ACCOUNTING FOR UNOBSERVABLES IN PRODUCTION MODELS: MANAGEMENT AND INEFFICIENCY. Antonio Alvarez, Carlos Arias and William Greene.

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1 ACCOUNTING FOR UNOBSERVABLES IN PRODUCTION MODELS: MANAGEMENT AND INEFFICIENCY Antono Alvarez, Carlos Aras and Wlla Greene March 0, 2004 ABSTRACT Ths paper explores the role of unobserved anageral ablty n producton and ts relatonshp wth techncal effcency. Prevous analyses of anageral ablty were based n strong assuptons about ts role n producton or the use of proxes. We avod these shortcongs by ntroducng anageral ablty as an unobserved rando varable n a translog producton functon. The resultng eprcal odel can be estated as a producton fronter wth rando coeffcents. Keywords: Manageral ablty, techncal effcency, producton fronter, rando coeffcents odel, axu sulated lkelhood. JEL codes: C5, D2

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3 ACCOUNTING FOR UNOBSERVABLES IN PRODUCTION MODELS: MANAGEMENT AND INEFFICIENCY ABSTRACT Ths paper explores the role of unobserved anageral ablty n producton and ts relatonshp wth techncal effcency. Prevous analyses of anageral ablty were based n strong assuptons about ts role n producton or the use of proxes. We avod these shortcongs by ntroducng anageral ablty as an unobserved rando varable n a translog producton functon. The resultng eprcal odel can be estated as a producton fronter wth rando coeffcents.. INTRODUCTION Manageent has always been consdered an portant factor of producton. However, the odelng of anageent s probleatc because t s unobservable. For ths reason t has been otted fro any producton odels. Ths ay be the source of portant probles because the osson of relevant varables can lead to based estates of the reanng paraeters of the producton functon (Grlches, 957). Econosts have coned the ter anageent bas to refer to ths proble. The lterature has suggested two strateges for avodng ths proble. Followng Mundlak (96) soe authors have used covarance analyss [Massell (987)] or slar tools, such as the wthn transforaton, to control for the effect of te nvarant anageent by sweepng t out of the estatng equaton. Other studes have used proxes for anageent [e.g. Dawson and Hubbard (985); Mefford (986)]. An alternatve strategy s to consder anageent a rando effect and odel t as part of the stochastc eleent of the producton functon. Ths s the approach followed plctly by the stochastc fronter producton functon lterature (Agner et al., 977) where the stochastc structure s coposed of two ters: a syetrc ter, that accounts for nose, and an asyetrc ter that accounts for techncal effcency (TE). In ths lterature, t has been coon to assue that the neffcency ter s pckng up (aong other thngs) dfferences n the level of anageral sklls.

4 These two strands of lterature have followed parallel but ndependent paths. On the one hand, the average producton functon lterature recognzes the role of anageent but seldo entons producton neffcency. On the other, the stochastc fronter lterature focuses on estatng techncal neffcency but does not provde an analytcal lnkage between neffcency and anageent. In ths paper we explore the analytcal lnkages between techncal effcency and anageent n the fraework of a translog producton odel where anageent s treated as an unobservable fxed nput. Startng fro an average producton functon we are able to derve an econoc-based stochastc producton fronter and an explct relatonshp between techncal effcency and anageent. Techncal effcency s shown to be a functon of the dfference between the current level of anageent and soe noton of optal anageent. We show that the assupton of fxed anageent leads to a te-varyng ndex of techncal effcency. Ths s a pertnent result snce the estaton of odels wth tenvarant techncal effcency s usually justfed on the assupton that anageent s fxed over te. The eprcal porton of the paper estates the producton functon wth a latent fxed nput descrbed above. The odel that eerges s a type of rando coeffcents odel. The estaton s carred out by axu sulated lkelhood. An portant aspect of ths estaton procedure s that t allows us to construct estates of both current and optal anageent, as well as the assocated techncal effcency. The structure of the paper s as follows: In secton 2 we develop a odel n whch anageral ablty s treated as an unobservable nput n a flexble producton functon. Secton 3 dscusses estaton ssues. Eprcal results based on a study of dary farers are presented n Secton 4. Soe conclusons are drawn n Secton A PRODUCTION MODEL WITH FIXED MANAGERIAL ABILITY One exaple of the abguous noton of effcency s the defnton posed by Farrell (957) who stated the techncal effcency ndcates the gan that can be acheved by sply gngerng-up the anageent. Farrell appeared to suggest that techncal neffcency s the result of a lack of anageral ablty. On the other hand, Lebensten (966) vewed techncal neffcency, whch he called X- effcency, as the result of a lack of otvaton or effort. In ths case, the soluton to neffcency calls for better organzaton of the work process or ore otvaton and supervson of eployees, all of whch are coonly consdered to be anageent functons [Mefford (986)]. 2

5 Our startng pont s a translog producton functon wth one te varyng varable nput, x t and anageral ablty,, whch s consdered a fxed nput. The flexblty of the producton functon relaxes ex-ante constrants n the roles of and x t, n the producton process. The translog producton functon can be wrtten as: ln yt = α + βx ln xt + βxx(ln xt ) + β + β + βx ln xt + vt where subscrpts and t denote frs and te, respectvely, and y t s the sngle output. The nputs n the translog odel are conventonally n logs, but snce, by constructon, anageral ablty s unobservable, we have entered t n the level for, snce the unts of easureent, wth the varable tself, are unobservable. We assue that v t, s a syetrc rando dsturbance wth zero ean. Therefore, the odel n corresponds to the typcal average producton functon. Its key feature for present purposes s the nteracton of anageent wth the varable nput. Wthout ths ter, the two anageent ters collapse nto an ndvdual effect and the odel does not dffer substantally fro the standard fxed or rando effects producton functon odel. We expect producton to be onotoncally ncreasng n ; greater anageral ablty should allow the agent to produce greater output for any aount of nput. Ths effect wll characterze the producton functon n f: ln y t = β + β + β ln x > 0 x t The assupton of onotoncty n anageral ablty ples that, for gven x t, hgher values of ply hgher levels of techncal effcency. The axal output for gven x t (the fronter) corresponds to a axal level of anageral nput hgher than that for coparable (sae x t ) but neffcent producers. The stochastc producton fronter would then be wrtten as: ln y α + β ln x + β (ln x ) + β + β + β ln x + v 2 2 t = ½ ½ x t xx t x t t where y t denotes effcent output. We can now establsh a lnk between techncal effcency and anageent. Ths follows fro the defnton of an output-orented ndex of techncal effcency as the rato of observed to potental output, whch n log ters s: 3

6 2 2 ( x ) ( ) ( ) ln TE = ln y -ln y = + ln x + β t t t β β t ½ 0 Note that when =, that s when the fr s usng the aount of anageent that defnes the fronter, lnte t =0, and, therefore, the fr s techncally effcent. Equaton (4) can be rewrtten as: ln TE = θ + θ ln x where t x t 2 2 ( - ) ½ ( - ) ( - ) θ = β + β θ x = β x Equaton shows that TE has two coponents. One, can be odeled as a te nvarant ndvdual effect (θ ). But the other ter, reflectng the nteracton of anageent wth nput use, has to be specfed as a te varyng coponent n the producton functon (θ x lnx t ). Therefore, an nterestng feature of expresson s that the pled techncal effcency wll be te varyng because t depends on x t even when the observed level of anageent and the one at the fronter are constant over te. Ths s so because the dfference between current anageent and effcent anageent nteracts wth the nput level. It s portant to note that only when β x = 0 s t possble to assocate each level of techncal effcency wth a level of anageent. Ths suggests that gven the specfcaton of t does not see approprate to odel techncal effcency as a fxed effect snce t depends on x t. Ths specfcaton of the odel departs fro the prevous lterature n ths feld that uses panel data to estate the producton functon (e.g., Schdt and Sckles, 984). Another nterestng feature of our easure of TE s that t s dfferent for frs wth the sae level of anageral nput f they use dfferent quanttes of the other nputs. The change n anageral nput necessary to ncrease TE n a gven aount dffers dependng on nput use as well. 2 2 Ths dea appeared n an early paper. Hall and Wnsten (959) claed that for slar frs (usng the sae aounts of nputs) ore anageent would ply ore output and therefore greater TE. In ths case, there s a clear drect relatonshp between anageral ablty and TE. However, thngs are less clear n the case of two frs usng dfferent nputs wth the sae level of TE. In ths case, any ncrease n techncal effcency would requre dfferent ncreases n the levels of anageent for each fr 4

7 It s nterestng to analyze the effects on TE of changes n anageral ablty and nput use. Those effects can be seen n the followng dervatves: lnte lnte ln x t t t = β + β + β = β ( ) ln x x t x The dervatve of TE wth respect to anageral nput corresponds exactly to the condton for onotoncty of producton wth respect to anageral ablty shown n expresson. Therefore, an ncrease n anageral ablty ncreases TE gven conventonal nputs f the producton functon s onotonc n anageral ablty. The dervatve of TE wth respect to the level of nput use s negatve f β x s postve because s saller than by defnton. Therefore, when β x s postve the ncrease n the use of conventonal nputs decreases TE for a gven aount of anageral ablty. In ths secton we have shown that the odel n rases nterestng ssues about the relatonshp between fxed anageent and techncal effcency. In partcular, t shows that TE s not necessarly a fxed effect, rather TE can vary over te and that the relatonshp between TE and anageral ablty depends on the aount of anageral ablty and conventonal nputs. In the next secton we dscuss the estaton of ths odel n ore general ters, wth ore than one observed nput. 3. ESTIMATION ISSUES Model cannot be drectly estated because the ndvdual level of anageent s unobservable. Prevous authors have dealt wth ths proble by ntroducng a proxy for anageent n a cost functon. [See Dawson and Hubbard (985), Alvarez and Aras (2003)]. Snce the use of a proxy ntroduces new abgutes (easureent error) nto the odel, we wll eploy a ore drect approach to the proble whch takes advantage of the panel nature of the data set we wll analyze. We wll now translate the odel n the precedng secton nto an eprcally estable for that ncludes anageral ablty as an nput. 3.. Stochastc Fronter Model wth Fxed Manageent 5

8 We consder producton wth K nputs, x,xk. As before, let denote the level of anageent that defnes the fronter and the actual anageent nput for fr. We contnue to eploy the translog for. The producton odel wll then be ln y = ln y - u t t t where u = ln y - ln y t t t K K K ln k xtk 2 ln ln kl x k k l tk x = = = tl = α + β + β + 2 K 2 ln - k x k tk vt u = t β + β + β + K ( ln ) ( ) k= ( ) = β + β + β k xkt In (7), u t now corresponds to the standard defnton of techncal neffcency, so that TE t = exp(-u t ). The assupton that producton s onotoncally ncreasng n anageral nput enters the defnton of the producton fronter. The aount of 0. produces the axu output gven nputs. The resultng producton fronter s the sae n sprt as the falar stochastc fronter odel save for the te varaton n the neffcency ter and the presence of the unobservable nput ( ) n the fronter producton functon. For estaton, a crtcal assupton s the orthogonalty of u t and the nput levels n. Whle lnx t does appear n the te varyng part of u t we assue that t does not nfluence ( ). Thus, u t s of the for a + ( ) g ( xkt ) wll, by vrtue of the presence of the freely varyng (, and each ter ( ) g ( x ) k kt ) be uncorrelated wth x kt. We note, ths s essentally the arguent of Zellner et al. (966), who argued that n a producton functon, the nput levels would be uncorrelated wth the devaton of output fro the optal output, even though they would obvously be correlated wth the actual output, tself. Although nvolves an unobservable varable, we can translate t nto an eprcally estable for. The result follows fro the fact that the unobservable can be seen as a rando effect n a panel data odel. For that purpose, we rewrte as follows: ( ) 2 ( ) ln y = α + β + β 2 t K K K k k xtk 2 kl x k k l tk xtl v t u = = = t + β + β ln + β ln ln +. 6

9 In ths for, the odel takes the appearance of a rando coeffcents stochastc fronter odel [See Tsonas (2002), Greene (2003)] although t dffers fro ore falar rando coeffcents odels n several respects. paraeter s the sae, Frst, the rando coponent of each rando. Second, the square of the rando coponent appears n the odel. Despte these coplcatons the odel s estable. The estaton of the odel has two portant requreents that rean to be consdered. Frst, the rando coeffcents odel requres as an dentfcaton condton that the rando coponents of the coeffcents be uncorrelated wth the explanatory varables. The rando coponent of the coeffcents n our odel s the level of anageent that defnes the fronter ( ), whch lkely s correlated wth at least soe of the nputs. (Note ths s a dfferent ssue fro the devaton of fro whch we argued above s uncorrelated wth the nputs). In order to avod ths proble, we take the approach suggested for rando effect odels by Chaberlan (984) (and borrow fro early work by Mundlak) and specfy = τ lnx + w where lnx s the vector of the eans of the logs of nputs, τ s a vector of paraeters to be estated (a constant ter wll not be dentfed) and w s a rando ter that follows a standard noral dstrbuton and whch we assue s uncorrelated wth the nputs. The second ssue concerns the stochastc specfcaton of u t. The estaton by axu lkelhood of the odel n requres a dstrbutonal assupton about u t. Greene (980) shows that the choce of ths assupton ay affect the results of the estaton. In ths paper, we odel the dstrbuton of u t as half noral, whch produces a rando coeffcents stochastc fronter odel n the sprt of Agner et al. (977). The pact that the specfc assupton has on the eprcal results s an nterestng queston that has been left for future research. 7

10 3.2. Estaton of the Rando Coeffcents Stochastc Fronter Model Ths secton wll descrbe the ethod used to estate the paraeters of the rando coeffcents stochastc fronter odel (RCSFM). A crucal aspect of the paraeterzaton of the odel s that each occurrence n of fronter anageent, as, 2, and log tk x appears attached to a free paraeter, and there s a free constant ter n the odel. As such, asde fro the regresson paraeters, τ, n = τ lnx + w, even though there s no free constant ter n, nether locaton paraeter, µ w = [ ] E w, nor scale paraeter, σ w 2 = [ ] [ ] [ ] 2 Var w = E w E w are dentfed fro observable data n the context of the odel. We assue, then, wth no loss of generalty, that w has ean zero and varance. Noralty s not strctly necessary, but noralzaton of both oents s. No free paraeters of the dstrbuton of w are estable. Fro, we defne ε = ( v - u ) t t t In what follows, t s useful to note explctly that εt wll be condtoned on condtonal densty for a sngle observaton n the half noral stochastc fronter odel s 2 ε t λε t f ( ε t ) = φ Φ σ σ σ where φ(z) and Φ(z) denote the densty and CDF of the standard noral varable, respectvely [See Agner et al., (977)]. The jont densty for T observatons on fr s f ( ε,..., ε ) = f ( ε ). T t= t Ths s the contrbuton to the condtonal lkelhood for fr, L. The uncondtonal contrbuton to the lkelhood functon s T = ( ε ) ( ) t= t L f g d where g( ) s the argnal densty of. Consstent wth the precedng dscusson, there are no new paraeters n ths densty we have assued that has a standard noral dstrbuton. The log lkelhood s log L( δ) = log L ( ) = δ N T. The 8

11 where we use δ to denote the full vector of paraeters n the odel. The axu lkelhood estates of the paraeters are obtaned by axzng wth respect to δ. Snce the ntegral n wll not have a closed for, t s not possble to axze drectly. We wll use the ethod of axu sulated lkelhood, nstead. [See Tran (2003), Greene (2003a) and Goureroux and Monfort (996) for dscusson] EMPIRICAL APPLICATION Our eprcal applcaton uses data fro a balanced panel of 247 dary fars located n Northern Span. We have data on these fars for a perod of sx years ( ). Snce the fars are specalzed n lk producton we consder only one output. The varables used n the estaton of the producton fronter are descrbed n Table. Table. Varables Used n Producton Analyss Mean Std. Dev. Mnu Maxu Mlk Mlk producton (lters) 3, ,28 Cows Nuber of lkng cows Labor Nuber of an-equvalent unts Land Hectares of land devoted to pasture and crops Feed Klogras of feedstuffs fed to dary cows ,73 We wsh to explore the eprcal consequences of restrctons on the role of anageent n the producton functon. For that purpose, we frst estate a conventonal (pooled) stochastc producton fronter wth four nputs and ncludng te-effects (ths s equvalent to estatng equaton wthout consderng anageent). 4 The results of the estaton of the producton fronter can be seen n the frst colun of estates n Table 2. The explanatory varables n the orgnal data were dvded by ther geoetrc ean. By dong ths, the frst order coeffcents can be nterpreted as output elastctes evaluated at the geoetrc ean of the saple. They are postve and sgnfcantly dfferent fro zero at conventonal levels of sgnfcance..table 2. Estated Fronter Models 3 We nclude a suary of the an features of the sulaton n the appendx. 4 All odels were estated usng LIMDEP 8.0 (Greene, 2002). 9

12 Varable Para. Stochastc Fronter Cows β (0.0223) Labor β (0.028) Land β (0.046) Feed β (0.02) Constant α.79 (0.028) Means of rando paraeters β k (0.03) (0.008) (0.003) (0.0060).676 (0.004) Rando Paraeters Model Manageent x nputs β k (0.0062) (0.0043) (0.0050) (0.0036) Manageent β (0.004) Manageent x Manageent β (0.007) Cows Cows β (0.58) Labor Labor β (0.0457) Land Land β (0.037) Feed Feed β (0.0964) Cows Labor β (0.0507) Cows Land β (0.0638) Cows Feed β (0.0753) Labor Land β (0.0263) Labor Feed β (0.0500) Land Feed β (0.04) Year 993 δ (0.020) Year 994 δ (0.026) Year 995 δ (0.07) Year 996 δ (0.02) Year 997 δ (0.025) λ.8657 (0.52) σ (0.0479) (0.073) (0.0334) 0.06 (0.05) (0.0209) (0.024) (0.0258) (0.048) (0.008) (0.043) (0.0044) (0.0044) (0.0044) (0.0043) (0.0044).969 (0.0465) (0.006) (0.003) Log L Standard errors n parentheses. Varable Means τ k (0.206) (0.0850) (0.268) (0.0707) 0

13 We now estate the rando coeffcents odel specfed n equaton followng the estaton procedure descrbed prevously. The results, whch were obtaned usng,000 Halton draws n each replcaton, can be seen at the rght n Table 2. The eans for the rando paraeters rean postve and sgnfcant at the geoetrc ean of the saple. The su of these coeffcents s close to.0, plyng a constant returns to scale technology. As expected, these eans dffer slghtly fro the coeffcents of the conventonal stochastc fronter. The coeffcents of anageent (β, β and βk ) are sgnfcantly dfferent fro zero at conventonal levels of sgnfcance. Ths can be nterpreted as evdence n favor of the rando coeffcents odel wth respect to the conventonal stochastc fronter approach snce the coeffcents of the producton functon change wth the level of anageent of the far. The coeffcents of the nteractons of anageent wth nputs (βk ) have a postve sgn for cows and land and a negatve sgn for feed and labor. Fnally, the negatve sgn of anageent squared can be reasonably nterpreted as evdence that anageent has a postve but decreasng effect on producton. The coeffcents of anageent at the fronter on nput eans (τ) are all postve wth the excepton of the coeffcent of feed. Ths result sees ntutve. as usng ore feed (cp) can be seen as a way of copensatng for anageent stakes due to a lack of anageral ablty. In suary, anageent plays a coplex role n producton far fro the splstc approach pled by a producton fronter wth fxed coeffcents. Therefore, the fxed coeffcents fronter s not a good nstruent to analyze fr behavor when anageent s unobservable. Ths result s portant n a nuber of settngs. For exaple, unobserved anageent s a key factor n explanng fr sze and growth (Jovanovc, 982). Another exaple s far polcy, where the level of anageent s portant to assess the effects of ncreasng the sze of fars (Dawson and Hubbard, 985; Alvarez and Aras, 2003). The effcency levels coputed n the odel wth fxed anageent can be coputed accordng to Jondrow et al. s (982) prescrpton. Neglectng the presence of the anageent varable for the oent, the coputaton s, = σλ φ( ( εt ) λ / σ) ( εt ) λ E ut εt ( 2 + λ ) Φ ( ( ε ) / ) t λ σ σ where ε t s defned n. Lke the other quanttes whch nvolve, ths ust be sulated. We wll copute ths fro the condtonal ean, gven the other data for far. The value

14 of can be coputed fro the condtonal dstrbuton of gven the data on far usng Bayes theore as follows: Let y denote the vector of logs of the outputs for far for the sx years. Let X denote the other data (nputs and year duy varables, lnear and quadratc ters n logs) for far. The condtonal dstrbuton of gven y s f ( y, X ) = = f g ( y, X ) ( ) f ( y X ) f g ( y, X ) ( ) y X f (, ) g( ) d The denonator s the contrbuton of far to the lkelhood functon for the saple (not the log lkelhood) n equaton. Thus, we can estate for far as the condtonal ean fro ths dstrbuton. Ths would be (5) E ( y, X ) = f ( y, X ) g( ) d f ( y, X ) g( ) d (6) Lke the lkelhood, tself, ths quantty cannot be coputed drectly, as the ntegrals wll not have a closed for. But, they can be sulated, n the sae fashon. Thus, the sulaton based estator of s R (/ R), r f ( y, r, X ) r = ˆ( y, X ). R ˆ (/ R) f ( y, X ) r =, r E = ˆ (7) Note, ˆf denotes the porton of the lkelhood functon for far, evaluated at the paraeter estates and the current draw of. Draws on coe fro the standard noral dstrbuton and, as before, are generated usng Halton sequences. Wth these estates n hand, estated neffcences for the fars are produced usng. An estate of the unobserved can be obtaned by substtutng the estate of n the defnton of u t n 2

15 equaton and solvng for. The resultng expresson s: 5 ± β + = K β (ln x ) k= k kt β 2 K 2 ( ) k 2 β + K β ln x 2β u β + β ln x β β k = k kt t = k kt Table 3 presents descrptve statstcs for the estates of u t based on produced fro the basc stochastc fronter odel and fro the rando coeffcents stochastc fronter odel, frst wthout accountng for the effect of anageent and then fro the rando paraeters odel that does nclude t. The dstrbuton for the latter set of estates has a uch saller ean and a uch tghter dstrbuton. Fgure below suggests the sae pattern. Ths suggests that not accountng for the effect of anageent on producton has soewhat nflated the estates of u t. One ght vew ths as a decoposton of the neffcency nto two parts, one explctly accounted for by the anageent effect and the other apparently due to other unexplaned factors. (8) Table 3. Descrptve Statstcs for Estated Ineffcences and Estated Manageent Mean Standard Devaton Mnu Maxu Estated Ineffcency Wthout Manageent Wth Manageent Estated Manageent Current anageent, Fronter anageent, In the quadratc forula for estatng, notce that depends on varables whch vary over te, snce t s a functon of the data and of u t. In order to obtan a te nvarant, we copute ths at the te varyng data, and take the far eans of the estated 's. Also, note that the u t that appears n s not the estator of u t coputed n that we use to do the coputaton. As such, the te nvarance s not preserved n our estate and we use the far average as an estate of the underlyng quantty. 3

16 Table 3 also shows descrptve statstcs for the estates of and. As expected, and are postvely correlated, as can be seen n Fgure 2. whch shows the far ean of the te varyng estated values for. The dagonal lne drawn n the fgure shows that current anageent s always less than the level of anageent at the fronter,. As expected, the relatonshp between and techncal neffcency s negatve. The relatonshp s shown n Fgure 3, where the te varyng estate s plotted aganst the te nvarant relatonshp s weak (R 2 s only 0.02) but statstcally sgnfcant. 6. The 5. CONCLUSIONS Ths paper explores the relatonshp between fxed anageral ablty and techncal effcency. For that purpose, fxed anageral ablty s ntroduced n the odel as an unobservable nput. The nteracton between the unobservable nput and conventonal nputs creates a great deal of dffculty n estatng the resultng odel. However, the odel can be cast as a stochastc fronter wth rando coeffcents under certan assuptons about the relatonshp between anageral ablty and conventonal nputs. We llustrate the feasblty of the proposed estaton procedure usng a saple of dary fars n Northern Span. The ean of the rando coeffcents are of the sae agntude as the coeffcents of a standard stochastc fronter, but anageral ablty s found to affect the value of soe rando nput coeffcents. The fxed coeffcents fronter provdes a easure of techncal effcency that can be related wth dfferent levels of anageent dependng on the crcustances of the far (nput use and output producton). The feasblty of estatng both the far current level anageent and the level of anageent that defnes the producton fronter s a clear advantage of the eprcal odel developed n the paper. The eprcal odel developed n the paper can be useful n analyzng fr polcy ssues snce anageent s consdered a key varable n assessng the effects of these polces. In fact, the odel avods both the splstc assuptons about anageent pled by fxed coeffcents fronters and the econoetrc probles assocated wth the use of proxes. 6 The estated regresson equaton s u ˆt = ( ) ( ) ˆ (estated standard errors n parentheses). 4

17 References Agner, D. and S.F. Chu, (968) "On Estatng the Industry Producton Functon", Aercan Econoc Revew, 58, Agner, D., C.A.K. Lovell and P. Schdt, (977) Forulaton and Estaton of Stochastc Fronter Producton Functon Models, Journal of Econoetrcs, 6, Alvarez, A. and C. Aras, (2003) "Dseconoes of Sze wth Fxed Manageral Ablty", Aercan Journal of Agrcultural Econocs, 85 (), Bhat, C., (999) Quas-Rando Maxu Sulated Lkelhood Estaton of the Mxed Logt Model, Manuscrpt, Departent of Cvl Engneerng, Unversty of Texas, Austn. Chaberlan, G., (984) Panel Data n Z. Grlches and M. Intrlgator, eds., Handbook of Econoetrcs, Asterda, North Holland. Dawson, P. and L. Hubbard, (985) "Manageent and Sze Econocs n the England and Wales Dary Sector", Journal of Agcultural Econocs, 38, Farrell, M.J. (957) The Measureent of Producton Effcency. Journal of the Royal Statstcs Socety A (20), Goureroux, C. and A. Monfort (996) Sulaton Based Methods: Econoetrc Methods, Oxford, oxford Unversty Press. Greene, W., (980), Maxu Lkelhood Estaton of Econoetrc Fronter Functons, Journal of Econoetrcs, 3, pp Greene, W., (2003a) Econoetrc Analyss, 5 th edton, Englewood Clffs, Prentce Hall. Greene, W., (2002) LIMDEP verson 8.0 User s Manual, Econoetrc Software, Inc. Greene, W., (2003b) "Reconsderng Heterogenety n Panel Data Estators of the Stochastc Fronter Model", Journal of Econoetrcs, forthcong. Grlches, Z., (957) Specfcaton Bas n Estates of Producton Functons, Journal of Far Econocs, 39(), Hall, M. and Wnsten, C., (959) The Abguous Noton of Effcency, The Econoc Journal, 69 (273), Jondrow, J., I. Materov, K. Lovell and P. Schdt, (982) On the Estaton of Techncal Ineffcency n the Stochastc Fronter Producton Model, Journal of Econoetrcs, 9, pp Jovanovc, B., (982) Selecton and the Evoluton of Industres, Econoetrca, 50, Lebensten, H. (966) "Allocatve Effcency versus X-Effcency", Aercan Econoc Revew, 56, Massell, B.F., (967) Elnaton of Manageent Bas fro Producton Functons Ftted to Cross-secton Data: A Model and an Applcaton to Afrcan Agrculture, Econoetrca, 35(3-4), Mefford, R.N. (986) Introducng Manageent nto the Producton Functon, The Revew of Econocs and Statstcs, Mundlak, Y., (96) Eprcal Producton Functons Free of Manageent Bas, Journal of Far Econocs, 43,

18 Orea, L. and Kubhakar, S. (2003) Effcency easureent usng a latent class stochastc fronter odel, Eprcal Econocs (forthcong). Ptt, M.M. and L.F. Lee, (98) "Measureent and Sources of Techncal Ineffcency n the Indonesan Weavng Industry", Journal of Developent Econocs, 9, Schdt, P. and R. Sckles, (984) Producton Fronters and Panel Data, Journal of Busness and Econoc Statstcs, 2, Tran, K. (2003) Dscrete Choce Methods wth Sulaton, Cabrdge, Cabrdge Unversty Press. Tsonas, E.G., (2002) Stochastc Fronter Models wth Rando Coeffcents, Journal of Appled Econoetrcs, 7, Zellner, A., J. Kenta and J. Drèze, (966) "Specfcaton and Estaton of Cobb-Douglas Producton Functon Models", Econoetrca, 34,

19 Densty U IN O M G T K er nel densty est ate for U IN O M G T Densty U IM G T K er nel densty est ate for U IM G T Fgure. Kernel Densty Estates for Ineffcency Estates. 7

20 Fgure 2. Plot of Current Manageent (MEANMI) vs. Manageent at the fronter (MANAGE) Fgure 3. Plot of Estated Ineffcency vs. Manageent at the fronter 8

21 APPENDIX. Let r denote the r-th rando draw fro the standard noral populaton of n a saple of R such draws. Then, the contrbuton to the sulated lkelhood functon for fr s L f (9) S R T = ( ε ) r = t = t r R The sulated log lkelhood that s axzed s δ R (20) log L S N R T ( ) = log f ( ε ) = r = t= t r Ths functon s sooth and twce contnuously dfferentable n the paraeters. (For condtons under whch axzaton of the sulated log lkelhood produces an estator wth the sae asyptotc propertes as the true MLE, see Goureroux and Monfort (996), Tran (2003) and Greene (2003)). The dervatves of the sulated log lkelhood are obtaned as follows: S S log L L / δ = S δ L = ( ) R T R Σr= ( Πt= f ( εt r) ) T r gt ( r) t Σ Π f ( ε ) Σ log f ( ε ) / δ R T T R r = t = t r t= t r R = ω r= = R = ω r= rg ( r ) where ω r s a set of nonnegatve weghts that su over r to one for each by constructon and (2) g t ( r ) s the vector of dervatves, ε δ. Let log f ( t r ) / x tr = [,,lnx tk,,, ½ lnx tk lnx tl,, r, ½ r 2,, ½lnx tk r ] so that ε t r = y t - β x t ( r ) = ε tr. For convenence, let h tr = φ(-λε tr /σ )/Φ( -λε tr /σ). The requred frst dervatves are β {( εtr / σ ) + htrλ} x tr 2 log f ( εtr) / σ = ( εtr / σ) + ( εtr / σ) λhtr σ (22) λ εtrh tr. 9

22 The ntegrals n and ts dervatves are approxated by obtanng a suffcent nuber of draws fro the populaton generatng. The law of large nubers s nvoked to nfer that the saple averages wll converge to the underlyng ntegral. Rando draws fro the populaton are suffcent for ths process, but not necessary. What s essental s coverage of the range of varaton of, not randoness of the draws. The ethod of Halton sequences [see Bhat (999), for exaple] s used to provde uch ore effcent coverage of the range, and n turn, uch faster estaton than the ethod of rando sulaton (See, as well, Tran (2003, pp ) for a dscusson of Halton sequences). Thus, r s the rth eleent of the Halton sequence for ndvdual. The eleents of the Halton sequence, H r are spread over the unt nterval, (0,). The draw of r s obtaned by the nverse probablty transfor. Thus, r = Φ - (H r ). The estated standard errors of the paraeter estators are coputed by usng the BHHH estator, as before, wth sulaton used for the dervatve vectors. Snce we are only ntegratng over a sngle denson, the gan n effcency, f ths applcaton s lke others, s on the order of ten fold - that s, the sae results are obtaned wth only about one tenth the nuber of draws needed. We have used,000 draws n our estaton, whch would correspond to several thousand draws were they produced wth a rando nuber generator nstead. 20