalzaton to the unt nterval of the measure u,astquantfes neffcency relatve to the most effcent frm. Interpreted ths way, the nonlnear exponental normalzaton of r may dstort the scale of u. The proposed measure r has a smlar nterpretaton but relatve to the least effcent frm, and t too may dstort the scale of u. Ether way, the alternatve measure, r,may prove useful to emprcsts, partcularly f bas and confdence nterval coverage are mportant(as we shall see). 3 Qan and Sckles (2008). Indeed, Qan and Sckles consder cross-sectonal (T =1)andrandom-effects estmaton of u. Hence, the current paper and the Qan and Sckles papers are complements. 5

9 Ultmately we subject these estmates to fnte sample smulatons and compare ther performance under a varety of assumptons on the dstrbuton of neffcency. (In all cases the dstrbuton has bounded support from above and below, so the mn-measure has a populaton nterpretaton.) However, t s useful to consder the theoretcal bases. In partcular, the bases of û and û are drectly comparable, even though they estmate dfferent measures. 4 The bas of the max-measure estmate s: b max = E(û ) u = E max j = E(max j = E(max j ˆα j max α j ) E(ˆα α ) j ˆα j ) max α j, j ˆα j ˆα (max α j α ) j snce E(ˆα j )=α j for each j. Notce that the bas s not frm-specfc. The bas wll be largest when there s much uncertanty over the dentty of the best frm (max j α j ) n the populaton. Per Horrace and Schmdt (1996), ths occurs when T s small or when the varablty of the u s large. Uncertanty over the best frm s also worse when there are many frms n the populaton (α ) close to beng best (max j α j ). Ths s lkely to occur when the dstrbuton of u s skewed to the rght: "mostly stars, few dogs". In ths paper, we use the Beta dstrbuton B(a, b) to model three cases of the dstrbuton of u : B(2, 8) "mostly stars, few dogs", B(8, 2) "mostly dogs, few stars", and the symmetrc dstrbuton B(2, 2) "few stars, few dogs". See Fgure 1. The dscusson above suggest that ceters parbus, the bas, b max, s small n the case of "mostly dogs, few stars". (It s nterestng to note that most Monte Carlos studes of the stochastc fronter model nvolve the truncated normal dstrbuton whch can only be skewed n the opposte drecton: "mostly stars, few dogs. See, for example, KKS, 2007.) 4 It s not entrely clear how to compare the theoretcal bas of the ˆr and ˆr. Also, per Km, Km and Schmdt (2007) coverage rates for bootstrap confdence ntervals on û (and û ) converted to ntervals on ˆr (and ˆr )are better than coverage rates on ˆr (and ˆr ) drectly, so understandng the bas of the û (and û ) helps us better understand the coverage rates of the preferred ntervals. 6

10 Smlarly, the bas of the mn-measure estmate s b mn = E(mn j ˆα j ) mn j α j. Here, bas s large n magntude when there s uncertanty over the worst frm n the populaton, whch wll be worse when the are many frms n the populaton (α ) close to beng worst (mn j α j ). Ths corresponds to the case where the dstrbuton of u s "mostly dogs, few stars". Whle we cannot know whch of the bases, b max or b mn, wll be larger n magntude n any emprcal analyss, t would be easy to speculate based on one s knowledge of the relatve frequences of dogs and stars that occur n the sample. Obvously when the relatve frequency of dogs and stars s equal, one would speculate that the bases be equal n magntude. These types of results are borne out n smulatons that follow. Table 1 reports the smulaton results on the bases b max and b mn. Ignorng regressors n equaton 1, smulatons are performed wth v t dn(0,σ 2 v) and u dstrbuted B(8, 2) or B(2, 2) or B(2, 8). Each Beta dstrbuton represents dfferent effcency scenaros as descrbed above. As s standard n SF model smulatons, we defne γ = Var(u)/(σ 2 v + Var(u)) and vary σ 2 v so that γ =0.1, 0.5 and 0.9. The γ s a "sgnal to nose rato" measure, so small γ ndcates a partcularly nosy experment. We focus on the cases where T s small and bas of the estmates of the mnand max-measures wll be largest, so we fx T =10. We consder four values of N =10, 20, 50 and 100. The bas analyss n Table 1 contans no surprses. Bas for both estmates s ncreasng n N (for fxed T ) and decreasng n γ as uncertanty over the best and worst frms n the populaton ncreases. Varyng the skew of the dstrbuton of neffcency also produces predctable results. The max-measure estmate outperforms our mn-measure estmate when the dstrbuton of neffcency s B(8, 2), whle the mn-measure estmate outperforms the max-measure estmate when the dstrbuton of neffcency s B(2, 8). Compare the of B(8, 2) and of B(2, 8) for measure u (frst row of results) to the of B(8, 2) and of B(2, 8) for measure u (second row). Ths near-perfect symmetry of the results across the two neffcency dstrbutons occurs every- 7

11 where n the table for obvous reasons. When the neffcency dstrbuton s symmetrc, B(2, 2), the estmates perform equally wth any dfferences n bas beng caused by samplng varablty of the smulatons. Compare for u to for u n the frst and second rows of results for B(2, 2). The mplcatons are clear: n an ndustry marked wth mostly stars and few dogs, the "mn" operator n the mn-measure estmate nduces a smaller bas than the max operator of the max-measure estmate. Put more generally, the mn-estmator s less based than the max-estmator when the ndustry under study has many effcent frms. (In compettve markets ths may be the relevant case.) In any emprcal exercse, f bas concerns outwegh the choce of the neffcency measure employed (u vs. u ), then the choce of estmator should be based on prevalng effcency market condtons n the ndustry under study. Knowledge of the dstrbuton of neffcency (up to locaton) s contaned n the dstrbuton of the estmated α and can be used used to nform these emprcal choces. 3 Two-Sded Measure We now consder a two-sded measure that ncorporates both the max operator and the mn operator. The motvaton of the two-sded measure s the ssue of scale. By scale we mean the way n whch estmators are normalzed (transformed) to the unt nterval. For the max-measure we have the normalzaton r, whch rescales (dstorts) effcency dfferences wth the exponental functon.5 Due to the non-lnearty of the exponental functon, techncal effcency dfferences between frms n the low range of br are smaller than those n the hgh range, for a gven dfference n bα,and are, therefore, not comparable. Hence, effcency dfferences n the low range of br are less nformatve. Ths creates a dstorton n the effcency dfferences for br when t serves as a normalzaton for bu. The normalzaton of the mn-measure, r, also nonlnearly rescales (dstorts) effcency 5 Ths dea of dstorton s based on the dea that r u, for small u. Obvously f output s n logarthms, then r s not smply a normalzaton; t s the true percentage of the output of frm relatve to the effcent frm for a fxed set of nputs. However, the normalzaton could magnfy the bas assocated wth estmatng u. If output s not n logarthms, then r s smply a normalzaton. 8

12 dfferences. An alternatve (or complementary) effcency measure that does not dstort effcency dfferences yet normalzes effcency scores on the unt nterval s the two-sded: wth estmate e α mn j α j max j α j mn j α j, be bα mn j bα j max j bα j mn j bα j. Compared to br or br, techncal effcency dfferences (be be j )arenotdstorted: be be j = bα bα j max j bα j mn j bα j. Snce bα s a consstent estmator n T for α u, then the dfference be be j s consstent for u ( u j ). Snce the denomnator s constant for each par of frms, effcency dfferences have the same scale across the sample. To demonstrate the dstorton nduced by br or br relatve to be we agan consder a beta dstrbuton for techncal neffcency. By consderng dfferent levels of skew, we are consderng dfferent levels of effcency dfferences between ranked sample realzatons at the hgh and low ends of the rank statstc. For example any ranked sample from the B(8, 2) or B(2, 8) dstrbutons wll have larger dfferences at one end of the rank statstc and smaller dfferences at the other (on average). The B(2, 2) dstrbuton wll have symmetrcal dfferences at ether end of the ranked sample, because the probablty mass s symmetrc about the mean. There are many ways that we could llustrate these dfferences and the dstortons created by normalzaton to the unt nterval. One way would be to use the dstrbuton of u to calculate the theoretcal dstrbutons of the transformatons r, r and e. Then the dstortons could be compared smply by comparng plots 9

13 of the dstrbutons. However, rather than calculate these dstrbutons (a not trval task), we smulate and estmate them usng kernel technques. We smulate each beta dstrbuton wth 100,000 draws of u. Wth ths many draws the maxmal draw s arbtrarly close to 1 and the mnmal draw s arbtrarly close to 0, so "uncertanty" over the populaton maxmum and mnmum s essentally zero, and any "bas" caused by ths uncertanty s mtgated. Our purpose s to get a farly accurate pcture of the dstrbuton and not to understand the effects of samplng varablty on effcency estmaton, whch we nvestgated n the last secton. We estmate the dstrbutons of u,r, r and e usng the Gaussan kernel and an arbtrarly selected bandwdth of 0.1. The estmated dstrbutons are n Fgures 2a-c for u B(8, 2), u B(2, 8), andu B(2, 2), respectvely. Obvously, the densty estmates are only approxmate at the boundares (there s no boundary bas correcton). However, ths s fne for the purposes of scale comparsons. Begnnng wth panel a n Fgure 2, we see that when the dstrbuton of u (thck dashed lne) s "mostly dogs", the estmated dstrbuton of the max-measure estmate, r, s farly close to that of u, whle that of the mn-measure, r, s not. In ths case the scale dstorton of the max-measure s small relatve to that of the mn-measure. Also, the two-sded measure, e, s comparable to the max-measure n terms of scale dstorton. The two-sded measure over-scales n the center of the dstrbuton whle the max-measure over-scales n the rght tal of the dstrbuton. Ths makes sense as the two-sded measure s (n some sense) a "mddle ground" between the max-measure and the mn-measure. The mn-measure, r, clearly over-scales n the left tal of the dstrbuton. Of course thngs are reversed n the "mostly stars" case of u B(2, 8), contaned n panel b of Fgure 2. Here the mn-measure outperforms the max-measure n terms of scale preservaton. Agan, the two-sded measure also preserves scale farly well and s comparable to the mn-measure. In panel c we see that the dstrbuton of the two-sded measure, e,s nearly dentcal to the dstrbuton of u (thck dashed lne), whle the max- and mn-measures exhbt large scale dstortons. (Agan, the reader s remnded that these are merely kernel densty 10

14 estmates wth no end-pont correcton.) Ths s not surprsng, gven the way the two-sded measure s constructed, but the pont should be clear on ts usefulness when scale preservaton of effcency scores s mportant. Regardless of the skewness of the neffcency dstrbuton, the two-sded measure relably preserves scale (or dfferences n the rank statstc), whle the performance of the max- and mn-measures s a functon of the dstrbutonal skewness. For completeness we now examne the bas of the two-sded measure wth a bref smulaton study. The smulated bas results for the estmator be n Table 2 use the same parameterzatons as the bas results of Table 1. Unlke the bu and bu estmates, the bas results for be are frm specfc, so average based across frms are reported. A few results are noteworthy. Frst, the drecton of the bas s a functon of the skewness of the effcency dstrbuton. For u B(8, 2) (mostly dogs) the bas s postve, for u B(2, 8) (mostly stars) the bas s negatve, and for u B(2, 2) (few stars or dogs) the bas s close to zero. Ths may suggest that for effcency dstrbutons wth centralzed mass or symmetrc effcency dstrbutons, the two sded estmator s the approprate choce. 6 Indeed, n the symmetrc case, the average bas for the two-sded measure s always smaller n absolute value than the bases n the one-sded measures n Table 1. (For these dfferent measures bas comparsons of estmates are not entrely meanngless, because all three measures are essentally untless percentages.) Second, bas s (not surprsngly) ncreasng n N and decreasng n γ for all levels of skewness. 4 Confdence Intervals Per Schmdt and Sckles (1984), β b converges to β for large N or T,whlebα converges to α for large T only. Therefore, when T s small (the usual panel case) asymptotc approxmatons for confdence ntervals on functons of α are napproprate, and a bootstrap method should be employed. See Km, Km and Schmdt (2007) for a detaled survey of methods for bootstrap confdence ntervals 6 Agan ths may not be a "choce" per se, but the two-sded measure may smply be a convenant way to report effcency scores, u. 11

15 on techncal effcency and a comprehensve smulaton of the coverage rates and confdence nterval wdths of a varety of bootstrap technques. Our purpose here s two-fold. Frst, we would lke to replcate the salent features of the Km, Km and Schmdt (KKS) confdence nterval smulatons, whle expermentng wth the skewness of the techncal neffcency dstrbutons usng our three parameterzatons of the beta dstrbuton. Second (and smultaneously), we extend the smulatons to nclude our mn-measure and the two-sded measure. Agan, all the estmates consdered are for dfferent measures and cannot be consder drect substtutes, but t s useful to emprcsts to know whch measures are better n a statstcal sense when nformaton on the skew of the effcency dstrbuton s known or can be approxmated from the fxed-effects estmates. For smplcty the underlyng data generaton mechansm for our confdence ntervals s dentcal to that of our bas analyss of secton 2. Our overall fndng s that the bas assocated wth max and mn operators erodes the coverage rates of the bootstrap confdence ntervals, so the relatonshp between coverage rates and dstrbutonal skewness s smlar to that between bas and skewness. The KKS smulaton study consders both drect bootstrap confdence ntervals from the dstrbuton of br and ndrect bootstrap confdence ntervals from the dstrbuton of bu,whchare transformed to confdence ntervals on r. When the ndrect and drect confdence ntervals are the same, the nterval s sad to be transformaton-respectng (Efron and Tbshran 1993, p.175). The emprcal advantage of transformaton-respectng confdence ntervals are obvous: the choce of estmator to report (transformed or not transformed) does not affect the coverage probabltes of the ntervals. Of all the bootstrap ntervals consdered by KKS, only the percentle bootstrap (percentle) s transformaton respectng. However, the bas-corrected wth acceleraton (BC a ) ntervals are approxmately transformaton-respectng (KKS, p.169). They fnd that the bas-corrected percentle (BC percentle) ntervals are generally not transformaton-respectng but conclude that they have better coverage rates than the other bootstrap confdence ntervals that they consder. They also fnd that, when the ntervals are not transformaton-respectng, the ndrect method for nter- 12

16 val constructon on br has better coverage rates than drect methods. Therefore, n what follows we only consder these three confdence nterval constructon technques and only for the ndrect method. Our bootstrap confdence nterval constructon procedures are EXACTLY those of KKS, so we do not detal ther procedures here. The reader s referred to the KKS study for detals on ndrectly constructng percentle, bas-corrected percentle and bas-corrected wth acceleraton ntervals. Whle our coverage rate results are slghtly dfferent than those of KKS, our overall fndngs are the same: for the ndrect method, the bootstrap BC percentle ntervals (Smar and Wlson, 1998) are generally better n terms of coverage rates than ether the percentle or the BC a ntervals. 7 Coverage results for the percentle and the two bas-corrected bootstraps for r and for r usng the ndrect method are reported n Table 3. Here the nomnal coverage rate s Generally speakng, the BC percentle coverage rates appear to be best for all scenaros consdered. (Ths s the general fndng of the KKS study.) When the dstrbuton of neffcency s symmetrc, B(2, 2), the coverage rates and nterval wdths for the r and for r measures are dentcal up to samplng varablty for all the bootstrap technques. Ths corresponds to the case where the bases of the two measures are the same (few stars, few dogs). Ths s partcularly clear when the sgnal to nose rato (samplng varablty) s large (small). Not surprsngly the coverage rates are always decreasng n N, as uncertanty over the best or worst frms n the sample s ncreasng (as s bas). Coverage rates are unformly better for r when neffcency s dstrbuted B(8, 2) and better for r when neffcency s dstrbuted B(2, 8). For example, when N = 100, γ =0.1, andb(8, 2) the BC percentle coverage rate for r and r are and n Table 3. However, when neffcency s dstrbuted B(2, 8) the respectve coverage rate are and Agan these results are drven by the relatve level of bas of the two estmates under the dfferent neffcency 7 A referee also ponted out that the dstrbutons of nterest for constructon of the bootstrap confdence ntervals are nvarant to the true values of the model parameters α and β. 13

17 regmes (mostly stars or mostly dogs). In terms of the dfferent nterval constructon technques, t appears that both bas-corrected technques have coverage rates of about regardless of the skew of the dstrbuton, the sze of N, or the sgnal to nose rato (except n the nosest cases). The uncorrected percentle bootstrap ntervals do surprsngly well relatve to the bas-corrected ntervals except n the nosest cases (large N and small γ). For example, for N = 100, γ =0.1, and B(2, 2) the percentle coverages for r and r are and 0.349, respectvely. However, even then the bas-corrected ntervals are not very mpressve. For example, the BC a ntervals are and 0.611, respectvely n ths case. The worst coverage probablty s the uncorrected percentle bootstrap n the nosest case (N = 100, γ =0.1) forr and B(8, 2). In ths case, the coverage rate s only For completeness the bootstrap confdence ntervals for the two-sded measure, e,areprovded n Table 4. Notce that the coverage rates for ths measure are farly stable across the dfferent techncal neffcency dstrbutons. For example, n the least nosy settng (N =10, γ =0.9) the coverage rates for the BC a ntervals are 0.847, and for B(8, 2), B(2, 2) and B(2, 8), respectvely. The nterval wdths are also about the same. Agan, ths s due to the fact that scale dstorton s mnmal for the two-sded measure across the dfferent levels of skew. Also, the uncorrected percentle bootstrap has generally hgher coverage rates than the bas-corrected ntervals, and n the least nosy cases t comes very close to achevng the nomnal coverage rate of Whle the measures n ths study are all dfferent, t s nterestng to note that, when neffcency s dstrbuted B(2, 2), the coverage probabltes on the two-sded measure (Table 4) are unformly hgher than those of the one-sded measures (Table 3). Ths llustrates the effects of scale dstortons of the exponental transformatons of the u and u when the dstrbuton s symmetrc. Ths s depcted n panel c of Fgure 2. 14

18 5 Rce Farm Applcaton The quntessental example of large N and small T n the stochastc fronter lterature s the Indonesan rce farm data set wth N =171and T =6. Ths partcular data set has been analyzed a number of tmes, startng wth Erwdodo (1990) and, most recently, wth Km, Km and Schmdt (2007). See Horrace and Schmdt (1996, 2000) for a detaled descrpton. In our example we gnore the ssue of bas caused by undentfable tme-nvarant nputs n fxed-effects estmaton (Feng and Horrace, 2007). The form of the producton functon and the parameter estmates are precsely those contaned n Horrace and Schmdt (2000), but what s mportant to know s that output s n logarthms of klograms of rce. Our purpose s to hghlght the dfferent effcency measures consdered. The dstrbuton of the bα n Fgure 3 s produced usng the ksdensty(x) command n MATLAB and a Gaussan kernel. It has a normalzed postve skewness of Therefore, the dstrbuton of the bu (up to locaton) s ts mrror mage and has a skewness of However, the dstrbuton n Fgure 3 s actually qute symmetrc (except for a small wggle n the rght tal). Table 5 presents effcency estmates for 7 of the rce farms. These are the ranked by bα and correspond to the 2 best farms, the 75%le farm, the medan farm, the 25%le farm and the two worst farms. Each entry n the last three columns contans the estmate and the ndrect 90% bas-corrected percentle confdence nterval based on 999 bootstrap replcatons. In any emprcal exercse a dscusson of potental bas s dffcult. However, gven our smulaton results and the fact that the dstrbuton of the bα s nearly symmetrc (or u have slght negatve skewness), perhaps the two-sded measure (or tradtonal measure) wll have a less based estmate than that of the the measure relatve to the least effcent farm r. Even though the measures are dfferent, they are all untless, so ther bases wll be untless and, perhaps, comparsons are not entrely unreasonable. However, for ths partcular data set the effcency measures are qute mprecse, judgng from the confdence ntervals. Ths s partcularly tellng, when one consders the 15

19 poor coverage rates of the bootstrap confdence ntervals that arose n our (and KKS s) smulaton study. The mplcaton beng that the confdence ntervals n Table 5 may only acheve 70-80% coverage rates, even after bas correcton. We now dscuss scale consderatons. The dfference n the bα of the two best farms s logponts. For the measure r we see that the second-best farm has techncal effcency 6.8 percentage ponts below the best farm ( ), a good approxmaton for the log-pont dfference n the bα. Ceters parbus, effcency dfferences suggest that the second-best farm wll produce 6.8% fewer klograms of rce (0.070 fewer log-ponts of rce) than the best farm. For r we see that the second-best farm s 27 percentage ponts below the best, so ts approxmaton for the the log-pont dfference n the bα s poor at ths end of the order statstc (where u u s large). Thngs are reversed at the other end of the order statstc. The log-pont dfference n the bα for the two worst frmssapproxmatedwellbyr and poorly by r. By defnton the two-sded measure, e, wll always approxmate these dfferences well, partcularly at the ends of the order statstc. All three measures approxmate the log-pont dfferences less precsely n the mddle of the order statstc, but t s clear what the two-sded measure s dong: t s a mddle ground between the two exponental measures. For the medan farm r =0.544, r =0.340 and e = Hence, for reportng purposes, the two-sded measure s a smple log-pont normalzaton that facltates dscusson of relatve effcency over the entre range of the order statstc and that approxmates well the percentage change of output (r and r )atbothendsoftheorderstatstc. 6 Conclusons The goal of ths research s to consder the performance of varous techncal effcency measures under dfferent skewness of the dstrbuton of techncal neffcency. We fnd that the tradtonal onesded estmate relatve to the sample maxmum, br, performs best n terms of bas and confdence nterval coverage rates when the dstrbuton of neffcency conssts of "few stars, mostly dogs." In 16

20 hghly compettve markets where neffcent frms are rare, estmators of the tradtonal measure may not be relable (large bas and poor nterval coverage). On the other hand, estmators of the tradtonal measure may be relable n markets where compettve forces are weak, and ths may be the emprcally relevant case for effcency measurement n general. That s, estmatng techncal effcency may only be meanngful n markets or ndustres where s mght already exst to a great extent (lke utlty ndustres, where captal barrers to entry lmt compettve forces). The proposed mn-measure, br, has small bas and better confdence nterval coverage when the neffcency dstrbuton has "mostly stars, few dogs, whch may correspond to hghly compettve ndustres. The majorty of economc theory would suggest that ths corresponds to the more frequently encountered case. Of course n compettve markets, techncal effcency estmaton may be dffcult from the start, but that does not dmnsh the potental mportance of the mn-meaure. For example, the margnal cost of the least effcent frm n the sample may equal the market prce, so a measure relatve to the least effcent frm n the ndustry may be useful. Estmates of the two-sded measure, e, are partcularly appealng when the dstrbuton of neffcency s symmetrc and when ssues of scale are mportant. That s, when the magntude of the dfferences of the u must be preserved, the two-sded measure normalzes scores to the unt ntervals wthout the nonlnear scale dstorton nduced by the exponent operator. We reterate that all the measures are dfferent, so comparsons between the measures are sometmes dffcult to nterpret. However, ths study adds a few new measures to the emprcst s toolbox that may prove useful n the future. Our example suggests that the neffcency dstrbuton of Indonesan rce farms s farly symmetrc; ths suggests that the two-sded estmator may be preferred n terms of bas and confdence nterval coverage of the measure. Symmetry asde, the two-sded measure necessarly preserves the scale of the bu, better than the tradtonal measure n ths (or any) partcular example. 17

21 References [1] Battese GE, Coell TJ Predcton of frm-level techncal effcences wth a generalzed fronter producton functon and panel data. Journal of Econometrcs 38: [2] Efron B, Tbshran RJ An Introducton to the Bootstrap. Chapman and Hall: New York, NY. [3] Entan T, Maeda Y, Tanaka H Dual models of nterval DEA and ts extenson to nterval data, European Journal of Operatonal Research 136: [4] Erwdodo Panel data analyss on farm-level effcency, nput demand and output supply of rce farmng n West Java, Indonesa. PhD dssertaton. Department of Agrcultural Economcs. Mchgan State Unversty. [5] Feng Q, Horrace WC Fxed-effect Estmaton of Techncal Effcency wth Tme-nvarant Dummes. Economcs Letters 95: [6] Hall P, Hardle W, Smar L On the nconsstency of bootstrap dstrbuton estmators. Computatonal Statstcs and Data Analyss 16: [7] Horrace WC, Schmdt P Confdence statements for effcency estmates from stochastc fronter models, Journal of Productvty Analyss, 7, [8] Horrace WC, Schmdt P Multple comparsons wth the best, wth economc applcatons. Journal of Appled Econometrcs 15: [9] Km M, Km Y, Schmdt P On the accuracy of bootstrap confdence ntervals for effcency levels n stochastc fronter models wth panel data. Journal of Productvty Analyss 28:

22 [10] Koop G, Osewalsk J, Steel MFJ Bayesan effcency analyss through ndvdual effects: hosptal cost fronters. Journal of Econometrcs 76: [11] Qan J, Sckles RC Stochastc Fronters wth Bounded Ineffcency. Unpublshed manuscrpt. Department of Economcs. Rce Unversty. [12] Satchacha P, Schmdt P., Estmates of Techncal Ineffcency n the Stochastc Fronter Model wth Panel Data: Generalzed Panel Jackknfe Estmaton. Unpublshed manyscrpt. Mchgan State Unversty. [13] Schmdt P, Sckles RC Producton fronters and panel data. Journal of Busness and Economc Statstcs 2: [14] Smar L Estmatng effcences from fronter models wth panel data: A comparson of parametrc, non-parametrc and sem-parametrc methods wth bootstrappng. Journal of Productvty Analyss 3: [15] Smar L, Wlson P Senstvty of effcency scores: How to bootstrap n nonparametrc fronter models, Management Scence 44:

23 Table 1. Bases of Estmates of the Mn- and Max-Measures. Bas Measure T γ N B(8, 2) B(2, 2) B(2, 8) u u u u u u u u u u u u u u u u u u u u u u u u

24 Table 2. Average Bas of the Two-sded Estmate, be Average Bas Measure T γ N B(8, 2) B(2, 2) B(2, 8) e e e

25 Table 3: 90% Indrect Bootstrap Confdence Intervals for Techncal Effcency Measures Percentle BCa BC percentle Measure T gamma N B(8,2) B(2,2) B(2,8) B(8,2) B(2,2) B(2,8) B(8,2) B(2,2) B(2,8) cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth r* r** r* r** r* r** r* r** r* r** r* r** r* r** r* r** r* r** r* r** r* r** r* r**

26 Table 4: 90% Indrect Bootstrap Confdence Intervals for the Two-sded Measure Percentle BCa BC percentle T gamma N B(8,2) B(2,2) B(2,8) B(8,2) B(2,2) B(2,8) B(8,2) B(2,2) B(2,8) cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth cover wdth

27 Table 5. Rce Farm Effcency Estmates. α %le r r e best [ 0.912, ] [ 0.493, ] [ 0.929, ] th [ 0.837, ] [ 0.463, ] [ 0.836, ] th [ 0.423, ] [ 0.000, ] [ 0.156, ] th [ 0.448, ] [ 0.078, ] [ 0.197, ] th [ 0.439, ] [ 0.000, ] [ 0.070, ] st [ 0.306, ] [ 0.000, ] [ 0.000, ] worst [ 0.275, ] [ 0.000, ] [ 0.000, ] Bracketed values are 90% bas-corrected percentle ntervals. 22

28 Feng, Fgure 1 Fgure 1. Skewness of Varous Beta Dstrbutons B(8,2) B(2,8) B(2,2)

29 Feng, Fgure 2 Fgure 2a. Effcency Dstrbuton Estmates Mostly Dogs B(8,2) u~b(8,2) r* r** e Fgure 2b. Effcency Dstrbuton Estmates Mostly Stars B(2,8) u~b(2,8) r* r** e Fgure 2c. Effcency Dstrbuton Estmates Few Stars or Dogs B(2,2) u~b(2,2) r* r** e Note: the curves u and e are ndstngushable.

30 Feng, Fgure 3 Fgure 3. Dstrbuton of α for 171 Rce Farms. Skewness =