Econometrics (10163) MTEE Fall 2010

Transcription

1 Economtrcs 063 MTEE Fall 00 Lctur nots for Mcro conomtrc part. Man rfrnc: Grn Wllam H Economtrc analyss. Uppr Saddl Rvr N.J.: rntc Hall. 6th Edton. Wllam Nlsson Dpartmnt of Appld Economcs Unvrstat d ls Ills Balars E-mal: llam.nlsson@ub.s Offc: DB57

2 Modls th Dscrt Dpndnt Varabls Th ordnary last squar s vry smpl to stmat and n many cass actually vry porful to gv mportant nformaton concrnng a rlatonshp. It can hovr happn that th assumptons for th modl ar not fulflld or that th data has a structur that can b takn advantag of to mprov th stmator. Th dpndnt varabl could follo from a dscrt dcson; - To go on vacaton or not. - To go by bus tran car or flyng. In som cass th qualtatv dpndnt varabl prsss an opnon; - 0 could man vry unhappy could man unhappy could man nutral tc. Count data s non-ngatv dscrt data; - Days stayng on vacaton numbr of patnts tc. durng a crtan tm fram. Somtms th data s contnuous but th many obsrvatons th 0 cnsurd data; - Epndtur on vacaton.. f you do not go on vacaton you ll not spnd anythng durng th prod for th study. W ll study many of ths modls n ths cours. Th da s to alays us th ordnary last squar as rfrnc mthod. It s ntrstng to kno;. Why may not OLS b approprat?. What modl ould b a bttr altrnatv? 3. Ho can stmat th altrnatv modl? 4. Ho can ntrprt th coffcnts and th rsults from th modl? 5. What ar th lmts and drabacks of th modls? 6. Can calculat a spcfcaton tst to assur that th modl n fact s approprat? Bfor dscussng th dffrnt modls hr as you can s th dpndnt varabl has a partcular form t s mportant to brfly dscuss th OLS modl and go though th us of qualtatv planatory varabls and ho to tst hypothss n rgrsson modls.

3 3 Ordnary last squar OLS rgrsson modl y... ε... n 3 3 K K y s th dpndnt pland varabl... K ar ndpndnt planatory varabls and rfrs to th obsrvatons n th sampl. ε s a random dsturbanc rror trm rsdual. Assumptons: Lnar functonal form. Dscuss: a Ho can varabls b allod to hav a nonlnar ffct? b Ho can ntrprt th coffcnts of th dpndnt and ndpndt varabls ar masurd n logarthm? log-lnar modl. Idntfcaton condton. Th planatory varabls ar lnarly ndpndnt and thr ar at last K obsrvatons. Dscuss: a If s a constant n th modl.. hat happn f anothr plantory varabl n fact also s constant n th sampl? 3 Dsturbanc has pctd valu zro. E[ ε ] 0. No plantory varabl contan nformaton to prdct th rsdual. 4 Sphrcal dsturbancs. Var[ ε ] σ for all... n and cov[ ε ] 0 for all j. A constant varanc s labld homoscdastcty. ε j Rsduals also cannot b autocorrlatd. 5 Nonstocastc rgrssors. 6 Normalty. Th dsturbancs ar normally dstbutd th zro man and constant varanc. ε ; N[0 σ I].

4 4 Dscuss: a What happn f clud rlvant plantory varabls? b What happn f nclud rrlavant varabls? c What happn f th dpndnt varabl only! s masurd th rror? d What happn f th plantory varabls s ar masurd th rror? What happn f try dffrnt spcfcatons ncludng and cludng plantory varabl untl th rsult confrm th conomc thory? f What can happn f som of th plantory varabls ar hghly corrlatd? g What happn f th plantory varabls ar not ognously dtrmnd.. thy ar n fact also a functon of varabls? h What happn f th sampl usd s not a random sampl.. th sampl could b slf-slctd?

5 5 Qualtatv planatory varabls Eplanatory varabls can b contnuous but mght also b dscrt or vn bnary. In survys t s for ampl common to collct background nformaton such as gndr ducatonal lvl rgon of brth natonalty tc. In ths cass t could b ncssary to cod th qualtatv nformaton nto varabls that can b usd n rgrsson analyss. A ay to cod th data s to attach a crtan valu to a spcfc outcom. All th omn could b codd to hl all th mn n th sampl could b codd to 0 or oppost!. Ths s a dummy varabl. All th a unvrsty dgr could b codd to hl all that has not rach that lvl ould hav 0. Dpndng on th purpos th qualtatv nformaton could b usd to crat svral varabls. Thos th scondary schoolng could for ampl also b codd to hl all th unvrsty dgr and prmary schoolng ould b codd to 0. It s vry mportant to hav a rfrnc group hn constructng dummy varabls. You cannot for ampl crat to varabls for gndr mal fmal or thr for ducaton prmary scondary unvrsty. Ths s somtms calld th dummy varabl trap and ould man prfct collnarly and th paramtrs cannot b stmatd. Many statstcal programs automatcally drop a varabl and notfy th usr about t. Qualtatv varabls th to optons Eampl: Wag quaton - th dummy for mal and fmal as rfrnc group f ag ducaton gndr α [ gndr] ε E mal E fmal δ E mal E fmal α α δ

6 6 fmal mal M M fmal E mal E 0 *0 * ε δ α δ α δ α Eampl: Wag quaton - th dummy for fmal and mal as rfrnc group mal fmal F F mal E fmal E 0 * * * ε δ α δ Whn ntrprtng th coffcnts you hav to b carful ho you hav dfnd th dummy varabl. 0 : H Modl 0 : H Modl * 0 0 δ δ ffct gndr no Tstng Ho th tst can b constructd ll b pland n th nt scton. Qualtatv varabls th mor than to optons [ ] φ α δ α α φ δ ε α Unvrsty E Scondary E prmary E prmary E Unvrsty E prmary E Scondary E rmary ScondaryUnvrsty EDUCATION EDUCATION :

7 7 E E E prmary Scondary Unvrsty α δ *0 φ *0 α δ * φ *0 α δ *0 φ * S 0 Scondary othrs U 0 Unvrsty othrs α δs φu ε 3 Ths ampl uss prmary schoolng as th rfrnc cas but t ould of cours b possbl to us scondary or unvrsty as rfrnc group. Ho can you cod th dummy varabls to drctly s th ag ffct of ach addtonal lvl n th ducatonal systm hl kpng prmary schoolng as th rfrnc cas? Qualtatv varabls and ntractons Th dummy varabls dfnd abov ar assumd to captur a shft n th dpndnt varabl. A qualtatv varabl could hovr also chang th slop of anothr rlatonshp. Mayb th payoff for hghr ducaton s dffrnt for mn and omn. Ths ffct can b achvd by ntractons. α δm γm ε 4 If s a dummy varabl th ntracton ould man that M ould b 0 f M 0 and/or 0 and M ould only occur f both and M. γ an ncrmntal ffct s asly ntrprtd n lnar modls. Anothr opton s to crat a st of mor dtald dummy varabls capturng all possbl optons - rmmbr that nd a rfrnc cas. An ampl ould b unvrsty dgr as a oman unvrsty dgr as a man scondary schoolng as a oman tc. lavng prmary schoolng for mn as a rfrnc cas. B carful about th dummy varabl trap! You should also assur that you hav nough obsrvatons n ach subgroup othrs you ll stmat your modl th lo prcson and mght vn gt surprsng coffcnts. Eampl: If you hav a small sampl mayb you only hav a f omn th prmary schoolng. If s a contnuous varabl th ntracton ould man that M ould b 0 f M 0 and f M. Whn you ntrprt th coffcnts you hav to b carful to clarfy that th varabl could hav an ffct both through δ and γ. It s asy to comput margnal ffcts for contnuous varabls or ncrmntal ffcts for dummy varabls n lnar modls. Not hovr that calculatng margnal ffcts for nonlnar modls such as logt and probt th ntractons s a lttl bt mor complcatd. Th ntracton ffct s a doubl drvat or a doubl dffrnc. S

8 8 Tstng hypothss Wald tst W [ c ˆ θ q] Var[ c ˆ θ q] θ aramtr to tst. Eampl: [ c ˆ θ q] Lts say that hav stmatd th paramtr ρ and ant to tst th hypothss ρ. c ˆ θ q ˆ ρ Estmatd Asymptotc Varanc: [ c ˆ θ q] Est.AsyVar [ ˆ ρ ] Est.Asy.Var [ρ ˆ] Rsults from rgrsson; ˆ ρ th standard rror W [3.57 ] [3.57 ] Compar to th crtcal valu 3.84 n th Ch-squard dstrbuton th on dgr of frdom at th 5 % sgnfcanc lvl. A vry common tst s to compar th coffcnt th zro.. s t sgnfcantly dffrnt from zro. Oftn th t-dstrbuton or th z-dstrbuton standard normal s usd. ˆ θ θ0 z s ˆ θ If th hypothss s that θˆ 0 θ 0 0 and hnc z or t s ˆ θ / s ˆ θ. z or t s than compard to th crtcal valu n th z or t dstrbuton. Wth a larg sampl t s.96 for a to-sdd tst at th 5% sgnfcanc lvl. Cho tst A Cho tst can b usd to compar coffcnts for to dffrnt groups. W can run thr dffrnt rgrssons; on for mn on for omn sparatly and fnally on for th complt sampl poold modl. RRSS URSS / k F ~ Fk n n k URSS / n k n k hr RRSS s th Rstrctd Rsdual Sum of Squars from th poold modl URSS s RSS RSS.. th sum of th Rsdual Sum of Squars for th

9 9 sparat modls. RSS ˆ ε ˆ ' ε hr ˆ ε s th stmatd rsduals for group on. Th othr rsduals sum of squars ar dfnd n th sam ay. k s th numbr of parmtrs n th modl and n and n ar th sampl szs for th dffrnt groups. Th F-dstrbuton s usd and th tst statstc s dstrbutd Fk n n -*k. Anothr quvalnt opton to stmat th URSS s to stmat th unrstrctd modl as on rgrsson only but usng an ntracton btn a dummy varabl for gndr and all of th ncludd varabls. Not that th Cho tst rqurs that th varanc s qual for th to groups. Not also that ths tst cannot b usd for th Logt and th robt modl! Lklhood-rato tst λ Lˆ Lˆ R U hr Lˆ R ndcats th constrand lklhood som paramtrs ar assumd to b zro for ampl and Lˆ U unrstrctd lklhood. LR [ln L R ln L U d ] χ [ J] Estmat to modls on hr som varabls ar droppd from th modl and hnc assumd to b zro and on full modl. Compar th stmatd loglklhoods. J ndcats th numbr of rstrcton.. ho many coffcnts ar assumd to b zro?. Compar LR th th crtcal valu n th Ch-squar dstrbuton. Rmark: Mak sur to stmat th modls for th sam sampls. B carful so that th sampl sz dos not ncras n th rstrctd modl du to mssng valus for varabls usd n th full modl.

10 0 Modls for Dscrt Choc Bnomal choc btn to altrnatvs Multnomal choc btn mor than to altrnatvs could b unordrd or ordrd outcoms. Lnar probablty modl r Y r Y F 0 F Th Lnar probablty modl assums F and hnc y ε. roblms: ε s htroscdastc; t dpnds on. ε s thr or th probablty -F and F rspctvly. Var ε W cannot assur that prdcton from th modl ll look lk probablts; s not ncssarly n th ntrval Th margnal ffcts ar constant hch n many cass ar unralstc!. robt and Logt modls To ovrcom th problms th th lnar probablty modl t s ncssary to fnd a modl consstnt th thory; lm lm r Y r Y 0 a Usng th normal dstrbuton ll gv th probt modl; r Y φ t dt Φ hr Φ. s th standard normal dstrbuton. b Usng th logstc dstrbuton rsult n th logt modl; r Y Λ

11 hr Λ. ndcat th logstc cumulatv dstrbuton functon. Th logstc dstrbuton has havr tals compard to th normal dstrbuton. Accordngly th logt modl tnd to produc largr probablts of y 0 hn s trmly small and smallr probablts of 0 hn s larg compard to th normal dstrbuton. Th modls can b partcular dffrnt f th sampl contans vry f obsrvaton th y 0 or y and or d varaton of planatory varabls. Rmark: aramtrs stmatd from th modls ar not margnal ffcts! Intrprtng th coffcnts and stmatng margnal ffcts a Th partal drvat of th robt modl th rspct to k s Φ φ k k hr φ s th standard normal dnsty. b Th partal drvat of th Logt modl th rspct to k s Λ Λ Λ k k As you s th drvatvs vary th th valus of. It s possbl to valuat th ffct at for ampl th mans of th rgrssors. Not that anothr opton s to calculat margnal ffcts for all ndvduals n th populaton and thn avragng th rsults. Dummy varabl For bnary ndpndnt.. dummy varabls you can calculat th probablty of th outcom hn d 0 and thn hn d. Th dffrnc s th margnal ffct. r[ Y * d ] r[ Y * d 0] hr * s th mans of all othr varabls n th modl. Somtms th dffrnc s shon th a fgur to ndcat ho th margnal ffct can b dffrnt th rspct to a thrd varabl. For ampl th probablty to go on vacaton th and thout chldrn can b dffrnt for dffrnt lvls of ncom. artal ffcts on nonlnar trms In som cass a varabl s ncludd th an tra squard trm to captur a nonlnar rlaton. For ampl f ant to stmat th probablty to go th doctor mght ant to nclud ag and ag-squard as to planatory varabls.

12 Small chldrn and old popl could for ampl nd to vst th doctor mor frquntly. Calculatng a margnal ffct for ag and ag-squard sparatly dos not mak sns as nthr can vary hl th othr s hld constant. W should accordngly calculat a margnal ffct that capturs both stmatd coffcnts. If ork th th robt modl; r Y Φ α ag ag 3ducaton 4ncom Φ ag φ α ag ag 3ducaton 4ncom * ag As you s t s ncssary to calculat th margnal ffct at a spcfc ag. If th ffct ndd s nonlnar t s of cours also ntrstng to study th margnal ffct at dffrnt ags. Th scal paramtr and th rato of coffcnts and ts conomc manng Th coffcnts of th stmatd modl ndcat th ffct of obsrvd varabls rlatv to th varanc of unobsrvd factors. Th logt formula s basd on th assumpton that th unobsrvd factors ar dstrbutd trm valu th varanc π / 6 hch mans an normalzaton for th scal of utlty. Not that th scal of utlty s rrlvant for bhavor. Th dffrnc btn to trm valu varabls s dstrbutd logstc th varanc π / 3. * Only th rato / σ s stmatd and th paramtrs ar not sparatly dntfd. σ s th scal paramtr. Not that th scal paramtr drops out f th rato of to coffcnts s studd. / σ σ * * * / / / / * Wllngnss to pay valus of tm and othr margnal rats of substtuton ar not affctd by th scal paramtr. Rmark: Not that th Cho tst to compar coffcnts for to dffrnt groups basd on to dffrnt modls cannot b usd du to th scal paramtr that could b dffrnt for th dffrnt groups. Th rato of to coffcnts can hovr b compard.

13 3 Multnomal logt modl Follong McFaddn 974; Th probablty that dcson makr n chooss altrnatv s n r V n nj ε n n > V r ε < ε V nj n ε V nj nj j j Th logt probablty s nj Vn j Vnj Wth utlty spcfd to b lnar n paramtrs: V nj nj hr nj s a vctor of obsrvd varabls rlatd to altrnatv j. Th logt probablts ar n nj j n Rmark: In Grn ths s calld th condtonal logt modl. Indvdual spcfc charactrstcs that do not vary ovr th chocs fall out of th probablty. If ths charactrstcs ar to b ncludd thy hav to b ntractd th th altrnatvs. Th multnomal logt modl s dntfd by a normalzaton 0.. th coffcnts for altrnatv 0 ar all zro. 0 r Y r Y j J 0 j k J k k k for j J Th coffcnts ar accordngly stmatd th on altrnatv as comparson group.

14 4 Intrprtng th coffcnts t odds ln log log Eampl: Multnomal logt modl If a subjct r to ncras hs scnc tst scor by on pont th multnomal log-odds for lo ss rlatv to mddl ss ould b pctd to dcras by 0.04 unt hl holdng all othr varabls n th modl constant. An ampl th thr varabls ould b; Y Y If ncras th valu of th on unt th odds ould b; Y Y Th odds-rato s; / 0 / Y Y Y Y Rlatv Rsk Rato - Th rlatv rsk ratos can b obtand by ponntatng th multnomal logt coffcnts cof.. Standard ntrprtaton of th rlatv rsk ratos s for a unt chang n th prdctor varabl th rlatv rsk rato of outcom m rlatv to th rfrnt group s pctd to chang by a factor of th rspctv paramtr stmat gvn th varabls n th modl ar hld constant. Th ponntal of a coffcnt can b ntrprtd as th ffct on th odds that a unts chang n th varabl ould hav. It s mportant to not that a chang n odds dos not man th sam chang n probablty. It s accordngly ntrstng to calculat prdctd probablts and margnal ffcts. A RRR abov blo ndcats that... has hghr lor probablty...compard to rfrnt group. Eampl: Condtonal logt modl...th odds of choosng a partcular commutng rout th an addtonal mnut n alkng tm ar p tms as hgh as choosng anothr

15 5 rout. Smlarly th odds of choosng a partcular rout th an addtonal mnut n atng tm ar p tms as hgh as usng anothr rout. An ncras n on cnt of n-vhcl cost ould mak th odds of choosng an altrnatv dcras by a factor of p But ths s th ffct of on cnt only. An ncras n 5 cnts of n-vhcl cost ould mak th odds of choosng an altrnatv dcras by a factor of p * Ths ampl cam from prtngcoffcntscondtonallogt&sourcb&otsd70zjzo_&sgbaep FSM99QnggTqcbk4R0DQ&hlca&sa&obook_rsult&rsnum0&ctrs ult#a67m hr you also can fnd ampl on ho to calculat prdctd probablts and margnal ffcts n th condtonal logt modl. roprts of th logt modl Th probablty for a choc s btn 0 and. Th sum of th probablts for th dffrnt chocs s. Th logt modl s consstnt th utlty mamzaton. Th rlaton of th logt probablty to rprsnt utlty s S-shapd hch s rlvant for polcy. A small ncras n utlty for an opton that s far ors or bttr than anothr opton ll hav a small ffct on th probablty to choos th opton. If th probablty s clos to 0.5 vn small changs n polcy can hav mportant ffcts. 3 Logt can captur systmatc varaton n tast. Th rror trms ar assumd to b ndpndnt and any unobsrvd part of th utlty cannot b corrlatd th unobsrvd utlty for anothr choc. Thus th modl has to b ll spcfd so that th logt modl ll b approprat. 4 Th substtuton btn th altrnatvs s proportonal. An ncrasd probablty for on altrnatv ll dcras th probablty for th othr altrnatvs proportonally. Ths follos from th proprty IIA Indpndnc from rrlvant altrnatvs. Th probablty rato of to altrnatvs s ndpndnt of th othr altrnatvs. IIA has to b fulflld othrs s th logt modl a msspcfcaton. Eampl; rd buss blu buss! Th advantag of IIA s that mportant smplfcatons can b don. Ho to tst IIA Hausman & McFaddn 984 f a choc st truly s rrlvant omttng t from th modl ll not chang th paramtrs systmatcally. ˆ ˆ χ [ Vˆ Vˆ ] s f s f ˆ ˆ s f hr s ndcat stmats from rstrctd subst and f stmats from th full st of chocs. Ch-squar th k dgrs of frdom knumbr of coffcnts

16 6 Consumr surplus Th consumr surplus s th utlty n Euro Dollar or that a prson rcvs n a choc stuaton. CSn / α n ma j Unj j hr α n s th margnal utlty of ncom of prson n: du ncom of prson n. U nj s not obsrvd but V nj can b usd. E CSn / α n E[ma j Vnj ε nj j] n / dy n α n th n Y th If ε nj s d ndpndntly dntcally dstrbutd trm valu and utlty s lnar n ncom.. α n dos not chang th ncom thn E CS n / α nln J j V nj C hr C s an unknon constant rprsntng absolut lvl of utlty hch cannot b masurd but s rrlvant from polcy prspctv. A chang n consumr surplus s E CSn / α n ln J Vnj j ln J Vnj j hr 0 and rfrs to bfor and aftr th chang. Th numbr of altrnatvs can chang as ll as attrbuts of th altrnatvs for ampl th ntroducton of a subay or ncrasd prc for bus. α can b calculatd from prc or cost varabls n n hch th ngatv of ts coffcnt s α n by dfnton. Important: Rmmbr th assumpton that α dos not chang th ncom. Goodnss-of-ft Lklhood rato nd n LRI ln L ln L 0 hr 0 ln L s th log lklhood from a modl th a constant only. LRI s boundd by 0 and. Not that th valus btn 0 and ar vry dffcult to ntrprt and you cannot compar LRI for to dffrnt sampls or th dffrnt sts of altrnatvs.

17 7 Th prcnt corrctly prdctd Th prcnt corrctly prdctd s somtms usd to masur goodnss-of-ft but ths should b avodd accordng to Tran 003. Th prcnt corrctly prdctd s th prcntag of th sampld dcson makrs for hch th altrnatv th hghst probablty basd on th stmatd modl also r th altrnatv chosn. Rmmbr that th rsarchr dos not hav prfct nformaton. In fact th nformaton s only nough to stat th probablty for an altrnatv not that th altrnatv th hghst probablty ould b chosn ach tm. Lmts of Logt and altrnatvs Th logt modl cannot handl random tast varaton and IIA proprty gvs lmtd substtuton pattrns. An altrnatv to us s th probt modl. A nstd logt could somtms b suffcnt to avod problms th th IIA assumpton. A nstd logt consst of to or mor lvls of dcsons or nsts. IIA s rqurd thn ach nst but IIA s not ncssary for altrnatvs n dffrnt nsts. A nstd logt can b stmatd squntally startng th th lor nsts. Th uppr modl s stmatd th nclusv valus ntrng as planatory varabls. Th nclusv valu s th log of th dnomnator of th lor modl. Th squntal stmaton crats to problms; th standard rrors n th uppr modl ar basd donard and som paramtrs can appar n svral sub modls and hnc tak dffrnt valus. Smultanous stmaton th mamum lklhood s mor ffcnt. Softar: EVs can stmat th Lnar probablty modl LS robt and Logt. You can also stmat condtonal logt multnomal logt and nstd logt by adaptng sampl programs provdd th EVs. EVs6/Eampl Fls/Sampl rograms/logl/. Multnomal Logt s pland undr Eampls n th usr s gud. You can us th data EVs6/Eampl Fls/EV6 Manual Data/Chaptr 30 /bnary to try bnary choc modls. Ths s th Spctor and Mazzo 980 data usd as an ampl n Grn. At you can fnd many data sts usd as ampls n Grn. Burntt 997 can b usd to study bnary choc modls. Grn and Hnshr 997 s usd as an ampl of a condtonal logt modl. In EVs you can us th command forcast to calculat prdctd probablts or th nd. In th usr s gud you can fnd nformaton on ho to to us dffrnt functons such as th standard normal dnsty logstc cumulatv dstrbuton functon tc. S Statstcal Dstrbuton Functons n th usr s

18 8 gud. You can for ampl us th command Gnrat Srs by Equaton to calculat margnal ffcts. In som cass you may nd to ork th matrcs to tst hypothss. Th chaptrs Assgnng Matr Valus and Matr Eprsson n th usr s gud can b hlpful. Th chaptr Basc Mathmatcal Functons s also usful f you for ampl ant to tak th ponntal or th logarthm.

19 9 Count data modls Data that taks th valus 0 s count data. Eampls ar numbr of patnts accdnts mdcal vsts durng a spcfc tm fram. A multpl rgrsson th a dpndnt varabl of ths knd ould n prncpl b possbl. Som othr mthods that tak nto account th dscrt natur of th data ar avalabl that could ork bttr than th ordnary last squar. Th osson rgrsson modl y s dran from a osson dstrbuton th paramtr λ hch s rlatd to rgrssors. y r Y y y 0... y! λ λ A log-lnar modl for λ s ln λ. Th pctd numbr of vnts pr prod s E y Var[ y ] λ So partal drvatv s E y λ Mamum lklhood can b usd to stmat th modl s Grn for th loglklhood functon tc. Wald tsts and Lklhood rato tst can b computd as pland bfor. Intrprtng th coffcnts Eampl For a on unt chang n th prdctor varabl th dffrnc n th logs of pctd counts s pctd to chang by th rspctv rgrsson coffcnt gvn th othr prdctor varabls n th modl ar hld constant. Such an ntrprtaton s not vry convnnt and t could b orth to transform th coffcnts to Incdnc Rat Ratos. IRR s obtand by ponntatng th posson rgrsson coffcnt. A coffcnt of ould man an IRR of and hnc th follong ntrprtaton;

20 0 Eampls If a studnt r to ncras hs mathnc tst scor by on pont hs rat rato for daysabs ould b pctd to dcras by a factor of hl holdng all othr varabls n th modl constant. 3 Fmals compard to mals hl holdng th othr varabl constant n th modl ar pctd to hav a rat.493 tms gratr for daysabs. 4 th coffcnt for lo znc status s Its ant-log s.9. W thrfor conclud that thos th lo znc status hav.9 tm as many pnumonas pr yar as thos th normal znc status. Spcfcaton tsts Not that th osson modl assums that th varanc of y quals ts man. If ths assumpton s not fulflld th osson modl s not a corrct spcfcaton. Svral tsts of ovrdsprson hav bn suggstd n th ltratur. Rgrsson basd tsts Camron and Trvd 990 suggst a rgrsson basd tst. In Grn th tst s pland as calculatng z y ˆ λ y ˆ λ and thn rgrssng t on thr a constant or λˆ thout a constant trm. A t-tst of hthr th coffcnt s sgnfcantly dffrnt from zro tsts th assumpton of ovrdsprson. A smlar tst s pland n th EV hlp-fl. Thn y ˆ λ y s rgrssd on ˆ λ. Lagrang multplr statstc Anothr opton s a Lagrang multplr statstc. Assumng that th altrnatv dstrbuton s th ngatv bnomal th statstc s LM ny ˆ λ ˆ λ It s only ncssary to stmat th osson modl to b abl to comput th statstc. Ths LM statstc s Ch-squard th on dgr of frdom. Ngatv bnomal rgrsson modl An altrnatv to th osson modl n cas of ovrdsprson s th ngatv bnomal rgrsson modl.

21 Th ngatv bnomal modl follos as a gnralzaton of th osson modl to nclud ndvdual unobsrvd ffcts. Th Ngbn II modl r th varanc s a quadratc functon of th man has condtonal man λ and condtonal varanc λ aλ hr a s a paramtr to stmat. A tst of th osson modl s to tst hthr a s sgnfcantly dffrnt from zro. Rmark: Both ths modls could b nconsstnt f a larg proporton of th obsrvatons ar zro. An tnson s zro-nflatd osson/ngatv bnomal modls to dal th ths. Hurdl modls can also b usd. Softar: EVs can stmat th osson and Ngatv Bnomal modls. You can also stmat zro nflatd osson modl by adaptng a sampl program provdd th EVs. EVs6/Eampl Fls/Sampl rograms/logl/zposs. At you can fnd McCullagh and Nldr 983 data on shp accdnts. Data on numbr of drogatory rports ar ncludd n Grn 99 hr th count varabl oftn s zro.

22 Cnsord rgrsson modls Lt us say that th dpndnt varabl s 0 for an mportant part of th sampl and contnuous for th rst. Thr s a qualtatv dffrnc btn th zro lmt and th non-lmt contnuous obsrvatons that th convntonal rgrsson modls dos not tak nto account. Th dstrbuton that appls to th sampl s a mtur of dscrt and contnuous dstrbutons. Tobt modl y * y 0 y y * f ε f y * 0 y * > 0 Th purpos of th study has to gud hch condtonal man functon that s rlvant. Choosng hch margnal ffcts to calculat ar also mportant. a Th man of th latnt varabl E [ y *] Eampl: rdct th nd for a n faclty a stadum? Th margnal ffct s E [ y*] b Th cnsord man E[ y ] Φ σλ σ φ / σ hr λ Φ / σ Eampl: prdct th numbr of tckts sold for an upcomng vnt. Th margnal ffct s E [ y ] r[ a < y* b] < hch undr cnsorng at zro and normally dstrbutd dsturbancs s

23 3 E[ y ] Φ σ OLS stmats ar usually smallr n absolut valu compard to a Tobt modl stmatd th mamum lklhood. In fact dvdng th OLS stmats by th proporton of nonlmt obsrvaton s oftn a good appromaton of th mamum lklhood stmats. McDonald and Mofftt 980 suggst a dcomposton of E[ y ]/. E[ y ] r[ y E[ y > 0] y > 0] E[ y y r[ y > 0] > 0] A chang n has to ffcts; It affcts th condtonal man n th postv part. It affcts th probablty that th obsrvaton ll fall n th postv part. Spcfcaton tst A draback th th Tobt modl s that a varabl that ncrass th probablty of a nonlmt obsrvaton also by assumpton ncras th man of th varabl. Eampl: Oldr buldngs may hav hghr probablty to hav frs but bcaus of th gratr valu of nr buldngs ncur smallr losss hn thy do. In ths cass t s possbl to stmat a mor gnral modl. Th modl s calld Hckman s to-stp stmator Hckt tc.. Dcson quaton: r[ y * > 0] Φ γ r[ y * 0] Φ γ z f z 0 y * > 0 f y * 0. Rgrsson quaton for th nonlmt obsrvatons: E[ y z ] σλ ε hr ˆ φ ˆ γ λ s th nvrs Mll s rato. Φ ˆ γ Estmat a probt modl and thn a truncatd rgrsson modl. Th scond stp ncluds λˆ as an addtonal planatory varabl. If γ / σ ths s th sam as th tobt modl. A lklhood rato tst can b usd to tst th rstrcton st by th tobt modl.

24 4 λ [log L T log L log LTR ] hr L T lklhood for th tobt modl sam coffcnts by assumpton L lklhood for th probt modl ft sparatly L lklhood for th truncatd rgrsson modl ft sparatly TR Th modl by Hckman can b stmatd th full nformaton mamum lklhood FIML. Th to stp procdur dos othrs produc nconsstnt standard rrors n th scond stp. That s th rgrsson dos not corrct for that nvrs Mll s rato s stmatd and th standard rrors ar to small. Hckman has shod that th covaranc matr can b corrctd. S Grn. Hckman s to stp stmator can b usd for mor gnral slcton problms.. not only to solv a problm th a cnsord dpndnt varabl. Softar: EVs can stmat th Tobt modl. You can also stmat Hckman s to-stp stmator by adaptng a sampl program provdd th EVs. EVs6/Eampl Fls/Sampl rograms/logl/hckman. You can us th data EVs6/Eampl Fls/EV6 Manual Data/Chaptr 30 /tobt_far to try a cnsord rgrsson modls. Ths data Far 978 s also usd as an ampl n Grn.