SPECIFICATION TESTS FOR THE COMPETING RISKS DURATION MODEL: AN APPLICATION TO UNEMPLOYMENT DURATION AND SECTORAL MOVEMENT
| Author | Jonathan Thomas,Stephen Pudney |
| Date | 01 August 1995 |
| DOI | http://doi.org/10.1111/j.1468-0084.1995.mp57003004.x |
| Published date | 01 August 1995 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 57, 3(1995)
0305-9049 $3.00
SPECIFICATION 'lESTS FOR THE COMPETING
RISKS DURATION MODEL: AN APPUCATION
TO UNEMPLOYMENT DURATION AND
SECTORAL MOVEMENT
Stephen Pudney and Jonathan Thomas
I. INTRODUCTION: COMPETING RISKS MODELS
Multi-state transition models of various types are widely used in the study of
economic behaviour at the micro level. In the context of a single spell, such
models amount to a particular specification for a joint distribution of two
variables: a spell duration, t, and the exit route, r. We assume that t is continu-
ously variable, whilst r is an integer variable taking values in the set { 1, ..., ql.
For example, in the particular application we consider in this paper, t is the
duration of a spell of unemployment, and r indexes the possible alternative
transitions that can mark the end of the spell: in our case the risks are a move
to either a job within the same 2-digit industrial sector or a different sector.
An important class of multi-state models is the competing risks (CR)
structure. Econometric applications include Burdett, Kiefer and Sharma
(1985), Han and Hausman (1990), Katz and Meyer (1990), Narendranathan
and Stewart (1990) and Thomas (1992, 1993). Corresponding to each
possible exit route (or risk), j, is a latent duration, t, which is interpreted as
the waiting time which would elapse before the episode ends via risk j, in the
absence of any other risks which might cause the episode to end before that
time. The observed duration, t, is the smallest of the t1, since the first termin-
ating event is the one that ends the episode and determines its duration.
Hence: tniin(t,...,t9)
rargmin{t1}
j{1...q}
CR models have much in common with discrete choice models. If we
interpret the latent durations as monotonically-decreasing transformations of
utilities, then the minimum duration corresponds to the maximum of the set
of utilities associated with the q exit routes. The important difference
between the two is that the maximum utility is not observable for discrete
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C Blackwell Publishers Ltd. 1995. Published by Blackwell Publishers, 108 Cowley Road. Oxford 0X4 1JF,
UK & 238 Main Street, Cambridge, MA 02142, USA.
(1)
324 BULLETIN
choices, whereas in CR models the minimum of the t1 is observed directly as
the completed duration. In this paper, we confine attention to the case of
independent risks, which is the assumption underlying most empirical appli-
cations. Under independence, we need specify only the marginal distribution
of each t1: the associated pdf and cdf are denoted f1( t x; O) and F1( t1 x1; Of),
where x1 is a vector of explanatory variables and O. is a vector of unknown
parameters. The distribution of t can be characterised in terms of its hazard
function, h1( t1 Ix1; Of):
fj(t»x,; 61)=h1(t11x1; O1) exp(-11(t»x1; Od)) 2
F1(t1 Jx1; 01)=1exp(-11(t1 jx1; Od))
where I( Ix1; O) is the hazard for risk j, integrated from O to t.
The joint distribution of t and r can be derived as follows.
f(r,t)=lirn Pr(tr(t,t+dt),{tj>tr,jr})/dt
dt -0
fr(t) fl exp(1,(t))
j'r
hr(t) exp (t))
=hr(t)exp(Io(t)) (3)
where 10(t) is the aggregated integrated hazard, and where we have left
implicit the dependence of the hazard function on the X1 and O. The joint
distribution (3) can be partitioned into marginal and conditional components:
Pr(rlt,x; 0)_hr(tIXr; Or) (4)
h0(tlx; O)
f(tlx;O)=h0(tlx;O)exp{-10(tlx;O)} (5)
wherex={x1 ...xq}, O{O ... Oq} and h0(tlx; O}=1h1(tIx1; 01)is the aggre-
gated hazard function.
Equations (4) and (5) make clear the possible restrictiveness of the CR
structure in empirical applications: the distributions of duration and exit
route are both constructed from the same primitive elements: the h1. Thus,
for example, if the variables x1 have coefficients O in the exit route distribu-
tion (4), then x1 is restricted to have the same coefficients in the expression (5)
for the duration distribution. Moreover, the list of explanatory variables that
determine the exit route is exactly the same as the list of variables determin-
ing the duration distribution.
There are other potential drawbacks of the basic CR model. As defined by
(3), the model makes no allowance for unobserved heterogeneity in the latent
© Blackwell Publishers Ltd. 1995.
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