Duration Models. Class Notes Manuel Arellano November 1989 Revised: December 3, 2008

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1 Duraion Models Class Noes Manuel Arellano November 1989 Revised: December 3, 28 1 Duraion daa Duraion daa appear in a diversiy of siuaions in economics, bu I will ofen refer o unemploymen duraion o provide a focus for he presenaion of he maerial. Suppose we selec a random sample of N persons enering unemploymen and wai unil hey find jobs. Then we record he number of weeks each person has been unemployed: 1, 2,..., N. We have observaions on he duraion of a spell of unemploymen for each of he individuals in he sample. In pracice we are more likely o have censored duraion daa. Tha is, for some individuals we do no observe i, possibly because hey have no found a job a he ime of he inerview, so ha all we know is ha i > for some paricular, or hey have found a job beween selecion and inerview and all we know is ha i lies wihin a cerain inerval < i <. There are enormous variaions in he duraion of spells of unemploymen from one individual o he nex (from a few weeks o five years or more) and i is imporan o model he causes for hese differences. In paricular i is imporan o know how he re-employmen probabiliy changes over he period of he spell and wha is he impac of he level of unemploymen benefis on hese probabiliies. More generally, duraion daa measure how long individuals remain in a cerain sae. Oher examples include: job urnover, marial insabiliy, ime o ransacions in sock markes, or duraions o realignmens of inernaional currencies. The analysis of duraion daa has a long radiion in biomerics and medical saisics (as well as in indusrial life esing and demography), from where mos of he erminology and he saisical models used by economericians originae. This is also he reason for he inimidaing names ha are used in his field: Failure ime daa, hazards, risks, survivor funcions... A ypical example in exbooks is of he form: you have a number of ras, which you injec wih a cancer producing subsance, hen ime o moraliy is measured (e.g. Kalbfleisch and Prenice). Remark on he noaion: The sandard noaion for a cross-secional explained variable in economerics is Y i. However, in duraion analysis we ofen use T i, which emphasizes he fac ha he variable measures ime in a sae. We may also have a ime series of duraions (e.g. ime o ransacion or ime o a price change), which naurally connecs wih he perspecive of poin processes. The purpose of his noe is o cover basic conceps in duraion analysis. Muliple spell daa, reamen effecs in duraion models, and poin processes are discussed in separae noes. 1

2 2 The hazard funcion 2.1 Hazard funcion for a discree random variable Le T be a discree duraion random variable aking on values {1, 2, 3,...} wih pmf : and cdf : p () =Pr(T = ) ( =1, 2,...) F () =Pr(T ) =p (1) + p (2) p (). The hazard funcion or exi rae from he sae is Pr (T = ) h () = Pr(T = T ) = Pr (T ) = Pr (T = ) 1 Pr (T 1) p () F () F ( 1) = = for >1 1 p (1)... p ( 1) 1 F ( 1) and h (1) = p (1) = F (1). Tha is, he hazard gives probabiliies of exi defined over he surviving populaion a each ime. Example: Suppose T is ime o moraliy (in years). Then p 1 = Pr(T = 1) ' h 1 = Pr(T = 1 T 1) ' 1. In his case he hazard may look as depiced in Figure 1. Figure 1: Moraliy hazard rae The hazard h () provides an alernaive way of characerizing he disribuion of T. Leussee how we can recover F () and p () from h (). For shorness we use he noaion p = p (), F = F (), and h = h (). For>1 Pr (T +1 T ) =1 h = 1 F 1 F 1. 2

3 Using his expression recursively we ge 1 F 1 = (1 h 1 ) 1 F 2 = (1 h 1 )(1 h 2 ) Therefore, 1 F =. Y (1 h s ) (1) Y F =1 (1 h s ) ( =1, 2,...), (2) which shows how he cdf of T can be obained from he hazards. Similarly, Y 1 p =(1 F 1 ) h = h (1 h s ) (3) Noe ha (1) is an inuiive probabiliy facorizaion. We have: 1 h = Pr(T +1 T ) 1 F = Pr(T +1) so ha Pr (T +1)=Pr(T +1 T )Pr(T ). Repeaedly using his facorizaion we obain (1): 1 Pr (T +1)=Pr(T +1 T )Pr(T T 1)... Pr (T 2) 2.2 Hazard funcion for a coninuous random variable We know ha a coninuous random variable T can be characerized by he pdf f () and by he cdf F (). An alernaive way of characerizing he disribuion of T is he hazard funcion defined as h () = f () 1 F (), which is paricularly useful when T represens duraion in a cerain sae. Noe ha h () gives he condiional densiy of T given T>: Pr ( T<+ T>) h () = lim 1 Noe ha Pr (T 2) = 1 h 1 =Pr(T 2 T 1) since we assume ha Pr (T 1) = 1. 3

4 where Pr ( T<+ T>)= Pr ( T<+ ) Pr (T >) = F ( + ) F (). 1 F () If T represens unemploymen duraion, h () d is he probabiliy of leaving unemploymen during (, + d) given ha he individual has been unemployed for periods. Noe ha h () is he closes link o he predicions of a heory of job search. Suppose an individual has a sequence of reservaion wages over ime w (), ha he faces a disribuion of wage offers G (w), and ha job offers arrive randomly a a rae ψ () (i.e. he probabiliy of a job offer in he inerval (, + d) is ψ () d). Then he probabiliy of exi during (, + d) is he probabiliy of geing a job offer imes he probabiliy of an offered wage greaer han he reservaion wage: h () d =[1 G (w ())] ψ () d I is herefore naural o sar by making economic assumpions abou h () raher han oher aspecs of he disribuion of duraions such as f (), F (), ore (T ). Recovering F () and f () from he hazard rae funcion (or inegraed hazard), which is given by H () = Z h (u) du (or R h (u) du if T is non-negaive as i would normally be). Moreover, Z f (u) H () = 1 F (u) du) =[ ln (1 F (u))] = ln [1 F ()] so ha F () =1 exp [ H ()] and similarly, f () =h ()exp[ H ()]. Firs le us inroduce he cumulaive hazard The comparison beween survival probabiliies for discree and coninuous duraions is as follows. In he discree case we have X ln [1 F ()] = ln [1 h (s)] whereas in he coninuous case we have ln [1 F ()] = Z [ h (s)] ds. Noe ha for small h (s) we have ln [1 h (s)] ' h (s). 4

5 3 The hazard funcion of some frequenly used disribuions 3.1 Consan hazard: The exponenial disribuion A simple model of he hazard funcion is o assume ha i is consan: h () =λ. In he conex of our example, a consan hazard means ha he probabiliy of leaving unemploymen in a given ime inerval is he same regardless of how long he individual has been unemployed. The consan hazard assumpion defines a family of probabiliy disribuions indexed by one parameer (λ). If he duraion variable is coninuous we obain he class of exponenial disribuions. The inegraed hazard is H () = Z λdu = λ so ha he log survivor funcion is a sraigh line hrough he origin: ln [1 F ()] = λ. Therefore, he cdf and pdf are given by F () = 1 e λ λ > f () = λe λ. The fac ha an exponenial random variable has a consan hazard is called he memoryless propery of he exponenial disribuion. The hazard funcion is also he inverse of he expeced value: E (T )= 1/λ. If he duraion variable is discree we obain he class of geomeric disribuions. The cdf and he pmf are given by F () = 1 (1 λ) p () = λ (1 λ) 1. These expressions resul from seing h = λ in (2) and (3). The expeced value is also 1/λ. 3.2 The Weibull disribuion This is a wo-parameer generalizaion of he exponenial disribuion, which allows for a hazard increasing or falling monoonically: F () = 1 e (λ)α λ >, α > f () = λ α α α 1 e (λ)α. 5

6 The hazard is given by h () =αλ α α 1. If α > 1 h () is monoone increasing; if α < 1 h () is monoone decreasing; if α =1h () is consan and reduces o he exponenial case. 4 Condiional models and he proporional hazard specificaion In economeric applicaions we are usually concerned wih he relaionship beween duraion ime and explanaory variables, and herefore we look a he condiional disribuion of duraions given a se of exogenous variables x, F ( x), soha h (, x) = f ( x) 1 F ( x). Tha is, we expec he hazard rae o differ beween members of he populaion. 4.1 The proporional hazard model The proporional hazard (PH) model (Cox, JRSS, 1972) specifies ha h (, x) =λ ()exp x β. Tha is, h (, x) facors ino a funcion of and a funcion of x, so ha wo differen individuals have re-employmen probabiliies ha are proporional for all. The model is widely used because of is simpliciy and sraighforward inerpreaion. λ () is called he base-line hazard funcion. Common specificaions for λ () are he ones considered earlier: λ () consan as in he exponenial case (in fac, λ () =1if he consan is subsumed in he x β index), or λ () =α α 1 as in he Weibull case. Linear represenaion of he proporional hazard model The cdf associaed wih he PH model saisfies 1 F ( x) =exp Z h (u, x) du =exp e x β where Λ () is he inegraed baseline hazard Λ () = Z λ (u) du. 6 Z h i λ (u) du =exp e xβ Λ ()

7 Therefore, we also have ln { ln [1 F ( x)]} = x β +lnλ (). Now, if he random variable T x has cdf F ( x), U F (T x) is uniformly disribued independenly of x. Moreover,hevariable V =ln[ ln (1 U)] ln { ln [1 F (T x)]} is an exreme value variae, independen of x, wihcdf F (r) =1 exp ( e r ). Thus, he PH model can be wrien as a linear regression model for a ransformaion of he duraion variable: ln Λ (T i )= x iβ + V i where he error erm is exreme value disribued independenly of x (see Fourgeaud, Gourieroux, and Pradel, 1988). In paricular, if λ () =1, Λ () = and he PH model reduces o he exponenial regression ln T i = x iβ + V i. On he oher hand, if λ () =α α 1, Λ () = α and we obain α ln T i = x iβ + V i or ln T i = x β i α + 1 α V i. 5 Likelihood funcions for censored and non-censored duraion daa This secion is based on Tony Lancaser s 1979 paper Economeric Models for he Duraion of Unemploymen. Le T i x i be a duraion variable of ineres wih densiy f ( i x i ). Ideally, we would like o observe a random sample of compleed duraions from enrans. This is he firs case. Case I: Compleed duraions (inflow sample) funcion of he sample is NY L = f ( i x i ) i=1 where Z i f ( i x i )=h( i,x i )exp h (u, x i ) du. For example, if h ( i,x i ) is PH wih λ () =1,hen f ( i x i )=e x i β exp ³ β i e x i. We observe { 1, 2,..., N }. The likelihood 7

8 Case II: Censored duraions (inflow sample) We have a random sample of enrans inerviewed periods laer. We observe i if i, oherwise we jus observe ha i > : L = Y f ( i x i ) Y 1 F xi i i > Case III: Censored sock sample We have a sample from he sock of unemployed individuals a a cerain period and inerview hem h periods laer. Le {d 1,d 2,...,d N } be he number of weeks hey have been unemployed a he ime of selecion. We observe i if i d i + h. Oherwise we jus observe ha i >d i + h. The likelihood is: L = Y d i < i d i +h f ( i x i ) 1 F (d i x i ) Y i >d i +h 1 F (d i + h x i ). 1 F (d i x i ) In a sock sample individuals wih a small d i are less likely o be sampled. Thus, conribuions o he likelihood are condiioned on T i >d i. For example, he likelihood conribuion of a censored of observaion is Pr (T i >d i + h T i >d i ). CaseIV:Inervalsocksample Same as in Case III bu now we never observe i. We only observe wheher d i < i d i + h or i >d i + h: L = Y d i < i d i +h F (d i + h x i ) F (d i x i ) 1 F (d i x i ) Y i >d i +h 1 F (d i + h x i ). 1 F (d i x i ) In his case, he likelihood conribuion of he individuals who found a job beween selecion and inerview is Pr (d i <T i d i + h T i >d i ). Clearly, he previous lis of cases is no exhausive. I is jus inended o illusrae how he consrucion of he likelihood makes he connecion beween he sample design and he underlying model of ineres. 6 Unobserved heerogeneiy 6.1 Inroducion: Unobserved heerogeneiy in a simple case Consider a discree duraion variable T and a (, 1) covariae X. The condiional hazard raes are assumed consan for all : h 1 () = Pr(T = T, X =1)=h 1 h () = Pr(T = T, X =)=h. Suppose ha h 1 >h and le p =Pr(X =1). Now consider he marginal (or aggregae) hazard: h () =h 1 Pr (X =1 T )+h Pr (X = T ) 8

9 For =1, h (1) = h 1 p + h (1 p), bu for =2he probabiliy of X =1in he surviving populaion wih T 2 is Pr (X =1 T 2) = = Pr (T 2 X =1)p Pr (T 2 X =1)p = Pr (T 2) Pr (T 2 X =1)p +Pr(T 2 X =)(1 p) (1 h 1 ) p (1 h 1 ) p +(1 h )(1 p) = p <p. 1+³ h1 h 1 h 1 (1 p) Therefore, we observe spurious sae dependence": h (2) <h(1). 6.2 The PH model wih unobserved heerogeneiy Lancaser (1979) argued ha in he specificaion of he hazard here may be omied explanaory variables, e.g. because some deerminans of he hazard canno be observed. He addressed he problem by inroducing a muliplicaive random effec in he PH specificaion: h (, x, v) =λ ()exp x β v where v is assumed independen of x wih E (v) =1and pdf g (v). h (, x, v) is now he hazard funcion condiional on x and v. The cdf of unemploymen duraion given x and v is Z F ( x, v) =1 exp h (u, x, v) du, while he cdf given x only is F ( x) = Z F ( x, v) g (v) dv. Lancaser assumed a Gamma disribuion for g (v). 6.3 Idenificaion of λ () and g (v) in he PH model If λ () 1 here is no ime dependence whereas if v 1 here is no unobserved heerogeneiy. In principle, one could hink ha he wo effecs are inerchangeable and he only reason why we can separae hem is because we impose paricular funcional forms for λ () and g (v). The inuiive argumen is as follows: Suppose λ () 1 and g (v) is non-degenerae, so ha each individual has a consan probabiliy of leaving unemploymen, bu hese probabiliies vary beween hem. Individuals wih high probabiliies leave early, and by his selecion process long duraions are 9

10 recorded for individuals wih low probabiliies of finding a job. This resul is apparenly he same as when probabiliies are he same for everybody bu declining wih he ime already elapsed: λ () < and v 1. This suggess ha i is no possible o disinguish beween ime dependence and unobserved heerogeneiy. In fac when here are no exogenous covariaes his is exacly he case. For example, i can be shown (c.f. Lancaser and Nickell, JRSS, 198; Elbers and Ridder, RES, 1982) ha he cdf given by F () =1 1 1+a can be obained from he following wo observaionally equivalen specificaions: Specificaion 1 : Noe ha H 1 () = ln(1+a) ( 1 v 1 G 1 () = v<1. F () =1 exp [ H 1 ()] = 1 exp [ ln (1 + a)] = a. Specificaion 2 : H 2 () = G 2 () = Z v 1 a e u/a du. In his case v has an exponenial disribuion. Noe ha F ( v) =1 exp [ vh 2 ()] = 1 exp ( v) F () =E v [1 exp ( v)] = 1 E v e v =1 1 1+a where we are using he expression for he mgf of he exponenial disribuion. Elbers and Ridder (1982) prove ha his is no he case when here are regressors. Specifically, hey prove ha here are no wo combinaions of heerogeneiy disribuions and ime dependence funcions given he same duraion disribuion. This resul is sronger han he more radiional resuls on idenificaion in parameric models and pioneered semiparameric idenificaion resuls in economerics. Why his is so can be seen inuiively in he previous example assuming ha we can observe he duraion disribuions for wo differen known values of exp (x β), say1 and φ. Forexp (x β)=1he 1

11 wo cdf s are idenical. The quesion now is wheher for exp (x β)=φ he wo specificaions also lead o he same cdf for duraions. I can be seen ha his requires ha (1 + a) φ =(aφ +1) for bu his only holds if φ =1. So inroducing sysemaic variaion in he hazard has as a consequence ha specificaions 1 and 2 are disinguishable. This idenificaion resul suggess ha i may be possible o devise semiparameric esimaors of PH models wih heerogeneiy ha do no require a specific disribuion for v. Heckman and Singer (1984) proposed a mehod along hese lines. 7 Esimaing discree duraion models The usual approach has been o model discree (or grouped) duraions using densiy funcion erms in a likelihood as if he spells were observed coninuously. Suppose T i is a discree random variable. The condiional hazard given covariaes x i is given by h (, x i )=Pr(T i = T i, x i ) ( =1, 2, 3,...). Since his is now a probabiliy i is naural o use discree choice models in order o specify he exi rae: Pr (T i = T i, x i )=F γ + x iβ where F is a cdf (e.g. normal or logisic). If γ and β were consan for all, henh (, x i ) would be consan for a given value of x and here would be no sae dependence. A possible way of inroducing sae dependence is o specify γ and β as polynomials in. For example, γ = γ + γ 1 (ln )+γ 2 (ln ) 2. More generally, γ and β could be reaed as unresriced duraion-specific parameers: X γ = γ j 1 (T i = j) T j=1 where T is near o he maximum spell observed in he daa. Such a model amouns o a sequence of T binary choice models for he surviving populaions a each. More specifically, for =1: Pr (T i =1 T i 1,x i )=Pr(y 1i =1 w 1i =1,x i )=F γ 1 + x iβ 1 11

12 where y 1i = 1 (T i =1)and w 1i = 1 (T i 1) (noe ha w 1i =1wih probabiliy one). Now for =2: Pr (T i =2 T i 2,x i )=Pr(y 2i =1 w 2i =1,x i )=F γ 2 + x iβ 2 where y 2i = 1 (T i =2)and w 2i = 1 (T i 2), ec. Hence, he log likelihood funcion for a sample of enrans can be wrien as: NX XT L = w i yi ln F γ + x iβ +(1 yi )ln 1 F γ + x ª iβ. i=1 =1 There are wo ypes of individual conribuions o he likelihood: A compleed spell T i = conribues a erm Pr (T i = x i ): ln Pr (T i = x i ) = ln " # Y X 1 (1 h si ) =lnh i + ln (1 h si ) h i 1 = lnf γ + x X 1 iβ + ln 1 F γ s + x iβ s. A censored spell a conribues a erm Pr (T i +1 x i ): " Y # ln Pr (T i +1 x i ) = ln[1 Pr (T i x i )] = ln (1 h si ) X = ln 1 F γ s + x iβ s. The amoun of generaliy in he specificaion of γ and β will be parly dicaed by he frequency a which duraions are repored bu also by he shape of empirical hazards. Thus, discree duraion models can be regarded as a sequence of binary models. This is convenien because sandard logi sofware can be used o esimae discree duraion models (Jenkins, 1995). Time-varying characerisics We ofen have explanaory variables whose values change during he course of he spell for a given individual. For example, changing levels of unemploymen insurance benefis over ime, or aggregae economic variables. In such cases, he inerpreaion of he hazard rae changes. h (, x i ()) canberegardedas: Pr (T i = T i, x i (1),...,x i ( )) = F (γ + x i () β ). Tha is, he hazard for he disribuion of T i condiioned on he enire {x i ()} process. This is acasewherex is aken as a sricly exogenous variable in duraion ime. Duraion models wih predeermined variables are also possible (see Bover, Arellano, and Benolila, 22, for a discussion). 12

13 Coninuous-ime duraion models wih ime-varying x s pose difficul problems because he x s are no coninuously observed. However, in discree duraion models hey jus involve a rivial exension of he basic models. Unobserved heerogeneiy Le u i be an unobserved covariae disribued independenly of x i wih cdf G. Then Z Z Y 1 Pr (T i = x i )= Pr (T i = x i,u) dg (u) = h i (u) [1 h si (u)] dg (u) where for example h i (u) =F γ + x iβ + u. Acommonspecificaion for G is a mass poin disribuion: u is assumed o ake on {ξ 1,...,ξ m } differen values wih probabiliies {p 1,...,p m }.Thus, Pr (T i = x i )= mx 1 Y h i ξj 1 hsi ξj pj. j=1 Boh he ξ j and he p j are reaed as addiional parameers o be esimaed. 8 Grouped duraion daa: linking discree and coninuous models In a coninuous model {T = } was inerpreed as an observaion from a coninuous process, hence conribuing a densiy funcion erm o he likelihood. In a discree model, {T = } is inerpreed as an observaion of a discree process. Alernaively, i can be regarded as a grouped (or ime-aggregae) observaion from a coninuous process. Tha is, i may be inerpreed as he observaion of he even { T < +1}, wheret is an underlying coninuous duraion random variable. In such case, he conribuion o he likelihood of a spell compleed beween and +1would be Pr ( T<+1)=F ( +1) F () and he conribuion of a spell censored a c: Pr (T >c)=1 F (c) where F is now he coninuous cdf of T. In his conex, he discree hazard rae of he previous secion would be reinerpreed as a grouped hazard : µ Pr (T +1) 1 F ( +1) =1 Pr (T ) 1 F () Pr ( T<+1 T ) = 1 Pr (T +1 T ) =1 = h exp R i +1 h (u) du 1 h exp R i =1 exp h (u) du 13 Z +1 h (u) du

14 where h (u) is he coninuous hazard funcion of T. Proporional hazards from grouped duraions Suppose ha h (u, x) is specified as: Then where h (u, x) =λ (u)exp x β. Pr ( T<+1 T, x) =1 exp γ =ln Z +1 λ (u) du e x β Z +1 λ (u) du =1 exp ³ e γ β +x = F γ + x β and F is he exreme value cdf F (r) =1 exp ( e r ). Thus, he discree-ime hazard akes he form of an exreme value disribuion. Noe ha his follows direcly from he PH specificaion wihou any furher disribuional assumpions. The more radiional approach was o use a Weibull specificaion for λ (u), which implies a paricular parameric funcion for γ (ha in his case would depend on a single parameer). Bruce Meyer (Economerica, 199) propose o censor any ongoing observaions a some large spell lengh T andhenrea γ, γ 1,...,γ T 1 as a vecor of T addiional parameers o be esimaed. From his perspecive, a parameric assumpion abou λ (u) can be regarded as puing resricions on he elemens of γ. We can use LR ess o es parameric specificaions of λ (u) agains he unresriced model. Meyer does so and rejecs he parameric models; however, here is no srong evidence of biases in he esimaes of β due o he Weibull assumpion (Meyer, 199, p. 776). Proporional hazards wih ime-varying covariaes Consider he specificaion h (u, x) =λ (u)exp x (u) β. In such case, assuming ha x (u) is consan for u<+1, i.e. ha he changes in he ime-varying covariaes occur a ineger poins, he discree ime hazard can also be wrien as he exreme value model: Z +1 Pr ( T<+1 T, x) =1 exp e x() β λ (u) du =1 exp ³ e β γ +x = F γ + x β. Noe hough, ha he resul requires ha β does no change wih. This is a convenien device for handling ime-varying covariaes in coninuous ime models, since he x s are only observed a given inervals. 14

15 The sraighforward inerpreaion of he discree hazard model wih he exreme value specificaion as a grouped hazard for he coninuous PH specificaion is an aracive feaure of he exreme value model. However, in some applicaions probi and logi have been seen o ouperform he exreme value probabiliy (c.f. Narendranahan and Sewar, J. of Applied Economerics, 1993). Esimaing a coninuous ime model has he advanage ha he parameers are free from he level of aggregaion or grouping in he daa, hence making he comparison wih esimaes from oher daa ses feasible. Neverheless, economic duraion daa is ofen more suied o discree or grouped specificaions han o coninuous ones, given he naure of he daa where he spells ake on only a small number of differen vales (as in weekly or monhly unemploymen duraions). General mapping beween discree and coninuous duraion models Consider a generic coninuous-ime hazard model h (u, x). Generalizing he previous argumen we have where Pr ( T<+1 T, x) =F [H (, x)] = 1 exp e H(,x) = Z +1 h (u, x) du. ³ e H(,x) To explore he connecion wih he logisic discree duraion model, define he funcion µ F Ã [H (, x)] 1 exp e H(,x)! ϕ (, x) =ln 1 F =ln [H (, x)] exp e H(,x), so ha Z +1 We have, h (u, x) du =ln ³1+e ϕ(,x). Pr ( T<+1 T, x) =Λ [ϕ (, x)] where Λ (r) is he logisic cdf. Thus, when we esimae a logisic model of he form Λ [x β ()], he index x β () can be regarded as an approximaing model for ϕ (, x). This is of ineres because, having esablished he connecion, we can use he logisic esimaes o calculae approximae derivaive effecs for he underlying coninuous-ime model. A derivaive effec for he coninuous-ime hazard inegraed beween and +1is D (, x) = Therefore, x Z +1 Z +1 h (u, x) du = h (u, x) du. x e ϕ(,x) ϕ(, x) = Λ [ϕ (, x)] 1+e ϕ(,x) x 15 ϕ(, x). x

16 So, if we have esimaed he model Λ [x β ()], he following holds as an approximaion: D (, x) ' Λ x β () β (). so ha More generally, for an arbirary cdf G and ϕ (, x) =G 1 [H (, x))] we have Z +1 x h (u, x) du = ln [1 Pr ( T<+1 T, x)] = ln {1 G [ϕ (, x)]}, Z +1 h (u, x) du = g [ϕ (, x)] ϕ(, x) ϕ(, x) = h g [ϕ (, x)] 1 G [ϕ (, x)] x x where h g is he hazard associaed wih G. The conclusion is ha if we have esimaed he model G [x β ()], he following holds as an approximaion: D (, x) ' h g x β () β (). 9 Muliple-exi discree duraion models Consider now a model in which here is more han one possible exi from unemploymen. This secion follows closely he presenaion in Bover and Gomez (24), and as in heir paper we disinguish beween exis o a permanen job and exis o a emporary job. If we have a discree duraion variable T and wo alernaives represened by he indicaors D 1 and D 2,wecandefine he following inensiies of ransiion o each of he saes: φ 1 () = Pr(T =, D 1 =1 T ) φ 2 () = Pr(T =, D 2 =1 T ) such ha he hazard rae from unemploymen is given by: φ H () =φ 1 ()+φ 2 (). Likewise, in order o see he discree duraion models as discree choice models, i is useful o inroduce sequences of exi indicaors a o a given alernaive: Y 1 = 1 (T =, D 1 =1),Y 2 = 1 (T =, D 2 =1) for =1, 2, 3... According o his noaion, φ 1 () =Pr(Y 1 =1 T ) and φ 2 () =Pr(Y 2 =1 T ). Alernaively, we can define exi raes o each of he saes condiional upon no exiing o he alernaive sae: h 1 () = Pr(Y 1 =1 T, Y 2 =) h 2 () = Pr(Y 2 =1 T, Y 1 =). 16

17 The relaionship wih he previous ransiion inensiies is given by: h 1 () = Pr (Y 1 =1 T ) Pr (Y 2 = T ) = and similarly φ 1 () 1 φ 2 () h 2 () = φ 2 () 1 φ 1 (). Thus, unlike he coninuous case, in he conex of discree duraion variables and muliple alernaives, we can choose beween modeling he inensiies φ j () or he condiional hazard raes h j (). For example, if T represens he duraion of unemploymen and exis 1 and 2 are permanen employmen and emporary employmen, respecively, φ 1 () is he probabiliy of exiing o permanen employmen a T = among hose who remain unemployed for a leas T periods. For is par, h 1 () is he probabiliy of exiing o permanen employmen a T = among hose who remain unemployed for a leas T and do no exi o emporary employmen a T =. A specificaion commonly used in muliple choice problems is he mulinomial logi model. In such case, he dependence of φ 1 () and φ 2 () on he explanaory variables x is specified by e x β 1 φ 1 () = 1+e x β 1 + e x β 2 φ 2 () = e x β 2. 1+e x β 1 + e x β 2 Noe ha, in accordance wih he relaionships given above, his specificaion for φ 1 () and φ 2 () implies ha e x β 1 h 1 () = 1+e x β 1 e x β 2 h 2 () =. 1+e x β 2 Tha is, if he ransiion inensiies are mulinomial logi, he condiional exi raes are binary logi wih he same parameers. As a resul, he use of he logisic specificaion leads o he same model in boh cases. Neverheless, having obained esimaes of he parameers β 1, β 2, we can obain wo differen measuremens of he effec of he explanaory variables on he probabiliies of exi o a specific alernaive depending on wheher changes in he φ j () or changes in he h j () are used. Specifically, for a coninuous variable and for he exi o alernaive 1 we can use ε φ1 k = φ 1 () x k. x k φ 1 () or else ε h1 k = h 1 () x k. x k h 1 (). 17

18 I can be easily shown ha he relaionship beween he wo elasiciies is given by ε h1 k = ε φ1 k + φ 2 () 1 φ 2 () ε φ 2 k ε h2 k = ε φ2 k + φ 1 () 1 φ 1 () ε φ 1 k. In addiion, in he logisic case: ε h1 k = β 1k [1 h 1 ()] x k where β 1k denoes he k-h coefficien of he vecor β 1. The differences beween he wo ypes of elasiciy may be greaer when he emporal aggregaion of he duraions is large. An alernaive way of consrucing derivaive effecs is o regard discree duraions as grouped observaions from a coninuous process, and relae he grouped ransiions o he underlying coninuous ransiion inensiies, along he lines of he discussion in he previous secion. Compeing risk models The models for he condiional probabiliies h 1 (), h 2 () are usually called compeing risk models. This name derives from he fac ha if we consider he exisence of wo laen duraion variables T1 and T 2, such ha he observed duraion is T =min(t 1,T 2 ) and T1,T 2 are independen, hen he condiional exi raes can be inerpreed as exi raes for he laen duraions: h 1 () = Pr(T 1 = T 1 ) h 2 () = Pr(T 2 = T 2 ). Tha is, o analyze exis o alernaive 1 we ake he exis o alernaive 2 as censored observaions, and vice versa. Noe ha irrespecive of wheher T1,T 2 correspond o well defined conceps (and in he case of exis o permanen or emporary conracs i is difficul o imagine ha hey do), h 1 (), h 2 () generally represen useful descripive characerisics for he duraions and exis observed. Esimaion of he parameers (β 1, β 2 ) of he logisic model We can consider wo differen mehods for esimaing he model. The firs consiss in he join esimaion of β 1 and β 2 by maximum likelihood, while he second consiss in separae esimaion of β 1 and β 2 by condiional maximum likelihood. Boh mehods provide consisen and asympoically normal esimaes of he parameers, alhough he firs esimaor is in general asympoically more efficien han he second. The advanage of he second is basically ha is compuaion is faser. Moreover, separae esimaors of he parameers corresponding o one of he alernaives are robus o specificaion errors in he regression index for he oher alernaive. 18

19 Consider a sample ha includes boh enrans ino unemploymen and he sock of he unemployed workers a he ime of inerview. Le c i be an indicaor ha akes he value 1 if he end of he period of unemploymen is observed, and if no; Ti denoes he observed duraion, and q i is he number of monhs in unemploymen a he ime of he firs inerview (for an enran q i =1). The join log-likelihood funcion is given by: NX TiX L (β 1, β 2 ) = (1 c i) ln [1 φ 1i () φ 2i ()] i=1 =q i T Xi 1 +c i ln [1 φ 1i () φ 2i ()] + D 1i ln φ 1i T i + D2i ln φ 2i T i. =q i Likewise, using he sequences of indicaors definedabovewecanexpressl (β 1, β 2 ) as L (β 1, β 2 )= max(t i ) X =1 L where L = NX i=1 1 T i q i {ci Y 1i ln φ 1i ()+c i Y 2i ln φ 2i () +(1 c i Y 1i c i Y 2i )ln[1 φ 1i () φ 2i ()]}, which shows ha L (β 1, β 2 ) can be regarded as he log-likelihood of a mulinomial logi model defined on he basis of he concaenaion of he samples surviving a each duraion. The join maximum ³ likelihood esimaors bβ1, β 2 b are defined as he values ha maximize L (β 1, β 2 ). In addiion, he condiional log-likelihood funcion for exi 1 is given by: NX T L c1 (β 1 ) = c Xi 1 i D 1i ln h 1i T i + D1i ln [1 h 1i ()] i=1 =q i TiX +[D 2i +(1 c i )] ln [1 h 1i ()] =q i wih a similar expression for he likelihood corresponding o exi 2, L c2 (β 2 ). Noe ha in L c1 (β 1 ) he exis o alernaive 2 are reaed as censored observaions, so ha formally i is a funcion wih exacly he same form as he likelihood wih a single exi of he previous secion. The implicaion is ³ ha he condiional maximum likelihood esimaors, eβ1, β 2 e defined as he maximizers of L c1 (β 1 ) and L c2 (β 2 ), respecively, can be obained as separae maximum-likelihood esimaes of wo binary logi models. 19

20 References [1] Bover, O., M. Arellano, and S. Benolila (22): Unemploymen Duraion, Benefi Duraion, and he Business Cycle, The Economic Journal, 112, [2] Bover, O. and R. Gómez (24): Anoher Look a Unemploymen Duraion: Exi o a Permanen vs. a Temporary Job, Invesigaciones Económicas, 28, [3] Elbers, C. and G. Ridder (1982): True and Spurious Duraion Dependence: The Idenifiabiliy of he Proporional Hazard Model, Review of Economic Sudies, 49, [4] Heckman, J. and B. Singer (1984): A Mehod for Minimizing he Impac of Disribuional Assumpions in Economeric Models for Duraion Daa, Economerica, 52, [5] Jenkins, S. (1995): Easy Esimaion Mehods for Discree-ime Duraion Models, Oxford Bullein of Economics and Saisics, 57, [6] Lancaser, T. (1979): Economeric Models for he Duraion of Unemploymen, Economerica, 47, [7] Lancaser, T. (199): The Economeric Analysis of Transiion Daa, Cambridge. [8] Meyer, B. (199): Unemploymen Insurance and Unemploymen Spells, Economerica, 58, [9] Narendranahan, W. and M. Sewar (1993): How does he benefi effec vary as unemploymen spells lenghen?, Journal of Applied Economerics, 8, [1] Van den Berg, G. (21): Duraion Models: Specificaion, Idenificaion and Muliple Duraions, in Heckman and Leamer (eds.), Handbook of Economerics, Vol. 5, Ch