Centre for Efficiency and Productivity Analysis

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1 Cetre or Ecec d Productvt Alss Workg Per Seres No. 4/3 Ttle Semrmetrc Estmto o Stochstc Froters A Bes Peled Aroch Authors Gholmre Hrgsht Dte: School o Ecoomcs Uverst o Queesld St. Luc Qld. 47 Austrl

2 Semrmetrc Estmto o Stochstc Froters A Bes Peled Aroch Gholmre Hrgsht * School o Ecoomcs Uverst o Queesld St Luc QLD 47 Austrl ABSRACT Almost ll revous roches to estmtg semrmetrc roter models where the uctol orm or the roducto cost ucto s ukow hve ee locl ormetrc e. kerel roches. I ths er we use eled e. sle roch. We show how ths roch c e led to vret o roter models cludg el models wth ed d rdom eects wth Bes rmework. We lso l our roch to deret multvrte settgs cludg ddtve d ddtve wth tercto models. The ltter s romsg model ecuse t s ver lele d does ot suer the severe curse o dmesolt rolem commo wth ull ormetrc uctos. We llustrte our method usg smulted emle. * I m deted to m suervsors Pro. Prsd Ro d Dr. Chrs O Doell or ther suort d gret suggestos.

3 . Itroducto DEA d stochstc roter lss re the two roches commol used to estmte roter uctos d ecec. DEA models re cosdered ormetrc whch mes there s o eed to sec uctol orm or the roducto ucto d the re usull o-stochstc whch mes the eect o ose d rdom errors re gored or mesured s ecec eects. Stochstc roter models tke ose d rdom errors to ccout ut the re usull rmetrc we hve to sec uctol orms or the roducto uctos d ecec dstrutos. There hve ee some eorts to rel rmetrc ssumtos stochstc roter models. Some studes lke Prk et. l Horrce d Gr & Steel hve ocused o the estmto o roter model wth rmetrc ler roducto ucto ut ukow uctol orm or the ecec dstruto. I other studes the ocus hs ee o the estmto o stochstc roter model wth ukow uctol orm or the roducto ucto. F et. l. 995 rovded two stge seudo mmum lkelhood roch to estmte such model. The roches o Ke & Smr 996 d Hederso re ol lcle to models wth el dt. Adms et. l. 999 estmte stochstc el roter model relg the rmetrc ssumto o the ecec dstruto d o sugrou o regressors mult-outut dstce ucto. Kumhkr & Tsos hve used locl lkelhood roch to estmte roter model. Ther model could e clled ull ormetrc model ecuse oth the rmeters o the roducto ucto d the ecec dstruto rem ed ol locl eghourhood. I ths er we ocus o the estmto o stochstc roducto cost roter wth ukow uctol orm we mke the usul rmetrc ssumtos out the ecec dstruto. I cotrst to ove studes we use eled e. sle roch to ormetrc estmto rther th locl e. kerel roch. Ulke the locl roches t s strghtorwrd to l our roch to deret stochstc roter models cludg models wth el dt ed d rdom eects models d eve the true ed eect model o Greee. It s lso eser to mose

4 3 ecoomc regulrt restrctos usg our roch comred to kerel-sed roches. Aother ovelt o ths er s the use o ddtve ucto wth terctos Serlch et. l. the stochstc roter. The ddtve model wth terctos s ver lele model; t s geerlto o some mortt uctol orms lke the trslog; d the curse o dmesolt rolem s ot severe comred to ull ormetrc ucto. Although t s ossle to estmte our model usg eled mmum lkelhood method we use ull Bes roch. There re some rgumets vour o the Bes roch to roter model estmto. As Koo d Steel.55 clm the theoretcl ustcto or ot d tervl estmtes o ececes sed o mmum lkelhood s ot strog ut the Bes roch rovdes te smle dstrutos or rm ececes d tht llows us to ot ot d tervl estmtes esl. It s lso ossle to mose curvture restrctos usg Bes roch eg. Cuest et. l.. Aother reso or usg Bes roch here s tht the Bes roch to ormetrc estmto elds the smoothg rmeters utomtcll. It s lso eser to mlemet Gs smlg lgorthm rther th mme comlcted lkelhood ucto wth m rmeters. We eg the er Secto wth troducto to the Bes roch to rmetrc roters d we show tht the lss c e eteded to uvrte semrmetrc model usg rorte ror orrowed rom the Bes ormetrc lterture. I Secto 3 we geerle the lss to multvrte ddtve d rtll ler ddtve models. We dscuss semrmetrc ed d rdom eects models Secto 4. There re deret roches to Bes ormetrcs d our methodolog s deedet o the roch used. We el oe o the oulr roches clled smoothg sles Secto 5. I Secto 6 we dscuss the estmto o more elorte multvrte models usg P-sles. I Secto 7 we l our method to smulted emle. I secto 8 the roosed method s led to rel dt emle.

5 4 Ths er s hevl sed o the Bes roch to rmetrc roter estmto d lso Bes semrmetrc estmto wth sles. We reer the reder to Koo & Steel d Tsos or detls o the Bes roch to estmtg roters d Hste & Tshr Gree & Slverm 994 Euk 999 Fhrmeer Koo & Porer Ruert & Crroll d Berr et. l. or ddtol ormto o sle d Bes semrmetrc estmto.. The Bes Aroch to Estmtg Uvrte Semrmetrc Froters We strt ths secto wth re revew o the Bes roch to rmetrc roter estmto d the we eted t to uvrte ormetrc uctos. The stochstc roter model m e seced s ollows: β ε where reresets the log o outut s vector o uts logs reresets ecec eects d ε s rdom error. The suscrt dees rms. The ollowg rmetrc ssumtos re mde the seccto o the ove model: The roducto ucto s ler the rmeters. hs kow dstruto.e. eoetl wth rmeter 3 u s dstruted s N I ull Bes roch our m s to ot the osteror β. Accordg to Bes s theorem we c wrte: β β β It c esl e show tht β N β d e. Ferde et. l. 997 hve show tht the osteror dstruto s ot well deed whe the usul o-ormtve rors re ssumed or - Other dstrutos lke hl orml tructed orml d gmm c lso e used.

6 5 d. The hve roosed the ollowg gmm ror or these rmeters: G G. It hs lso ee show tht roer or ouded ror the orm o E I β where E I s the dctor ucto or the ecoomc regulrt codtos s good ror or β. Here we ssume tht E I s equl to oe or ll vlues o β. 3 Wth the ove ormto d usg the ssumto tht d ε re d we c ot the ollowg osteror: / e } { e β β For urther erece we must e le to drw rom the ove dest. But ths osteror s ot stdrd oe d we c ot drw rom t drectl. However we c derve the ollowg codtol dstrutos: } { ' S β N where ' ' S ' / β β β G } { N β β G β The ove codtol dstrutos re ll stdrd dstrutos d drwg rdom umers rom them s rl es. Seccll Gs smler wth dt ugmetto c e set u to geerte smle o vlues or the rmeters. The smle c e used to ot eecttos stdrd errors d codece tervls or the rmeters. Etedg the ove lss to semrmetrc uvrte roter models s rl strghtorwrd. Let the semrmetrc uvrte stochstc roter model e deed s: - We dee the Gmm dstruto s e Γ 3 - We c mose ecoomc codtos lke curvture restrctos lettg IE e oe or those vlues o β whch sts the restrcto d ero otherwse.

7 6 ε The derece etwee d s tht we ow hve ukow orm or regresso ucto. I we c sec rorte ror or {. } t wll e ossle to set u Bes roch smlr to the rmetrc cse. I we choose o-ormtve ror we get ucto whch terolte the dt d tht s ot we eect rom regresso estmto we usull eect tht the estmted ucto stses some degree o smoothess I the Bes semrmetrc lterture the ollowg ror hs ee roosed or : N K where K s mtr deed deretl deret roches to Bes ormetrcs 4. The ove ror eles the roughess o t relects our ror ele tht the estmted must ot e too rough or wggl. We do t sec the mtr K here we wll come ck to K whe dscussg smoothg sles d P-sles the et sectos. For the momet we hve d etr rmeter. The usg Bes s Theorem we c wrte the osteror s: The rst term o the rght hd sde c e derved esl s β N. For the secod term we use the ror dscussed ove. For d we g use gmm rors such tht G G d IG. The our osteror c e wrtte s e { } K'K e e / / Ag the ove dest ucto s ot stdrd oe d so we c t drw drectl rom t. However we c esl ot the ollowg codtol dstrutos: 4 There re umer o roches to Bes ormetrc estmto: smoothg sle regresso sle P-sle Koo d Porer s roch Fhrmeer s roch. We wll dscuss smoothg sle d P-sle more detl lter ths er. 6 -The ddtve regresso model hs ee dscussed detl Hste d Tshr 99. The stdrd requetst method or estmto o ddtve regresso s ckttg.

8 7 } { / S S N where } / { K I S ' / G / ' K G } { N G A Gs smler wth dt ugmetto c e set u sequetll drwg rom the ove codtol dstrutos. Notce tht we do t eed to use thg deret rom the rmetrc cse; ll tht we eed to do s geertg rdom umers rom tructed orml d gmm dstrutos. 3. Eteso to Addtve d Prtll Ler Addtve Models I ths secto we eted the revous lss to two secl multvrte uctos: the ddtve d rtll ler ddtve uctos. We wll dscuss less restrctve multvrte models lter Secto 6. The stochstc roter model wth ddtve roducto structure c e wrtte s 6 ε A tlor-mde Gs smlg lgorthm whch c e terreted s Bes ckttg roch Hste d Tshr 998 c e setu or Bes estmto o ddtve models. Assumg N K s ror dstruto or we c setu our Gs smlg rocedure drwg rom the ollowg codtol dstrutos.

9 8 } { / N S S ' / G / ' G K } { N G The ove Bes lss c e esl eteded to the ollowg rtll ler ddtve model where w s vector o vrles tht s relted to ler sho. ε β w 4. Bes Semrmetrc Froters wth Rdom d Fed Eects It s cresgl commo to use el dt stochstc roter lss. Two deret models the ed eects d rdom eects models hve ee roosed or stochstc roter models wth el dt. The Bes roch to rmetrc ed d rdom eects estmto hs ee dscussed Koo d Steel. Here we descre the semrmetrc Bes ltertve to rdom d ed eect models. Cosder the ollowg rdom eects model or ossl ulced el dt set t t t ε ε ε N Gmm T t t Assumg verse gmm ror or d s eore t s ot dcult to show tht our osteror wll e

10 9 e e } { e / / T T t t t T K'K Y where T T. Ths osteror dstruto c e used or urther erece usg Gs smler. The codtol dstrutos re: } { / S Y S N } { / T t t t T G / ' K T G [ ] T t t t N T T T G Note tht or osteror lss we ol eed to drw rdom umers rom tructed orml d Gmm dstrutos. The eteso o the ove el dt model to the multvrte ddtve cse wth Gs smler s strghtorwrd. The ed eects model c e wrtte s the ollowg rtll ler model: t t t D ε α where D reresets the dumm vrle ssocted wth the -th rm. Ths model s othg more th rtll ler regresso model d c e estmted usg Gs smlg esl.

11 5. Smoothg sles d Bes semrmetrcs Cosder the ollowg uvrte estmto rolem: ε ; ε N where the uctol orm or s ukow. Oe method o estmtg such model s mmg o ollowg eled sum o squre crtero J { } λ { } d over ll uctos such tht the tegrl ests. The tegrl reresets roughess elt d λ s the smoothg rmeter. Lrger vlues o λ result smoother curve. The soluto ucto to the ove mmto rolem hs ee show to e turl cuc sle wth kots t ech o the uque vlues o. Let {. }. The usg the cuc sle ture o t c e show tht the elt term c e wrtte s λ { } d λ K where K s mtr o rk - d s deed Gree d Slverm The usg mtr lger t s es to show tht the smoothg sle mmer J s Sλ where S λ I λk d {... }. I the Bes roch to smoothg sles the ollowg rtll mroer Guss ror s gve to : N K where K s the geerled verse o K d / λ. Kowg K we c do Bes lss o the stochstc roter model s dscussed Sectos d 3. Geerltos to ddtve d rtll ler ddtve multvrte roducto uctos re strghtorwrd wth the rmework dscussed the revous sectos ut or less restrctve multvrte roducto uctos we use P-sle roch.

12 6. P-sles d multvrte stochstc roters So r we hve roosed Bes estmto method or roter model wth ddtve roducto structure whch s ot ver lele. The urose o ths secto s to eted our lss to less restrctve orms o multvrte roter models. We c cosder ull ormetrc multvrte ucto ut s s well kow the ormetrc ecoometrcs lterture there s curse o dmesolt wth multvrte ormetrc ucto. However there s orm o multvrte roducto ucto clled ddtve wth tercto model whch s ver lele d s geerlto o some mortt uctos lke the trslog geerled Leote d qudrtc uctol orms. Ths orm c e wrtte s 7 :... α < The rest o ths secto cosders Bes methods to estmte multvrte roter model wth the ove structure. We use P-sle estmtor whch s smler th smoothg sles whe estmtg these models. There re two geerl roches to sle ttg smoothg sle d regresso sles. Smoothg sles use ll the oservtos s kots. Cosequetl whe the umer o oservtos s lrge the ecome comuttoll mrctcl d geerlg them to multvrte ucto estmto ecet or the ddtve d rtll ler model s ot strghtorwrd. We c t regresso sles usg ordr lest squres oce the kots hve ee selected ut kot selecto rocedures re comlcted d comuttoll tesve Smth d Koh 996. P-sles troduced Elers & M 996 d Ruert & Crroll come etures o smoothg sles d regresso sles such w tht ulke regresso sles the loctos o kots re ot crucl d the hve r ewer rmeters th smoothg sles. The ollowg dscusso s sed o Ruert d Crroll s troducto to P-sles d we reer the reder to ther ers or more ormto. 7 - For detcto uroses we eed to ut some restrcto o comoets o the tercto models. Ths hs ee dscussed Serlch & et l. d Che 993. Here ecuse we re mostl terested estmto o ececes we do t dscuss them more detl.

13 Suose we wt to estmte the ollowg ormetrc model: ε Let ' β β β... β k β d cosder the ollowg regresso sle model 8 K β β k k k β β κ where u ui d κ <... < κ K re ed kots. I the P-sle roch u we llow K to e lrge d ed ut we ut elt o the K { βk k } the set o ums the dervtve o β such tht our eled lest squre crtero wll e ' Xβ Xβ λβ'kβ where X s mtr wth κ κ t s -th row d K s k dgol mtr whose rst two dgol elemets re d the remg dgol elemets re. Smle clculto shows tht the eled lest squre mmer β wll e βλ X'X λk X' The eteso to ddtve uctos s strghtorwrd. Suose we hve vrte ddtve ucto o the ollowg orm: The we c wrte the regresso sle ucto s K M β β β β β k kk β K m m k m µ Dee X XXX3 where X X k... kk 8 - Here or ese o llustrto we use ler ss. Other ss lke oloml d B-sle should e used rel rctce.

14 3 d X3 µ... µ M. The our eled lest squres crtero wll e ' Xβ Xβ β'kβ where K s lock dgol mtr wth locks λ I K λ I I3 M d I M s dett mtr o dmeso M ote tht we hve used deret smoothg rmeters λ or deret vrles. The ove lss c e eteded to multvrte ormetrc models usg tesor roduct sles. We dscuss vrte model here ut geerlto to the multvrte cse s strghtorwrd. Suose B {} B B s the set o our ss uctos where s vector o oes B { } d B { κ... κ }. The suscrts l d l l k deote ler d ecewse ler. B c lso e deed the sme w or vrle. The tesor roduct regresso sle ss s deed B B B whch s the set o ll roducts d where B B. Let B l [ Bl B] [ B Bl] The we c wrte our regresso sle s ollows β X β β X β X3 β3 X4 4 where X X X3 d X4 re equl to X X B l X 3 B d X 4 B. We c dee the eled lest squres crtero l l usg deret smoothg rmeters or β β3 d β 4 It s cler tht the dmeso o the ss grows geometrcll wth the crese the umer o vrles llustrtg the curse o dmesolt. So t mght ot e rctcl to estmte ull ormetrc model or more th two vrles. Isted we roose usg tercto model whch s ver lele d where the curse o

15 4 dmesolt s ot tht severe. Wthout loss o geerlt we c wrte ddtve model wth terctos s ddtve model o vrte uctos s ollows... < Comg our lss o ddtve d vrte models we c ot the regresso sle or the tercto model. We see tht wth rorte deto o X ll the ove models c e wrtte ollowg regresso sle orm P... β X β Xβ where X X... X P ' β β... β P. ' The ove lss shows tht we c wrte our multvrte roter model s β ε whch s ver smlr to the rmetrc cse the ol derece eg tht here we hve to sec secl ror or β. We use the ollowg ror used Berr et. l. : β N K where K s lock dgol mtr wth locks I... I. Usg ths ror we c ot the ssocted osteror the sme w s we dd revous sectos. Ag the osteror s ot stdrd oe ut we c derve the ollowg codtol dstrutos d l Gs smlg or urther lss:

16 5 β N { S S / } Xβ' Xβ β G / β β IG m / β N{ X β } ' K β 9 β G 7. A Smulted Emle I ths secto we estmte ddtve roter model usg smulted dt. We geerted dt usg the ollowg model: 5 e ε 5 where.. seq{.4} uorm 4 Gmm 5 ε Norml.4. We hve chose ver oler uctol orm or so tht t shows the three well-kow stges o roducto; or ler ucto hs ee seced. We used smoothg sle to estmte ove ddtve model s dscussed Secto 3. For Bes lss we eed to sec rors or the rmeters. We used: The results re ot ver sestve to moderte chges these rors. Strtg vlues or the rmeters were oted usg smle COLS estmtor. Posteror lss ws sed o reltos; the rst were ecluded rom l lss s ur- erod. The mes o the osteror smles were used s ot estmtes o the rmeters. We hve summred the results severl grhs. Frst the tted vlues o d hve ee comred wth smulted dt o d 9 - m s the dmeso o vector B

17 6 Fgure. As we see oth seem to t the rel dt ver well. The dshed le reresets the orgl smulted dt d the thck le reresets the estmted ucto Fgure. Ftted d smulted vlues o d lotted gst d Fgure 3 comres tted vlues o outut wth orgl dt o outut. The estmted outut seems to e roer roducto roter sor s most dt re elow t. Fgure 3. Ftted vlues comred wth rel dt o outut I Fgure 4 we hve comred the estmted vlues o ececes wth the orgl dt o. As we see our estmtes ollow the rel dt ut there re some sgct dereces. These dereces re ot ueected roter lss. I we

18 7 estmte rmetrc roter usg stdrd methods we see smlr dereces etwee estmted ececes d true ececes Fgure 4. Estmted vlues o comred to rel dt. The dshed les show the rel dt A Rel Dt Emle I ths secto we l our method to rel dt. The dt set cossts o oservtos o mor rvtel-owed Tes electrc utltes oserved ull over 8 ers rom 966 to 985 d cludes ormto o lour ctl d uel uts or electrcl ower geerto outut. Ths dt set hs ee lred used Kumhkr 996 Schmdt d etl We ssume ollowg rdom eect ddtve stochstc roter L K F ε t t t 3 t t where L K d F rereset lour ctl d uel resectvel. A Bes P-sle roch s used to estmte the ove model. equ-dstce ots were chose s kots or ech vrle. The results o estmto o 3 c e see gures 5 to 7. The she o s ot wht we usull eect ut ervous studes wth the sme dt set corm ths egtve reltosh other rmetrc studes.e. Schmdt d etl the hve oud egtve coecets or L. or two other uts the roducto ucto hs regulr she d t s ot r rom eg ler. - The dt hs ee dowloded rom Jourl o Aled Ecoometrc Dt Archve

19 8 Fgure 5 the grh o versus L Fgure 6 the grh o 3 versus F Fgure 7 the grh o versus K

20 9 I gure 8 we comre the ormetrc clculted sed o the method we hve roosed d rmetrc estmtes drw rom Schmdt d etl. 999 o ececes o te rms. The lue d red rectgles rereset ormetrc d rmetrc estmtes resectvel. As we see the tter s the sme or oth cses ut the ormetrc estmtes ths rtculr emle re hgher

21 8. Cocluso I ths er we roose eled roch to estmtg semrmetrc stochstc roters where the uctol orm or roducto cost ucto s ssumed ukow. We use Bes roch d show tht orrowg rorte ror rom the Bes ormetrc lterture we c esl geerle the Bes rmetrc stochstc roter to the semrmetrc cse we lso see tht we do t eed secl ecoometrc tools other th techques or drwg rom tructed d gmm dstrutos wth Gs smlg set-u. Throughout the er t s show tht our roch s lcle to deret stochstc roter models cludg models wth cross sectos ed d rdom eects d deret orm o semrmetrc multvrte roducto uctos cludg the terestg cse o ddtve wth tercto uctos.

22 Reereces Adms R. A. Berger d R. Sckles 999 Semrmetrc roches to stochstc el roters wth lctos the kg dustr Jourl o Busess d Ecoomc Sttstcs Ager D.J. C.A.K. Lovell d P. Schmdt 977 Formulto d estmto o stochstc roter roducto uctos. Jourl o Ecoometrcs Che Z. 993 Fttg multvrte regresso uctos tercto sle models Jourl o Rol Sttstcl Socet B Coell T.J. D.S. Ro. d G.E. Bttese 998 A troducto to ecec d roductvt lss Kluwer Acdemc Pulshers Bosto. Cuest R.A. C. J. O Doell T. J. Coell d S. Sgh Imosg curvture restrcto o roducto roter CEPA workg er Uverst o New Egld. Elers P. H. C. d Mr B. D. 996 Flele Smoothg Wth B-sles d Peltes wth dscusso. Sttstcl Scece 89-. Euk R. L. 999 Normetrc regresso d sle smoothg Secod Edto Mrcel Dekker Ic. New York. Ferde C. J. Osewlsk d M.F.J. Steel 997 O the use o el dt stochstc roter models Jourl o Ecoometrcs 79: F Y. Q. L d A.Weersk 996 Semrmetrc estmto o Stochstc Producto Froter Jourl o Busess d Ecoomc Sttstcs Gree P.J. d Slverm B. 994: Normetrc regresso d geerled ler models. Chm d Hll Lodo Gr J.E. d M. Steel Semrmetrc Bes Ierece or Stochstc Froter Models Hste T. d Tshr R. 99: Geerled ddtve models. Chm d Hll Lodo. Hste T. d Tshr R. : Bes ckttg. Sttstcl Scece Hederso D.J. Normetrc kerel mesuremet o techcl ecec htt://

23 Koo G. d D. Porer Bes vrts o some clsscl semrmetrc regresso techques Koo G. M.F.J. Steel d J. Osewlsk 995 Posteror lss o stochstc roter models usg Gs smlg Comuttol Sttstcs Koo G. d M.F.J. Steel Bes lss o stochstc roter models. I A Como to Theoretcl Ecoometrcs Bltg B. ed. Blckwell Kumhkr S.C. d G. Tsos Normetrc Stochstc Froter Models htt:// Prk B. R. Sckles d L. Smr 998 Stochstc el roters: semrmetrc roch Jourl o Ecoometrcs Prk B. d L. Smr 994 Ecet semrmetrc estmto stochstc roter model Jourl o the Amerc Sttstcl Assocto Ruert D. d R.J Crroll Stll-dtve eltes or sle ttg. Austrl d New Zeld Jourl o Sttstcs Smth M. d Koh R. 996 Normetrc regresso v Bes vrle selecto Jourl o Ecoometrcs Serlch S. D. Tøsthem d L. Yg Normetrc estmto d testg o tercto ddtve models Ecoometrc Theor Tsos G. A Itroducto To Ecec Mesuremet Usg Bes Stochstc Froter Models Glol Busess & Ecoomcs Revew