Chapter 5: Duration Models
Joan Llull
Microeconometrics. IDEA. Fall 2017
joan.llull [at] movebarcelona [dot] eu
Introduction
Chapter 5. Fall 2017
2
Duration analysis
Duration data:
how long
has an individual been in a state
when
exiting
from it (e.g. weeks unemployed).
Examples
: unemployment duration, marriage duration, life ex
pectancy, rms exit from the market,.
..
We want to analyze:
Why
durations di er
across individuals?
How and why do exit probabilities
vary over time
?
Long tradition in
biometrics
: surviving probabilities, hazard
functions,.
..
Chapter 5. Fall 2017
3
Duration data
We
denote
these durations for the
N
observations as
t
1
,t
2
,...,t
N
.
These data are typically
censored
. E.g.:
Individuals may not nd a job before the interview (we observe
that
t >
¯
t
, but not
t
).
Individuals may nd a job between selection and interview, and
we cannot interview them (we observe
t
< t <
¯
t
)
One of the main motivations for duration analysis:
dealing with
censoring
.
Chapter 5. Fall 2017
4
Figure I
.
Two examples of censored observations
i. Example 1
0
1
2
3
4
Individual
Jan90
Jul90
Jan91
Jul91
Jan92
Date
ii. Example 2
Jan90
Jul90
Jan91
Jul91
Jan92
Date
Note:
Black lines represent the time when the individual was unemployed. A dot indicates that
the individual is still unemployed at that date, but we do not have further information about
him/her. Vertical red dashed lines in Example 2 are interview dates.
Chapter 5. Fall 2017
5
The Hazard Function
Chapter 5. Fall 2017
6
The hazard function
Hazard function
: probability (or density) of exiting at
t
condi
tional on being alive.
It can be
timevarying
(e.g. mortality) or
constant
.
We analyze
discrete
and
continuous
hazard functions.
Why do we
care
?
Theoretically appealing (e.g. job arrival rate).
Empirically convenient (binomial discrete choice+censoring).
We start from
unconditional
hazards and then add regressors.
Chapter 5. Fall 2017
7
Figure II
.
Mortality Hazard Rate
0.0
0.2
0.4
0.6
0.8
1.0
Hazard function
0
20
40
60
80
100
Age
Note:
The line depicts the hazard mortality rate, i.e. the probability of dying at age
a
conditional
on survival until that age.
Chapter 5. Fall 2017
8
Hazard function for a discrete variable
Probability mass
function for
t
:
p
(
τ
) = Pr(
t
=
τ
)
for
τ
= 1
,
2
,...
.
Cdf
for
t
:
F
(
t
) =
p
(1) +
p
(2) +
...
+
p
(
t
)
.
Hazard function
for
t
:
h
(
τ
) = Pr(
t
=
τ

t
≥
τ
) =
Pr(
t
=
τ
)
Pr(
t
≥
τ
)
=
p
(
τ
)
1

F
(
τ

1)
=
F
(
τ
)

F
(
τ

1)
1

F
(
τ

1)
.
We can
recover
p
(
t
)
and
F
(
t
)
from
h
(
t
)
recursively:
1

h
(
t
) =
1

F
(
t
)
1

F
(
t

1)
⇒
F
(
t
) = 1

t
Y
s
=1
(1

h
(
s
))
,
p
(
t
) = (1

F
(
t

1))
h
(
t
) =
h
(
t
)
t

1
Y
s
=1
(1

h
(
s
))
.
(Interpretation of these expressions)
Chapter 5. Fall 2017
9
Hazard function for a continuous variable
Pdf
function for
t
:
f
(
τ
) = lim
dt
→
0
Pr(
τ
≤
t<τ
+
dt
)
dt
.
Cdf
for
t
:
F
(
t
) =
R
t
0
f
(
u
)
du
.
Hazard function
for
t
:
h
(
τ
) = lim
dt
→
0
Pr(
τ
≤
t < τ
+
dt

t
≥
τ
)
dt
= lim
dt
→
0
Pr(
τ
≤
t < τ
+
dt
)
Pr(
t
≥
τ
)
±
dt
=
f
(
τ
)
1

F
(
τ
)
.
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