ESTIMATION METHODS FOR DURATION MODELS. Brian P. McCall. University of Minnesota. and. John J. McCall. University of California, Los Angeles

Transcription

1 Draf: Commens Welcome ESTIMATION METHODS FOR DURATION MODELS by Bran P. McCall Unversy of Mnnesoa and John J. McCall Unversy of Calforna, Los Angeles and Unversy of Calforna, Sana Barbara JANUARY Bran P. McCall and John J. McCall. All rghs reserved.

2 TABLE OF CONTENTS 1. Inroducon 2. Hazard Funcons 3. Counng Processes and Marngales 4. Paramerc Mehods for Connuous-me Daa wh Covaraes 5. The Cox Regresson Model 6. Dscree-Tme Duraon Daa 7. Mul-Spell Dscree-Tme Models 8. Compeng Rsks Models 9. Dscree-me Lfe Hsory Models 1. Specfcaon Dagnoscs for Duraon Models 2

3 Absrac Ths paper s a seleced overvew of economerc mehods for duraon models and wll appear n he forhcomng book The Economcs of Search by he auhors. The focus of he paper s on marngale mehods for connuous me daa and general mehods for he analyss of dscreeme daa ncludng mul-spell models and general lfe-hsory models. 3

4 1. Inroducon Many emprcal ess of search heory employ duraon daa (see Devne and Kefer, 1991). For example, he sandard model of job search n a saonary envronmen model mples ha unemploymen duraons have an exponenal dsrbuon. In hs chaper we develop some sascal ools used o analyze duraon daa (for more horough reamen see Lancaser, 199 or Klen and Moeschberger, 1997). Duraon analyss s also referred o as survvor analyss, where he duraon of neres s he survval me of a subjec (e.g. person or machne). Much of he recen sascal analyses of duraon daa focus on he hazard funcon. The hazard funcon s relaed o he probably of exng he nal sae whn a shor nerval, condonal on havng survved up o he sarng me of he nerval. In many applcaons, hazard funcons are condonal on a se of covaraes. An mporan feaure of he hazard funcon s ha can be made o depend on covaraes ha change over me. In secon 2 of hs chaper we shall revew he basc defnon of a hazard funcon and s relaon o he probably densy and cumulave dsrbuon funcon. Secon 3 hen gves a bref descrpon of counng process heory and marngales. Ths framework s useful for analyzng duraon daa ncludng mulple spell duraon daa (See Andersen and Borgan, 1985, Arjas, 1989, Flemng and Harrngon, 1991, and Anderson, Borgan, Gll and Kedng, 1992 for more horough dscussons). Paramerc esmaon mehods for connuous me duraon models wh covaraes are presened n Secon 4 whle he sem-paramerc Cox regresson model s dscussed n Secon 5. Secon 6 presens esmaon echnques for grouped or dscree-me duraon daa. In many suaons we are neresed n sudyng an ndvdual s movemen hrough several labor marke saes over me. Afer exendng dscree-me mehods o a mul-spell 4

5 framework n secon 7 and d compeng rsks models n secon 8, secon 9 presens he general esmaon mehods for dscree-me lfe hsory daa. The chaper concludes wh a bref dscusson of some specfcaon dagnosc mehods ha can be derved from he counng process approach. 2. Hazard Funcons Ths secon presens a bref overvew of hazard funcons. Inally we wll focus on models whou covaraes. Laer secons of he chaper wll hen nroduce boh me-consan and me-varyng covaraes no he hazard framework. Le T? represen a posve random duraon varable, whch has some probably dsrbuon n he populaon; denoes a parcular value of T. In survval analyss, T s he lengh of me ha an ndvdual lves. In many economc applcaons T s he duraon of an unemploymen spell or he duraon of job enure. The cumulave dsrbuon funcon (c.d.f.) of T s defned as F( ) P ( T ),. The survvor funcon s defned as S( ) 1 F( ) P ( T ). Thus, S() represens he probably ha an even has no occurred by me or ha he ndvdual has survved pas. Throughou hs secon we asume ha T s connuous and denoe he df probably densy funcon (p.d.f.) of T by f ( ) ( ). For d >, P(?T<+ T?) s he probably of leavng he nal sae n he nerval [,+ ) gven survval unl me. The 5

6 hazard funcon for T s defned as P( T T ) ( ) lm. (1) Thus, he hazard funcon s he nsananeous rae of leavng per un me (he escape rae) From equaon (1) folows ha, for smal, P( T T ) ( ). The hazard can hen be used o approxmae a condonal probably n much he same way ha he hegh of he p.d.f. of T can be used o approxmae an uncondonal probably. We can express he hazard funcon n erms of he densy and c.d.f. very smply. Frs, wre ( ) ( ) ( ) ( ) P T F F P T T. P( T ) 1 F ( ) When he c.d.f. s dfferenable, we can ake he lm of he rgh hand sde, dvded by approaches zero from above:, as F( ) F( ) 1 f ( ) f ( ) ( ) lm h 1 F ( ) 1 F ( ) S( ) Because he dervave of S() s -f(), we have d log S( ) ( ) (2) d and, usng F() =, we can negrae (2) o ge 6

7 F( ) 1 exp ( s) ds, (3) Sragh forward dfferenaon of (3) gves he p.d.f. of T n erms of he hazard funcon: f ( ) ( )exp ( s) ds. Therefore, all probables can by compued usng he hazard funcon. For example, for all pons a 1 < a 2, 1 F ( a ) P Ta Ta s ds a2 2 ( 2 1) exp ( ) 1 F ( a1 ) a and a2 P( a1t a 2 Ta 1) 1 exp ( s) ds. a In many emprcal applcaons he shape of he hazard funcon s of prmary neres. In he smples case, he hazard funcon s consan: ( ), for all. In hs case he ex process s memoryless: he probably of ex n he nex nerval of me does no depend on how much me has been spen n he curren sae. The sandard connuous-me model of saonary job search wh a consan offer arrval raeand wage dsrbuon G mples a consan re-employmen hazard rae r ( ) (1 G ( w )) 7

8 For a consan hazard funcon, equaon (3) mples ha F( ) 1 exp( ) whch s he c.d.f. of he exponenal dsrbuon. When he hazard funcon s no consan we say ha exhbs duraon dependence. Assumng ha?(?) s dfferenable, he hazard exhbs posve duraon dependence a me f d?()/d > and negave duraon dependence a me f d?()/d <. If d?()/d > for all we say he process exhbs posve duraon dependence. Wh posve duraon dependence, he probably of exng he nal sae ncreases he longer one s n he nal sae. Example 1: Webull dsrbuon. A popular paramerc dsrbuon used n emprcal analyss s he Webull dsrbuon. The random varable T s sad o have a Webull dsrbuon= f s c.d.f. s gven by F( ) 1 exp( ) where? and?are non-negave parameers. The p.d.f. s gven by ( ) f ( ) S( ) 1 and he hazard funcon s 1 f ( ) exp( ). When?= 1, he Webull dsrbuon reduces o he exponenal wh?=?. If?> 1, he hazard s monooncally ncreasng, so he hazard everywhere exhbs posve duraon dependence; for? < 1, he hazard s monooncally decreasng. Graphs of he Webull hazard funcon for dfferen values ofare presened n 8

9 Fgure1. Hazard Fgure 1 Webull Hazard Funcon Tme () Alpha=.8 Alpha=1. Alpha=1.2 Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 9

10 Example 2: Log-logsc dsrbuon: The random varable T has a log-logsc dsrbuon f s c.d.f. s gven by 1 F( ) 1 1 and s hazard funcon s gven by 1 f ( ) ( ) (4) S( ) 1 for> and>. To examne wheher he hazard funcon exhbs posve or negave duraon dependence n some ranges we dfferenae (4) wh respec o : ( 1) ( ) ( 1) ( 1) 2 2 For 1, ( ) for all and for> 1 for ( ) 1/ 1 and ( ) f 1/ 1. Graphs of he log-logsc hazard funcon for dfferen values ofare presened n Fgure 2. 1

11 Hazard Fgure 2 Log-Logsc Hazard Funcon Tme () Alpha=.8 Alpha=1. Alpha=1.2 Alpha=1.5 Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 11

12 3. Counng Processes and Marngales The heory of counng processes and her accompanyng marngales s useful n developng esmaon procedures for duraon daa. Here we wll be conen wh gvng a cursory overvew of he counng process approach. More dealed dscussons can be found n Bremaud (1981), Flemng and Harrngon (1991), and Anderson e al (1992). Recall ha a counng process s a process ha couns he number of evens ha occur a random mes. Le T n, n=,1,2, be a sequence of posve random varables such ha 1) T = 2) T n < T n+1 P a.s. 3) lmt n n P a.s. Condon 3) s needed o nsure ha he counng proces doesn blow up o nfny n fne me. The counng process N() s hen defned by N( ) I Tn. n1 Thus, N() =, N( ) almos surely, and he sample pahs of N() are pecewse consan, rgh connuous and non-decreasng wh jumps of sze 1. A flraon or hsory denoed by { F, } s a sequence of sgma algebras ndexed by ha measure he accumulaed nformaon up o me. As me progresses nformaon ncreases and so F s F for s <. Tha s AF s A F. The hsory jus before me s denoed by F and s he sgma algebra generaed by all ses n F s for all s<. Defnon: A process X() s adaped o { F, } f X() s F measurable for all. 12

13 Before connung we shall revew some resuls from marngale heory. Defnon: A rgh-connuous sochasc process X() wh lef-hand lms s sad o be a marngale wh respec o he hsory F f ) X() s adaped o F ) X() s negrable ( E( X ( ) ) ) for all ) For all s, E(X() F s )=X(s) P-a.s. X() s called a submarngale f we replace ) by a) For all s, E(X() F ) X(s) P-a.s. and X() s called a supermarngale f we replace ) by b) For all s, s E(X() F ) X(s) P-a.s. s The nex heorem s mporan for dervng marngales assocaed wh sochasc processes. Doob-Meyer Decomposon Theorem Le X() be a rgh-connuous non-negave submarngale wh respec o hsory F. Then here exss a rgh-connuous marngale M() and an ncreasng rgh-connuous predcable process A() such ha E( A( )) X ( ) M ( ) A ( ) P a.s. and Corollary: Le N() be a counng proces wh assocaed nensy proces (). Then M ( ) N( ) ( ) ds s a F marngale. 13

14 Defnon: A process X() s sad o be predcable wh respec o{ F, } f X() s measurable for all. Anoher useful heorem s he followng: F - Theorem: If V() s a predcable process such ha E[ V ( s) ( s) ds P a.s. hen V ( s) dm ( s) V ( s) dn( s) V ( s) ( s) ds (5) s a F - marngale. Le C be a censorng me and defne Y() o be he sochasc process Y ( ) I { C }. Thus, Y() equals one up unl and ncludng he me a whch an observaon s censored and equals zero, hereafer. We assume ha hs sochasc process s measurable wh respec o Furher defne Z() o be he sochasc process Z( ) I { T1 }. Usng hs heorem and defnng wh V() = Z()Y() can hen be shown ha F. Y ( s) Z( s) dn( s) Y ( s) Z( s) ( s) d s a F - marngale. Thus, ( ) ( ) ( ) ( ) ( ) (6) E Y s Z s dn s F E Y s s ds F for all >. Ths resul enables us o derve an esmaor for he negraed hazard funcon() : ( ) ( s) ds. From (6) we have ha 14

15 E Y ( )( Z ( ) dn ( ) F Y ( ) ( ) E Z ( ) F d. (7) Suppose we have a random sample of sze N and le Y (), Z (), and N () denoe he sample pahs of Y(), Z(), and N() for he h ndvdual, =1,,N. Then appealng o he law of large numbers gves: N Y ( ) Z ( ) dn ( ) ( ) d Y ( ) Z ( ) N N 1 1 N. Hence, N Y ( ) Z ( ) dn ( ) Y ( ) Z ( ) dn ( ) 1 1 ( ) d N 1 Y ( ) Z ( ) N R( ) where R() s he number a rsk se a me and ncludes al hose who have no been censored or have faled by : N. 1 R( ) Y ( ) Z ( ) Thus, we have Y ( s) Z ( s) J ( s) N ˆ * ˆ( ) ( ) s ds dn ( s ) dn ( s ) 1 R( s) R( s) (8) 15

16 where and ( ) 1 N ( ) ( ) ( ) ( ) * dn s Z s Y s dn s 1 J s f R(s) > and J(s) = f R(s) = wh he convenon ha /=1. The esmaor ˆ( ) n (8) s referred o as he Nelson-Aalen esmaor of he negraed hazard funcon and s a sep funcon ha s consan a all mes excep falure mes and jumps up by 1/R() a me when a falure occurs a me. The negraed hazard funcon of he sngle duraon varable T 1 equals * * ( ) ( s) ds Z( s) ( s) ds. Noe hs negraed hazard funcon s sochasc because of Z(). Thus, N J ( s) Y ( s) Z ( s) dn ( s) J ( s) Y ( s) Z ( s) d( s) 1 1 N N * * J ( s) dn ( ) ( ) ( ) ( ) s J s Y s d s 1 1 N 1 J s Y s dm * ( ) ( ) ( s) N (9) So, J ( s) Y ( s) Y ( s) J ( s) Z ( s) Y ( s) J ( s) N N N * * ˆ( ) ( ) ( ( ) ( )) d ( ) s d N s s dm s 1 R( s) 1 R( s) 1 R( s) (1) Whch shows ha for all, ˆ( ) s an unbased esmaor of ( s) d( s) where ( s) = P(J(s) = 1), snce 16

17 N N * 1 1 J ( s) Y ( s) J ( s) J ( s) d( s) Y ( s) Z ( s) d( s) R ( s) d ( s) R( s) R( s) R( s) J ( s) d( s) Furhermore, as N, P( J ( s) 1) 1 a.s. and, hence, ˆ( ) s a conssen esmaor of ( ). The nex wo corollares are useful applcaons of he Doob-Meyer Decomposon Theorem: Corollary 1: Le M() be a rgh-connuous marngale wh respec o F and assume ha E M 2 ( ) for all. Then here exss a unque rgh connuous predcable process called he predcable quadrac varaon of M() and denoed by <M,M>() such ha <M,M>()=, E M, M ( ) for all and M 2 ( ) M, M ( ) s a rgh-connuous marngale. Corollary 2: Le M () be a rgh-connuous marngales wh respec o F and assume ha E M 2 ( ) for all, =1,2. Then here exss a unque rgh connuous predcable process called he predcable covaraon process of M 1 () and M 2 () and denoed by <M 1,M 2 >() such ha <M 1,M 2 >()=, E M1, M 2 ( ) for all and M1( ) M 2( ) M1, M 2 ( ) s a rgh- 17

18 connuous marngale and. M1, M 2 ( ) s he dfference of wo ncreasng rgh-connuous predcable processes. Fnally, we have Theorem: Le V 1 () and V 2 () be bounded predcable processes and M 1 () and M 2 () marngales wh respec o F such ha M 2 ( ), =1,2. Then V 1( ) dm 1( ) V 2( ) dm 2( ) V 1( ) V 2( ) d M 1, M 2 ( ) (11) s a marngale. Fnally we presen a heorem ha relaes he compensaor of M 2 () o he compensaor of M(). Theorem: Le N() be a counng process wh compensaor A(). Assume ha almos all sample pahs of A() are connuous and ha E M 2 ( ). Then <M,M>()=A(). Or n oher words, M 2 () A s a marngale. Skech of Proof: Inegraon by pars shows ha 2 2 M M s dm s M s s ( ) 2 ( ) ( ) ( ). Now, snce M() = N() A() we have 18

19 M ( s) N( s) A( s). Subsung no above yelds 2 M M s dm s N s A s s ( ) 2 ( ) ( ) ( ) ( ) 2 s 2 2 M ( s ) dm ( s) N ( s) 2 M ( s ) dm ( s) N ( ) where he second equaly follows from he assumpon ha A() has no jumps P-a.s. and he hrd equaly follows from he fac ha N() s a counng process and so N( ) N ( s). s Snce M()= N()-A() we hen have 2 M A M s dm s M ( ) ( ) 2 ( ) ( ) ( ). Now M(s - ) s a predcable process, so boh erms on he rgh hand sde are marngales whch shows ha A() s he compensaor of M 2 (). Q.E.D. Nex, we have Theorem: If N (), =1,,N are ndependen counng proceses wh compensaors, A (), defned by ( ) s ds and H () are F predcable funcons hen 19

20 N 1 M ( ) H ( s) d N ( ) A ( ) 1) E(M()= for all s an F marngale wh 2) N 2 var( ( )) ( ) ( ) 1. M E H s s ds Applyng hs heorem o he Nelson-Aalen esmaor we have ˆ * * Var ( ) J ( s) d( s) Var dm ( s) J ( s) R( s) 2 2 J ( s) * * J ( s) E, ( ) ( ) dm M E R s d s R( s) R ( s) J ( s) E ( s) d( s) R ( s) hs can be esmaed by J ( s) R ( s) * dn ( s) 2. For large n, ˆ 2 nj ( s) 1 R ( s) ( s) * * * lm ne ( ) J ( s) d( s) lm E d( s) d( s) n n Fnally, noe ha 2

21 * J ( s) * n ˆ( ) J ( s ) d ( s ) n dm ( s ) R( s) n n 1 nj ( s) * 1 nj ( s) * d M ( ) ( ) s dm ( ) s n R s 1 n 1 R( s) whch from he marngale cenral lm heorem leads o he resul ha he Nelson-Aalen esmaor s asympocally normally dsrbued. Summarzng our resuls for he Nelson-Aalen esmaor J ( s) * ˆ( ) dn ( s ) we have: R( s) Theorem: 1) ˆ( ) s an unbased esmaor of J s d 2) ˆ( ) s a conssen esmaor of ( ). 3) n * ˆ( ) ( ) ( ) ( s ). s asympocally normally dsrbued wh mean and varance 1 s ( ) * d ( s). ) Kaplan Meer Esmaor of he Survvor Funcon The Nelson Aalen esmaor can be used o derve an esmaor for he survvor funcon S(). Noe ha snce 21

22 * df( ) d( ) 1 F ( ) we have * * S( ) 1 1 F ( s ) d( s) 1 S( s ) d( s) So, we can hnk of an esmaor of S() has beng defned recursvely usng Sˆ ( ) 1 Sˆ ( s ) dˆ( s) (12) where ˆ( ) s he Nelson-Aalen esmaor of he negraed hazard funcon. Subsung he defnon of he Nelson-Aalen esmaor no (12) yelds: 1 * ˆ ˆ ( ) ( ) S dn J dn ds( ) S ( ) R ( ) R( ) * ˆ( ) f ( ) 1 * f dn ( ) Snce S ˆ() 1 we hen have dn * ( s) 1 Sˆ( ) 1 1 s R( s) ( ) R. (13) 22

23 Before urnng o models wh covaraes, we presen an example usng joblessness spell daa from he February 1996 Curen Populaon Survey s Dsplaced Worker Supplemen (CPS- DWS). In he CPS-DWS workers who have been dsplaced from a job n he prevous hree years are asked how many weeks ook before hey were re-employed. Joblessness duraon daa are rgh censored f he spell was ongong a he me of he survey. For convenence we also censor all spells afer 1 weeks. The Nelson-Aalen esmae of he negraed or cumulave hazard funcon s presened n Fgure 3. Snce he negraed hazard s dsconnuous s no possble o drecly esmae he baselne hazard. However, applyng kernel smoohng echnques an esmae can be derved. Ths s presened n Fgure 4. 23

24 Fgure 3 Nelson-Aalen Cumulave Re-employmen Hazard Esmae Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 24

25 Fgure 4 Smoohed Re-employmen Hazard Esmae Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 25

26 4. Paramerc Mehods for Connuous-me Daa wh Covaraes ) Tme-Consan Covaraes Usually n economcs we are neresed n hazard funcons condonal on a se of covaraes or regressors. When hese do no change over me, hen we smply defne he hazard condonal on he covaraes. Thus he condonal hazard s P( T T, x) ( ; x ) lm (14) where x s a vecor of explanaory varables. All of he formulas nroduced n secon 2 above connue o hold provded he c.d.f. and densy are defned condonal on x. Ofen we are neresed n he paral effecs of he x j on?(;x), whch are defned as he paral dervaves for connuous x j and dfferences for dscree x j. Whle he duraons defned by (14) refer o some nernal me unl he occurence of an even, he mpac of calendar me can be modeled by ncorporang suable covaraes no x. An especally mporan class of models wh me-consan regressors consss of he proporonal hazards model. A proporonal hazards model can be wren as ( ; x) ( x) ( ) where?(?) > s a non-negave funcon of x and? () > s called he baselne hazard. The baselne hazard s common o all ndvduals n he populaon; ndvdual hazards dffer proporonaely based on a funcon?(x) of observed covaraes. Typcally,?(?) s 26

27 parameerzed as ( x) exp( xβ ) where?s a vecor of parameers. ) Tme-Varyng Covaraes Sudyng hazard funcons s more complcaed when we wsh o model he effecs of mevaryng covaraes on he hazard funcon. For one hng makes no sense o specfy he dsrbuon of he duraon T condonal on he covaraes a only one me perod. Neverheless, we can sll defne he approprae condonal probables ha lead o a condonal hazard funcon. Le x() denoe he vecor of regressors a me. For?, le X(),? denoe he covarae pah hrough me : X()={x(s):?s?}. We defne he condonal hazard funcon a me by P( T T, X( )) ( ; X ( )) lm. (15) The proporonal hazard form s commonly used when covaraes are me-varyng: ( : X( )) ( x ( )) ( ). Usually() = exp[x()?]. Below, we shall focus on echnques prmarly for flow samplng. Wh flow samplng, he sample consss of ndvduals who ener he sae a some pon durng he nerval [, ] and we record he lengh of me each ndvdual s n he nal sae. Flow daa are usually subjec o 27

28 rgh censorng. Tha s, afer a ceran amoun of me ( ), we sop followng he ndvduals n he sample, whch we mus do n order o analyze he daa. For ndvduals who have compleed her spells n her nal sae we observe he exac duraon. Bu for hose sll n he nal sae, we only know ha he duraon lased a leas as long as he rackng perod. ) Maxmum Lkelhood Esmaon wh Censored Daa For a random draw from he populaon, le e? [, ] denoe he me a whch ndvduals eners he nal sae (he sarng me), le * denoe he lengh of me n he nal sae (he duraon), and le x denoe he vecor of observed covaraes. We assume ha connuous condonal densy f( x ;?),?, where? s he vecor of unknown parameers. Whou rgh censorng we would observe (e,, x ) and esmaon would proceed by * has a condonal maxmum lkelhood esmaon. To accoun for rgh censorng we assume ha he observed duraon s s obaned as c * mn(, ) where c s he censorng me for ndvdual. In some cases, c s consan across. For example f you were o rack all ndvduals whose duraon sars a he same calendar me and rack hem for up o 2 years hen he common censorng me would be 14 weeks. We assume ha, condonal on he covaraes, he rue duraon dsrbuon s ndependen of he sarng pon e and he censorng me c. H x e c H x (16) * * (,, ) ( ) where H(??) denoes he condonal dsrbuon. Under assumpon (16), he dsrbuon of * gven (x,e,c ) does no depend on (e,c ). Therefore, f he duraon s no censored, he densy 28

29 of = gven (x,e,c ) s smply f( x ;?). The probably ha s censored s P c F c * ( x ) 1 ( x ; θ) Le d be a complee spell ndcaor (d = 1 of uncensored, d = f censored), he condonal lkelhood for observaon can be wren as d f ( x ; θ) [1 F ( x; θ)] (1 d ) For a random sample of sze N he maxmum lkelhood esmaor of?s obaned by maxmzng N { d log[ f ( x ; θ)] (1 d ) log[1 F ( x ; θ)]} 1 For example, he Webull dsrbuon wh covaraes has condonal densy f 1 ( ; ) exp( ) x θ xβ exp[ exp( xβ ) ] where x conans uny as s frs elemen for all. v) Unobserved Heerogeney One way o oban more general duraon models s o nroduce unobserved heerogeney no farly smple duraon models. In addon, we somemes wan o es for duraon dependence condonal on observed covaraes and unobserved heerogeney. 29

30 The key assumpons used o ncorporae unobserved heerogeney are ha: 1) Unobserved heerogeney s ndependen of he observed covaraes. 2) Unobserved heerogeney dsrbuon s known up o a fne number of parameers 3) Unobserved heerogeney eners he hazard funcon n a mulplcave fashon. Before movng o a more general framework we wll consder he model by Lancaser (1979). For a random draw from he populaon s assumed ha he hazard funcon has he Webull form condonal on he observed covaraes x and unobserved heerogeney v : ( ;, v ) v exp( ) 1 x xβ (17) where x 1?1 and v >. To denfy he parameers?and?we need o normalze he dsrbuon of v so ha E(v ) = 1. In Lancaser (1979), was assumed ha he dsrbuon of v = Gamma(?,?) so ha E(v ) = 1 and Var(v ) = 1/?. In he general case where he c.d.f. of gven (x,v ) s F( x,v,?) we oban he dsrbuon of * gven x by negrang ou he unobserved effec. Because v and x are ndependen, he c.d.f. of * gven x s G( x ; θρ, ) F( x, v ; θ) k( v; ρ) dv 3

31 where s assumed ha he densy of v, k(?;?) s assumed o be connuous and depend on he unknown vecor of parameers?. Wh censorng and flow daa we should assume H v e c H v v e c K v * * ( x,,, ) ( x, ) and K( x,, ) ( ) Suppose ha he unobserved heerogeney dsrbuon has a gamma dsrbuon and ( ; x, v ) v ( )exp( xβ ) Then, F( x, v ) 1 exp v exp( x β) ( s) ds 1 exp v exp( xβ ) ( ) where ( ) ( s) ds. Now he densy of v s k v v v v 1 ( ) exp( ) / ( ) where Var(v )=1/?and?(?) s he Gamma Funcon. Thus, 31

32 1 ( x;, ) 1 exp( exp( x β) ( )) exp( ) / ( ) G v v v dv exp( ) ( ) 1 x 1 β. Why would we nroduce heerogeney when he heerogeney s assumed o be ndependen of he observed covaraes? In many nsances n economcs, such as job search heory, we are neresed n esng for duraon dependence condonal on he observed and unobserved heerogeney, where he unobserved heerogeney eners he hazard mulplcavely. As shown by Lancaser (1979), gnorng mulplcave heerogeney n he Webull model resuls n asympocally underesmang?. Therefore, we could very well conclude ha here s negave duraon dependence condonal on x, whereas here s no duraon dependence condonal on x and v. Reurnng o our example usng he CPS-DWS we esmae boh Webull and Webull-Gamma models conrollng for a number of covaraes ncludng he weekly benef amoun an ndvdual s qualfed o receve (WBA) and an ndcaor for UI recep (UI) and he neracon of he wo (UI???WBA). 1 Fgures 5 and 6 dsplay he esmaed cumulave hazard and survvor funcon, respecvely, when he covaraes are se o her sample means. To nvesgae he mpac of UI recep, Fgure 7 porrays he dfference n he esmaed survvor funcon for a UI recpen and UI non-recpen who boh qualfy for weekly benefs of $2 per week, and whose remanng covaraes are fxed a her sample means. As can be seen from he fgure, he survvor funcon 1 Conrols for gender, race maral saus, age, educaon, mmgran saus, regon of counry, ndusry of los job, enure n los job, bluecollar-whecollar saus, reason for dsplacemen, weekly wage n los job and year of dsplacemen were also ncluded. The sample ncludes only hose who are mpued o be elgble for UI benefs. 32

33 of he non-recpen decreases much more rapdly ndcang ha hey fnd jobs more quckly han UI recpens. Fgures 8 hrough 1 presen graphs for he Webull-Gamma model. Fgure 5 Webull Model: Cumulave Hazard Esmae Cumulave Hazard Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 33

34 Fgure 6 Webull Model: Survvor Funcon Esmae Survval Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 34

35 Fgure 7 Webull Model: Survvor Funcon Esmae by Unemploymen Recep Survval Joblessness Duraon Non Recpen Recpen Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 35

36 Fgure 8 Webull-Gamma Model: Cumulave Re-employmen Hazard Esmae Cumulave Hazard Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 36

37 Fgure 9 Webull-Gamma Model: Survvor Funcon Esmae Survval Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 37

38 Fgure 1 Webull-Gamma Model: Survvor Funcon Esmae by Unemploymen Insurance Recep Survval Joblessness Duraon Non Recpen Recpen Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 5. The Cox Regresson Model ) Daa wh no es: The models presened above mpose paramerc assumpons on he baselne hazard funcon. In many nsances economc heory provdes lle help n denfyng a parcular paramerc class. However, f he rue baselne hazard funcon does no belong o he assumed paramerc class of funcons, esmaes wll generally be based. Cox (1972, 1975) developed a echnque for obanng esmaes of hewhou mposng any paramerc form on he baselne hazard. Ths echnque s referred o as Cox regresson. The model was developed wh 38

39 connuous me duraon daa n whch he probably of wo duraons endng a he same me equals zero. In mos applcaons however, duraon daa s grouped o some exen and he model has been modfed o accommodae es n he daa. Frs, however, we wl look a he case of no es. The Cox regresson model assumes ha he condonal hazard funcon follows a proporonal hazards model: ( ; x ) ( ) exp( xβ ) The benef of he Cox regresson mehod s ha makes no assumpons abou he form of he baselne hazard funcon? (). In fac he esmaon mehod parals ou he baselne hazard so ha doesn appear n he maxmand; he only parameers ha appear are he regreson coeffcens. Thus, Cox regresson s a sem-paramerc esmaon mehod. Cox regresson esmaon reles on formng a rsk pool or rsk se a each falure me n he daa. The rsk se a falure me??ncludes all ndvduals,, wh and c greaer han or equal o?. Thus a?an ndvdual s n he rsk se f he even has no occurred before ha me or hey have no been censored. The paral lkelhood funcon s hen consruced by consderng he condonal probables of falure a each falure me. For example suppose ha here are 5 observaons n he daa such ha Obs. d x

40 Le R( j ) be he rsk se a he (ordered) falure me j. Thus, R(3) = {1,2,3,4,5}, R(6) = {3,4,5}, R(9) = {4,5} and R(11)={5}. A each even me j, we consder he condonal probably ha he even occurs for he parcular observaon among hose observaons remanng n he rsk se jus before j, condonal on one even occurrng a. For any observaon,, n he rsk se a me j he probably of falure a j approxmaely equals?( j ;x ). Thus, f observaon j fals a me j, hen he condonal probably of observng j equals exp( xj βj ) exp( xβ j j ) or smply where R jr( j). exp( xβ ) exp( xβ ) R ( j ) R j An alernave way o hnk abou he consrucon of he paral lkelhood s o hnk of drawng balls from an urn a each falure me. A ball s ncluded n he urn for ndvdual a me j only f ndvdual has no been censored or has no faled before me j. Insead of equal probables he relave probably of drawng he ball assocaed wh ndvdual equals exp( x β. ) Thus probably of drawng ndvdual j equals exp( xj βj ) exp( xβ ). R ( j ) falure mes: The paral lkelhood s formed by he produc of hese condonal probables over all K exp( xj β) PL( β) j1 exp( ) x β R j or K log PL( β) xβ j log exp( ) xβ (18) j1 R j 4

41 Esmaes are obaned by maxmzng (18) wh respec o Leˆβdenoe he value of ha maxmzes (18). Then, he frs order condons for a maxmum sae ha he vecorˆβmus sasfy: x exp( ˆ x β) K ( ˆ R j s β) xj ˆ j1 exp( ) x β R j The vecor funcon s s usually referred o as he score funcon. From our dscusson of counng processes s clear ha M ( ) Y ( s) Z( s) dn ( s) Y ( s) Z( s) ( s) exp( ) ds x β s a marngale. Defnng he predcable funcon H j (, βx, 1, x2,, xn ) as j 1 2 n j R j H (, βx,, x,, x) x R j x exp( xβ ) exp( xβ ) we have 41

42 n j j 1 2 n j j j j1 s ( β) H ( s, βx,, x,, x) Y ( s) Z ( s) dn ( s) n j1 n j1 n j1 n j1 H ( s, βx,, x,, x) Y ( s) Z ( s) dm ( s) Y ( s) Z ( s) ( s)exp( xβ ) j 1 2 n j j j j j H ( s, βx,, x,, x) Y ( s) Z( s) dm ( s) j 1 2 n j H (, βx,, x,, x) Y ( s) Z ( s) ( )exp( xβ ) j 1 2 n j j H j ( s, βx, 1, x 2,, x n ) Y ( s) Z( s) dm j ( s) (19) snce n j1 H ( s, βx,, x,, x) Y ( s) Z ( s) ( s)exp( xβ ) ds j 1 2 n j j j n R j x j j j j j1 exp( ) x β R j n j1 x exp( xβ ) Y ( s) Z ( s) ( s)exp( xβ ) ds exp( x j β) x exp( xβ ) R j Yj ( s) Z j ( s) x j exp( x j β) exp( ( s) ds xβ) R j exp( x j β) x exp( xβ ) R j j exp( x x j β) ( s) ds. R exp( ) j R j xβ R j Usng (19) and appealng o resuls from laws of large numbers and Marngale Cenral Lm Theory, can be shown ha he esmaes are boh conssen and n - asympocally normal wh he varance-covarance marx equal o 42

43 1 2 log( PL( β)) E ββ whch can be conssenly esmaed by k log( PL ( ˆ)) k log( ( ˆ)) log( ( ˆ j β PLj β PLj β)) or j 1 j 1 ββ β β where he vecorˆβdenoes he vecor of Cox regresson esmaes and exp( ˆ ˆ xj β) PLj ( β). exp( xβ ) R j ) Daa wh Tes Wh es n he daa he exac paral lkelhood becomes more complex alhough here are some approxmaon mehods ha reduce he complexy and perform well as long as he number of es are smal. Suppose a each even me j, d j evens occur and led j be he se of ndvduals for whch he even occurs a me j. Reurnng o he urn analogy hen nsead of drawng one ball from he urn a me j, we draw d j balls whou replacemen. For a parcular sequence s={j(1),, j(d j )} of draws he probably of observng ha sequence equals exp( β ) d j x j( q) j( q) q1 R ( j ) { j(1),, j( q1)} exp( xβ ). (2) Thus, he probably equals 43

44 d j 1 exp( xj ( q) βj ( q) )! D exp( xβ ) d j sp ( j ) q1 R ( j ) { j(1),, j( q1)} where P(D j ) represens he se of permuaons of he ndces of he D j ndvduals who fal a me j. Snce he consrucon of he exac paral lkelhood wh es can be que complex, several approxmaons have been suggesed. Perhaps he mos well know s ha by Breslow(1974) who essenally subsues samplng wh replacemen for samplng whou replacemen n he urn analogy. Defne s j kd k x k. Then he Breslow approxmaon o he paral lkelhood equals: K exp( s) jβ PLB ( β) d. j j1 exp( ) x β R j An alernave approxmaon by Efron (1977) adjuss he denomnaor of he Breslow approxmaon o he paral lkelhood by subracng a erm for he number of balls drawn from he urn. Bu nsead of deducng he probably weghs for he acual balls drawn and hen averagng over all permuaons Efron (1977) deducs he average probably wegh where he 44

45 average s over all d j n D j. The Efron (1977) approxmaon o he paral lkelhood equals: K PLE ( β) j1 d k1 R j exp( sj β). k1 exp( x β) exp( xβ ) d D j Snce he Cox regresson paral lkelhood elmnaes he baselne hazard funcon does no produce esmaes of he baselne hazard funcon. The esmaes of, however, can be used o esmae he cumulave baselne hazard, ( ), usng an esmaor ha weghs he daa usng exp( xˆ β: ) * ˆ( ) dn ( s ) n 1 J ( s) Y ( s) Z ( s)exp( xβ ˆ). An esmae of he baselne survvor funcon, S (), hen equals: ˆ Sˆ ( ) exp ( ). Reurnng o or joblessness example presened above, Fgures 11 and 12 presen esmaes of he cumulave hazard and survvor funcons when covaraes are fxed a he sample mean whle Fgure 13 presens esmaes of he survvor funcon for a UI recpen and non-recpen who qualfy for $2 per week n benefs and whose oher covaraes are fxed a he sample mean. 45

46 Fgure 11 Cox Regresson Model: Cumulave Hazard Funcon Esmae Cumulave Hazard Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 46

47 Fgure 12 Cox Regresson Model: Survvor Funcon Esmae Survval Joblessness Duraon Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. 47

48 Fgure 13 Cox Regresson Model: Survvor Funcon Esmae by Unemploymen Insurance Recep Survval Joblessness Duraon Non Recpen Recpen Source: February 1996 Curren Populaon Survey Graph produced by Saa 8.2 Saa Corp. College Saon, TX. ) Srafed Cox Regresson In some crcumsances he proporonal hazards assumpon may be napproprae. If he suspec varable s a caegorcal varable, hen one can relax he proporonal hazards assumpon for ha varable by esmang a srafed model. Suppose he varable w has H caegores and ha, a pror, you suspec ha he hazards are non-proporonal across he H caegores. Then he srafed Cox regresson nvolves maxmzng he paral lkelhood funcon: 48

49 h H K exp( xh β) j SPL( β) h1 h j 1 exp( ) x β R j h where x ncludes all oher varables excep w. 6. Dscree-me Duraon Daa ) Tme-Consan Covaraes Mos duraon daa avalable n economcs s grouped. Tha s duraons are only known o fall no a ceran me nervals, such as weeks, monh, or even years. For example, unemploymen duraon daa are ypcally grouped no weeks. One economerc approach ha s aken when analyzng grouped duraon daa summarzes he nformaon on sayng n he nal sae or exng ha sae n each me nerval usng a sequence of bnary responses. Essenally, we have a panel daa se where he duraon of an ndvdual deermnes a vecor of bnary responses. These n conjuncon wh he covaraes can be hough of as creang an unbalanced panel where he number of observaons per ndvdual equals K = mn(d, C ) where D equals he number of perods unl he even occurs and C equals he number of perods unl he observaon s (rgh) censored. In addon o allowng us o rea grouped duraons, he panel daa approach has a leas wo advanages. Frs, n a proporonal hazard specfcaon leads o smple mehods for esmang flexble hazard funcons. Second, because of he sequenal naure of he daa, me varyng covaraes are easly nroduced. Through ou hs dscusson we wll assume flow samplng. We dvde he me lne no 49

50 M + 1 nervals, [, k 1 ), [k 1, k 2 ),?, [k M-1, k M ), [k M,?), where k m are known consans. For example, we mgh have k 1 = 1, k 2 = 2, k 3 =3, and so on, bu unequally spaced nervals are also feasble. The las nerval s chosen so ha any duraon fallng no s censored a k M : no observed duraons are greaer han k M. For a random draw from he populaon, Le y m represen a bnary ndcaor equal one f he even occurs n he m h nerval and zero oherwse. For each person, we observe ( y,, y ) whch s an unbalanced panel daa se of lengh K. Noe ha he srng of bnary 1 K ndcaors s eher a sequence of all zeros or a seres of zeros endng wh a one where he former sequence s observed when he observaon s censored and he laer sequence s observed when he seres of zeros ends because an even occurred. Le k k I C k =1, K. Wh me nvaran covaraes, each draw from he populaon s,( y, ),,( y, ), x 1 1 k k. We assume ha a paramerc hazard funcon s specfed as?(; x,?), where? s a vecor of unknown parameers. Le T denoe he me unl ex from he nal sae. Whle we do no fully observe T, eher we know whch nerval falls no, or we know wheher was censored n a parcular nerval. Thus we can compue p( ym ym 1, m 1, x,), p( ym1 ym 1, m 1, x,), m=1, M. To compue hese probables n erms of T, we assume ha he duraon s condonally ndependen of censorng: T s ndependen of C gven C. Thus, 5

51 km P( ym1 ym 1, m 1, x) P ( km 1T k m Tk m1 ) 1 exp ( s;, ) ds x θ km1 km 1 m ( x, θ), m=1,, M where m ( xθ, ) exp ( s;, ) ds. xθ km1 Therefore, P( y 1 y,, x,) ( x, θ). m m1 m1 m We can use hese probables o consruc he lkelhood funcon. If, for observaon, uncensored ex occurs n nerval m, he lkelhood funcon s k1 m ( x, θ) [1 k ( x, )]. θ (21) m 1 The frs erm represens he probably of remanng n he nal sae for he frs k - 1 nervals, and he second erm s he (condonal) probably ha T falls no nerval k. If he duraon s censored n nerval k, we know only ha ex dd no occur n he frs k -1 nervals, and he lkelhood consss of only he frs erm n expresson (21). If d s a censorng ndcaor equal o one f duraon s uncensored, he log lkelhood for observaon can be wren as k1 log( L ) log[ ( x, θ)] d log[1 ( x, θ )] (22) m k m1 Thus, for a sample of sze N, he log lkelhood funcon s N N k1 x θ x θ (23) log( L) log( L ) log[ (, )] d log[1 (, )] m k 1 1 m1 51

52 To mplemen condonal MLE, we mus specfy a hazard funcon. One hazard funcon ha s popular s a pecewse-consan proporonal hazard: for m=1,?, M, ( ; x, θ) ( xβ, ), k k. m m-1 m Wh a pecewse consan hazard and ( x, β) = exp( xβfor ) m=1,?, M we have m ( x, ) exp[ exp( xβ ) m ( kmk m1 )] where k m are known consans (ofen k m = m). Alernavely one could assume an underlyng connuous baselne hazard and defne k km1 m ( ), m=1,, M m s ds and ( x, θ) exp[ exp( xβ ) ]. (24) m om Whou covaraes, maxmum lkelhood esmaon of he mn (24) leads o a well known esmaor of he survvor funcon. We can movae he esmaor from he represenaon of he survvor funcon as a produc of condonal probables. For m=1,?, M, he survvor funcon a me k m can be wren as S( k ) P ( Tk ) P( Tk Tk 1) m m m r r r1 52

53 Now, for each r=1, 2,?, M le N r denoe he number of people n he rsk se for nerval r: N r s he number of people who have neher lef he nal sae nor been censored by k r-1. Therefore, N 1 s he number of ndvduals n he nal random sample; N 2 s he number of people who dd no ex he nal sae n he frs nerval, less he number of ndvduals censored n he frs nerval, and so on. Le E r be he number of people observed o leave n he r h nerval. A conssen esmaor of P(T > k r T > k r-1 ) s ( N ) re r, r=1,2,, M. (25) N r I follows from equaon (25) ha a conssen esmaor of he survvor funcon a me k n s m ( N ) ˆ( ) re r S km, m=1, 2,,M. r1 Nr Ths s he dscree-me Kaplan-Meer esmaor of he survvor funcon (a pons k 1, k 2,?, k m ). We can derve hs Kaplan-Meer esmaor by maxmzng he lkelhood funcon N k1 d L m (1 k ). (26) 1 m 1 wh respec o, m m=1, M where d s an ndcaor varable ha equals one f he ndvdual spell s no censored. Takng he log of (26) gves 53

54 N k1 ln( L) ln( m ) d ln(1 k ) 1 m 1 and hen rearrangng erms yelds M ln( L) ln(1 m) ln( m ) m1 S ( m) D ( m) where S(m) denoes hose ndvduals who survvor pas m and D(m) denoes he se of ndvduals for whch he even occurs durng m. Ths reduces o M # # ln( L) S ( m) ln( m ) D ( m)ln(1 m ) m1 where S # (m) (D # (m)) denoes he number of ndvduals n S(m) (D(m)). Dfferenang he log lkelhood and seng o zero yelds 1 1 ( ) ( ) ˆ 1ˆ # # S m D m m m or ˆ. # # # ( S ( m) D ( m)) m S ( m) Solvng for ˆm hen yelds ˆ m S ( m) N E # m # # S ( m) D ( m) Nm m Now m m ˆ N re r S( a ) ˆ m r r1 r1 Nr whch was he Kaplan Meer esmaor above. 54

55 ) Tme-Varyng Covaraes For he populaon, le x 1, x 2,?, x M denoe he oucomes of he covaraes n each of he M me nervals and le X = (x 1, x 2,?, x M ), where we assume ha he covaraes are consan whn he nerval. In general we wll le X r = (x 1, x 2,?, x r ). We assume ha he hazard a me condonal on he covaraes up hrough me depends only on he covaraes a me. If pas values of he covaraes maer, hey can smply be ncluded n he covaraes a me. The condonal ndependence assumpon on he censorng ndcaors s now saed as D( Tk Tk, x, ) D ( Tk Tk, x ), m=1,2,,m m m1 m m m m1 m Ths assumpon allows he censorng decson o depend on he covaraes durng he me nerval (as well as pas covaraes, provded hey are eher ncluded n x m or do no affec he dsrbuon of T gven x m ). Under hs assumpon, he probably of exng (whou censorng) s P( y 1 y, x, ) P ( k T k Tk, x ) m m1 m m m1 m m1 km 1 exp ( s; m, ) ds 1 m (, ). x x km1 m m (27) we can use equaon (27) along wh P( ym ym 1, xm, m ) m ( xm, ) o consruc he log lkelhood for person as n (22) and he sample log lkelhood (23). ) Unobserved Heerogeney 55

56 Unobserved heerogeney can also be added o hazard models for grouped daa. For example, addng unmeasured heerogeney o (24) and leng? = exp(v ) gves ( x,, θ) exp[ exp( xβ ) ] (28) m m m om Now he survvor funcon assocaed wh () equals m m m S( m; X m,, θ ) r ( r,, ) exp[ exp( r ) or ] exp exp( r ) or x θ r1 xβ r1 xβ r1. (29) Le?have c.d.f. G(?). Then m S( m; Xm, ) exp exp( xr β) or dg( ). r1 Now, f G s a gamma dsrbuon wh E(?) = 1 and Var(?) =? 2 hen (29) becomes 1 m 2 2 ( ; m, ) 1 exp( r ) or. r1 S m X θ xβ (3) We can hen use (3) o form he log lkelhood funcon by nong ha he probably of he even endng n perod m equals S(m-1;X m-1,?) - S(m;X m,?). Leng d equal 1 f even occurs and oherwse (censored) we have he log-lkelhood funcon 56

57 N Xk 1 θ Xk 1 θ Xk θ 1 ln( L) (1 d ) ln[ S( k 1;, )] d ln[ S( k 1;, ) S ( k ;, )]ln where 2 θ α, β, wh α ( 1, 2,, M ). Oher unobserved heerogeney dsrbuons are: Inverse Gaussan: 1 S m m 2 ( ; Xm, θ) exp 1 (1 2 exp( ) ) 2 r or xβ r1 1 2 Sable Dsrbuon (Hougaard,1986): 1 S m c m 2 ( ; Xm, θ) exp 1 (1 exp( ) ) 2 r or xβ r1 1 c Mass-Pon Dsrbuon: J m S( m; Xm, θ) p j exp jexp( xβ r ) or j1 r1 where J equals he number of mass pons and p j1. Raher han fxng he mean of he J j1 mxng dsrbuon o 1 for hs dsrbuon, emprcal mplemenaon s easer by nsead fxng? 1 = Mul-spell Dscree-me Models Suppose ha nsead of a sngle duraon we have mulple duraons. For example, we may be neresed n examnng consecuve unemploymen duraons. The survvor funcon for he g h spell sasfes 57

58 m g g g g m m r r 1 r1 S ( k ) P( T k ) P( T k T k ) We shall assume ha P( T k T k ) ( x, ) exp[ exp( xβ ) ] g g g g g g g g g m m1 m m m m or P( T k T k ) 1 ( x, ) 1 exp[ exp( xβ ) ]. g g g g g g g g g m m1 m m m m Thus, m g g g g g g g m m x r β r r1 S ( k ) P( T k ) exp[ exp( ) ] We wll have daa for up o G spells for an ndvdual. If an ndvdual complees he g h spell wh K g = k g ha spell conrbues g k 1 g g g g g g g g P( K k ) expexp( r ) r expexp( g ) xβ xβ g k k r 1 o he lkelhood funcon. If hey are censored n he g h spell a me g k hen ha spell conrbues g k g g g g g P( K k ) exp[ exp( x r β) r ] r1 o he lkelhood funcon. Le he spell ndcaor varables v g =1 f an ndvdual eners he g h 58

59 spel and zero oherwse, g=1,,g and defne he censor varables d g = 1 f he ndvdual complees he g h spel and zero oherwse, g=1,,g. Then each ndvdual conrbues g G k 1 g g g g g g L exp[ exp( r ) r ] 1exp[ exp( g ) g ] x β x β k k g1 r1 o he lkelhood funcon whch s L L x β g N N G k 1 d g g g g g exp[ exp( r )] gr1 exp[ exp( xg β) g ] k k 1 1 g1 r1 and he log-lkelhood funcon equals g d g v g g g g g g g x r β r x g β g k k ln( L) ln exp[ exp( ) ] 1exp[ exp( ) ] 1 g1 r1 g N G k 1 d g N G k 1 g g g g g g g g v exp( x k β) r d ln 1exp[ exp( x g β) g ] k k 1 g1 r1 g g v g v Under he assumpon ha spell duraons are ndependen, he lkelhood decomposes and one can oban esmaes for hs model by esmang G sngle spell models where all ndvduals wh v g = 1 are ncluded n he g h esmaon. If we nsead assume ha g = and for al g hen we can sack observaons for each spel whn each ndvdual and g m m esmae a sngle spell duraon model wh log lkelhood 59

60 g N G k 1 g g g g ln( L) v exp( x r β) r d ln 1exp[ exp( x g β) g ] k k 1 g1 r1 An nermedae case would be a model ha resrcs g = for all g bu allows he baselne hazard parameers o be spell dependen. N g G k 1 1 g1 r1 x β ln( L) v exp( ) d ln 1exp[ exp( xβ ) ] g g g g g g r r g g k k Ths case s smlar o he connuous duraon Cox regresson model ha srafes he sacked daa by spell. One could employ sngle spell dscree duraon mehods o esmae such a model by sackng he daa and ncorporang (G-1)? M me-varyng covaraes ha are of he form x I ( Kk ) I ( Sg ) where S s a varable denong he parcular spell. gm Esmaon becomes more complcaed f we assume ha for each duraon g g g g P( T k T k 1) has he form m m P( T k T k 1) ( x, θ) exp[ exp( xβ ) ] g g g g g g g g g g g m m m m m m where processes g are unobserved random varables whch are assumed ndependen of he covarae g x m, g=1,,g. In general, he g may be correlaed wh each oher. Denoe hs jon dsrbuon of he G x 1 vecor ξby G( ξ;) where we have assumed ha he dsrbuon can be parameerzed by he Q x 1 vecor. The uncondonal log lkelhood funcon s obaned by negrang ou ξ. Thus we have g v g N G 1 g k d g g g g g g g g ln( L) lndg exp( exp( r ) r1 exp[ exp( g ) g ] ( ; ) x β x β ξδ (31) k k 1 g 1 r 1 6

61 Esmaes are obaned by maxmzng () wh respec o he g, mg, and. Maxmum lkelhood esmaon may prove compuaonally nensve snce he negral n (31) s ypcally mulvarae. One may assume a mass-pon specfcaon for G where here are M ypes of ndvduals n he populaon and each ype as a unque G x 1 vecor of locaon parameers. Le p q denoe he proporon of he q h ype n he populaon, q=1,,q. Then he loglkelhood (31) becomes g N Q G k 1 ln( L) ln pq exp( q exp( r ) r 1exp[ q exp( ) ] x β x β k k 1 q 1 g1 r 1 g d g g g g g g g g g g g v Ths lkelhood s hen maxmzed wh respec o g g, m, g=1,,g and ξ q, and p q q=1,,q where Q pq1. q1 8. Compeng Rsk Models In many cases spells end for dfferen reasons. Indvduals may qu a job or be lad off, a person may de because of cancer or a hear aack, re-employmen may occur no a job ha s par-me or full-me. In such cases he explanaory varables may have dfferng effecs on he relave probables of spells endng for parcular reasons. Compeng rsks framework s mean o allow for hs possbly. Whle whou regressors s no possble n general o dsngush models wh correlaed rsks from hose wh ndependen rsk, Heckman and Honore (1989) have derved suffcen condons on he regressors whch enable such denfcaon. We 61

62 shall assume ha such regressors exs. Also for smplcy n he dscusson below we focus exclusvely on he case of wo rsks. Exenson o cases where he number of rsks exceeds wo s sraghforward. Compeng rsk models assume ha here are wo laen duraon varables, T 1 and T 2, whch represen he me unl he occurrence of he ype 1 and ype 2 evens, respecvely. Wha s observed, however, s only T= mn(t 1, T 2 ) and an ndcaor I{T=T 1 }. Ths s referred o as he denfed mnmum. Thus, we know no only how long ook before a leas one of he wo ypes of evens occurred bu also whch one was. For example, f you were lad off from your job afer T monhs of enure, we know ha you hadn qu and weren lad of before hs me. We also know he reason for job spel endng (.e. layof). Wha we don know s when (f ever) you would have qu your job had you no been lad off frs. We assume ha duraon daa are dscree and proceed by specfyng he jon survvor funcon for he wo laen duraons T 1 and T 2 whch s denoed by S (T 1, T 2 ). In parcular we assume ha P( T k r T k r1 ) r ( xr, ) exp[ exp( xr β ) r ] (32) and P( T k r T k r1 ) r ( xr, ) exp[ exp( x rβ ) r ] (33) where we assume ha he varables 1 and 2 are unobserved and ndependen of he observed explanaory varables. Correlaed rsks arse n hs model o he exen ha 1 and 2 are 62

63 correlaed. From (32) and (33) he laen survvor funcon sasfes X x β xβ 1 1 S(,,, ) exp[ exp( ) ] exp[ exp( ) ] exp exp( x β) exp( xβ ) where X{ x, x,, x }. Le G be he dsrbuon funcon for he unobservables 1 and 1 2 max( 1, 2 ) 2. Then he uncondonal survvor funcon sasfes 1 2 k k S( k, k X) exp exp( x β) exp( xβ ) dg(, ). (34) To consruc he lkelhood funcon n hs case suppose for he h ndvdual he fal a me due o cause 1. Then, P( kmn( k, k ), I ( kk ) 1 X,, ) X S( k 1, k 1 X,, ) P( k 1 T k T k 1 T k, X,, ) P( k 1 T k T T T k 1 k 1 T k, X,, ) S k k1 X,, ) S ( k1, k X,, ) S( k1, k1,, ) P( k 1 T k T T T k 1 T k 1, X,, ) = ( 1, S( k 1, k 1 X,, ) S( k, k X,, ) S( k 1, k X,, ) S( k, k 1 X,, ) S k k S k k A k where ( 1, 1 X,, ) (, 1 X,, ) ( X,, ) 63

64 A k S k k S k k S k k S k k ( X,, ) 1 2 ( 1, 1 X,, ) (, X,, ) ( 1, X,, ) (, 1 X,, ). In a smlar manner we have P k k k I k k S k k S k k A k ( mn( 1, 2 ), ( 2 ) 1 X, 1, 2 ) ( 1, 1 X, 1, 2 ) (, 1 X, 1, 2 ) ( X, 1, 2 ) Le 1 c ( c 2 ) be an ndcaor varable ha equals 1 f he h person spell ends for reason 1 (2) and le 3 c be an ndcaor varable ha equals one f he spell s rgh censored. Then he log lkelhood funcon for he compeng rsks model Log(L) sasfes: N ln(l) c ln S( k1, k1 X,, ) S ( k1, k X,, ) A ( k X,, ) dg(, ) =1 c ln S( k1, k1 X,, ) S ( k, k1 X,, ) A ( k X,, ) dg(, ) c ln S( k1, k1 X,, ) dg(, ) As we wll dscuss laer, hese mehods were used by McCall (1996, 1997) o nvesgae he reemploymen paerns of dsplaced workers n he Uned Saes. In parcular, McCall (1996) used a compeng rsks framework o model joblessness duraons ha end due o re-employmen no par-me or full-me jobs and how parcular parameers of he US unemploymen nsurance (UI) sysem affec no only how long an ndvdual remans unemployed bu also he ype (par-me versus full-me) of job ha hey are re-employed no. In he Uned Saes, many unemployed ndvduals who qualfy for UI benefs do no fle a clam. McCal s (1996) analyss focused only on dsplaced workers who qualfed for benefs. To allow for he possbly ha changes n he parameers of he UI sysem wll affec he choce o fle a UI clam (see Anderson and Meyer, 1997 and McCall, 1995), McCall (1996) 64 (35)

65 modeled he clam flng choce and allowed for he possbly ha unobservable deermnans 1 of ha choce may be correlaed wh unobservable deermnans (.e. and 2 n (35) ) of he re-employmen raes no par-me and full-me jobs. Thus UI recep was modeled by he dchoomous varable UI whch equals 1 f an ndvdual fles a clam and oherwse where Pr(UI=1) has he funconal form u P(u=1)=1-exp[ exp( zδ )] where z s a vecor of explanaory varables, s a vecor of parameers and varable ha s uncorrelaed wh X and z. However, u s an unmeasured u may be correlaed wh 1 and 2 n (35) and u (along wh s neracon wh some varables n X) are added as explanaory varables n (35). Denoe hese varables by v. The lkelhood funcon for hs model selecvy correced compeng rsks model s ln(l) = N u u u 1 u u z z X X X =1 c ln P(u 1, ) P(u, ) S( k 1, k 1,u,, ) S( k 1, k,u,, ) A( k,u,, ) dg(,, ) u c S kk S k k1 X,u,, ) A ( k X,u,, ) dg(,, ) u u u 1u ln P(u 1 z, ) P(u z, ) ( 1, 1 X,u,, ) (, c ln P(u 1 z, ) P(u z, ) S( k1, k1 X,u,, ) dg(,, ) u u 1 2 u 1u u (36) McCall(1996) used a mass-pon specfcaon for he unobserved heerogeney dsrbuon G whch assumes ha here are M ypes of ndvduals n he populaon wh ype m havng he unque rple 1 2 u ( m, m, m) of locaon pons and composng p m of he populaon, m=1,,m, wh M pm1. For hs specfcaon, he lkelhood n (36) becomes m1 65