WORKING PAPER SERIES 2014-EQM-03

Transcription

1 Ma 04 WORKING AER SERIES 04-EQM-03 A Note on arameterzng Input stance Functons: oes the Choce of a Functonal Form Matter? Rolf Färe Unverst of Aransas Mchael Vardanan IESEG School of Management (EM CNRS) IESEG School of Management lle Catholc Unverst 3 rue de la gue F lle Tel: 33(0) Fa: 33(0)

2 A Note on arameterzng Input stance Functons: oes the Choce of a Functonal Form Matter? Rolf Färe epartment of Economcs Unverst of Aransas Busness Buldng Room 40 Faettevlle AR 770 USA Emal: gferrer@walton.uar.edu Mchael Vardanan IÉSEG School of Management (EM CNRS) arvs de la éfense 9044 ars a éfense France Emal: m.vardanan@eseg.fr Aprl 04 Abstract: We use a Monte Carlo eperment to compare the quadratc and translog functonal forms n terms of ther ablt to appromate nown fronters that possess conve curvature. Unle some of the estng smulaton studes that have studed ths topc we fnd that both functonal forms provde a relable appromaton to a true fronter. Our results lend support to estng eplanatons concernng the translog form s nnate propenst to eld conve rather than concave fronter estmates. Kewords: stance Functons arameterzaton JE Classfcaton: 4 C63

3 Two recent smulaton studes comparng the ablt of dstance functons to represent producton technolog have concluded that quadratc drectonal output dstance functon (Chambers et al ) fares better than translog Shephard (970) output dstance functon. Färe et al. (00) performed ths comparson n the quantt space whereas Chambers et al. (03) mplemented t n the prce space. As a possble eplanaton both papers menton the translog functon s ntrnsc propenst to produce fronter estmates that possess conve curvature a problem when modeg output dstance functons but a useful feature for parameterzng nput dstance functons (Shephard 953). In ths paper we perform a seres of Monte Carlo eperments to compare a quadratc drectonal nput dstance functon to a translog Shephard s nput dstance functon n terms of ther ablt to appromate dfferent famles of true producton technologes. We rel on the frst- and second-order dervatves of these functons n order to determne whch of them performs best. In the net secton of ths short note we gve a quc overvew of the two functonal forms whose performance we assessed n ths stud. Secton descrbes our smulaton desgn and our most nterestng results. Secton 3 concludes.. The Functonal Form Alternatves In ths secton whch bulds on Chambers et al. (03) and Färe et al. (00) we ntroduce the functonal forms that we consder n ths paper. These parametrc forms are derved from the two representatons of the underg technolog namel Shephard s (953) nput dstance functon and uenberger s (99) beneft functon whch we term the drectonal We cte these reference papers repeatedl throughout the tet due to ther mportance n motvatng the present stud.

4 nput dstance functon (Chambers et al. 996) and the condton that the are of the form generalzed quadratc. et be an nput vector and N an output vector so that the nput M requrement set n terms of these vectors s () N : can produce. We assume that meets the standard propertes such as non-emptness closeness convet and dsposablt. Shephard s nput dstance functon s defned as sup : and we note that under wea dsposablt of nputs () and () 0 where tells us that the dstance functon full represents the technolog and states that t s homogeneous of degree + n nputs. The second representaton of ; g sup : g ndcates how s projected towards the boundar of s n terms of the drectonal nput dstance functon N where g g 0 s the drectonal vector whch. Ths dstance functon satsfes () ; g 0

5 and (v) g; g ; g 0.e. ; g le Shephard s dstance functon full represents the technolog. However t satsfes the translaton propert (v) rather than homogenet (). Net we show how homogenet and translaton propertes nfluence the choce of functonal forms. We sa that a functon F : s a generalzed quadratc functon f () Fq a0 ah q ajh q h q j j where a and h :. Färe and Sung (986) showed that f a functon s a j generalzed quadratc and homogeneous t s ether (3) q a0 a q aj q q j F j.e. translog or (4) F q a r r r 0 ajq q j j

6 .e. the quadratc mean of order r (enn 974 ewert 976). Snce the latter functonal form has onl second order terms we choose the translog functon (3) when parameterzng Shephard s nput dstance functon. If a generalzed quadratc functon meets the translaton propert there est onl two solutons to the resultng functonal equaton (Färe and undberg 006) namel (5) q F a a q a q q 0 j j j.e. the quadratc functon or (6) F I I q aj ep q ep q j 0. j Consstent wth estng studes that compared the performance of the two dstance functons we wll use the quadratc functonal form to parameterze the drectonal nput dstance functon.. Monte Carlo Smulaton esgn and Results We assume that two nputs produce a sngle socall desrable output and consder two famles of true technologes. Those belongng to the frst faml have a ffth-order poomal structure.e. Here we have chosen g.

7 (7) : The second group conssts of so-called logarthmc technologes gven b (8).5 : ep. 0 ep We closel follow the desgn of Färe et al. (00) and Chambers et al. (03) snce our man goal s to test whether ther man conclusons reman unchanged when the true fronter s conve. Varng the vectors of coeffcents and enables us to alter the curvature of the true fronter. We consder three such varatons for both famles of true technologes labeled b 3 and 3 whose correspondng coeffcent values are summarzed n Table. We choose the coeffcents so that the fronters of and are relatvel flat and subsequentl change them to add more curvature. Normalzng the outputs wth an arbtrar quantt allows us to obtan the plots of these fronters smlar to those gven n Fgure. We then randoml draw quanttes of the frst nput and mpose assumptons on the parameters of underg dstrbutons to ensure that the technolog s well-behaved. For the poomal technologes these quanttes are generated as ~ Gamma and two assumptons are placed on the dstrbuton parameters. Tang 5 and 0. 5 elds relatvel unequal quanttes of and for nearl all observatons whereas assumng 8 and 0. 5 gves relatvel balanced nput values. In the case of the translog technologes s drawn from the unform dstrbuton on the nterval Both outputs are assumed to be standard unform and the sample sze equals and 500 observatons gvng us a total of

8 twent-seven true models. Fnall we complete the data generatng process b assumng that when the true technolog s poomal and when t s translog where ~ 0 and ~ 0 dsturbance term respectvel. represent techncal neffcenc and the conventonal The net step of the eperment nvolves usng these data to estmate dstance functons. Snce we can assume that ep ep ep the specfcaton error can be added as. After rearrangng pluggng ths result n the epresson representng dstance functon s homogenet assumng the followng result: and rearrangng agan we have (9) ep. After tang the log of both sdes and assumng the translog form for the normalzed dstance functon ths specfcaton can be estmated usng the Agner et al. (977) method. 3 Its estmated parameters can subsequentl be used to recover the coeffcents of the assocated translog Shephard s nput dstance functon ep : 3 Reg on Shephard s nput dstance functons homogenet to obtan sutable econometrc specfcatons s common n the lterature. See Atnson et al. (003a 003b) and Grossopf et al. (997) for more detals.

9 (0) 0 j j As far as the drectonal dstance functon s concerned we can frst assume that ; g and then add the two-sded error to the rght-hand sde of that equaton. Combnng ths result wth the epresson for the functon s translaton propert tang and assumng g seres of rearrangements (Färe et al. 005): 4 g elds the followng composed-error specfcaton after a () g g. Ths normalzed dstance functon s parameterzed usng the quadratc functonal form and estmated usng the same method. 5 Its parameter estmates eld the coeffcents of the correspondng quadratc drectonal dstance functon gven b () ; g o j j j 4 Unle n the prevous case an nfnte number of sutable econometrc specfcatons can be obtaned b varng the assumptons placed on the mappng vector g whch s located n the thrd quadrant suggestng that the nputs are beng contracted. 5 Chambers et al. (03) reled on the same method whle Färe et al (03) chose parametrc ear programmng algorthm of Agner and Chu (968).

10 We use the vectors of mamum-lelhood parameter estmates ˆ ˆ ˆ and ˆ ˆ recover quadratc and translog nput set fronters n order to assess whether ether of these famles of estmates fares better than the other. We assume that 0 normalze both ˆ to outputs wth the sample average and use to get the relevant best-practce quanttes of ˆ ˆ ˆ ˆ and ~ ˆ ˆ ˆ for the quadratc and translog estmates respectvel. Fnall we use the pars and ~ ˆ together wth the dstance functons parameter estmates to compute the assocated margnal rate of techncal substtuton (MRTS) and the Morshma (967) elastct of substtuton at each observaton and then compare them to the true MRTS and elastct for both parameterzatons. 6 The true MRTS and elastct can be obtaned usng the epressons () and () and are the negatve of and respectvel where for the poomal technologes and for the log technologes. The estmated MRTS can be nterpreted as the relatve shadow prce of nputs [Färe and rmont (995)] and s gven b where 6 In addton to these two benchmars Färe et al. (00) and Chambers et al. (03) also compute the Eucldean dstance between the true and estmated fronter ponts. The subsequentl average across these three dscrepances before assessng the results usng a sngle crteron whch s based on that average. Here we use onl two benchmars and choose to compare the translog and quadratc functons ablt to appromate the true MRTS separatel from elastct.

11 denotes ether g ; or. Consequentl the dfference between the estmated and true MRTS s ; ~ ; ~ ˆ ˆ ˆ g g and ˆ ; ˆ ; ˆ ˆ ˆ ˆ g g where equals ether or dependng on the tpe of true technolog. Our frst crteron s based on these dscrepances and s gven b K. The estmated Morshma elastct of substtuton s the log dervatve of the shadow prce of nputs wth respect to the log of the rato of nput quanttes. Snce the fronter of the nput set s conve ths elastct must be postve. Smlar to Färe et al. (00) and Chambers et al. (03) the dfference between the estmated and true elastct equals ; ~ ; ~ ; ~ ; ~ ˆ ˆ ˆ g g g g and. ˆ ˆ ˆ ˆ ˆ ˆ ˆ (4) (3) (5) (6)

12 Note that the true elastct of substtuton correspondng to the true log models s gven b and that as before s ether or. Our second crteron assesses the average dfference between the true and estmated curvature of the fronter and s gven b K. We formulate four econometrc specfcatons for ever true technolog and estmate them usng the Agner et al. (977) approach. Three of them rel on the quadratc functon and are defned b rotatng the drectonal dstance functon s mappng vector from 3 g and then to g 3 g to a setup allows us to see f the appromaton qualt depends on the drecton of nput contracton. The fourth specfcaton reles on the translog Shephard s dstance functon that assumes a proportonal reducton of both nputs. Appromaton qualt crtera that are based on the MRTS and elastct are reported for each of the 08 cases n Tables and 3 respectvel. Fgures and 3 contan a selecton of fronter estmates obtaned usng nput vectors and ~ ˆ. Even though both functons can sometmes volate the global convet of a true fronter we chose not to mpose an curvature condtons snce our man focus s the comparson of the functons nnate modeg propertes. We start b comparng the overall performance of the quadratc drectonal dstance functon and the translog Shephard s dstance functon. The MRTS-based dscrepances suggest that the translog functonal form whose correspondng benchmar quanttes are reported n the bottom panel of Table fares better than the quadratc functon n appromatel 85% (46 out of 54) of cases when the nown technolog s poomal and n about 56% (7 out of 7) of cases when t has a log confguraton. In tpe-a models where the translog fronter estmates often volate local convet the translog functon outperforms the quadratc n all 7 cases. Ths s

13 perhaps our most notable result whch contrasts sharpl wth the fndngs of Färe et al. (00) who report that n the case of the true poomal technologes the quadratc functon s global behavor s clearl superor to that of the translog functon as well as those of Chambers et al. (03) who menton that the quadratc parameterzatons are overall better than translog n appromatng both tpes of true technologes. However as shown n Table 3 the advantage swngs bac n the quadratc functon s favor when modeg the curvature. It domnates the translog functon n 6% of cases when the true technolog s poomal and almost ever tme when t has a log structure. Table 3 also suggests that whle most of the quadratc specfcatons that assume g 3 beat the translog functon the latter can outperform the quadratc functon when the mappng vector s rotated toward g 3. Consstent wth the results of prevous related studes a quadratc specfcaton whose mappng vector s most n e wth the approach used to add neffcenc to the true models appears to domnate an of ts quadratc counterparts. In other words snce ths neffcenc component was smpl added to the second nput t s the specfcaton that assumes a predomnantl southern drecton of contracton that does the best job of tracng a true technolog. Another sgn that the translog functon ma be better at appromatng conve than concave fronters s ts large sample performance whch tpcall mproves n true poomal models both when t s used to model the MRTS and the elastct of substtuton. 7 However ths result no longer holds n true log models where the translog functon s sample sze related performance mostl deterorates and whose correspondng translog fronter estmates usuall 7 Ths too dffers from the results of Färe et al. (00) who report precsel the opposte for concave fronters.

14 volate global convet of the true fronter (Fgure 3). Sample sze-related performance of the quadratc functon s ver good but t depends on the drectonal vector n both tpes of true models. Table suggests that as the number of observatons ncreases the MRTS-based benchmar decreases n out of 8 cases when g 3 nearl alwas when g ever tme when g 3 and. Elastct dscrepances assocated wth ths functonal form also decrease wth an ncrease n sample sze n the overwhelmng majort of cases. The last two results are consstent wth fndngs reported n estng smulaton studes suggestng that the quadratc functon fares relatvel well regardless of whether the nown fronter s conve or concave. Fnall the last set of fndngs that contradct the conclusons of our reference studes concerns the translog functon s handg of an ncrease n the true curvature. Benchmar values n columns and 4 (tpe-a models) as well as columns 8 and 0 (log models) of the bottom panel of Table 3 suggest that appromaton qualt usuall mproves when we add more curvature to a conve fronter. B contrast the quadratc functon alwas fares best when the true fronter s relatvel flat. 3. Concluson Recent smulaton studes b Färe et al. (00) and Chambers et al. (03) have compared the quadratc and translog functonal forms and found that the former domnates the latter when used to appromate the concave fronter of the output set. Ther authors have suggested that the e reason ma be the translog functon s nherent propenst to eld globall conve fronter estmates even when the true fronter s concave. We nvestgate ths possblt b estmatng a selecton of conve fronters of the nput set and show that the translog functon

15 does behave much better. Although the quadratc functon s overall performance remans adequate ts domnance of the translog form has dmnshed. For eample we found evdence that the translog form sometmes outperforms quadratc even when the true technolog has a poomal structure. The performance of ether functon can be rather uneven and t depends on the characterstcs of the nown technolog. To put ths analss n a more general contet we note that our conclusons are consstent wth the results of smulaton studes b Wales (977) and Gule et al. (983) who have compared the performance of varous functonal forms ncludng the translog but dd not consder the quadratc functon. Even though the translog form s clearl mperfect at modeg conve fronters t can sometmes outperform other functonal forms ncludng those that are better than translog at appromatng concave fronters. Whenever possble we recommend usng both of these forms n emprcal studes n order to model a true technolog as best as possble.

16 References Agner Chu SF (968) On Estmatng the ndustr producton functon. Am Econ Rev 58: Agner ovell CAK Schmdt (977) Formulaton and estmaton of stochastc fronter producton functon models. J Econom 6: 37 Atnson SE Cornwell C Honeramp O (003a) Measurng and decomposng productvt change: stochastc dstance functon estmaton versus data envelopment analss. J Bus and Econ Stat ():84 94 Atnson SE Färe R rmont (003b) Stochastc estmaton of frm neffcenc usng dstance functons. South Econ J 69(3):596 6 Chambers RG Chung Y Färe R (996) Beneft and dstance functons. J of Econ Theor 70: Chambers RG Chung Y Färe R (998) roft dstance functons and Nerlovan effcenc. J Optm Theor and Appl 98: Chambers RG Färe R Grossopf S Vardanan M (03) Generalzed Quadratc Revenue Functons. J Econom 73: enn MC (974) The relatonshp between functonal forms for the producton sstem. Canadan J Econ 7: 3 ewert E (976) Eact and superlatve nde numbers. J Econom 4:5 45 Färe R Sung KJ (986) On second-order Talor s seres appromaton and ear homogenet. Aequatones Mathematcae 30:80 86 Färe R Grossopf S Noh W Weber W (005) Characterstcs of a pollutng technolog: theor and practce. J Econom 6():469 49

17 Färe R undberg A (006) arameterzng the shortage functon. Mmeo Färe R Martns-Flho C Vardanan M (00) On functonal form representaton of multoutput producton technologes. J rod Anal 33:8 96 Grossopf S Haes KJ Talor Weber W (997) Budget constraned fronter measures of fscal equalt and effcenc n schoog. Rev Econ and Stat :6 4 Gule K Kno ovell CA Scles RC (983). A comparson of the performance of three fleble functonal forms. Int Econ Rev 4:59 66 uenberger G (99) Beneft functons and dualt. J Math Econ :46 48 Shephard RW (953) Cost and producton functons. rnceton Unverst ress rnceton Shephard RW (970). Theor of cost and producton functons. rnceton Unverst ress rnceton Wales TJ (977). On the fleblt of fleble functonal forms. J Econom 5:83 93

18 Table arameters efnng the True Technolog oomal Technologes ogarthmc Technologes

19 Table Appromaton Crteron Based on the Margnal Rate of Techncal Substtuton rectonal Input stance Functon K=50 K=00 K=500 K=50 K=00 K=500 K=50 K=00 K=500 A A 3A B B 3B g = (3 ) g = ( ) g = ( 3) Shephard s Input stance Functon K=50 K=00 K=500 A A 3A B B 3B

20 Table 3 Appromaton Crteron Based on the Morshma Elastct of Substtuton rectonal Input stance Functon K=50 K=00 K=500 K=50 K=00 K=500 K=50 K=00 K=500 A A 3A B B 3B g = (3 ) g = ( ) g = ( 3) Shephard s Input stance Functon K=50 K=00 K=500 A A 3A B B 3B

21 Fgure Fronters of the Input Set Correspondng to the True oomal and og Technologes

22 Fgure Fronter Estmates Correspondng to Selected oomal Technologes A 3B

23 Fgure 3 Fronter Estmates Correspondng to Selected og Technologes 3