ECON 721: Lecture Notes on DurationAnalysis

Petra E. Todd

Fall, 2013

2

Contents

1 Two state Model, possible non-stationary 1

1.1 Hazard function . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Expected duration . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The Exponential Distribution . . . . . . . . . . . . . . . . . . 51.5 Show that a distribution with any form of hazard function can

be tranformed into a constant hazard . . . . . . . . . . . . . . 61.6 Introducing Covariates . . . . . . . . . . . . . . . . . . . . . . 61.7 Variety of Parametric Hazard Models . . . . . . . . . . . . . . 71.8 How would you estimate? . . . . . . . . . . . . . . . . . . . . 10

2 Multi-state models 11

2.1 Stationary Models . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Alternative way of computing the likelihood if you only

have information on number of spells (not on length ofspell) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Nonstationary Models . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Applications (with nonstationarity) . . . . . . . . . . . 162.2.2 The Problem of Left-Censored Spells (Nickell, 1979) . . 18

2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Cox’s Partial MLE 21

4 Nonparametric Identification - Heckman and Singer (1981)result 23

5 Mixed Proportional Hazard Models 25

i

ii CONTENTS

6 Nonparametric Estimation of the Survivor Function: TheKaplan-Mier Estimator 27

7 Competing risks model 31

Chapter 1

Two state Model, possiblenon-stationary

reference: Lancaster’s Econometric Analysis of Transition Data

Assume people can be in one of two possible states (e.g. married or unmar-ried, alive or dead, employed or unemployed, pregnant or not)Probability of leaving between t, t+ dt having survived to t

Pr(t T t+ dt|T � t)

1.1 Hazard function

Define the hazard function as

h(t) = limdt!0

P (t T t+ dt | T � t)

dt

By Bayes rule

P (t T t+ dt | T � t) =P (t T t+ dt , T � t)

P (T � t)

=F (t+ dt)� F (t)

1� F (t) = S(t)

1

2 CHAPTER 1. TWO STATE MODEL, POSSIBLE NON-STATIONARY

The denominator, S(t), is called the survivor function. f(t) is the densityof exit times, and F (t) is the associated cdf.

h(t) = limdt!0

F (t+ dt)� F (t)

dt

1

S(t)=

F

0(t)

S(t)=

f(t)

S(t)

Therefore,

h(t) = �d log(S(t))

dt

and

�Z

t

0

h(s)ds =

Zt

0

d log(S(t))

ds

ds+ C

= log S(s)|t0 + C

= log S(t)� logS(0) + C

= log S(t) + C

1.2. EXAMPLES 3

The initial condition S(0) = 1 (everyone survives in the beginning), impliesthat C = 0.

The survivor function can be written as:

S(t) = 1� F (t) = exp{�Z

t

0

h(s)ds}

Using that h(t) = f(t)S(t) we get

f(t) = h(t)S(t) = h(t) exp{�Z

t

0

h(s)ds}.

If exit is certain, then

limt!1

S(t) = 0,

which implies

limt!1

Zt

0

h(s)ds = 1,

if not, then h(t) is called a defective hazard.

1.2 Examples

4 CHAPTER 1. TWO STATE MODEL, POSSIBLE NON-STATIONARY

1.3 Expected duration

What is the expected total duration in a state conditional on surviving totime s?

g(t|t � s) =f(t)

S(s)p.d.f.

e(s) =

Z 1

s

t

f(t)

S(s)dt =

1

S(s)

Z 1

s

tf(t)dt

Integrate by parts:

limm!1

Zm

s

tf(t)dt = limm!1

⇢tF (t) |m

s

�Z

m

s

F (t)dt

�

= limm!1

{mF (m)� sF (s)�Z

m

s

F (t)dt�m+ s+

Zm

s

dt}

= limm!1

{�mS(m) + sS(s) +

Zm

s

S(t)dt}

= sS(s) +

Z 1

s

S(t)dt

1.4. THE EXPONENTIAL DISTRIBUTION 5

Thus, expected durations as of time s and as of time 0 are:

e(s) = s+1

S(s)

Z 1

s

S(t)dt

e(0) =

Z 1

0

S(t)dt

Result: Mean duration is the integral of the survivor function.

1.4 The Exponential Distribution

When the hazard is independent of how long the state has been occupied,the integral is

Zt

0

h(s)ds = ✓t

The survivor function is

S(t) = exp{�✓t}, ✓ > 0, t � 0

and the pdf is

f(t) = ✓ exp{�✓t}

This is the exponential probability density function. We can say that T isdistributed exponental with parameter ✓.

6 CHAPTER 1. TWO STATE MODEL, POSSIBLE NON-STATIONARY

1.5 Show that a distribution with any form of

hazard function can be tranformed into a

constant hazard

Pr(T � t) = S(t) = exp{�Z

t

0

h(s)ds}

Let

Z(T ) =

ZT

0

h(s)ds

Because h(s) > 0, Z(T ) is a monotonic transformation.

Pr(T � t) = Pr(Z(T ) � Z(t)) = exp{�Z(t)},

This expression is the survivor function for a random variable at point Z(t),so we can conclude that Z has an exponential distribution.

This shows that the integrated hazard is a unit exponential variate.

A distribution with any form of the hazard function can be transformed intoa constant hazard (i.e. exponential form) by a suitable transformation of thetime scale.

1.6 Introducing Covariates

time-invariant: sex, race, perhaps educationtime-varying: age, business cycle e↵ects, month

1.7. VARIETY OF PARAMETRIC HAZARD MODELS 7

h(t; x) =P (t T t+ dt | T � t, x)

dt

=f(t; x)

S(t; x)

and

S(t; x) = exp{�Z

t

0

h(s; x)ds}

We could do all the estimation within x cells.

Alternatively, we could specify a functional form for how the hazard func-tion depends on observables x, such as

S(t; x) = exp{�Z

t

0

h(s; x)ds} = exp{�x�}

1.7 Variety of Parametric Hazard Models

Would like to use economic theory to guide in selection of appropriate func-tional forms, depending on context

Weibull family

f(t) = ↵�

↵

t

↵�1 exp{�(�t)↵}h(t) = ↵�

↵

t

↵�1

S(t) = exp{�(�t)↵}� = exp{��x}

This hazard depends on time (unless ↵ = 1) but is restricted to be monotonic.↵ = 1 gives the exponential hazard.

8 CHAPTER 1. TWO STATE MODEL, POSSIBLE NON-STATIONARY

Proportional hazard

h(x, t) = k1(x)k2(t)

This form implies that hazards for two people with x = x1 and x = x2 arein the same ratio for all t, but not necessarily monotonic.k2 is called the baseline hazard.

A key advantage of this model is that we can estimate k1(x) without havingto specify k2(t).(moreonthislater)

If k1(x) = k1(x(t)), then only get proportional hazard model if covariatesvary same way for all.

Box-Cox Model

h(t) = k1(x)↵ exp{�t(�)}

1.7. VARIETY OF PARAMETRIC HAZARD MODELS 9

Box-Cox transformation:

t

(�) =t

� � 1

�

t

(1) = t� 1

lim�!0

t

� � 1

�

= lim�!0

e

� ln t � 1

�

= lim�!0

e

� ln t ln t

1(by l

0hopital

0s rule)

= ln t

As �! 0, we get

h(t) = k1(x)↵ exp{�t(�)}

= k1(x)↵ exp{� ln t}

= k1(x)↵t� � Weibull

The Box-Cox generates Weibull as a special case. Still, monotonic though

dt

�

dt

= t

��1, which is of constant sign

Generalized Box-Cox

Flinn and Heckman (1982) suggest using instead

h(t) = k1(x)↵ exp{�1t(�1) + �2t(�2)}

(for example, �1 = 1 and �2 = 2). This form now allows for a nonmonotonichazard.

10CHAPTER 1. TWO STATE MODEL, POSSIBLE NON-STATIONARY

1.8 How would you estimate?

If you observe all exit times, the likelihood is

L(t;�1,�2,↵, �1, �2) = ⇧n

i=1[1� S(ti

)]

= ⇧n

i=1[1� exp{�Z

t

0

h(s, x)ds}]

If there is no closed form solution for the survival function, then evaluatingthe likelihood requires numerical integration.

Chapter 2

Multi-state models

reference: Amemiya, Chapter 11

�

i

jk

(t)�t = Prob(person i observed in state k at time t+�t|in state j at time t)

If �ijk

(t) = �

i

jk

, then the model is stationary.

Will start with stationary models, then do nonstationary.

2.1 Stationary Models

The probability that a person stays in state j in period (0, t) and then movesto k in period (t, t+�t) (event A) is given by

P (A) = (1� �j

�t)t/�t

�

jk

�t (treating t/�t) as an integer),

where

�

j

=MX

k=1

�

jk

� �jj

is the probability of exiting j.

11

12 CHAPTER 2. MULTI-STATE MODELS

Note that

limn!1

(1� 1

n

)n = e

�1

Now, using stationarity, one can show that

lim�t!0

(1� �j

�t)t/�t = exp{��j

t}

Thus,

P (A) = exp{��j

t}�jk

�t

Because �t does not depend on parameters, we can drop it and regard

exp{��j

t}�jk

as the likelihood.

Example 1:

Suppose M=3 and event history is

state 1 in period (0, t1)

state 2 in period (t1, t1 + t2)

state 3 in period t1 + t2 to t1 + t2 + t3

then back to state 1

The likelihood in this case is given by

L = exp(��1t1)�12 exp(��2t2)�23 exp(��3t3)�31

2.1. STATIONARY MODELS 13

exp(��1t1) is the probability of surviving t1 periods in state 1.

�12 is the probability of transiting to state 2

exp(��2t2) is the probability of surviving in state 2

�23 is the probability of transiting to state 3, etc...

If instead we observe the person leaving state 3 but do not know where theperson went, then replace �31 by �3

L = exp(��1t1)�12 exp(��2t2)�23 exp(��3t3)�3

Now suppose that we terminate observation at time t1 + t2 + t3 withoutknowing whether a person continues to stay in state 3 or not (right censoring),then would drop term �3 altogether.

Example 2: Two State model

M = 2, �1 = �12,�2 = �21

state #1 is unemployment, state #2 is employment

Suppose an individual experiences r completed unemployment spells of lengtht1, t2, ..., tr. Then

L = �

r

1e��T (contribution of r unemp spells to the likelihood)

T =rX

j=1

t

j

For overall likelihood, would also need the corresponding part for employmentspells.

14 CHAPTER 2. MULTI-STATE MODELS

Here,

F (t) = 1� P (T > t) = 1� e

��t (cdf of a r.v. signifying duration of a spell)

f(t) = �e

��t (density of observed duration spell)

� =f(t)

1� F (t)=�e

��t

e

��t

hazard rate

��t =f(t)�t

1� F (t)= Prob(leaves unemployment in t, t+�t|has not left up to t)

� is the hazard rate.

Example #3:

Suppose we observe one completed unemployment spell of duration t

i

for theith individual:

L = ⇧N

i=1fi(t

i

) = ⇧N

i=1�i exp(��it

i

)

where �i is the probability of exiting unemployment and exp(��iti

) is theprobability of being unemployed t

i

periods.

Now suppose that individuals 1..n complete their unemployment spells ofduration t

i

but that individuals n+ 1..N are right censored at time t

⇤i

.

The likelihood is then given by

L = ⇧n

i=1fi(t

i

)⇧N

i=n+1[1� F

i(t⇤i

)]

Note how the duration model with right censoring is similar to a standardTobit model.

Previously, the likelihood depended on the observed duration t1, t2, ..., tr onlythrough r (number of spells) and T (total length of time in state). This im-plies that r and T are su�cient statistics, which is a property of a stationarymodel.

2.1. STATIONARY MODELS 15

2.1.1 Alternative way of computing the likelihood ifyou only have information on number of spells(not on length of spell)

Assume that there are two completed unemployment spells and that the thirdspell is incomplete.

T = total unemployment time

Probability of observing two completed spells and one censored spell in totaltime T is

P (0 t1 < t, 0 < t2 T � t1, t3 � T � t1 � t2)

=

ZT

0

f(z1)

⇢ZT�z1

0

f(z2)

Z 1

T�z1�z2

f(z3)dz3

�dz2

�dz1

=(�T )2e��T

2

The probability of observing r completed spells in total time T is

Pr(r, T ) =(�T )re��T

r!, which is poisson

The likelihood is given by

L = ⇧N

i=1�rii

exp{��i

T

i

}

Now, assume that � depends on individuals’ characteristics xi

�

i

= exp{↵ + �

0x

i

}

Amemiya, Chapter 11, derives MLE estimators for ↵, �.

16 CHAPTER 2. MULTI-STATE MODELS

2.2 Nonstationary Models

Relax assumption that �ijk

(t) = �

i

jk

(constant hazard rate)

Distribution function of duration under a nonstationary model

F (t) = 1� exp[�Z

t

0

�(z)dz]

f(t) = �(t) exp[�Z

t

0

�(z)dz]

We can write the likelihood function in the integral representation as before.Suppose that

�

i(t) = g(xit

; �)

We need to specify x

it

as a continuous function of t.

2.2.1 Applications (with nonstationarity)

Tuma, Hannan and Groeneveld (1979)

Study marriage duration. Divide sample period into 4 subperiods and assumethat the hazard rate is constant between subperiods.

�

i(t) = �

0P

x

i

, t 2 T

p

p = 1, 2, 3, 4

T

p

is the pth subperiod

2.2. NONSTATIONARY MODELS 17

Lancaster (1979)Studies unemployment duration

F (t) = 1� exp(��t↵) (Weibull distribution,

nonstationary because it depends on t)

�(t) = �↵t

↵�1 (hazard function)

↵ > 1 =) @�

@t

> 0 (increasing hazard)

↵ = 1 =) @�

@t

= 0 (constant hazard,=exponential)

↵ < 1 =) @�

@t

< 0 (decreasing hazard)

Lancaster introduced covariates by specifying

�

i(t) = ↵t

↵�1 exp(�0x

i

), x

i

is constant over time

Lancaster found a pattern of decreasing bazard, but he would that the hazarddecreased less when more covariates were included. Thus, he was concernedthat the finding of negative duration dependence might be due to omittedunobservables. This led him to consider an alternative specification for thehazard rate that explicitly incorporated unobservables.

µ

i(t) = v

i

�

i(t), v

i

unobservable, assumed to be

iid gamma(1,�2)

v

i

is a proxy for unobservable, exogenous variables

Heckman and Borjas (1980)Study of unemployment duration

l =lth unemployment spell experienced by an individual

�

il(t) = ↵t

↵�1 exp(�0l

x

il

+ v

i

),

where v

i

is an unobservable that needs to be integrated out to obtain themarginal distribution function of duration.

Flinn and Heckman (1982)

18 CHAPTER 2. MULTI-STATE MODELS

Use a modified version of the Box-Cox hazard:

�

il(t) = exp[�0l

x

il

(t) + c

l

v

i

+ �1t

�1 � 1

�1+ �2

t2 � 1

�2]

If you set �1 = 0 and �2 = 0, get a Weibull modelHere, x

il

(t) is assumed to be exogenous.

2.2.2 The Problem of Left-Censored Spells (Nickell,1979)

The problem occurs if individuals are not observed at the start of their un-employment spells.

2.3 Examples

Three Cases considered in the literature:

(i) s observed, t not observed(ii) both s and t observed(iii) t is observed but s is not observed

Case (i):

Analyzed by Nickell (1979) in studying unemployment

Assume that the beginning of the spell (s) is observed, end of the spell (t)not observed. We observe that individual is unemployed at time t.

Also, assume that the P [U started in (�s ��s,�s)] does not depend on s

(constant entry rate). For su�ciently small �s,

g(s)�s = P [U started in (�s��s,�s)|U at 0]

=P [U at 0 |U started in (�s��s,�s)]�sP [U started in (� s��s,�s)]R1

0 (Numerator)ds = Pr(U at 0)

2.3. EXAMPLES 19

-s Start of spell

0 Time of interview

t End of spell

The denominator integrates the numerator over all possible dates when thespell could have started. Assuming that P [U started in (�s��s,�s)] doesnot depend on s:

=P [U at 0 |U started in (�s��s,�s)]�sR1

0 (Numerator)ds

=[1� F (s)]�sR10 [1� F (s)]ds

=[1� F (s)]�s

ES =R10 sf(s)ds

(show by integration by parts)

so,

g(s) =[1� F (s)]

ES

.

Case (ii):

Lancaster (1979)

Assume that both s and t are observed Need the joint density g(s, t) =

g(t|s)g(s). The density g(s) was derived above.

20 CHAPTER 2. MULTI-STATE MODELS

Let X denote total unemployment duration. First, evaluate

P (X > s+ t|X > s) =P (X > s+ t ,X > s)

P (X > s)(*)

=P (X > s+ t)

P (X > s)

=1� F (s+ t)

1� F (s).

Therefore,

Pr(X < s+ t|x > s) = 1� 1� F (s+ t)

1� F (s)=

F (s+ t)� F (s)

1� F (s)

Di↵erentiating with respect to t gives

g(t|s) = f(s+ t)

1� F (s),

which is the Pr(spell ends at time s+t given that it started at time s). Com-bining with the earlier results, we get

g(s, t) =f(s+ t)

ES

.

Case (iii):

Flinn and Heckman (1982):

t is observed but s is not observed

obtain g(t) by integrating g(t, s) with respect to s :

g(t) =1

ES

Z 1

0

f(s+ t)ds

=1� F (t)

ES

.

Chapter 3

Cox’s Partial MLE

Consider proportional hazard model of the form

�

i(t) = �(t) exp{�0x

i

}

and assume that data are right censored (observe start of spell but do notobserve length of spell for some individuals).

Let ti

, i = 1, 2, ..., n be completed durations and let ti

, i = n + 1, n + 2, n +3, ..., N be censored durations.

The likelihood is given by

L = ⇧n

i=1 exp{�0x

i

}�(ti

) exp[� exp(�0x

i

)

Zt

0

�(z)dz]⇥

⇧N

i=n+1 exp[� exp(�0x

i

)

Zt

0

�(z)dz].

Through some algebraic manipulation, can obtain

⇧n

i=1 exp(�0x

i

)�(ti

) exp{�Z 1

0

[X

h2R(t)

exp(�0x

h

)]�(t)dt},

where R(t) = {i|ti

� t}.

21

22 CHAPTER 3. COX’S PARTIAL MLE

Cox (1975) suggested and Tsiatis (1981) proved that the likelihood could bedecomposed into two components:

L = L1 ⇥ L2

=

"⇧n

i=1

exp(�0x

i

)Ph2R(t) exp(�

0x

h

)

#⇥

2

4⇧n

i=1

X

h2R(t)

exp(�0x

h

)�(ti

)

3

5 exp

0

@�Z 1

0

[X

h2R(t)

exp(�0x

h

)]�(t)dt

1

A

and that a consistent and asymptotically normal estimator for � can be ob-tained by maximizing L1 (the partial MLE).

This means that � can be estimated without specifying �(t).This is a remarkable result, given that L1 is not a proper likelihood.

See Amemiya, Ch. 11, for an interpretation of the di↵erent components.

Chapter 4

Nonparametric Identification -Heckman and Singer (1981)result

Ask what features of hazard functions can be identified from the raw data(i.e. G(t|x)).

Denote unobservables by ✓.

Would like to infer properties of G(t|x, ✓) without having to impose strongparametric assumptions, either on µ(✓) or h(t|x, ✓).

Assume that x(t) is constant.

Heckman and Singer (1981) show that if G(t|x) exhibits positive durationdependence, then it must be that h(t|x, ✓) also exhibits positive durationdependence over some interval of ✓ values in those intervals of t.

Also, show that omitting omitted variables leads to tendency towards nega-tive duration dependence.

Consider hazard of the form

h(t|x, ✓) = (t)�(x)✓,

23

24CHAPTER 4. NONPARAMETRIC IDENTIFICATION - HECKMANAND SINGER (1981) RESULT

which gives the proportional hazard model with ✓(t) = ✓, x(t) = x (time-invariant unobserved heterogeneity and time invariant regressors).

Let h(t|x) be the conditional hazard, not controlling for unobservables.

Let F (t|x, ✓) and F (t|x) be the conditional distributions.

h(t|x) =

Rf(t|x, ✓)dµ(✓)R

[1� F (t|x, ✓)]dµ(✓)

=

Rh(t|x, ✓)(1� F (t|x, ✓))dµ(✓)R

[1� F (t|x, ✓)]dµ(✓)

@h(t|x)@t

=

R@h(t|x,✓)

@t

(1� F (t|x, ✓))dµ(✓)R[1� F (t|x, ✓)]dµ(✓)

+ nonpositive term.

This shows that ignoring unobservables will bias the hazard downwards.

Chapter 5

Mixed Proportional HazardModels

Consider the Cox Proportional Hazard model:

h(t|X;↵, �) = �(t)exp(�0x)

Cox (1972) observed that it is possible to estimate parameters � withoutspecifying �(t) using the partial likelihood approach discussed earlier. It is asemi parametric estimation approach, because �(t) is left unspecified.

Now, suppose we want to introduce unobservables and want to be flexibleabout the way in which regressors enter. Let the unobservables be distributed�, which is unknown to the econometrician. The model that includes unob-servables is called the Mixed Proportional Hazard Model.

h(t|X,U ;↵, �) = �(t)exp(z(x, �))exp(U)

Elbers and Ridder (1982) and Heckman and Singer (1984)

Showed that this model, with regressors, is non parametrically identified un-der some restrictions on �. Elbers and Ridder (1982) assumed thatE(exp(U)) <1 (Heckman and Singer (1984) consider alternative assumptions).

25

26 CHAPTER 5. MIXED PROPORTIONAL HAZARD MODELS

Honore (1993)

Studied a multi-spell generalization where T1 and T2 are the length of mul-tiple spells. He showed the model can be identified under weaker conditionson �.

Hahn (1994)

derives the semi parametric e�ciency bound, which provides a bound on theattainable statistical accuracy (it was first introduced by Stein (1956) andhas been derived for a number of di↵erent models).

Because semi parametric estimation must be at least as di�cult as any para-metric sub model (model allowed under the semi parametric model), it followsthat the asymptotic variance of any

pN -consistent estimator is no smaller

than the supremum of the Cramer-Rao lower bounds for all parametric submodels. The infimum of the information matrix for �, the inverse of theCramer-Rao lower bound, gives the semi parametric version of the informa-tion matrix, called the semi parametric information bound.

Hahn (1994) shows that for the single-spell Weibull mixed proportional haz-ard model, the information matrix is singular, which implies that there can-not exist a

pN -consistent estimator. He also shows this to be the case for the

multi-spell version of the model. Thus, even though the model is identified,there is no

pN -consistent estimator.

Ridder and Woutersen (2003)

Present new conditions for the mixed proportional hazard model under whichparameters are identified and under which the information matrix is nonsin-gular. The paper also presents an estimator that converges at a

pN rate. The

key additional assumption is that the baseline hazard needs to be boundedaway from 0 and 1 near t = 0.

Chapter 6

Nonparametric Estimation ofthe Survivor Function: TheKaplan-Mier Estimator

Allow for right censored exit times

No regressors, but could accommodate regressors by doing everything withinx cells.

Does not allow for unobservables

N possibly right-censored exit times

M N distinct exit times t(1),t(2), t(3), ..., t(M), where multiple people canexit at the same time

n

j

=number leaving at time t

j

.

S(t) = 1� number leaving before t

N

= 1� n1 + n2 + ...+ n

k

N

, k = maxj

such that tj

< t

27

28CHAPTER 6. NONPARAMETRIC ESTIMATION OF THE SURVIVOR FUNCTION: THE KAPLAN-MIER ESTIMATOR

We can write

S(t) =N � n1 � n2 � ..� n

k

N

=

✓N � n1

N

◆✓N � n1 � n2

N � n1

◆✓N � n1 � n2 � n3

N � n1 � n2

◆...

⇥✓N � n1 � n2 � ...� n

n

N � n1 � ...� n

k�1

◆

=⇣1� n1

N

⌘✓1� n2

N � n1

◆✓1� n3

N � n1 � n2

◆...

⇥✓1� n

n

N � n1 � ...� n

k�1

◆,

where the ratios represent the number leaving of those who survive (the riskset).Thus, the hazard function is:

✓

j

=n

j

N � n1 � ...� n

j�1

The survivor function a time t (which would be used to handle right censor-ing) can be written as:

S(t) = ⇧tj<t

(1� ✓j

) (called a product limit estimator)

The term ✓

j

is the hazard rate. (prob of leaving at date t

j

given survivedup to that date.)

The Kaplan Mier estimator of the survivor function looks like a step function:

29

t

1.0

Kaplan-Mier Survivor Function

30CHAPTER 6. NONPARAMETRIC ESTIMATION OF THE SURVIVOR FUNCTION: THE KAPLAN-MIER ESTIMATOR

Chapter 7

Competing risks model

J causes of failure 1..JT

j

latent failure time from cause j

Observe duration to first failure and associated cause. That is, observe time

of death and cause of death. (e.g. observe death from cancer or heart disease)

(T, I) = {minj

(Tj

), argminj

(Tj

)}

In application, there may be considerable content in models with regressors.

For example, the goal may be to study how smoking, blood pressure andweight a↵ect the marginal distribution of time to death attributable to heartattack or cancer.

In models without regressors, need to make functional form assumptionsabout the joint distribution of failure times.

Heckman and Honore (1989) study identification in models with regressors.

31

32 CHAPTER 7. COMPETING RISKS MODEL

For example, in Cox proportional hazard model:

S(t|x) = exp{�z(t)'(x)} ('(x) usually e

x�),

we can assume that each potential failure time has a proportional hazardspecification.

Can specify joint survivor function of T1, T2 conditional on x.

S(t1, t2|x) = K[exp{�Z1(t1)'1(x)}, exp{�Z2(t2)'2(x)}]

Theorem:

Assume that (T1, T2) has the joint survivor function as given above. Then'1,'2, Z1 and Z2 are identified from the observed minimum of (T1, T2) underthe following assumptions:

(i) K is c1 with partial derivatives K1 and K2 and for i = 1, 2 the limit asn!1 of K(⌘1n, ⌘2n) is finite for all sequences ⌘1n ! 1, ⌘2n ! 1 for n!1.

K is strictly increasing in each of its arguments in all of [0 1]⇥ [0 1]

(ii) Z1(1) = 1, Z2(1) = 1,'1(x0) = 1,'2(x0) = 1 for some fixed point inthe support of x.

(iii) The support of {'1(x),'2(x)} is (0 1)⇥ (0 1)

(iv) Z1 and Z2 are nonnegative, di↵erentiable, strictly increasing func-tions, except that we allow them to be 1 for finite t.

Sketch of Proof:

Define:

Q1(t) = pr(T1 > t, T2 > T1) (die from cause 1 by time T1)

Q2(t) = pr(T2 > t, T1 > T2) (die from cause 2 by time T2)

Q

01(t) = �K1[exp{�Z1(t)'1(x)} exp{�Z2(t2)'2(x)}] exp{�Z1(t)'1(x)}Z 0

1(t)'1(x)

33

The ratio of Q01 at two points x and x0 (in the support of X) and using the

assumptions on x0 (assumption (iii)) gives:

=�K1[exp{�Z1(t)'1(x)} exp{�Z2(t2)'2(x)}] exp{�Z1(t)'1(x)}Z 0

1(t)'1(x)

�K1[exp{�Z1(t)'1(x0)} exp{�Z2(t2)'2(x0)}] exp{�Z1(t)'1(x0)}Z 01(t)'1(x0)

Takelimt!0

above expression='1(x)

By a symmetric argument, we can identify '2(x).

Heckman and Honore also provide approaches to identify K,Z1(t), Z2(t).