Estimating a Semi-Parametric Duration Model without Specifying Heterogeneity

Transcription

1 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety Jerry A. Hausman and Temen M. Woutersen MIT and Johns Hopkns Unversty Draft, August 2005 Abstract. Ths paper presents a new estmator for the mxed proportonal hazard model that allows for a nonparametrc baselne hazard and tme-varyng regressors. In partcular, ths paper allows for dscrete measurement of the duratons as happens often n practce. The ntegrated baselne hazard and all parameters are estmated at regular rate, N, where N s the number of ndvduals. A hazard model s a natural framework for tme-varyng regressors f a flow or a transton probablty depends on a regressor that changes wth tme snce a hazard model avods the curse of dmensonalty that would arse from nteractng the regressors at each pont n tme wth one another. Keywords: Mxed Proportonal Hazard Model, Tme-varyng regressors, Heterogenety. 1. Introducton The estmaton of duraton models has been the subject of sgnfcant research n econometrcs snce the late 1970s. Snce Lancaster (1979), t has been recognzed that t s mportant to account for unobserved heterogenety n models for duraton data. Falure to account for unobserved heterogenety causes the estmated hazard rate to decrease more wth the duraton than the hazard rate of a randomly selected member of the populaton. Moreover, the estmated proportonal effect of explanatory varables on the populaton hazard rate s smaller n absolute value than that on the hazard rate of the Comments are welcome, jhausman@mt.edu and woutersen@jhu.edu. We thank Su-Hsn Chang, Matthew Hardng, and Marcel Voa for research assstance. We have receved helpful comments from Bo Honoré, Moshe Buchnsky and semnar partcpants at Harvard-MIT, UCLA, Texas A&M, Rce Unversty, Yale Unversty, UC Santa Barbara, the Unversty of Maryland, and the Unversty of Vrgna. 1

2 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 2 average populaton member and decreases wth the duraton. To account for unobserved heterogenety Lancaster proposed a parametrc Mxed Proportonal Hazard (MPH) model, a generalzaton of Cox s (1972) Proportonal Hazard model, that specfes the hazard rate as the product of a regresson functon that captures the effect of observed explanatory varables, a base-lne hazard that captures varaton n the hazard over the spell, and a random varable that accounts for the omtted heterogenety. Lancaster s MPH model was fully parametrc, as opposed to Cox s sem-parametrc approach, and from the outset questons were rased on the role of functonal form and parametrc assumptons n the dstncton between unobserved heterogenety and duraton dependence 1. Ths queston was resolved by Elbers and Rdder (1982) who showed that the MPH model s sem-parametrcally dentfed f there s mnmal varaton n the regresson functon. A sngle ndcator varable n the regresson functon suffces to recover the regresson functon, the base-lne hazard, and the dstrbuton of the unobserved component, provded that ths dstrbuton does not depend on the explanatory varables. Sem-parametrc dentfcaton means that sem-parametrc estmaton s feasble, and a number of sem-parametrc estmators for the MPH model have been proposed that progressvely relaxed the parametrc restrctons. Nelsen et al., (1992) showed that the Partal Lkelhood estmator of Cox (1972) can be generalzed to the MPH model wth Gamma dstrbuted unobserved heterogenety. Ther estmator s sem-parametrc because t uses parametrc specfcatons of the regresson functon and the dstrbuton of the unobserved heterogenety. The estmator requres numercal ntegraton of the order of the sample sze, whch further lmts ts usefulness and makes t mpractcal for most stuaton n econometrcs. Heckman and Snger (1984) consdered the non-parametrc maxmum lkelhood estmator of the MPH model wth a parametrc baselne hazard and regresson functon. Usng results of Kefer and Wolfowtz (1956), they approxmate the unobserved heterogenety wth a dscrete mxture. The rate of convergence and the asymptotc dstrbuton of ths estmator are not known. Another estmator that does not requre the specfcaton of the unobserved heterogenety 1 Heckman (1991) gves an overvew of attempts to make ths dstncton n duraton and dynamc panel data models.

3 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 3 dstrbuton was suggested by Honoré (1990). Ths estmator assumes a Webull baselne hazard and only uses very short duratons to estmate the Webull parameter. Han and Hausman (1990) and Meyer (1990) proposes an estmator that assumes that the baselne hazard s pecewse-constant, to permt flexblty, and that the heterogenety has a gamma dstrbuton. We present smulatons and a theoretcal result that show that usng a nonparametrc estmator of the baselne hazard wth gamma heterogenety yelds nconsstent estmates for all parameters and functons f the true mxng dstrbuton s not a gamma, whch lmts the usefulness of the Han-Hausman-Meyer approach. Thus, we fnd t mportant to specfy a model that does not requre a parametrc specfcaton of the unobserved heterogenety. Horowtz (1999) was the frst to propose an estmator that estmates both the baselne hazard and the dstrbuton of the unobserved heterogenety nonparametrcally. Hs estmator s an adaptaton of the sem-parametrc estmator for a transformaton model that he ntroduced n Horowtz (1996). In partcular, f the regressors are constant over the duraton then the MPH model has a transformaton model representaton wth the logarthm of the ntegrated baselne hazard as the dependent varable and a random error that s equal to the logarthm of a log standard exponental mnus the logarthm of a postve random varable. In the transformaton model the regresson coeffcents are dentfed only up to scale. As shown by Rdder (1990) the scale parameter s dentfed n the MPH model f the unobserved heterogenety has a fnte mean. Horowtz (1999) suggests an estmator of the scale parameter that s smlar to Honoré s (1990) estmator of the Webull parameter and consstent f the fnte mean assumpton holds so that hs approach allows estmaton of the regresson coeffcents (not just up to scale). However, the Horowtz approach only permts estmaton of the regresson coeffcents at a slow rate of convergence and t s not N 1/2 consstent, where N s the sample sze. In practce, there may be three obstacles for applyng Horowtz (1999) MPH estmator. Frst, the duratons need to be measured at a contnuous scale n order to estmate the transformaton model. Ths condton often does not hold n economc data, e.g. unemployment duraton data as dscussed n Han and Hausman (1990). Second, lke the transformaton model, the

4 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 4 MPH estmator does not allow for tme-varyng regressors. Fnally, the estmator reles on arbtrarly short duratons to estmate the scale parameter and, therefore, converges slowly. Thus, the regresson coeffcent estmates, whch are often of prmary nterest, are often not estmated very precsely 2. In ths paper, we derve a new estmator for the mxed proportonal hazard model (wth heterogenety) that allows for a nonparametrc baselne hazard and tme-varyng regressors. No parametrc specfcaton of the heterogenety dstrbuton nor nonparametrc estmaton of the heterogenety dstrbuton s necessary. Intutvely, we condton out the heterogenety dstrbuton, whch makes t unnecessary to estmate t. Thus, we elmnate the problems that arse wth the Lancaster (1979) approach to MPH models. In our new model the baselne hazard rate s nonparametrc and the estmator of the ntegrated baselne hazard rate converges at the regular rate, N 1/2, where N s the sample sze. Ths convergence rate s the same rate as for a duraton model wthout heterogenety. The regressor parameters also converge at the regular rate. A nce feature of the new estmator s that t allows the duratons to be measured on a fnte set of ponts. Such dscrete measurement of duratons s mportant n economcs; for example, unemployment s often measured n weeks. In the case of dscrete duraton measurements, the estmator of the ntegrated baselne hazard only converges at ths set of ponts, as would be expected. It may be argued that the bas n the estmates of the regresson coeffcents s small, f the estmates of the MPH model ndcate that there s no sgnfcant unobserved heterogenety. The problem wth ths argument s that estmates of the heterogenety dstrbuton are usually not very accurate. Gven the results n Horowtz (1999) ths fndng should not come as a surprse. The smulaton results n Baker and Melno (2000) show that t s emprcally dffcult to fnd evdence of unobserved heterogenety, n partcular f one chooses a flexble parametrc representaton of the baselne hazard. However, Han- Hausman (1990) and applcatons of ther approach have found sgnfcant heterogenety usng a flexble approach to the baselne hazard. Bjwaard and Rdder (2002) fnd that the 2 It should be noted that the slower than N 1/2 convergence of Horowtz (1999) estmator s a property of the model. Hahn (1994) and Ishwaran (1996a) show that no estmator can converge at rate N 1/2 under the assumptons of Horowtz (1999). Horowtz (1999) assumes that the frst three moments of the heterogenety dstrbuton exst and Ishwaran (1996b) shows that the fastest possble rate of convergence s N 2/5 for that case and Horowtz (1999) estmator converges arbtrarly close to that rate.

5 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 5 bas n the regresson parameters s largely ndependent of the specfcaton of the baselne hazard. Hence, falure to fnd sgnfcant unobserved heterogenety should not lead to the concluson that the bas due to correlaton of the regressors and the unobservables that affect the hazard s small. Because t s emprcally dffcult to recover the dstrbuton of the unobserved heterogenety, estmators that rely on estmaton of ths dstrbuton may be unrelable. Therefore, we avod estmatng the unobserved heterogenety dstrbuton. Nevertheless, we can dentfy and estmate the regresson parameters and the ntegrated baselne hazard. We fnd the removal of the requrement to estmate the heterogenety dstrbuton a major advantage of our approach 3. Our estmator s related to the estmator by Han (1987). Han derves an estmator, up to scale, of the regresson coeffcents. However, Han s estmator cannot handle tme-varyng regressors and we estmate the regresson coeffcents when tme-varyng regressors are present as well as the scale of the regresson coeffcents. In partcular, by estmatng the regresson coeffcents up to scale, each regresson coeffcent can be nterpreted as the elastcty of the hazard wth respect to ts regressor. Smlarly, Chen s (2002) estmator of the transformaton model cannot handle tme-varyng regressons and only gves the transformaton functon up to scale. Whle Horowtz s (1999) estmator s not subject to the lmtaton of estmatng the regresson coeffcents up to scale only, t converges slowly and t s not N 1/2 consstent whch makes standard nferences technques napplcable unless N s very large. A hazard model s a natural framework for tme-varyng regressors f a flow or a transton probablty depends on a regressor that changes wth tme snce a hazard model avods the curse of dmensonalty that would arse from nteractng the regressors at each pont n tme wth one another. A nonconstructve dentfcaton proof for the duraton model wth tme-varyng regressors can be produced usng technques smlar to Honoré (1993b) and Honoré (1993a) gves such a proof. In partcular, Honoré (1993a) does not assume that the mean of the heterogenety dstrbuton s fnte 4. Rdder and Woutersen (2003) argue that t s precsely the fnte mean assumpton that makes the dentfcaton 3 An uncondtonal approach s also used, n another context, by Heckman (1978) who develops uncondtonal tests to dstngush true and spurous state dependence. 4 nor does Honoré (1993a) assume a tal condton as n Heckman and Snger (1985).

6 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 6 of Elbers and Rdder (1982) weak n the sense that the model of Elbers and Rdder (1982) cannot be estmated at rate N 1/2. As n Honoré (1993a), we do not need the fnte mean assumpton whch gves an ntutve explanaton why we can estmate the model at rate N 1/2. Ths paper s organzed as follows. Secton 2 dscusses the mxed proportonal hazard model (wth heterogenety) and presents our estmator. Secton 3 shows that our estmator converges at the regular rate and s asymptotcally normally dstrbuted. Secton 4 adjust the objectve functon for the case of an endogenous regressor. Secton 5 shows that msspecfyng the heterogenety yelds nconsstent estmates, even f the baselne hazard s nonparametrc. Secton 6 presents an emprcal example and secton 7 concludes. 2. Mxed Proportonal Hazard Model Lancaster (1979) ntroduced the mxed proportonal hazard model n whch the hazard s a functon of a regressor X, unobserved heterogenety v, and a functon of tme λ(t), θ(t X, v) =ve Xβ 0 λ(t). (1) The functon λ(t) s often referred to as the baselne hazard. The popularty of the mxed proportonal hazard model s partly due to the fact that t nests two alternatve explanatons for the hazard θ(t X) to be decreasng wth tme. In partcular, estmatng the mxed proportonal hazard model gves the relatve mportance of the heterogenety, v, and genune duraton dependence, λ(t), see Lancaster (1990) and Van den Berg (2001) for overvews. Lancaster (1979) uses functonal form assumptons on λ(t) and dstrbutonal assumptons on v to dentfy the model. Examples by Lancaster and Nckell (1980) and Heckman and Snger (1984), however, show the senstvty to these functonal form and dstrbutonal assumptons. We avod theses functonal form and dstrbutonal assumptons and consder the mxed proportonal hazard model wth tme-varyng regressors, θ(t x(t),v)=ve x(t)β 0 λ(t) (2) where x(t) s a set of regressors that can vary wth tme, v denotes the heterogenety and s ndependent of x(t) and λ(t) denotes the baselne hazard. We also use x(t) to denote the sequence of the regressors x(s) for s =0to s = t. The mxed proportonal hazard

7 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 7 model of equaton (2) mples the followng survval probabltes, P (T t x(t),v)= F (t x(t),v)=exp( v Z t P (T t x(t)) = E v { F (t x(t),v)} = E v {exp( v 0 e x(s)β 0 λ(s)ds) and Z t 0 e x(s)β 0 λ(s)ds)}. (3) In appled work, duraton are measured dscretely and to fx deas we assume that the duraton are measured on a weekly scale. We also assume that the regressors could only change at the begnnng of the week. Let the regressor x 1 denote the vector of regressors of ndvdual durng week 1, x 2 the regressors of ndvdual durng week two etc. We now can wrte equaton (3) as follows, P (T t x(t)) = E v { F tx (t x(t),v)} = E v {exp( v e x sβ 0 +δ 0,s )}, s=1 where t s a natural number, δ 0,s =ln{ R s s 1 λ(s)ds} and we normalze δ 0,1 =0. Ths specfcaton s smlar to Han-Hausman (1990) who specfy δ 0,s n a smlar manner, but who specfy and estmate v parametrcally, a requrement we remove n ths paper. Kendall (1938) proposes a statstc for rank correlaton. If we are nterested n the rank correlaton between T and the ndex Xβ, then Kendall s (1938) rank correlaton has the followng form, Q(β) = 1 N(N 1) X X 1{T >T j }1{X β>x j β}. j Han (1987) proposes an estmator that maxmzes Q(β), the rank correlaton between T and the ndex Xβ. Under certan assumptons, ncludng that the T only depends on X through the ndex Xβ, maxmzng Q(β) yelds an estmate for β up to scale, excludng the ntercept whch cannot be estmated. 5. However, Kendall s (1938) rank correlaton cannot be used for the case of tme-varyng regressors snce t s unclear whch regressor one should use. We therefore propose the followng modfcaton of the rank correlaton. In partcular, n our model, the expectaton does depend on an ndex, although t has a more complcated form. Defne Z (l; β,δ) = 5 For ths reason, Han (1987) estmates β/ β ; alternatvely, a nonzero coeffcent of a regressor could be normalzed to be one n absolute value, e.g. β 1 =1.

8 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 8 P l s=1 ex sβ+δ s. We propose mnmzng the followng objectve functon, 1 X X LX KX Q(β,δ) = [1{T l} 1{T j k}]1{z (l; β,δ) <Z j (k; β,δ)}. N(N 1) j l=1 k=1 (4) Thus, Z (l; β,δ) s the ndex durng the l th perod. The ntuton for ths objectve functon s the followng. We are comparng two dfferent ndvduals as does Han s objectve functon. However, we are now also takng account of the outcome n each perod through the parameters for the ntegrated hazard functon, δ. The probablty that ndvdual survves perod l s larger than the probablty that ndvdual j survves perod k f and only f Z (l; β 0,δ 0 ) <Z j (k; β 0,δ 0 ). Vce versa f Z (l; β 0,δ 0 ) >Z j (k; β 0,δ 0 ). Thus, we use the outcomes for ndvduals and j together wth these probabltes to yeld an objectve functon that permts dentfcaton of the parameters β 0 and δ 0, wthout the restrcton of only up to scale as n the Han approach. Consder the expectaton of the objectve functon, E{Q(β,δ)} = P P P L P K j l=1 k=1 E[{e vz (l;β 0,δ 0 ) e vz j(k;β 0,δ 0 ) } 1{Z (l; β,δ) <Z j (k; β,δ)}]. N(N 1) Ths expectaton of the objectve functon s mnmzed at the true value of the parameters. To see ths, suppose that Z (l; β 0,δ 0 ) >Z j (k; β 0,δ 0 ) so that e vz (l;β 0,δ 0 ) <e vz j(k;β 0,δ 0 ). Thus, {β,δ} = {β 0,δ 0 } mnmzes [E v {e vz,(l;β 0,δ 0 ) e vz j(k;β 0,δ 0 ) }1{Z (l; β,δ) <Z j (k; β,δ)} Z] for each set {, j, k, l} and therefore the expectaton of the sum. 6 Note that our approach focuses on the probablty than an ndvdual survves perod l (measured from tme 0) whch permts a convenent treatment of the heterogenety n comparson wth the tradtonal approach that focuses on the hazard functon. By only usng comparsons measured from tme t =0we are able to condton out the heterogenety dstrbuton. The more tradtonal hazard approach consders the probablty of survval condtonal on ndvdual survvng up to perod l whch requres an explct treatment of the heterogenety dstrbuton. The defnton of Q(β,δ) that s gven above contans a double sum so that the number of computatonal operatons for calculatng Q(β,δ) s N 2 (note that L and K are fxed). 6 Intheappendx1weshowthatthetruevalueunquely mnmzes the expectaton of the objectve functon.

9 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety 9 In order to reduce the number of computatonal operatons to be of the order N ln N, we use the rank operator. In partcular, let d r =1{T r} for the vector T of length N. Let d be constructed by stackng the vectors d r vertcally for all r =1,..., K. Now both d and Z are of dmenson NK 1. If a regressor s contnuously dstrbuted condtonal on the other regressors, then we can re-wrte Q(β,δ) usng these vectors and the rank functon, Q(β,δ) = 1 NKX d(j)[2 Rank(Z(j)) NK]. N(N 1) j=1 The computatonal burden to calculate 7 Q(β,δ) s proportonal to N ln(n). Note that we have dentfcaton of β rather than dentfcaton only up to an unknown scale coeffcent, whch s the usual outcome of most prevous approaches to the problem. Also, note that by focussng on survval from the begnnng of the sample, we have elmnated the requrement to specfy the heterogenety dstrbuton snce no survval bas (dynamc sample selecton) occurs n our sample comparsons. Our dentfcaton s somewhat smlar to the nonconstructve dentfcaton result of Elbers and Rdder (1982) n the sense that we also assume a contnuously dstrbuted regressor. However, our dentfcaton results dffers n two mportant ways. Frst, our dentfcaton proof s constructve n the sense that t suggests an estmator. Second, our dentfcaton result does not rely on an teratve procedure. An teratve procedure typcally precludes N 1/2 consstency LargeSamplePropertes In ths secton, we derve the large sample propertes of our estmator. Suppose that x k = {x 1,..., x k },x 1,..., x k arescalars. Wesaythatthedenstyoftheregressorspostve around x k f at least one element of x k s contnuously dstrbuted and the densty s postve around all contnuously dstrbuted elements of x k. We assume that we observe {T,x } where T s a natural number and T [0,K], K>1. For example, we observe unemployment duraton, whch s measured n weeks, and want to estmate the ntegrated baselne hazard at the end of each week. We assume the followng. 7 SupposewehaveanorderedvectoroflengthN 1; calculatng the rank of a new, N th observaton s ln(n). We can see ths by observng that havng 2(N 1) elements to begn wth would requre us to compare the new observaton to the medan of the 2(N 1) elements; we are then back to comparng the new element to N 1 observaton. Thus, the extra cost s ln(n). The summaton then yelds the rate N ln(n). 8 Indeed, Hahn (1994) shows that the dentfcaton result of Elbers and Rdder (1982) holds for sngular nformaton matrces, so that no N estmator exsts.

10 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety10 Assumpton 1 (Tme-varyng Regressors): Let () {T,v,x} be a random sample, x = {x 1,..., x K },x 1,...,x K are scalars, {T,x} be observed and K 2; () v and x are ndependent, () Pr(T l x) = E v exp{ v P l s=1 exsβ+δs } for l =1,..., K; (v) δ 1 s normalzed to be zero, let {β,δ} Θ, whch s compact; (v) let G be a K by K matrx and let the element G lk be equal to one f apar{x l,x k } R 2K such that Pr(T l x l )=Pr(T k x k ) where the densty of the regressor s postve n an arbtrarly small neghborhood around x l or x k and let G lk be zero otherwse; let the matrx G represent a connected graph; (v) ether (a) x r = x c f x 1 = x 2 =... = x r = x c (thus, x 1 = x c ),Pr(x K = x c ) > 0 for some x c R, let G be a K by K matrx and let the element G lk be equal to one f apar{x c,x k } R K+1 such that Pr(T l x c )=Pr(T k x k ) where the densty of the regressor s postve n an arbtrarly small neghborhood around x c or x k and let G lk be zero otherwse; let the matrx G represent a connected graph; or (b) x l,1 x l,2,x l,3,..., x l,k,x l 1,x l 2,... s contnuously dstrbuted for all l, and Pr(T l x l )=Pr(T k x k ) where x l,1 s n the nteror of the support of x l,1 x l,2,x l,3,..., x l,k,x l 1,x l 2,... for all k; (v) apar{x 1,x 0 1,x 0 2} R 3,x 0 1 6= x 0 2, such that 0 < Pr(T 1 x 1,x 2 )=Pr(T 2 x 0 1,x0 2 ) < 1 where the densty of the regressor s postve n an arbtrarly small neghborhood around x 1 or {x 0 1,x 0 2}. Condtons ()-(v(a)) ensure dentfcaton up to scale and condton ()-(v) ensures complete dentfcaton 9. Condton () s satsfed f the data generatng process s the mxed proportonal hazard model of equaton (2) wth exogenous regressors that can change at the begnnng of each perod. Condton (v) assumes that ether (a) the regressor are constant wth postve probablty or (b) a regressor s contnuously dstrbuted condtonal on the other regressors and earler realzatons of the regressors. The substantal restrcton of condton (v) s that one of the regressors vares wth tme; the theorem below stll holds f condton (v) holds after relabellng the perods. For example, one can label week 1 through 8 as perod 1 so that condton (v) holds whle the other condton hold before or after relabellng. 9 Matrces wth only zeros and ones can be represented by graphs; a connected graph means that, nformally speakng, you can travel from one pont to any other pont but not necessarly drectly. Condton (v) s consderably weaker than a condton that a regressor has a postve condtonal densty on the whole real lne.

11 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety11 Theorem 1: Let assumpton 1 hold. Let δ be contaned n a compact subset D of R K and normalze δ 1 =0. Let {ˆβ,ˆδ} = argmn β,δ Then Q(β,δ) where 1 X X KX KX Q(β,δ) = [1{T l} 1{T j k}]1{z (l) <Z j (k)}. N(N 1) j l=1 k=1 {ˆβ,ˆδ} p {β,δ}. Suppose that the regressor s a vector nstead of a scalar. The easest way to prove dentfcaton for that case s by notng that one can dentfy the regressor up to scale usng only observatons of the frst perod. In partcular, the parameter vector could be estmated up to scale usng the maxmum rank correlaton estmator (MRC). Rank correlaton was ntroduced by Kendall (1938) and Han (1987) proposed the MRC estmator. Han (1987) assumes that one regressor has nfnte support condtonal on all other regressors. We weaken ths assumpton. In partcular, f all regressors are dstrbuted contnuously, then we only requre one regressor s contnuously dstrbuted condtonal on the others wthout any support restrctons. In order to estmate β up to scale, we assume the followng. Assumpton 2: Let () β be contaned n a compact subset B e of R q () Pr(T 1 x 1 )=G(x 1 β,v) where G(.,.) s a strctly monotonc decreasng functon n ts frst argument; () {T,v,x} be a random sample (v) let x 1 = {x 1,1, x 1 }, let S denote that support or a subset of the support of x 1, let S1 denote the nteror of the support of the contnuously dstrbuted x 1,1 condtonal on x 1 and let, for all x 1 S 1, there be an x 1,1 S 1 such that 0 < Pr(T 1 x 1,1, x 1 )=p<1 for some p; (v) let S denote the support of x condtonal on x S 1 and assume that ths support of x, S, s not contaned n any proper lnear subspace of R q, (v) β 1 6=0, and (v) v and x are ndependent.

12 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety12 Proposton 1 Let assumpton 2 hold. Let {ˆκ} = argmn Q(κ) where κ: κ 1 =1 Then Q(κ) = 1 N(N 1) X X [1{T 1} 1{T j 1}]1{Z (1) <Z j (1)}. j ˆκ p β/ β 1. Ths proposton states that β can be estmated up to scale under weaker support condtons than presented by Han (1987). In partcular, f all regressors are dstrbuted contnuously, then we only requre that x 1,1 s contnuously dstrbuted on a small nterval wthout assumng that t has support over the whole real lne 10. If regressor has a dscrete dstrbuton and the support of the contnuously dstrbuted varables s small, then we can frst condton on the regressor wth the dscrete dstrbuton and dentfy the whole model usng theorem 1 and proposton 1. We then can try to dentfy the coeffcent on the regressor usng the objectve functon of equaton (4). Alternatvely, we can check whether ths objectve functon emprcally dentfes the parameters. Suppose that none of the regressors vares wth tme (most lkely, ths would be due to qualty of the data) and that we want to estmate β up to scale. We can then use the objectve functon of equaton (4). Besdes the mld support condton, ths objectve functon can also handle known censorng ponts that depend on the regressors whle Han s (1987) objectve functon cannot handle such censorng. Choosng G(x 1 β)=e v exp( ve x1β ) n proposton 1 and combnng the theorem 1 and proposton 1 yelds a consstency result for {ˆβ,ˆδ 2,..., ˆδ K }. Thus, nstead of estmaton of β up to scale, the objectve functon Q(β,δ) permts estmaton of the β, ncludng the scale. Theorem 2 (Consstency): Let assumpton 1-2 hold. Let Pr(T l x) =E v exp( P s=l s=1 vexsβ+δs ). Let δ be contaned 10 To see ths, consder choosng S such that x 1 β 1,0 < x β 0 <x 1 β 1,0 for some x 1 and x 1 S 1 and note that there must exst such an x 1 and x 1 snce S 1 contans an nterval.

13 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety13 n a compact subset D of R K and normalze δ 1 =0. Then {ˆβ,ˆδ} p {β,δ}. 3.1 Asymptotc Dstrbuton In ths subsecton, we derve the asymptotc dstrbuton of our estmator. As before, we use the followng objectve functon, where θ = {β,δ}, 1 X X LX KX Q N (θ) = [1{T l} 1{T j k}]1{z (l) <Z j (k)}. N(N 1) In the appendx, we show that Q N (θ) = j P N l=1 k=1 LX 1{T l}k[1 2 ˆF Z {Z (l)}]. (5) l=1 where ˆF K j k=1 Z {Z (l)} = N 1 K 1{Z j(k) <Z (l)}. Note that E[ ˆF Z {Z (l)} Z (l)] = F Z {Z (l)} where F Z s the cumulatve dstrbuton functon of Z (l) for l =1,..., K and =1,...,N. Let Q 0 (θ) be twce contnuously dfferentable at θ 0 wth respect to θ and let H denote the second dervatve dvded by the constant K and evaluated at θ 0,.e. H = 1 K θθq 0 (θ 0 ). We assume the followng. Assumpton 3 (Interor): Let θ 0 =(β,δ) Interor( Θ), where Θ s compact. Let f Z {Z (l)} denote the densty of Z (l). Assumpton 4: Let () the second dervatve H be nonsngular; () let f Z (z) be dfferentable and let f Z (z) Z θ <M for all θ, df Z{z} dz <M for all z and for some M<. Assumpton 4 s a standard regularty condton and supports an argument based on a Taylor expanson We cannot mmedately apply Sherman (1993) snce he requres that Q N (θ 0 ) Q 0 (θ 0 )=O p (N 1 ), an assumpton that s volated for our objectve functon. Therefore, we apply Newey (1991) and Newey and McFadden (1994, lemma 2.8 and secton 7).

14 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety14 Theorem 3 (Asymptotc Normalty) Let assumpton 1-4 hold. Then N{ˆθ θ} d N(0,H 1 ΩH 1 ) where Ω = E[D N (θ 0 )D N (θ 0 ) 0 ] and P LX D N (θ) = 2 [ N l=1 1{T l}f Z {Z (l)} Z (l) θ LX E[ l=1 1{T l}f Z {Z (l)} Z (l) ]]. θ The functon D N (θ) s an approxmate dervatve and an nfluence functon n the termnology of Newey and McFadden (1994). It allows to vew the asymptotc behavor of an estmator as an average, multpled by N. Moreover, bootstrappng an asymptotcally normally dstrbuted estmator that can be represented by an nfluence functon yelds a consstent varance-covarance matrx and consstent confdence ntervals, see Horowtz (2001, theorem 2.2) 12. In the applcaton, we bootstrap the estmator. The matrx Ω = E[D N (θ 0 )D N (θ 0 ) 0 ] can be estmated usng a sample analogue where f Z {Z (l)} can be estmated usng a second order kernel that omts observaton. In order to estmate H let e denote the th unt vector, ε N a small postve constant that depends on the sample sze, and Ĥ the matrx wth, jth element Ĥ j = 1 4ε 2 [ ˆQ(ˆθ+e ε N +e j ε N ) ˆQ(ˆθ e ε N +e j ε N ) ˆQ(ˆθ+e ε N e j ε N )+ ˆQ(ˆθ e ε N e j ε N )]. N Lemma 1 (Newey and McFadden, Estmatng H) Let the condtons of theorem 3 be satsfed. Let ε N 0 and ε N N. Then Ĥ p H. Theorem 3 requres the regressors to be exogenous. Sometmes a regressor can qualfy as an exogenous regressor, even f ts value depend on survval up to a certan pont. For example, a treatment that s randomly assgned wth probablty p h to ndvduals who survved h perods may appear to be endogenous snce t depends on survval. However, n ths duraton framework, we can relabel the treatment as f t s gven at the begnnng of the spell wth probablty p h and consder the randomly assgned treatment 12 Horowtz (2001, theorem 2.2) averages g n(x ).

15 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety15 exogenous 13. In the next secton, we consder endogenous regressors, such as randomly assgned treatment wth partal complance. Our estmates of {δ 1,..., δ K } mply an estmate for the the ntegrated hazard. In partcular, suppose that we measure survval at {0, 1,..., K}, e.g. weekly unemployment data, then Xs=t dλ(t) = exp(ˆδ s ) where t {0, 1,..., K}. s=1 We defne the average hazard on the nterval [a, b) to be the value λ for whch R b a λ(s)ds = Λ(b) Λ(a). Ths gves an expresson for the average hazard, dλ(s) =exp(ˆδ t ) for t 1 <s<t. If the duraton are measured on a fne grd, then one could also approxmate the hazard by numercally dfferentatng the ntegrated hazard Λ(t). d Thus, we can estmate the ntegrated hazard rate at each pont and also approxmate the hazard rate at each pont. Ths dffers consderably from Chen (2002), who only estmates the logarthm of the ntegrated hazard up to a unknown scalar, so that we do not know whether the hazard s ncreasng or decreasng. 4. An Endogenous Regressor The last secton dealt wth exogenous regressors. However, some regressors are endogenous n the sense that the regressor depends on the unobserved heterogenety. Ths stuaton occurs often n panel data and the geness of the problem and an approach to a soluton to the problem are dscussed n e.g. Mundlak (1961), Hausman and Wse (1979) and Hausman and Taylor (1981). For example, n the Natonal Supported Work Demonstraton 14 data, long term unemployed ndvduals are randomly offered tranng but some choose not to partcpate. Thus, there s a partal complance problem and the treatment ndcator can depend on unobserved heterogenety. See also Heckman, LaLonde, and Smth (1999). The duraton model of ths paper gves a natural framework to handle survval 13 In partcular, ndvduals that do not survve up to perod h wll be assgned treatment wth probablty p h ; an alternatve s to use a weghtng functon that gves the weths p h and (1 p h ) to both possble outcomes. 14 Ham and LaLonde (1996) dscuss ths data

16 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety16 selecton and tme-varyng regressors. As dscussed n the last secton, the model can handle survvor selecton wthout usng nstrumental varables. However, some treatment are not just endogenous n the sense of dynamc or survval selecton but are also endogenous n the sense that the treatment stll depends on the unobserved heterogenety v, even after condtonng on survval. Let R {0, 1} denote the treatment assgnment and let X {0, 1} denote actual treatment. Let R be randomly assgned among the ndvduals that are unemployed at tme h. Suppose that an ndvdual can refuse treatment, that s, we can observe R =1and X =0for a partcular ndvdual. The refusal of treatment, or equvalently, the choce of partcpatng, can potentally depend on the unobserved heterogenety v or on the observed regressors. If the probablty of X depends on v, the dstrbuton p(v X =1)s dfferent from p(v X =0). In partcular, let X be a functon of R, v and other exogenous regressor and random nose. Snce the dstrbuton of v depends on X, we have, n general, E v {e vz (l) Z (l),x =0} 6= E v {e vz j(l) Z j (l) =Z (l),x =1}. Therefore, E v {e vz (l) Z (l),x} may not be decreasng n Z (l). Therefore, we need to adjust the objectve functon Q(β,δ) that was ntroduced above, 1 X X LX KX Q(β,δ) = [1{T l} 1{T j k}]1{z (l; β,δ) <Z j (k; β,δ)}. N(N 1) j l=1 k=1 (6) One can vew the ndcators 1{T l} and 1{T j k} as estmators of survval functons. In order to deal wth the self selecton nto treatment, we replace these ndcators by other estmates of survvor functons. In partcular, we replace 1{T l} and 1{T j k} by survvor functons that have the same unobserved heterogenety dstrbuton so that we do not have to explctly model the dstrbuton of the heterogenety. Suppose that ndvduals are treated at the begnnng of perod h. In order to avod survval bas, we condtononsurvvaluptoh andalsoonthendexath 1, Z(h 1). Let R denote the treatment ntenton 15, X the actual treatment and R, X {0, 1}. For now, we assume that R =0mples X =0and that P (X =1 R =1)>P(X =1 R =0). Suppose that there are ndvduals that could have dfferent values of R or X but that have dentcal values of 15 The support of any nstrument can be reduced to two ponts.

17 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety17 exogenous regressors. For example, they became unemployed at the same tme and also have the same exogenous regressors but can have dfferent values of R or X. Let these groups be denoted by G 1,..., G M and let κ be an estmate of the odds rato, κ = 1 ˆp ˆp where ˆp = 1{T h}1{r=1} 1{T h}. For each group, we construct the followng dstrbuton functons, ˆF g,11 = P P K g l=h 1{T l}1{x =1} P K l=h 1{T h}1{x =1} and P g ˆF g,00 = = P P K g l=h [1{T l}1{r =0}1{X =0} κ 1{T l}1{r =1}1{X =0}] P P K g l=h [1{T h}1{r =0}1{X =0} κ 1{T h}1{r =1}1{X =0}] P P K g l=h 1{T l}1{x =0}[1{R =0} κ 1{R =1}] P K l=h 1{T h}1{x =0}[1{R =0} κ 1{R =1}]. P g We use these functons nstead of 1{T l} and 1{T j k} n equaton (6). Defne Z g,11 (l) as the ndex of the the ndvduals of group g wth R = X =1. Smlarly, Z g,00 (l) s the ndex of group g for whch R = X =0. Defne P KX KX Q g 1(β,δ) = [ M ˆF g,11 (l) ˆF g,00 (k)]1{z g,11 (l) <Z g,00 (k)}. l=h k=h For the perods 1, 2,..., (h 1) wecanusethesamestatstc 16 as n earler sectons, Q 2(β,δ) = 1 N(N 1) X X h 1 X h 1 X [1{T l} 1{T j k}]1{z (l; β,δ) <Z j (k; β,δ)}. j l=1 k=1 Moreover, snce R =0mples X =0we can also defne R as n the last secton. In partcular, Let R = R for all ndvduals who have been assgned treatment and for all others we assgn R =0wth probablty 1 ˆp,.e. P (R =0 R =0)=1and P (R =0 T <h)=1 ˆp. We then defne P P j KX KX Q 3(β,δ) = N 1(R =0) N 1 1(R j =0) [1{T l} 1{T j k}]1{z (l; β,δ) <Z j (k; β,δ)}. l=1 k=1 We then mnmze the followng objectve functon, Q (β,δ) =w 1 Q 1(β,δ)+w 2 Q 2(β,δ)+(1 w 1 w 2 )Q 3(β,δ) where 0 <w 1 < 1 and 0 <w 2 < 1. Let the random assgnment or nstrumental varable assumptons mentoned above hold and let the specfcaton assumptons of theorem 2 16 If we calculate the rank of the ndces Z k for k =1,..., (h 1) and =1,..., N, then we can use N(h 1) Q(β, δ) = j=1 d(j)[2 Rank(Z(j)) N(h 1)]. 1 N(N 1)

18 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety18 hold so that Q 1 (β,δ) has a maxmum at the true value of the parameters. If M, the number of groups, s fnte, than Q 1(β,δ) does not pont dentfy any parameters. Ths complcates the choce of the weghts. Suppose that the lmt of Q 1(β,δ), Q 2(β,δ) and Q 3 (β,δ), together, dentfy the parameters but that Q 1 (β,δ) does not play a role n local asymptotcs. Let the assumptons of theorem 2 and 3 hold for the objectve functons Q 2(β,δ) and Q 3(β,δ). In that case, consstency and asymptotc normalty follows from theorem 2 and 3 for any 0 <w 1 < 1 and 0 <w 2 < 1. In partcular, after choosng w 1 = w 2 = 1 3 to get an ntal estmate, one can calculate the asymptotc varancecovarance matrx usng theorem 2 and lemma 1 usng Q 2(β,δ) and Q 3(β,δ). The rato w 2 /(1 w 1 w 2 ) can then be chosen to mnmze a functon of the asymptotc varance. In a second step, one can mnmze Q (β,δ) usng the estmate the rato estmates for the weghts 17. For fnte M, one can use Q 1(β,δ) wthout usng Q 2(β,δ) or Q 3(β,δ) to derve bounds but ths s beyond the scope of the paper. The objectve functon Q 1(β,δ) can be nterpreted as condtonng on both survval up to the end of perod h as well as Z(h) whch removes possble dependence between treatment assgnment and the unobserved heterogenety term. Ths data generatng process resembles the data of Ham and LaLonde (1996); see also Heckman, LaLonde, and Smth (1999). We can extend the analyss n a straghtforward manner to the stuaton of noncomplance n both treatment and control ndvduals, so that R =1and X =0for a partcular ndvdual and R =0and X =1for another ndvdual. However, snce the latter stuaton s relatvely unlkely to occur n practce, we leave the detals as an exercse. 5. Gamma Mxng Dstrbuton Han and Hausman (1990) and Meyer (1990) use a flexble baselne hazard and model the unobserved heterogenety as a gamma dstrbuton. In ths secton we dscuss the senstvty of the estmators of the MPH model to msspecfcaton of the mxng dstrbuton. In partcular, msspecfyng the heterogenety yelds nconsstent estmators and havng a flexble ntegrated baselne hazard Λ(t) does not compensate for a falure to control for 17 An easer but computatonally more ntensve way s to determne the asymptotc varance usng bootstrap and to try several values for w 1 [0, 1], w 2 [0, 1].

19 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety19 heterogenety. We llustrate ths usng two examples. Example 1: Suppose we estmate the followng hazard model, θ(t v, x) =φ x λ(t). The functon λ(t) s nonparametrc and one could (ncorrectly) thnk that the flexblty of ths functon compensates for the lack of unobserved heterogenety. Ths model mples that the followng survvor functon, P (T t x) = F (t x) =exp( φ x Λ(t)). Suppose we observe F (t x) for x =0, 1 and all t 0. Wedefne F 0 (t) = F (t x =0)and estmate Λ(t), ˆΛ(t) = ln F (t x =0) For a gven ˆΛ(t) = ln F (t x =0), the MLE of φ can be derved (see appendx) and t can be shown that 1 plm ˆφ = N E[ln{ F 0 (T )} x =1]. where F 0 s the survval functon for x =0. Suppose that v Gamma(α, α), so that ³ α ³ F 0 (t) = and ln F0 (t) =α ln. Note that φ x Λ(T )= Z v where Z 1+ Λ(t) α 1+ Λ(t) α has an exponental dstrbuton wth mean one. Ths yelds 1 plm ˆφ = N E[α ln{1+z/(φvα}]. where v Gamma(α, α). Note that φ only appears n the denomnator of an argument of a logarthmc functon. Ths does not bode well for consstency. Usng N = 10, 000 we fnd the followng, True φ True α plm ˆφ φ =2 α =1 ˆφ =1.46 φ =2 α =2 ˆφ =1.09 φ =10 α =1 ˆφ =4.04 φ =10 α =2 ˆφ =3.20 Example 2: Suppose we estmate the followng hazard model, θ(t v, x) = ve xβ λ(t) where v has a gamma dstrbuton. The functon λ(t) s nonparametrc and ths tme one could (ncorrectly) thnk that the flexblty of ths functon compensates for the restrctve assumpton that v has a gamma dstrbuton.

20 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety20 Suppose the data s generated by a hazard model, θ(t v,x) =ve x λ(t) where p(v) =e c v, v c and c 0. Thus, v s an exponental stochast to whch the nonnegatve number c s added and the true value of β equals one. Consder estmatng ths model under the assumpton of gamma heterogenety. Wthout loss of generalty, we can wrte the ntegrated baselne hazard as follows, Λ(t) =H(t) d where H(t) s unrestrcted and d>0. Horowtz (1996) and Chen (2002) show how to estmate H(t) at rate N. Suppose that the condtons of Horowtz (1996) or Chen (2002) are fulflled and that one frst estmates H(t) usng one of these methods. Estmatng d s then lke estmatng a Webull model. In the appendx, we show that the nconsstency of β does not depend on the dstrbuton of the regressors. Usng N =10, 000, we found the followng, c β γ v δ v β; γ v =2,δ v = For c =0, correct specfcaton, all parameters can be consstently estmated; the last column gves estmaton results for β for γ v =2and δ v =1. The smulaton results show that the nconsstences ncreases wth c. Note that the asymptotc bas n the examples above does not depend on the shape of the hazard. The followng lemma gves a reason for the asymptotc bas. Lemma 2: Let θ(t v, x) =ve xβ λ(t) where v x. Letv c T 0 Gamma(γ v,δ v ). If c =0, then F (t x) decreases at a polynomal rate. If c>0, then F (t x) decreases at an exponental rate. The lemma states that the survvor probablty as a functon of tme decreases at a polynomal rate f the unobserved heterogenety dstrbuton s a gamma dstrbuton but that

21 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety21 the survvor probablty decreases at an exponental rate f the unobserved heterogenety dstrbuton s a shfted gamma dstrbuton. As the examples show, msspecfcaton of the heterogenety dstrbuton cannot, n general, be corrected by a flexble baselne hazard. The estmator presented n ths paper does not rely on specfyng or estmatng the heterogenety dstrbuton whch explans ts better performance n terms of asymptotc bas and consstency. 6. Emprcal Results We estmate our new duraton model on a sample of 15,491 males who receved unemployment benefts begnnng n 1998 n a data set called the Study of Unemployment Insurance Exhaustees publc use data. The study was desgned to examne the characterstcs, labor market experences, unemployment nsurance (UI) program experences, and reemployment servce recept of UI recpents. 18 The study sample conssts of UI recpents n 25 states who began ther beneft yearn 1998 and receved at least one UI payment, and s desgned to be natonally representatve of UI exhaustees and non-exhaustees. The data descrpton s: The data come from the UI admnstratve records of the 25 sample states and telephone ntervews conducted wth a subsample of these UI recpents. Telephone ntervews were conducted n Englsh and Spansh between July 2000 and February 2001 usng a two-stage process. For the frst 16 weeks, all 25 partcpatng states used mal, phone, and database methods to locate sample members, who were then asked to complete the survey. The second stage, conducted n 10 of the sample states, added feld staff to help locate nonrespondng sample members. The admnstratve data nclude the ndvdual s age, race, sex, weekly beneft amount, frst and last payment date, the state where benefts were collected, and whether benefts were exhausted. (op. ct.) The survey data contan ndvdual level nformaton about labor market and other actvtes from the tme the person entered the UI system through the tme of the ntervew. However, we lmt our econometrc study to the frst 25 weeks of unemployment 18 The followng descrpton follows from whch has further detals of the sample desgn and results.

22 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety22 due to the recognzed change n behavor n week 26 when UI benefts cease for a sgnfcant part of the sample, see e.g. Han-Hausman (1990). The data nclude nformaton about the ndvdual s pre-ui job, other ncome or assstance receved, and demographc nformaton. We use two ndcator varables, race and age over 50 n our ndex specfcaton. We also use the replacement rate whch s the weekly beneft amountdvdedbytheui recpent s base perod earnngs. Lastly, we use the state unemployment rate of the state from whch the ndvdual receved UI benefts durng the perod n whch the ndvdual fled for benefts. Ths varable changes over tme. Table 1 gves the means and standard devatons for the varables we use n our emprcal specfcaton: Table 1 here We frst estmate the unknown parameters of the model usng the gamma heterogenety specfcaton of Han-Hausman (1990) and Meyer (1990) (HHM). Ths specfcaton allows for a pecewse constant baselne hazard, whch does not restrct the specfcaton snce unemployment duraton s recorded on a weekly bass. However, t does mpose a gamma heterogenety dstrbuton on the specfcaton whch can lead to nconsstent estmates as we dscussed above. We estmate the model usng a gradent method and report the HHM estmates and bootstrap standard errors n Table 2. Table 2 here The estmates of the parameters, as reported n table 2, should not depend on how many weeks of data we use (6, 13 or 24 weeks). However, the coeffcents dffer sgnfcantly. We fnd sgnfcant evdence of heterogenety n the two larger samples, whle n the 6 perod sample we do not estmate sgnfcant heterogenety. We also fnd the expected negatve estmates for all of the coeffcents wth the state unemployment rate a sgnfcant factor n affectng the probablty of extng unemployment. When comparng

23 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety23 the estmates of the β across the 3 samples, the scalng changes dependng on the varance of the estmate gamma dstrbuton. Thus, the ratos of the coeffcents should be compared. The ratos of the coeffcents across samples reman smlar wth the results for the 13 perod and 24 perod very close to each other. We now turn to estmate of the new duraton specfcaton, whch does not requre estmaton of a heterogenety dstrbuton usng the same samples as above. Optmzaton of the objectve functon can now create a problem because of ts lack of smoothness. Usual Newton-type gradent methods or conjugate gradent (smplex) methods do not work n ths stuaton. To date we have found that generalzed pattern search algorthms perform best. 19 We use the pattern search routne from Matlab to estmate the parameters. See the Appendx for further detals of our computatonal approach. The basc dea s to begn wth the gamma heterogenety estmates and to construct a boundng box around each parameter estmates of 3 standard devatons. We then fnd new estmates and ncrease the boundng box untl we do not fnd an ncrease n the objectve functon. The routne converges relatvely rapdly. We estmate standard errors usng a bootstrap approach. In Table 3 we gve the estmates of the new duraton model. We also check our pattern search results usng a genetc optmzaton approach that s also dscussed n the appendx. The genetc optmzaton approach has the advantage of not dependng on ntal values. However, t has the dsadvantage of takng much longer to solve so t cannot be used feasbly to bootstrap the results to estmate the standard errors. However, the results of the pattern search algorthm and the genetc optmzaton algorthm are very smlar as we descrbe n the appendx. Table 3 here Agan we fnd that all of the estmated coeffcents have the expected negatve sgns. The coeffcents are also estmated wth a hgh degree of statstcal precson, although ths 19 Further research would be helpful here. We have also used gradent algorthms on a smoothed objectve functon to obtan ntal estmates and then employed Nelder-Mead routnes to fnd the optma. However, the pattern search algorthms appear to work best. See e.g. Audet and Denns (2003) for a recent survey of pattern search algorthms.

24 Estmatng a Sem-Parametrc Duraton Model wthout Specfyng Heterogenety24 fndng may be a result of our large sample sze of 15,491 ndvduals. We agan fnd that the rato of coeffcents remans relatvely stable across the three dfferent samples wth the excepton of the replacement rate whch becomes ncreasngly larger wth respect to the state unemployment rate as the sample length ncreases. The change n the estmated coeffcent for the replacement rate for the 24 week sample appears to arse because most recpents unemployment nsurance termnates after 26 weeks. Han-Hausman (1990) found a sgnfcant change n behavor at week 26. As ndvduals start to approach week 26 the sze of the replacement rate has a dmnshed effect on ther behavor as they foresee the end of ther unemployment benefts begnnng to draw near. In Fgures 1 and 2 we plot the survval curves for the 13 week and 24 week gamma heterogenety estmates and for the estmates from the new model. We ft the survval curves usng a second order local polynomal estmator whch takes account of the standard devatons of the estmated perod coeffcents n Table 2 and The estmated local polynomal survval curves ft the data well for all specfcatons. Fgure 1 here Fgure 2 here We fnd that the results of the new model gves extremely smlar results for the 6 perod data and the 13 perod data. Indeed, a Hausman (1978) specfcaton test on the slope coeffcents s 0.42 wth 4 degrees of freedom. Thus, we fnd that the new model s not senstve to the number of perods used to estmate the model. For the 24 perod model we fnd the coeffcents agan very close to the other results except for the coeffcent of the replacement rate. A Hausman test now rejects the equalty of the slope coeffcents wth a value of 234.3, based essentally on the change n the replacement rate coeffcent. However, snce most ndvduals unemployment benefts run out n the 26th week, the 20 We explan our approach n more detal n the appendx.