We study the non-parametric identification of a mixed proportional hazard model with lagged duration dependence when data provide multiple outcomes per individual or stratum. We show that the information conveyed by the within strata variation can be exploited to non-parametrically identify lagged duration dependence in more general models than in the literature.identification, parallel data, lagged duration dependence, mixed proportional hazard models, multiple spells

Picchio, Matteo

English

Research Papers in Economics

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Forschungsinstitut 
zur Zukunft der Arbeit
Institute for the Study 
of Labor 
Lagged Duration Dependence in
Mixed Proportional Hazard Models
IZA DP No. 6089
October 2011
Matteo Picchio 
Lagged Duration Dependence in 
Mixed Proportional Hazard Models 
 
 
 
 
Matteo Picchio 
CentER, ReflecT, Tilburg University, 
Sherppa, Ghent University and IZA 
 
 
 
 
 
 
Discussion Paper No. 6089 
October 2011 
 
 
 
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ABSTRACT 
 
Lagged Duration Dependence in Mixed Proportional Hazard Models
* 
 
We study the non-parametric identification of a mixed proportional hazard model with lagged 
duration dependence when data provide multiple outcomes per individual or stratum. We 
show that the information conveyed by the within strata variation can be exploited to non-
parametrically identify lagged duration dependence in more general models than in the 
literature. 
 
 
JEL Classification:  C14, C41 
  
Keywords:  lagged duration dependence, mixed proportional hazard models, identification, 
multiple spells, parallel data 
 
 
Corresponding author: 
 
Matteo Picchio  
Department of Economics 
Tilburg University 
PO Box 90153 
5000 LE Tilburg 
The Netherlands 
E-mail: m.picchio@uvt.nl   
 
                                                 
* The author acknowledges financial support by Stichting Instituut GAK, through ReflecT, the Research 
Institute for Flexicurity, Labor Market Dynamics and Social Cohesion at Tilburg University and by 
Fonds Wetenschappelijk Onderzoek (FWO). 1 Introduction
The identiﬁcation of lagged duration dependence in a single-risk mixed propor-
tional hazard (MPH) model is shown in Honoré (1993). Regressor variation and
independence between regressors and individual heterogeneity (along with mixed
proportionality) are assumptions required for identiﬁcation. Frijters (2002) sheds
further light on this issue by proving identiﬁcation without exploiting regressor
variation and without imposing restrictions on the tail of the unobserved hetero-
geneity distribution. The price to pay is that the same baseline hazard function
must be imposed in the initial spell and in the subsequent spell. This assump-
tion is too restrictive in many empirical studies, for instance when analysing the
impact of unemployment duration on subsequent employment stability, the time
until development of HIV infection and outbreak of AIDS, or mother’s age until
childbirth and child survival.
In this paper, we show that if parallel data (Hougaard, 2000) are available, i.e.
multiple realizations of subjects belonging to the same stratum related through
unobserved components,1 the assumption on spell-constant baseline hazard func-
tions can be relaxed without needing regressor variation and restrictions on the
mixing distribution. Parallel durations are observed for example when focusing
on siblings as they share genetic material and family background (family ﬁxed ef-
fect), employees at the same workplace as they share the same environment (ﬁrm
ﬁxed-effect), and individuals experiencing repeated unemployment and employ-
ment spells as their labour market career is affected by time-invarying personal
characteristics (individual ﬁxed effect).
There is increasing data availability and interest to apply survival analysis with
parallel durations. As a matter of fact, the identiﬁcation result in this paper can be
useful in many empirical frameworks with lagged duration dependence, as far as
the grouping results from multiple realizations for the same individual, household,
ﬁrm, geographical area, school, etc. In economics, applications that exploit mul-
tiple durations generated by a single unit are, for example, Doiron and Gørgens
(2008) and Cockx and Picchio (2011a,b) who analyse employment stability after
unemployment events, Bonnal et al. (1997) who study the impact of training pro-
grams on subsequent labour market performance, and Lindeboom and Kerkhofs
(2000) and Frederiksen et al. (2007) who focus on, respectively, sickness absen-
1As in Abbring and van den Berg (2003a,b), by stratum we mean a single individual for whom
the outcome process is observed (at least) twice or a group of individuals sharing similar unob-
served components.
1teeism and job tenure with strata at ﬁrm level. Further examples can be found
in biostatistics: Guo and Rodríguez (1992), Sastry (1997), and Ridder and Tu-
nalı (1999) study child mortality with strata at household level; Therneau and
Hamilton (1997) investigate several approaches to recurrent events data, such as
recurrent infections in AIDS patients or multiple infarcts in coronary study.
2 MPH Models with Lagged Duration Dependence
The MPH model with lagged duration dependence is characterized in Honoré
(1993) by the following hazard functions
1(t1jx;v) = z
0
1(t1)1(x)v1; (1)
2(t2jt1;x;v) = z
0
2(t2)2(x)h(t1)v2; (2)
where ts 2 <+ is the duration of the sth spell, z0
s is the baseline hazard, a function
of the elapsed duration in spell s, s(x) is the systematic part, a function of a set of
observed characteristics x. h(t1) is the lagged duration dependence function, i.e.
the effect of the duration of the ﬁrst spell on the hazard function of the subsequent
spell. Finally, v1 and v2 are non-negative time-constant terms, with distribution
function G(v1;v2). They capture unobserved heterogeneity that could determine
the duration of the ﬁrst and second spells, respectively. This model could be
used, for instance, to investigate the impact of unemployment duration on future
(un)employment durations.2
Honoré (1993) proves the non-parametric identiﬁcation of model (1)-(2) as-
suming that the unobserved heterogeneity is orthogonal to x and under a ﬁnite
moment condition on the unobserved heterogeneity distribution G. However, the
orthogonality condition might be too stringent and not plausible in many applica-
tions.
Frijters (2002) attempts to avoid the orthogonality condition by considering
the following MPH model
1(t1jv) = z
0(t1)v; (3)
2(t2jt1;v) = z
0(t2)h(t1)v: (4)
2The outcome variable t2 might not necessarily be a duration outcome, but it might represent
any other non-negative outcome variable, such as starting wages (Cockx and Picchio, 2011b),
earnings (Arni et al., 2009), working hours, prices, etc. The hazard function in (2) would then
fully characterize the corresponding distribution function.
2He shows that z, h, and G are non-parametrically identiﬁed without requiring re-
gressor variation and any assumption on the mixing distribution. Nevertheless, a
critical price is paid in terms of model ﬂexibility: in model (3)-(4), the baseline
hazards and the unobserved heterogeneity terms have to be the same in the initial
spell and in the subsequent spell. This type of model has been estimated since
the 1980s to understand whether unemployment duration affects the duration of
subsequent unemployment spells (e.g., Heckman and Borjas, 1980). Nonethe-
less, this restriction might not be natural for a lot of applications; for example,
investigations into the way unemployment duration t1 affects subsequent job or
employment stability t2. This arises the question of under which conditions we
can allow for different baseline hazards and different unobserved heterogeneity
components in each spell without requiring the variation of orthogonal regressors.
In what follows we show that if data provide information on (t1;t2) at least
twice in a stratum which is characterized by a single realization of (v1;v2), it is
possible to allow the baseline hazards and the ﬁxed-effects to be spell-speciﬁc
without needing variation of exogenous regressors and assumptions about the
mixing distribution.
The MPH model with lagged duration dependence that we study is

k
1(t
k
1jv) = z
k0
1 (t
k
1)v1; (5)
2(t
k
2jt
k
1;v) = z
k0
2 (t
k
2)h
k(t
k
1)v2; (k = 1;2) (6)
where the superscript k = 1;2 identiﬁes the recurrence of the outcome variables
within a stratum. In a panel data framework k would be the kth time we observe
the outcome variables (t1;t2) for a given individual. In twins or matched pairs
studies (Holt and Prentice, 1974), k is sibling’s identiﬁer. Note that model (5)-(6)
is more ﬂexible than the one in (3)-(4), in that the baseline hazards are allowed to
be different over-spells and within-stratum. Furthermore, differently from model
(1)-(2), the regressors x do not enter the model speciﬁcation, so that the analysis
can be thought of as conditional on x.
We assume that conditional on (v1;v2), (t1
1;t1
2) and (t2
1;t2
2) are independent.
With hazard functions speciﬁed as in (5) and (6), the joint survivor function of
3(t1
1;t1
2;t2
1;t2
2) is
S(t
1
1;t
1
2;t
2
1;t
2
2) =
Z
<2
+
exp
n
 v1
h X
k=1;2
z
k
1(t
k
1)
i
 v2
h X
k=1;2
z
k
2(t
k
2)h
k(t
k
1)
io
dG(v1;v2)
= LG
h X
k=1;2
z
k
1(t
k
1);
X
k=1;2
z
k
2(t
k
2)h
k(t
k
1)
i
; (7)
where LG(s1;s2) is the Laplace transform of G. In the identiﬁcation analysis
that follows, S(t1
1;t1
2;t2
1;t2
2) is observed and taken to be known, as well as the
subsurvival function @S(t1
1;t1
2;t2
1;t2
2)=@tk
2, for k = 1;2.
3 Identiﬁcation Result
Theorem 1 Functions G, zk
1, zk
2, and hk, with k = 1;2, in (7) are uniquely iden-
tiﬁed from the distribution of \k=1;2(T k
1 ;T k
2 ) under the following assumptions:
A1 zk
1(t) and zk
2(t), for k = 1;2, are non-negative, differentiable, and strictly
increasing 8t 2 <+. z1
1(t0) = z1
2(t0) = 1 for some ﬁxed t0 2 <+.
A2 h1 and h2 are non-negative on <+. h1(t) = h2(t) = 1 for some ﬁxed
t 2 <+.
Proof. Under Assumption A1, from the marginal distribution of (T 1
1;T 2
1) we can
identify z1
1 and z2
1 by invoking Theorem 1 in Honoré (1993). Identiﬁcation of
the remaining functions is shows in steps. In step (a), identiﬁcation of lagged
duration functions h1 and h2 is proven. Step (b) concerns identiﬁcation of the
integrated baseline hazards z1
2 and z2
2. Finally, step (c) deals with identiﬁcation of
the individual heterogeneity distribution G.
(a) From a large data set we can compute the subsurvival function
@S(t1
1;t1
2;t2
1;t2
2)
@t1
2
= z
10
2 (t
1
2)h
1(t
1
1)Ds2LG
h X
k=1;2
z
k
1(t
k
1);
X
k=1;2
z
k
2(t
k
2)h
k(t
k
1)
i
;
(8)
where Ds2LG(s1;s2)  @LG(s1;s2)=@s2. We can also compute the subsur-
vival function
@S(t1
1;t1
2;t2
1;t2
2)
@t2
2
= z
20
2 (t
2
2)h
2(t
2
1)Ds2LG
h X
k=1;2
z
k
1(t
k
1);
X
k=1;2
z
k
2(t
k
2)h
k(t
k
1)
i
:
(9)
4If we divide the subsurvival function in (8) by the one in (9), the compo-
nent related to the ﬁrst derivative of the Laplace transform drops out. This
is the advantage of having variation within strata. Consider an arbitrary
(t1
2;t2
1;t2
2) 2 <3
+ and pick (t1
1;t) 2 <2
+. From the the ratio
@S(t1
1;t1
2;t2
1;t2
2)=@t1
2
@S(t1
1;t1
2;t2
1;t2
2)=@t2
2
@S(t;t1
2;t2
1;t2
2)=@t1
2
@S(t;t1
2;t2
1;t2
2)=@t2
2
=
z10
2 (t1
2)h1(t1
1)
z20
2 (t2
2)h2(t2
1)
z10
2 (t1
2)h1(t)
z20
2 (t2
2)h2(t2
1)
=
h1(t1
1)
h1(t)
we get identiﬁcation of h1 up to a constant. Identiﬁcation of h2 is similarly
yielded considering an arbitrary (t1
1;t1
2;t2
2) 2 <3
+ and picking (t2
1;t) 2 <2
+.
(b) Taking the ratio of (8) over (9) and solving with respect to z2
2 with the nor-
malization z1
2(t0) = 1 yield
z
2
2(t
2
2) =
h1(t1
1)
h2(t2
1)
Z t2
2
0
Z t0
0
@S(t1
1;1;t2
1;2)=@1
@S(t1
1;1;t2
1;2)=@2d
1
 1
d
2: (10)
Since h1 and h2 have already been identiﬁed, ﬁxing (t1
1;t2
1;t0) 2 <3
+ and
letting t2
2 vary over <+ give identiﬁcation of z2
2.3 Similar computations yield
z
1
2(t
1
2) = z
2
2(t
2
2)
h2(t2
1)
h1(t1
1)
Z t1
2
0
Z t2
2
0
@S(t1
1;1;t2
1;2)=@2
@S(t1
1;1;t2
1;2)=@1d
2
 1
d
1;
which identiﬁes z1
2 for arbitrary (t1
1;t2
1;t2
2) 2 <3
+.
(c) All the functions entering the Laplace transform LG have already been iden-
tiﬁed. Thereby, we can trace LG on a non-empty open set by appropriately
varying (t1
1;t1
2;t2
1;t2
2). As LG is real analytic, it is uniquely determined on
<4
+. Uniqueness of the Laplace transform implies identiﬁcation of G and
concludes the proof.
This theorem states that assumptions on regressor orthogonality and on the
moments (or the tail) of the mixing distribution are not necessary for model iden-
tiﬁcation with within strata variation. Finally, the result can be extended to cover
the identiﬁcation of a model where the baseline hazards and lagged duration func-
tions are conditional on x and the mixing distribution G depends on x.
3If t1
1 = t2
1 = t, this step of the proof is like Honoré’s (1993) Theorem 1. Note however that
it can be repeated for all (t1
1;t2
1) 2 <2
+. Since all the solutions to equations (10) should be the
same, potentially testable overidentifying restrictions arise, similarly to Abbring and van den Berg
(2003a) and Melino and Sueyoshi (1990).
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7

Lagged Duration Dependence in Mixed Proportional Hazard Models

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