Economcs Letters 77 (00) 101 107 www.elsever.com/ locate/ econbase Technologcal neffcency and the skewness of the error component n stochastc fronter analyss Martn A. Carree a,b, * a Erasmus Unversty Rotterdam, Rotterdam, The Netherlands b Faculty of Economcs and Busness Admnstraton, Maastrcht Unversty, P.O. Box 616, 600 MD Maastrcht, The Netherlands Receved 0 August 001; accepted 0 March 00 Abstract Ths paper concentrates on negatvely skewed one-sded dstrbutons as an explanaton of the occurrence of postve (negatve) skewness n the case of stochastc producton (cost) fronter analyss. It takes as an example the bnomal dstrbuton that can have negatve or postve skew and derves the method-of-moments estmators. 00 Elsever Scence B.V. All rghts reserved. Keywords: Producton fronter; Cost fronter; Skewness; Bnomal dstrbuton JEL classfcaton: C1; D4 1. Introducton A popular econometrc technque to estmate the extent of frm neffcency s stochastc fronter analyss. A poneerng publcaton on the econometrc estmaton of stochastc fronters s Agner et al. (1977). They suggest an estmaton procedure n whch a producton fronter s estmated along wth a two-part composed error term. The frst part of the error term conssts of conventonal statstcal nose and s usually assumed to be normally dstrbuted. The second part represents frm neffcency and s assumed to follow a one-sded dstrbuton. Several dstrbutons have been proposed for the one-sded dstrbuton ncludng the half-normal dstrbuton, the truncated normal dstrbuton, the exponental dstrbuton and the two-parameter gamma-dstrbuton (Greene, 1990). Each of these one-sded dstrbutons has a postve skewness. L (1996) consders the case of the symmetrc one-sded unform dstrbuton. *Tel.: 11-4-88-76; fax: 11-4-88-4877. E-mal address: m.carree@mw.unmaas.nl (M.A. Carree). 0165-1765/ 0/ $ see front matter 00 Elsever Scence B.V. All rghts reserved. PII: S0165-1765(0)00119-
10 M.A. Carree / Economcs Letters 77 (00) 101 107 A common problem n the use of the stochastc producton fronter analyss s that the estmated skewness of the resduals s postve. Green and Mayes (1991) report that for a sample of 151 UK ndustres, % showed a postve skewness of the combned resdual and that for a sample of 140 1 Australan ndustres a smlar problem was encountered n 5% of the cases. A postve skewness s consdered problematc because t cannot be reconcled wth a one-sded dstrbuton of neffcences that s postvely skewed. Green and Mayes argue that, apart from possble msspecfcaton of the producton functons, ths ether ndcates super effcency (all frms n the ndustry are effcent) or the napproprateness of the technque of fronter producton functon analyss to measure neffcences. They overlook one mportant addtonal possblty: that of negatvely skewed one-sded dstrbutons of neffcences. In ths paper we consder ths possblty and show that t has mportant consequences for the nterpretaton of the skewness of the error term as a measure of technologcal neffcency.. The model of stochastc producton fronter analyss We consder the followng producton fronter model for a sample of N frms: y5 a 1 x9 b 1 5 1,...,N (1) wth 5 v u beng the composte error term. The commonly made assumpton for the statstcal nose term v s that t s..d. N(0, s ). The u $ 0 represents the technologcal neffcency of frm. The two parts of the error term are assumed to be ndependently dstrbuted. The y and the x-vector stand for the output and the nputs used n the producton process, respectvely. The composte error term has an expected value equal to E 5Eu and a thrd central moment equal to: E( E ) 5 E(v u 1 Eu ) 5E(u Eu ). () Therefore, a postvely skewed dstrbuton of the neffcences u mples that the adjusted error term E has a negatve skewness. Now t has been common practce to use a postvely skewed one-sded dstrbuton. In fact, n case one wants the one-sded error u to have an unbounded range, then most well-known dstrbutons are n fact postvely skewed. Examples nclude the gamma dstrbuton (ncludng the exponental dstrbuton), the Posson dstrbuton, the negatve bnomal dstrbuton, the truncated normal dstrbuton and the half-normal dstrbuton. But there s at least one well-known dstrbuton defned on [0, `) that may show negatve skewness: the Webull dstrbuton. In case one allows for the one-sded error to have a bounded range, a longer lst of 1 Another example s Mester (1997) who apples the stochastc cost fronter analyss and fnds that out of 1 US bank dstrcts three have negatvely skewed resduals. She carefully remarks that her fronter model wth normal half-normal error term does not ft the data n these dstrcts (p. 8). Although we concentrate upon the producton fronter case, the arguments are smlar for the cost fronter model n whch the costs of frm are determned by the cost fronter and an error term of the form 5 u1 v wth the one-sded error term u $ 0 capturng cost neffcences. Johnson et al. (1994, p. 6) show that a Webull dstrbuton has a postve skewness for parameter values up to.60 and a negatve coeffcent of skewness for hgher parameter values. L (1996, p. ) does not recognze ths possblty and argues that a one-sded error component wth unbounded range always has a postve skewness.
M.A. Carree / Economcs Letters 77 (00) 101 107 10 well-known dstrbutons wth (possble) negatve skewness becomes avalable. In the current paper we wll examne the bnomal b(n, p)-dstrbuton. There s no partcular reason to choose ths dstrbuton apart from that t allows for both postve and negatve skewness. A smple method-of-moments (MM) estmator for the bnomal one-sded dstrbuton can be derved by usng the corrected OLS approach (see e.g. Greene, 1990; Olson et al., 1980). Ths approach mples that frstly the parameters of the producton functon (1) are estmated usng least squares and that secondly the estmated resduals are used to estmate the parameters of the dstrbutons of u and v. The corrected OLS procedure leads to consstent estmators of the 4 parameters of the producton functon and of the composed error term dstrbuton. Frst defne j k j k r 5 u Eu so that Er 5 0. Because r and v are ndependent, we have that Er v 5 Er Ev wth j k and k postve ntegers. In addton, because v s dstrbuted symmetrcally we have Ev 5 0fk s an odd postve nteger. From the error decomposton E 5 v r we fnd: E( E ) 5 Ev 1 Er E( E ) 5Er () (4) 4 4 4 E( E ) 5 Ev 1 6Ev Er 1 Er. (5) 4 4 For a normally dstrbuted v we nsert Ev 5 s and Ev 5 s. By combnng Eqs. () and (5) we have: 4 4 E( E ) (E( E )) 5 Er (Er ). (6) From Eqs. (4) and (6) an MM-estmator for a two-parameter dstrbuton can be derved n analogue to Greene (1990). For the bnomal dstrbuton we have that Er 5 np(1 p), Er 5 np(1 4 5 p)(1 p) and Er 5 (np(1 p)) 1 np(1 p)(1 6p 1 6p ). From the thrd central moment t s obvous that the bnomal dstrbuton has a postve skewness for p between zero and one half and a negatve skewness for p between one half and unty. After replacng the kth central moments of k wth the sample analogues mk 5 o e /N we have the followng two equatons that determne MM-estmates for n and p: m 5np(1 p)(1 p) m m 5 np(1 p)(1 6p 1 6p ). (7) 4 That s, the values of p determne the sgns of the sample moments m (kurtoss adjusted for the value for normalty). Assume that m ± 0 and defne x 5 (m m )/m. (skewness) and m4 m 4 4 The method-of-moment estmators for the two parameters of the bnomal dstrbuton are derved usng the second, thrd and fourth sample moments. The use of hgher-order moments makes the estmators vulnerable to outlers and may lead to poor small sample propertes. Greene s (1990) MM-estmators for the two-parameter Gamma-dstrbuton suffer from ths problem as well. Hoskng (1990) proposes to use L-moments nstead of the standard measures of skewness and kurtoss to acheve relatvely small senstvty to outlers. However, the dervaton of estmators based upon L-moments s beyond the scope of the present paper. 5 See for example Johnson et al. (199, p. 107).
104 M.A. Carree / Economcs Letters 77 (00) 101 107 4 Fg. 1. Method-of-moments estmator p as a functon of x 5 (m m )/m. Accordng to Eq. (7) ths should be equal to (6p 6p 1 1)/(p 1). From ths we derve the two possble values of p as a functon of x: 1 1 1Π]] 1 1 1Π]] p15] 1] x 1] x 1 p5] 1] x ] x 1. (8) 6 6 6 6 For values of x less than 1 only the p1-soluton s allowed. For values of x n excess of 11 only the p-soluton s allowed. For values of x n between 1 and 11 the sgns of m and m4 m determne whch of the two solutons s approprate. That s, f skewness s postve (m. 0) then the p1-soluton wll be chosen otherwse the p-soluton. In Fg. 1 the graphs of p1 and p as a functon of x are gven. Not all combnatons of the emprcal values for m and m4 m allow for MM-estmates. In fact, n case m m. m. 0or m m. m. 0 there are no vald MM-estmates for p. To 4 4 derve the MM-estmator for Ev 5 s usng Eq. () we also requre that m should not be less than np(1 p) after nsertng the MM-estmates of p and n. It s a queston of emprcs whether these volatons, whch would ndcate the mplausblty of the one-sded dstrbuton to be of a bnomal type, are encountered.. What do negatve and postve skewnesses actually measure? Emprcal studes usng the producton fronter approach have been assumng postvely skewed one-sded dstrbutons (and, hence, negatvely skewed adjusted composte error terms). As a consequence, when a postve value of m was found, the only logcal concluson could be that there had been unfortunate samplng from a dstrbuton that had n fact a populaton skewness below zero. As Monte Carlo studes have shown, ths s a possblty that may occur relatvely frequently n case the one-sded dstrbuton has a small varance n comparson wth the symmetrc error dstrbuton (see e.g. Fan et al., 1996; Green and Mayes, 1991). Waldman (198) showed that resortng to a maxmum lkelhood procedure nstead of a corrected OLS procedure does not resolve the problem. In
M.A. Carree / Economcs Letters 77 (00) 101 107 105 Fg.. Probabltes of b(0, 0.5). fact, he has shown that, n case of a postve m, the ML estmator for the stochastc fronter model s smply OLS for the slope vector and the absence of any effcences (varance of u s zero). When an ndustry showed postve skewness of the resduals t was therefore assumed that there were lttle f any neffcences. Green and Mayes (1991) argue that a postve skew mples that establshments n the ndustry are super effcent, rather than neffcent (p. 58). In contrast to the concluson of super effcency n case of a postve skewness, the example of the bnomal dstrbuton shows that a postve skewness suggests a one-sded dstrbuton that has low probabltes for small neffcences and hgh probabltes of large neffcences. For the bnomal dstrbuton t ndcates that p s between one half and one. Hence, only a small fracton of the frms or plants attan a level of productvty close to the fronter whle a large fracton attans consderable neffcences. See Fg. n whch we have n equal to 0 (neffcency categores) and p equal to 0.75. The case of a negatve skewness mples that only a small fracton of frms are laggng behnd. See Fg. n whch we have p equal to 0.5. Fgs. and can also be nterpreted as two stages n an ndustry characterzed by the cycle of Fg.. Probabltes of b(0, 0.75).
106 M.A. Carree / Economcs Letters 77 (00) 101 107 nnovaton and mtaton. Assume that the productvtes n an ndustry are characterzed by Fg.. In case one frm acheves an mportant nnovaton by whch t can ncrease productvty, t becomes domnant (n terms of productvty), and Fg. may emerge. Other frms wll then seek to mtate the successful frm and Fg. may be restored. Ths process of transent domnance n an ndustry would lead to a cyclcal tme seres pattern of postve and negatve skewness of resduals of the stochastc producton fronter analyss: nnovaton leads to postve skew, mtaton leads agan to negatve skew. What s the more lkely nterpretaton of a postve skewness of the composte error term n stochastc producton fronter analyss? On the one hand, t may be an unfortunate draw and the ndustry may be characterzed by super effcency (or at least, the symmetrc error term v domnates the one-sded dstrbuted u ). On the other hand, the large majorty of frms may be qute neffcent (lke n Fg. ). The two nterpretatons are completely dfferent, ether ndcatng no effcences or large neffcences. An argument aganst the frst nterpretaton s that relatve productvtes of plants are persstent over tme (e.g. Baley et al., 199). In case there would have been no neffcences (.e. the error term s determned completely by statstcal nose 5 v ) one would not expect such persstence, unless the statstcal nose has strong autocorrelaton. 4. Concluson An mportant methodologcal problem n stochastc fronter analyss has been the occurrence of resduals beng skewed n the wrong drecton. In the case of producton fronters, many tmes postvely skewed resduals have been found, whle n the case of cost fronters, negatve skewnesses have been qute common. The tradtonal soluton to the problem has been to argue that there are no neffcences and to put the varance of the one-sded dstrbuton equal to zero. Ths soluton fals to be convncng. Ths paper suggests a dfferent soluton: the one-sded dstrbuton of neffcences may be negatvely skewed (n case of producton fronters) or postvely skewed (n case of cost fronters). Ths does not mply that the tradtonal soluton argung for unfortunate samplng s mpossble, but that a better approach to the stochastc fronter analyss, n whch a comparson s made of several ndustres (or regons or tme perods), s to use a dstrbuton allowng for postve and negatve skewness. Acknowledgements The author s grateful to the Royal Netherlands Academy of Arts and Scences (KNAW) for fnancal support and to Peter Brouwer for our dscussons on the applcaton of stochastc producton fronter models to Dutch constructon ndustry data. References Agner, D.J., Lovell, C.A.K., Schmdt, P., 1977. Formulaton and estmaton of stochastc fronter producton functon models. Journal of Econometrcs 6, 1 7.
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