Controlling for heterogeneity in the Illinois bonus
experiment∗
By Marcel Voia†
May 2003
Abstract
In this paper I consider a re-employment bonus experiment that was conducted in
Illinois. We answer questions such as: “How much money would be saved if this program
would be introduced generally?’ or “How much does the average unemployment duration
decrease if this program would beintroduced generally?”. To answer to these questions I
estimate a structural hazard model that allows for heterogeneity and partial compliance,
then I simulate the distribution of unemployment durations if the program would be
introduced generally.
1
Introduction
In this paper we re-analyze a segment of the data from the Illinois bonus experiment in a
way which was not done before, even though these data have been analyzed with increasing sophistication by Woodbury and Spiegelman (1987) and Meyer (1996). Given that the
experiment provided exogenous differences in individual incentives, Meyer use them to test
labor supply and search theories of unemployment. He also examined the effect of the fixedamounts bonus on different wage level groups and he examine predictions about timing of
exits from unemployment. In his analysis he did not allow for unobserved heterogeneity or
selective compliance and he concluded that the experimental evidence did not support the
desirability of a permanent program. Fortunately, Meyer gives a clear description of potential
problems of his estimates if there would be unobserved heterogeneity.
“While the Claimant Experiment group and control group are drawn from
identical populations, over time the individuals who have not ended their UI receipt in the control group and Claimant Experiment will become less comparable.
∗
I am gratful to Tiemen Woutersen, my supervisor, and I gratefully acknowlegde stimulating suggestions
from Geert Ridder and Todd Stinebrickner. All errors are mine.
†
Correspondence addresses: University of Western Ontario, Department of Economics, Social Science
Centre, London, Ontario, N6A 5C2, Canada. Email: mvoia@uwo.ca.
1
This change occurs because the effects of the Claimant Experiment may interact
with unobservable differences between individuals”.
Meyer proves this statement in a theorem (Meyer, 1996, appendix A). He also notes
“Thus, estimates of the effect of the Claimant Experiment without heterogeneity controls should be biased in the negative direction (..). Unobserved heterogeneity would also tend to diminish an effect of the experiment on the hazard
just before the offer expires.”
The Illinois bonus experiment is an interesting experiment about unemployment duration.
In particular, a randomized group of people who became unemployed was offered the chance
to participate in a program in which unemployed people were offered US $500 bonus if they
found a job within 11 weeks and kept it for at least four months. Some people declined to
participate even if they receive an offer to participate in the program, this group is called the
non-compliance group. We consider analyzing the characteristics of this group.Surprisingly,
nobody has answered simple policy questions such as, “How much money would be saved
if this program would be introduced generally?” or “How much would the average unemployment duration decrease if this program would be introduced generally?”. The reason
why these questions have not been asked (or answered) is that one has to deal with partial
compliance, which complicates the analysis of the data from a randomized experiment. That
is, some people were randomly offered to participate but declined. The people that decline
to participate from the treatment group are not doing it at random, they have some characteristics, which are important to the re-employment hazard. In this paper we estimate a
structural hazard model that allows for heterogeneity, then we simulate the distribution of
unemployment durations if the program would be introduced generally. Our findings confirm
unobserved heterogeneity and are in complete agreement with these quotes of Meyer. Also,
the results give more optimistic conclusions about monetary incentives than Meyer (1996).
Following the reasoning of two decades of econometric analysis of duration models, (see
Lancaster (1990) and Van den Berg (2000) for overviews), we argue that one needs to control
for unobserved heterogeneity to assess causality in a duration model. Randomly assigned
unemployed people are given the choice to participate. Using this intention to treat framework, we show that the integrated hazard estimator consistently estimates the treatment as
a function of time. Given that the experiment consists of a $ 500 bonus which is given to
people who find a job within the first 11 weeks after becoming unemployed, it follows from
Mortenson (1987) that you might observe a discrete jump in the hazard after 11 weeks. The
integrated hazard estimator does not require smoothness and can therefore accommodate
such a jump. In the biostatistics literature, Tsiatis (1990) and Robin and Tsiatis (1991) censor all their observations to deal with partial compliance. They hereby discard a large part of
the data and cannot make statements about the hazard over the intervals that were censored
away. The integrated hazard estimator avoids such censoring and base inference on the whole
dataset. Thus, our estimator can be viewed as an application of the integrated hazard principle as developed by Woutersen (2000). We consider the duration of unemployment as the
response variable to be analyzed, because it has a direct implication for the hazard rate, and
we use a semiparametric model to analyze the duration, that is the mixed proportion hazard
2
(M P H) model. In our analysis of causation we use the Potential Outcome Model, which
was introduced by Neyman (1923) and Fisher (1935), with major contributions made by Cox
(1958), Cochran (1965) and Rubin (1974,1977). Heckman (2000) argues that the statistical
Potential Outcome Model is a version of the econometric causal model and counterfacuals
are the base of causal inference in statistics. He also argue that the use of joint probability
of counterfacuals help us to better explain the distributional impacts of public policies, thus
counterfacuals help to move beyond comparisons of aggregate overall distributions and to
consider how people in different portions of an initial distribution are affected by a specific
public policy. The Potential Outcome Model was initially linked to the counterfactual notion
of causation by Glymour (1986). Galles and Pearl.(1998) give an axiomatic characterization
of causal couterfactuals in comparing logical properties of counterfactuals in Structural Equation Models and Kluve (2001) concentrates his paper on the counterfactual-based nature of
the Potential Outcome Model and pointed out which counterfactual causal question can be
asked and answered within a model.
Given that in our case the potential outcome model is applied to the non-compliers, to
construct counterfactual duration of unemployment for the non-complience group (the duration of unemployment of the non treated if they were treated), we use preliminary estimates
for the baseline hazard. These estimates were obtained from the compliance group. Also, we
extend the analysis to see how the duration of unemployment changes if the experiment is
applied to the overall population.
This paper is organized as follows. Section ?? gives a description of the data. Section
?? applies the integrated hazard principle to simple examples. Section ?? discusses counterfactuals in duration analysis and shows how to approximate counterfactual distributions.
Section ?? gives estimation results. Section ?? concludes.
2
Description of the Data
In our analysis we use the same data as in Meyer (1996) from the Illinois experiment, which
was conducted by the Illinois Department of Employment Security. The experiment randomly
assigned those in the eligible population to one of three groups, which are identified by Meyer
(1996) as control group, Claimant Experiment, and Employer Experiment. The goal of the
experiment was to explore if the unemployment duration is reduced when a bonus is paid to
Unemployment Insurance (UI) beneficiaries (treatment 1) or to their employers (treatment 2)
relative to a randomly selected control group. We concentrate our attention to the Claimant
Experiment (treatment 1), which consists of a random sample of new claimants for U I that
received a $500 bonus if they found a job (of 30 hours or more per week) in less than
11 weeks after filing for U I and held that job for at least 4 months. The size of the bonus
reflected a balancing of the experiments’ budget constraint (a maximum of $750,000 in bonus
payments) against an arbitrary judgement about how small a bonus could be to generate a
response (5% of annual wage or 4 weeks of U I payments). The 11 week period was chosen
to be approximately 40% of the potential duration of benefits in Illinois, which is 26 weeks.
The minimum of 4 months employment was required to avoid fraudulent hire and to avoid
payment of bonuses to seasonal workers and employers. Eligibility criteria excluded younger
and older claimants in order to reduce the number of complicated factors such as: special
3
programs for young people, and incentives to retire early for older workers which can influence
the job-finding behavior of those enrolled in the experiment. Three variables can be used
to control for the size of the sample: the number of sites (22), the length of the enrollment
period, and the proportion of claimants selected at any given site. More sites permitted a
shorter enrollment period, originally designed to be 13 weeks, ultimately 16. Thus, in the
treatment 1 group 4186 individuals were selected, 3527 (84%) (compliance group) agreed to
participate in the experiment, while 659 (16%) (non-compliance group) did not agree. The
individuals from the control group were excluded from participating in the experiment, they
actually did not know that the experiment took place.
3
Estimation using the integrated hazard, examples
In this section, we derive an estimator using the identification results of Elbers and Ridder
(1982) and the ‘integrated hazard principle’ of Woutersen (2000). The integrated hazard
principle suggests to use those parameter values (or functions) as estimates for which the
integrated hazard is an unit exponential variable. In this section we illustrate with simple
examples how this principle can be applied; in the next section we proof that the integrated
hazard principle yields a consistent estimator for the re-employment bonus experiment. Let
Z denote the integrated hazard which has a unit exponential distribution.
Z T
θ(s, x)ds ∼ ε(1).
(1)
Z=
0
R
f (s;x)
θ(s; x)ds = − ln(1 − F (s; x)) where F (s; x) is uniformly distribProof: θ(s) = 1−F
(s;x) ⇒
uted between 0 and 1.
We can use equation (1) to estimate parameters of the hazard function. To illustrate how
this is done, consider almost the simplest problem.
Example 1. Let T1 , ..., TN be independent durations with hazard θ(t) = λ so Z = λT . The
integrated hazards are independent unit exponentials: Z = λT ∼ ε(1). Equating the sample
analogue of the integrated hazard to one gives:
PN
i=1 λti
= 1.
N
This suggests an estimate for λ,
N
λ̂ = ( PN
i=1 ti
),
which is the maximum likelihood estimator.
Equation (1) can also be used for censored duration data. Suppose a duration is right
censored if it lasts longer than c where c is exogenous. Let Z 0 denote the integrated hazard
of this potentially censored observation and let the indicator d be zero if the observation is
censored and one otherwise. Then the expectation of Z 0 equals the expectation of d, i.e.
EZ 0 = Ed where d = 0, 1.
4
(2)
Proof: See appendix 1.
P 0
P
z
Equation (2) suggests to censor the durations as c and use g(λ) = Ndi − N i as one of the
moment equations.
Example 2. Let T1 , ..., TN be independent durations with the following hazard
½
λ1 if t ≤ c
θ(t) =
λ2 if t > c
The indicator di is one if ti ≤ c and zero otherwise. We can write the realization of the
integrated hazard of individual i as
zi = di λ1 ti + (1 − di )λ1 c + (1 − di )λ2 (ti − c).
Using the moment equations g1 (λ) =
estimates
λ̂1 =
λ̂2 =
P
i zi
− 1 and g2 (λ) =
N
P
i
N
di
−
P
0
i zi
N
gives the following
P
i di
P
i ti
i {d
P
+ (1 − di )c}
i (1 − di )
P
.
i (1 − di )(ti − c)
(3)
These estimates are, again, the maximum likelihood estimates.
The maximum likelihood estimator of example 1 and 2 is consistent and efficient and one
can therefore argue that the integrated hazard ‘should’ yield the same estimating function.
This equivalence result holds for any number of intervals over which the hazard is supposed
to be constant 1 . Consider, however, the mixed proportional hazard model as introduced
by Lancaster (1979) with hazard θ = ηφ(x)λ(t) where η is the realization of a random
(unobserved heterogeneity), φ(x) a function of a regressor x and λ(t) denotes the baseline
hazard. Joint estimation of φ(x), λ(t) and the distribution of η has to deal with one of
the following complications. If a parametric mixing distribution is chosen then λ(t) can be
estimated by a parametric or nonparametric technique. However, Heckman and Singer (1985)
show that the estimates are very sensitive to the choice of the mixing distribution. The other
option is to estimate the mixing distribution nonparametrically. As Horowitz (1999) shows,
however, a deconvolution problem cannot be avoided and the rate of convergence would be
very slow for all parameters (and functions). These two choices are less than attractive
and we, therefore, avoid this kind of joint estimation. We thus need moment functions
whose expectation does not depend on the mixing distribution. As ‘building blocks’ for these
moment functions we use the indicator d of equation (2) and what we call the semi-integrated
hazard, i.e.
Z
1 ti
θi du.
si =
η 0
An extension of example 1 shows how si can be used for estimation.
1
If the number of intervals increases with N then integrated hazard estimator is equivalent to the nonparametric estimator of Kaplan and Meier (1958).
5
Example 3. Suppose we observe N independent durations and that the treatment R is
randomly assigned to half of the population. Let the hazard have the following form
θ = η i γ R where R = 0, 1
where η is the realization of a mixing distribution. The random assignment ensures independence of η and X. Let the realizations denoted by ti , i = 1, ..., N. The semi-integrated
hazard has the following form
Z
1 ti
si =
θi du = γ R ti R = 0, 1
ηi 0
Consider the following moment function.
1 X R
(si − s1−R
)
i
N
i
1 X
{(γti )R − t1−R
}.
i
N
g(γ) =
=
i
This moment function is monotonic in γ and has expectation zero at γ = γ 0 , the true value
of γ.
1
γ
γ
1
1
Eg(γ) = ( )E − E = ( − 1)E .
γ0 η
η
γ0
η
Therefore, the parameter γ can be consistently estimated.
Our next, example writes the treatment effect γ as a function of the regressor x, φ(x),
and assumes that the baseline hazard is piece-wise constant. In example 2 we censored the
data at c to derive an estimator that was equivalent to maximum likelihood estimator. In
general, we can artificial censor durations to create more moments. Suppose that we censor
the durations of the treatment group at c, and denote the semi-integrated hazard by s0 , then
Es0 = E
where smax =
1
η
Rc
0
1
z0
= (1 − e−ηsmax )
η
η
θi du. Similarly, the indicator d has the following expectation
Ed = (1 − e−ηsmax ).
Suppose the regressor xi has two different values, x0 and x1 . and let N0 and N1 denote
the number of individuals with x = x0 and x = x1 , respectively. We can censor all the
observations with x = x0 at c0 and those with x = x1 at c1 . Consider choosing the censoring
point c1 such that
1 X
1 X
di −
di = 0.
(4)
N 0 x=x
N1 x=x
0
1
This expectation of this equation is satisfied if smax is the same for x = x0 and x = x1 , i.e.
smax (c0 , x0 ) = smax (c1 , x1 )
6
(5)
Suppose we have a MPH model and that the assumptions of Elbers and Ridder (1982) hold so
that φ(x0 ) 6= φ(x1 ) for time-invariant x. Then c0 6= c1 and equation (5) yields restrictions on
the parameters. In fact, artificially censoring the data many times gives as many restrictions.
Suppose that the regressor x is not relevant for the hazard rate so that φ(x0 ) = φ(x1 ) and
the identification condition of ER fails. This touches upon what ER consider the ‘main
contribution’ of their paper:
The main contribution of our paper is, that we prove that time dependence
and unobserved heterogeneity can be distinguished (..). Surprisingly, this is due
to the variation of individual probabilities with the explanatory variable x.
Equation (5) sheds some light on the need of “variation of the individual probabilities the
explanatory with x”. Let φ(x0 ) = φ(x1 ) then equation (5) holds for c0 = c1 so that the
function φ(x) is not identified through this restriction. This is in accordance with ER since
the MPH model is not identified in that case. Thus, we can use equation (4) to derive a
moment function. Let N0 and N1 denote the number of individuals with x = x0 and x = x1 ,
respectively. We write our first moment function as a function of c1 2 .
P
P
di
x=x0 di
− x=x1 .
(6)
g1 (c1 (θ)) =
N0
N1
To improve efficiency of the estimator, we also use a second moment function that is a
function of the parameters and c1 .
P
P
P
P
x=x0 di
x=x1 si
x=x1 di
x=x0 si
g2 (θ, c1 (θ)) =
−
.
(7)
N0
N1
N1
N0
Note that
P
x=x0
N0
di
is just the average of the indicators and not a function of the parameters.
At the true value of the parameter of interest, θ0 , E
P
x=x0
PN0
si
x=x0
si
=
P
1
−ηsmax )
x=x0 η (1−e
P
x=x1
N0
si
. The
distribution of η does not depend on x and therefore E N0
= E N1 . This implies
that the expectation of g2 (θ0 , c1 (θ0 )) is zero.
P
P
1
−ηsmax )
−ηsmax )
x=x1 η (1 − e
x=x0 (1 − e
Eg2 (θ0 , c1 (θ0 )) =
N0
N1
P
P
1
−ηs
max
(1 − e
) x=x0 η (1 − e−ηsmax )
− x=x1
N1
N0
= 0.
Example 4 illustrates the use of the moment function of equations (6) and (7).
Example 4.
Assume the following hazard model
θ = ηφ(x)λ(t)
2
Since the moment function is an implicit function of the parameters, one could call it a minimum distance
procedure.
7
where x is a time invariant regressor and φ(x0 ) 6= φ(x1 ) and η a realization of a mixing
distribution. Let λ(t) be a piecewise constant function that allows for a different baseline
hazard before and after c. For convenience, we refer to the individuals with x = x0 as the
group. We firstPcensor the treatment
treatment group and those with x = x1 as the control
P
di
di
x=x
0
and d¯1 (c) = x=x1 . Assume that
and control group at c and calculate d¯0 (c) =
N0
N
P1
P
di
di
0
1
− x=x
,
d¯0 (c) < d¯1 (c), otherwise relabel. Using the moment function g1 (c1 (θ)) = x=x
N0
N1
¯
¯
¯
¯
or, equivalently, g1 (c1 (θ)) = d0 (c)− d1 (c1 ) yields a censoring point c1 and d0 (c) < d1 (c)
implies c1 < c0 . Note that
Z
1 c0
θi du = φ(x0 ) ∗ c0
s0,max =
η 0
Z c1
1
s1,max =
θi du = φ(x1 ) ∗ c1 .
η 0
In large samples, the moment function g1 (c1 (θ)) yields s0,max = s1,max . This suggest the
1)
following estimator for φ(x
φ(x0 ) .
c1
φ(x1 )
= .
(8)
φ(x0 )
c
The sample is presumably finite and we use the moment function of equation (7) to improve
efficiency
g2 (θ, c1 (θ)) = d¯0 (c)s̄1 (c1 ) − d¯1 (c1 )s̄0 (c)
P
s0
P
s0
1 i
0 i
and s̄0 (c) = x=x
. Note that both g1 (c1 (θ)) and g2 (θ, c1 (θ)) are
where s̄1 (c1 ) = x=x
N1
N0
just functions of φ(x1 ) and φ(x1 ) since the durations were censored at (or before) c. Therefore,
misspecification of the baseline hazard for t > c has no effect on the consistency.
Suppose that the baseline hazard is normalized to have the unit value before c and the
value λ after c. Possible moment functions for estimating λ are
g3 (c1 (θ)) = d¯0 (c0 ) − d¯1 (c) or
g4 (θ, c1 (θ)) = d¯0 (c0 )s̄1 (c) − d¯1 (c)s̄0 (c0 )
g5 (θ) = s̄1 (c) − s̄0 (∞)
Note that this pair {g3 , g4 } resembles {g1 , g2 }. However, this time, the ‘treatment’ group is
1)
censored at c. One can test whether φ(x
φ(x0 ) is constant over time by allowing for a separate
φ(x1 )
φ(x0 )
1)
for t > c. Alternatively, one could estimate φ(x
φ(x0 ) as a function of all moment functions
(joint estimation of all parameters of interest) to increase the efficiency of the estimate.
The random distribution is unrestricted in example 4 while the baseline hazard was piecewiseconstant.
The next section is using the integrated likelihood principle to show how we can consistently
estimate the counterfactual durations, that is the duration of the non-treated as they were
treated.
8
4
Counterfactuals in duration analysis
The counterfacuals were introduced in econometric analysis to make inference of causal effects in treatments by analyzing those aspects of a counterfactual theory of causation in
terms of possible worlds, which are of great importance for empirical social scientists. Thus,
counterfactuals characterizes a possible world with minimum deviation from the actual world.
Three approaches are used in literature to model causation:
a) Structural Equation Model represented by the following type of equations
Y = Xβ + ε,
where β capture all causal connection between Y and X.
b) Potential Outcome Model of causation, which is used to explain the effect on the outcomes of some particular treatment (treatment group) relative to other particular treatment
(control group).
c) Directed Acyclic Graphs, which is a recent development and uses graphical approaches
to causation.
Counterfacuals become very important when we look at the differences between the factual
and counterfacual questions. Also, the choice of the counterfacual has to be close enough
to the data so that statistical methods provide empirical answers and the research questions
are well defined.
Thus, one of the most important conditions necessary to have a counterfactual causal question
is the relativity condition (the effect of one treatment is always relative to the effect of
another treatment or causal inference about a treatment in the actual world is based on
the counterfactual relation to what would have happened under exposure to a treatment in
the alternative possible world). Rubin (1980, 1986) considered that the basic assumption
necessary to assess potential outcomes and infer meaningful causal statements is the stableunit-treatment-assumption, which states that the outcome of a treatment for a given
unit is the same independent of the mechanism that is used to assign the treatment to that
unit. He also argue that one of the most important causal parameters to be analyzed is the
average causal effect of a treatment or the average treatment effect (of a treatment in
the actual world versus a treatment in the possible world), and is defined as the difference
between the expected value of the unit-level of outcomes in the actual world and possible
world
AT E = E (Ya ) − E (Yp ) .
To make the model applicable we need to have satisfied the independence assumption, which
can be obtain by looking at a randomized experiment in which units are randomly assigned to
different treatments. Thus, the initial population and the subpopulations in the treatments
do not differ from each other on average. The Illinois bonus experiment satisfy all the above
conditions and thus a counterfactual analysis on the noncompliers in the treatment group
can infer results about the applicability of the program to the overall population.
9
4.1
Counterfactuals for uncensored spells
To motivate our choice for a M P H model, we estimate the unobserved heterogeneity using
the following estimator
1
b
ηi = .
si
Regressing b
η i on the observables (age-average(age), (age-average(age)) squared, log(base period earnings)-average(log(BPE), dummy for race, dummy for sex), we can make inference
about our choice for the hazard model. Looking at the results from table 1 we can conclude
that there is correlation between the b
η i and the observables. This result suggest that the
best hazard model necessary to estimate the unemployment duration is M P H model.
4.1.1
I. Consider the following model of the hazard rate
θ (t) = η i φ (t) λ (t) ,
where vi is the heterogeneity, λ (t) is the baseline hazard, φ (t) = e−βxt , with
φ (t1 ) = 1,
½
λ1 if t1 ≤ 11
λ (t1 ) =
λ2 otherwise
where t1 is the duration of the non treated group, and
½
φ if t2 ≤ 11
φ (t2 ) =
1 otherwise
½
λ1 if t2 ≤ 11
,
λ (t2 ) =
λ2 otherwise
where t2 is the counterfactual
½ duration (duration of the
½ non-treated as they were treated).
1 if t1 ≤ 11
1 if t2 ≤ 11
and di2 =
.
Define the indicator di1 =
0 otherwise
0 otherwise
We have the semi-integrated hazard (the semi-integrated hazard is defined as the integrated
hazard divided by η i ) for the first spell equal to
s1 = λ1 t1 d1 + 11 (1 − d1 ) λ1 + (1 − d1 ) λ2 (t1 − 11) ,
and for the second spell
s2 = φλ1 t2 d2 + 11φ (1 − d2 ) λ1 + (1 − d2 ) λ2 (t2 − 11) .
Equalizing s1 = s2 we get the counterfactual duration t2 . Thus, we have
t2 (φλ1 d2 + (1 − d2 ) λ2 )+11 (1 − d2 ) (φλ1 − λ2 ) = 11 (1 − d1 ) (λ1 − λ2 )+t1 (λ1 d1 + (1 − d1 ) λ2 ) ,
and
t2 =
=
11 (1 − d1 ) (λ1 − λ2 ) − 11 (1 − d2 ) (φλ1 − λ2 ) + t1 (λ1 d1 + (1 − d1 ) λ2 )
φλ1 d2 + (1 − d2 ) λ2
11 [λ1 ((1 − d1 ) − φ (1 − d2 )) + λ2 (d2 − d1 )] + t1 (λ1 d1 + (1 − d1 ) λ2 )
.
φλ1 d2 + (1 − d2 ) λ2
10
Given that φ, λ1 , λ2 , are unknown, we can get an estimate for the counterfactual duration t2 ,
by replacing φ, λ1 , λ2 with their estimates, which are obtained solving the moment conditions
of Ridder and Woutersen (2001) . Thus,
³
´
h ³
´
i
b1 d1 + (1 − d1 ) λ
b1 (1 − d1 ) − φ
b (1 − d2 ) + λ
b2 (d2 − d1 ) + t1 λ
b2
11 λ
b
t2 =
.
b2
bλ
b1 d2 + (1 − d2 ) λ
φ
To get the distribution of the counterfactual duration t2 we use the Bootstrap method.
The Bootstrap Bootstrapping is characterized by using simulations generated from an
estimated model (estimated probability distribution). Given that, both theory and practice have validated the bootstrap for a wide range of applications, we use bootstrap to
estimate features of the distribution: as the standard errors and confidence intervals for
our contrafactual estimates. Although the methodology is nonparametric, the bootstrap
can be used to solve both parametric and nonparametric problems. Consider the following
model X1 , X2 , . . . , Xn ∼ iidD and we wish to estimate some parameter θ: If θ happens to
parametrize a family of distributions, then we have a parametric problem; otherwise it is
nonparametric (D -unknown).
For our situation we use the nonparametric bootstrap. Suppose that we have constructed
an estimator for theta, called b
θ; this is a function of the random variables X1 , X2 , . . . , Xn .
Our objective, is to develop confdence intervals for θ via estimating the distribution of b
θ − θ.
We can accomplish this objective by simulating bootstrap samples (pseudo-data). A
single bootstrap sample is obtained by sampling randomly, with replacement, n observations
b as an estimate
from the original sample. To do this we examine the empirical distribution D
of the unknown distribution D, by forming the Empirical Distribution Function (edf ) of the
data. We construct the edf as the Cumulative Distribution Function (cdf ) F of the data.
Thus using the realizations x1 , x2 , . . . , xn of the data we construct edf as
n
1X
b
F :=
1{Xi <x} ,
n
i=1
where F (x) is the cdf of the probability model, that is
F = Pr[X1 ≤ x].
Once Fb is constructed, we use it to generate new random variables from this distribution by
inverting the edf and plugging uniform random variables into the resulting function. This
is a bit problematic because the edf is not monotonic, and thereby is not easily invertible.
In practice, a “generalized inverse” is constructed. Thus, we use this method to generate at
least B = 1000 new data sets and we compute the statistic on each pseudo-data set:
´
³
(1)
(1)
(1)
b
= b
θ X1 , X2 , . . . , Xn(1)
θ
´
³
(2)
(2)
(2)
b
θ
= b
θ X1 , X2 , . . . , Xn(2)
(B)
b
θ
..
.³
´
(B)
(B)
b
= θ X1 , X2 , . . . , Xn(B) ,
11
which gives us B new estimates of θ, which are intimately related to the original b
θ. Now,
we can estimate the distribution of b
θ − θ. To estimate the the bias and the variance of the
estimator we use
B
³ (∗) ´
1 X b(b) b
b
=
bias θ
θ −θ
B
b=1
¶2
B µ
³ (∗) ´
X
(b)
(∗)
1
b
=
θ −b
,
θ
V ar b
θ
B−1
b=1
holds for bootstrap quantities under the probability mechanism Fb and
where the asterisk
(∗)
P
(b)
b
b
θ = B1 B
b=1 θ .
To determine the confidence interval. Thus, we form the edf of the data generated by
(1)
(2)
(B)
b
θ, b
θ −b
θ, . . . , b
θ −b
θ and compute its quintals x
bα which are estimates of the quintals
θ −b
xα . The estimate for 1 − α confidence interval will be determined as
(∗)
θ−x
b α2 ].
[b
θ−x
b1− α2 ≤ θ ≤ b
¡ ¢
Alternatively, to compute the V ar b
t2 , we linearize b
t2 . Thus,
b
t2 =
and
11 [λ1 ((1 − d1 ) − φ (1 − d2 )) + λ2 (d2 − d1 )] + t1 (λ1 d1 + (1 − d1 ) λ2 )
φλ1 d2 + (1 − d2 ) λ2
³
´
³
´
³
´
b
b
∂b
t2
b1 − λ1 + ∂ t2 |λ ,λ ,φ λ
b2 − λ2 + ∂ t2 |λ ,λ ,φ φ
b−φ ,
+
|λ1 ,λ2 ,φ λ
b1
b2 1 2
b 1 2
∂λ
∂λ
∂φ
¡ ¢
V ar b
t2 =
Ã
!2
!2
³ ´
³ ´
b
∂
t
∂b
t2
2
b1 +
b2
|λ1 ,λ2 ,φ V ar λ
|λ1 ,λ2 ,φ V ar λ
b1
b2
∂λ
∂λ
Ã
!2
³ ´
∂b
t2
b ,
|λ1 ,λ2 ,φ V ar φ
+
b
∂φ
Ã
b1 , λ
b2 , and φ.
b Thus, this method is not recomwhich require distribution assumptions for λ
mended here.
Having the variance of the counterfactual duration we can compute confidence intervals
for t2 and we can get the counterfactual distribution.
4.1.2
II. Now consider the following M P H model
ηex1 β Λ (t1 ) = z ⇔ ex1 β Λ (t1 ) = s,
where s is the semi-integrated hazard for the observed spell of unemployment for the nontreated group. Suppose x2 is the counterfactual treatment, then we can find the counterfactual duration, t2
ηex2 β Λ (t2 )
z ⇔ ex2 β Λ (t2 ) = s
µ
¶
³ s ´
³
´
z
−1
−1
−1
(x1 −x2 )β
=
Λ
e
Λ
(t
)
.
=
Λ
⇒ t2 = Λ
1
ηex2 β
ex2 β
=
12
Assume three breaking points c1 , c2 , c3 . We have the first spell hazard at each breaking point:

η exi1 β , if t1 ≤ c1


 i xi1 β
ηie
λ1 , if c1 < t1 ≤ c2
θ (t1 ) =
,
xi1 β λ , if c < t ≤ c
η
e

2
1
3

 i xi1 β 2
ηie
λ3 , if c3 < t1
and second spell hazard

η exi2 β ,


 i xi2 β
ηie
λ1 ,
θ (t2 ) =
x
β
i2
ηe
λ ,


 i xi2 β 2
ηie
λ3 ,
Define the indicator dlj , with
points) as
½
dl1 =
½
dl3 =
if
if
if
if
t2 ≤ c1
c1 < t2 ≤ c2
c2 < t2 ≤ c3
c3 < t2
l = 1, 2 (for the two spells) and j = 1, 2, 3, 4 (for the breaking
1 if tl ≤ c1
0 otherwise
1 if c1 < tl ≤ c2
0 otherwise
½
1 if c3 < tl
, dl4 =
0 otherwise
, dl2 =
1 if c2 < tl ≤ c3
0 otherwise
½
.
Then we have the integrated hazard for the first spell defined as
z1 = ηex1 β (c1 t1 d11 + (c1 + λ1 (t1 − c1 )) d12 +
+ (c1 + λ1 (c2 − c1 ) + λ2 (t1 − c2 )) d13 +
+ (c1 + λ1 (c2 − c1 ) + λ2 (c3 − c2 ) + λ3 (t1 − c3 )) d14 )
= ηi ex1 β (c1 (1 − λ1 ) (d12 + d13 + d14 ) + c2 (λ2 − λ3 ) (d13 + d14 )
+c3 (λ3 − λ4 ) d14 + t1 (c1 d11 + λ1 d12 + λ2 d13 + λ3 d14 )),
and for the counterfactual spell as
z2 = ηex2 β (c1 t2 d21 + (c1 + λ1 (t2 − c1 )) d22 +
+ (c1 + λ1 (c2 − c1 ) + λ2 (t2 − c2 )) d23 +
+ (c1 + λ1 (c2 − c1 ) + λ2 (c3 − c2 ) + λ3 (t2 − c3 )) d24 )
= ηex2 β (c1 (1 − λ1 ) (d22 + d23 + d24 ) + c2 (λ2 − λ3 ) (d23 + d24 )
+c3 (λ3 − λ4 ) d24 + t2 (c1 d21 + λ1 d22 + λ2 d23 + λ3 d24 )).
Given that z1 = z2 we can find the counterfactual duration, t2 by solving the following
equation
³
´
t2 = Λ−1 e(x1 −x2 )β Λ (t1 ) ,
which gives
c1 d11 + λ1 d12 + λ2 d13 + λ3 d14
ti1
c1 d21 + λ1 d22 + λ2 d23 + λ3 d24
c1 (1 − λ1 ) (d12 + d13 + d14 ) + c2 (λ1 − λ2 ) (d13 + d14 ) + c3 (λ2 − λ3 ) d14
+e(x1 −x2 )β
c1 d21 + λ1 d22 + λ2 d23 + λ3 d24
c1 (1 − λ1 ) (d22 + d23 + d24 ) + c2 (λ1 − λ2 ) (d23 + d24 ) + c3 (λ2 − λ3 ) d24
−
.
c1 d21 + λ1 d22 + λ2 d23 + λ3 d24
t2 = e(x1 −x2 )β
13
The condition that t2 < t1 is that xi1 − xi2 < 0. An estimate of the counterfactual duration
can be obtained by replacing λ1 , λ2 , λ3 , with their estimates, which are obtained, again,
solving the moment conditions of Ridder and Woutersen (2001) . Thus,
b1 d12 + λ
b2 d13 + λ
b3 d14
c1 d11 + λ
b
t2 = e(x1 −x2 )β
t
b1 d22 + λ
b2 d23 + λ
b3 d24 i1
c1 d21 + λ
³
´
³
´
³
´
b2 (d13 + d14 ) + c3 λ
b3 d14
b1 (d12 + d13 + d14 ) + c2 λ
b1 − λ
b2 − λ
c1 1 − λ
+e(x1 −x2 )β
b1 d22 + λ
b2 d23 + λ
b3 d24
c1 d21 + λ
³
´
³
´
³
´
b1 (d22 + d23 + d24 ) + c2 λ
b1 − λ
b2 − λ
b2 (d23 + d24 ) + c3 λ
b3 d24
c1 1 − λ
−
,
b1 d22 + λ
b2 d23 + λ
b3 d24
c1 d21 + λ
and the variance is obtained using again either the bootstrap method or by solving
¡ ¢
V ar b
t2 =
Ã
Ã
!2
!2
³ ´
³ ´
b
∂b
t2
∂
t
2
b1 +
b2
|λ1 ,λ2 ,λ3 V ar λ
|λ1 ,λ2 ,λ3 V ar λ
b1
b2
∂λ
∂λ
!2
Ã
³ ´
∂b
t2
b3 .
|λ1 ,λ2 ,λ3 V ar λ
+
b
∂φ
Again, having the variance of the counterfactual duration we can compute confidence intervals
for t2 and we can get the counterfactual distribution.
4.1.3
Problems in estimation
If x1 = x2 we have
x1 = x2 ⇔ Λ−1 (Λ (t1 )) = t1
³
´
t2 = Λ−1 e∆xβ Λ (t1 ) = Λ−1 (Λ (t1 )) = t1 ,
or when x1 = 0, given that ex2 β Λ (t2 ) = ex1 β Λ (t1 ) we have
x2 β + ln (Λ (t2 )) = ln (Λ (t1 ))
1 Λ (t2 )
.
x2 = − ln
β Λ (t1 )
4.2
MPH model examples
Suppose at the beginning of week 25, 18 people are unemployed.
hazard
¡ Assume the 1baseline
¢
is 1 for all periods
(λL−1 =
¢ λL = 1). In week 25, 6 find a job PLeave,L−1 = 3 . In week 26,
¡
2 find a job PLeave,L = 16 .
I. Consider PLeave,L = Fraction of leaves during last period = 1 − e−ηλL , and assume η i = η,
then
ln (1 − PLeave,L )
⇒η=−
.
λL
14
Using just data on the last period, we can find η
ln (1 − PLeave,L )
⇒ η=−
λ
¶L
µ
1
= − ln 1 −
6
5
= − ln = ln 6 − ln 5 = 0.182.
6
II. Assume η i = η H , η L , similarly, having PLeave,L , PLeave,L−1 , λL−1 , λL , we can find η H ,
p,given that with probability p, η = η H and with probability 1 − p, η = 0.
Thus, we have that
¡
¢

−η H λL−1 p
(1)
 PLeave,L−1 =
¡ Pr (Leave ¢at L−1 |ηH ) Pr (η H ) = 1 − e
=
Pr
(Leave
at
L|η
,
L
)
Pr
(η
PLeave,L = PLeave,L|L−1 PLeave,L−1
.
−1
H
H) P
¢Leave,L−1
¡

p
p
−η
λ
L
H
= Pr (Leave at L|ηH , L−1 ) 3 = 1 − (fraction that leave at L−1 ) e
(2)
3
Solving this system of two equations with two unknowns we can find p and η H . Thus, we
have from the first equation that:
p= ¡
PLeave,L−1
1
¢= ¡
¢.
−η
λ
−η
L−1
H
3 1 − e H λL−1
1−e
Replacing p in the second equation we get e−ηH = 13 .
Thus η H = − ln (0.3333) = 1.0986 and
1
1
¢=
¡
3 1 − e−ηH λL−1
3 − 3 13
p = 0.5.
p =
5
Estimation Results
Table1. Regression results for unobserved heterogeneity on observables (for
experiment)
Valid cases: 8134 Dependent variable: b
ηi
R-squared: 0.025
V ariable
Estimate Standard t − value P rob > |t|
Error
Constant
0.0798
0.0011
70.3992
0.000
age − age
−0.00057 0.0001
−6.9751
0.000
(age − age)2
0.00003
0.00001
4.2421
0.000
log(BP E) − log(BP E) 0.0057
0.0016
3.4454
0.001
black
−0.0132
0.0013
−9.6831
0.000
male
0.0010
0.0012
0.8406
0.401
log(wb_d) − log(wb_d) −0.0136
0.0028
−4.8235
0.000
15
claimant bonus
Correlation
with b
ηi
−
−0.0728
0.0209
−0.0244
−0.1102
0.0133
−0.0607
Table2. Regression results for unobserved heterogeneity on observables (for treated observations)
Valid cases: 4183 Dependent variable: b
ηi
R-squared: 0.023
V ariable
Estimate Standard t − value P rob > |t| Correlation
Error
with b
ηi
Constant
0.0829
0.0016
51.4813
0.000
−
age − age
−0.0006
0.0001
−5.5307
0.000
−0.0781
2
(age − age)
0.00003
0.00001
3.1328
0.002
0.0127
log(BP E) − log(BP E) 0.0053
0.0023
2.2966
0.022
−0.0162
black
−0.0134
0.0019
−6.8638
0.000
−0.1083
male
−0.0004
0.0017
−0.2389
0.811
0.0038
log(wb_d) − log(wb_d) −0.0103
0.0039
−2.5990
0.009
−0.0472
Looking at the results from tables 1and 2 we can conclude that there is correlation between
the b
η i and the observables. This result suggest that the best hazard model necessary to
estimate the unemployment duration is M P H model.
Table 3. Regression results for treatment effect
Valid cases: 8134 Dependent variable: duration of unemployment
Total SS: 1293806.222 Degrees of freedom: 8132
R-squared: 0.003
V ariable
Estimate Standard t − value P rob > |t|
Error
Constant
17.2816
0.1947
88.7369
0.000
X = 1 (control group) 1.3391
0.2794
4.7922
0.000
Table 4. Results for the average treatment effect for treated versus non-treated
Control Group T reated
Causal ef f ect
(X = 1)
All(X = 0) AT E = E (Ynt ) − E (Yt )
Benef it weeks
18.6207
17.28
1.3391
Standard errors 0.2028
0.2000
0.2739
N (individuals)
3951
4183
−
Comparing tables 3 and 4 we observe that we get the same treatment effect and standard
errors by regressing the duration of unemployment on the dummy for treatment, or by
computing the average treatment effect and bootstrapping it to get standard errors.
Table 4. IV Regression results for the treatment effect on the treated
Valid cases: 8134 Dependent variable: duration of unemployment
Total SS: 1293806.222 Degrees of freedom: 8132
R-squared: 0.004
V ariable
Estimate Standard t − value P rob > |t|
Error
Constant
17.0565
0.2120
80.4336
0.000
−1 0
0
b
0.3286
5.4845
0.000
X = (R R) R X 1.8023
We observe that when we use IV (an instrument for the treatment) we get the treatment
effect on the treated. In this case, we observe a slight increase in the R − squared, and we
16
also observe that the effect of the treatment effect on the treated is increasing from 1.3391
weeks to 1.8023 weeks. This result help us to conclude that if the program will be applied
to the overall population we’ll get smaller unemployment durations for the unemployed and,
thus, the U I claims we’ll be lower.
The fallowing results are referring to the counterfactual duration for non-compliers.
5.1
Assumption 1
We assume that people who have not find a job after 26 weeks will never find a job. Thus,
we censor the observations that exceed 26 weeks to 26 and we compute counterfactuals for
the non-compliance group. For the individuals from the non compliance group we have 658
individuals. To derive the counterfactual durations we use estimates for the baseline hazard
obtained by Woutersen and Ridder (2001) . Looking at the average duration of unemployment (table 7) for non-compliers we observe that is smaller than the average duration of
unemployment for the control group (both groups being not treated), also, the distribution
of the unemployment duration of non-compliers (fig.4), is different from the distribution of
unemployment duration for control group (fig.2).in the sense that the duration of unemployment is all the time below the one from the control group. Thus, we can infer that
non-compliers have different incentives to find a job in the absence of a treatment, and, their
unobserved heterogeneity is different than the ones from the control and compliance groups
(see fig 2,3 4). The individuals from the noncompliance group seem more willing to find a
job quicker. Thus, we expect that if they will be treated they will have a smaller duration of
unemployment than the compliers in the treatment group (average counterfactual duration
for non-compliers would be smaller than the average duration of unemployment for compliers, see table 7.) The counterfactual durations and they standard errors obtained using
bootstrap method are presented in the Table 6.
17
Table 6. Counterfactuals durations and standard errors for the non-compliance group
weeks of unemployment counterf actuals f or 26 breaking points Standard errors
1
0.15
0.0223
2
1.15
0.0003
3
2.05
0.00001
4
3
0.0002
5
3.95
0.0244
6
4.85
0.0243
7
5.8
0.1305
8
6.7
0.1227
9
7.65
0.0485
10
8.55
0.0486
11
9.5
0.0600
12
10.4
0.0723
13
11.35
0.0728
14
12.25
0.0837
15
13.2
0.1684
16
14.15
0.3385
17
15.05
0.1642
18
16
0.1202
19
16.9
0.1085
20
17.85
0.1082
21
18.75
0.1139
22
19.7
0.1326
23
20.6
0.1121
24
21.55
0.1106
25
22.5
0.1147
26
22.95
0.1437
We observe that for the individuals from the non-compliance group that found a job within 26
weeks, their duration of unemployment would decrease if they would receive the treatment,
and that people that have shorter durations of unemployment, less than 11 weeks, will have
shorter durations if they will get the treatment than people that have higher duration of
unemployment and they get the treatment.
In Table 7 we present the average unemployment duration for non-compliers when they are
not receiving the treatment and the average unemployment duration for non-compliers when
they would get the treatment and we compare with the other groups.
18
Table 7 . Average unemployment durations and standard errors obtain using Bootstrap
method for 5000 replica
Number of individuals in the claimant experiment is 8134
Control Group
T reated
Counterfactuals
(X = 1)
All(X = 0) Compliance(R = 1) (R = 0) N on − compliance
Benef it weeks
18.6207
17.28
17.056
18.48
16.41
Standard errors 0.2028
0.2000
0.2103
0.4789
0.3877
N (individuals)
3951
4183
3525
658
658
Table 8. Results for the average treatment effect (AT E) for the non-compliers
Counterfactuals
Causal ef fect
N on − compliance(R = 0) N on − compliance AT E = E (Ya ) − E (Yp )
Benef it weeks
18.48
16.41
2.0719
Standard errors 0.4789
0.3877
0.6447
N (individuals)
658
658
658
We observe that if we would have treated the non-compliance group, their average duration of
unemployment will be smaller (with higher standard errors) than the one from the compliance
group. This result is what we expected to see. Looking at the causal effect (the average
treatment effect for non-compliers), we observe that their average unemployment duration
decreased by about 2 weeks if they were treated, which implies that we get a better treatment
effect that the one we estimate by looking at the results of IV regression (treatment effect
on the treated). Thus, if the program will be applied to overall population, we’ll get a better
treatment effect than the one estimated by Meyer. This result help us to conclude that if the
bonus will be applied to overall population the U I claims will be lower and thus the program
is more efficient than initially thought.
19
Distribution of unemployment duration for individuals from the claimant
experiment
60
Unemployment duration
50
40
30
20
10
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
# of individuals
Figure 1:
Distribution of unemployment duration for individuals from the control group
60
Unemployment duration
50
40
30
20
10
0
0
500
1000
1500
2000
2500
# of individuals
Figure 2:
20
3000
3500
4000
4500
Distribution of unemployment duration for individuals from compliance group
60
Unemployment duration
50
40
30
20
10
0
0
500
1000
1500
2000
2500
3000
3500
4000
# of individuals
Figure 3:
Distribution of unemployment duration for individuals from non-compliance group
45
Unemployment duration
40
35
30
25
20
15
10
5
0
0
100
200
300
400
# of individuals
Figure 4:
21
500
600
700
5.2
5.2.1
Assumption 2 (censored spells)
Estimating θ.
First we smooth the data by assuming
it uniformly distributed in the interval before leavR
f (s;x)
ing. We have θ(s) = 1−F (s;x) ⇒ θ(s; x)ds = − ln(1 − F (s; x)) where F (s; x) is uniformly
distributed between 0 and 1. We consider the following hazard model
θ = ηφ(x)λ(t)
where x is a time invariant regressor and φ(x0 ) 6= φ(x1 ) and η a realization of a mixing
distribution. Let λ(t) be a piecewise constant function that allows for a different baseline
hazard before and after c. For convenience, we refer to the individuals with x = x0 as the
treatment group and those with x = x1 as the control group. We censor the treatment and
control group at c0 and c1 respectively. We choose the censoring points in such a way that the
survival probabilities are equal. Thus, we obtain a sequence c0j and c1j ,where j = 1, 2, . . . , 25,
by equating
Pr (x = 1, j) = Pr(x = 0, c0j )
Pr (x = 0, j) = Pr(x = 1, c1j ).
We have that
1 − Pr(x = 0, c0j ) = e−c0j φ(x0 )
1 − Pr(x = 1, c1j ) = e−c1j φ(x1 )
And given that
s0,max =
s1,max =
Z
1 c0
θi du = φ(x0 ) ∗ c0
η 0
Z
1 c1
θi du = φ(x1 ) ∗ c1 .
η 0
In large samples s0,max = s1,max . This suggest the following estimator for
φ(x1 )
φ(x0 ) ,
φ(x1 )
c0
= .
φ(x0 )
c1
Using moments of the form
gj = s0j (c0j , λ0j ) − s1j (c1j , λ1j ), j = 1, 2, . . . , 25,
which are linear in parameters λ, we estimate the hazard rates for the control and treatment
group.
Having estimates for the baseline hazard we can compute the counterfactuals for the non
treated group. To estimate the heterogeneity we consider the following cases:
Results for counterfactual duration and average durations for non compliers will be provided.
22
6
Conclusion
In this paper we use a new estimator for estimating the duration of unemployment, and
it allows that λ(t) to be discontinuous. This estimator can deal with selective compliance
so that we can use it to evaluate the re-employment bonus experiment of Illinois. We find
that allowing for selective-compliance and unobserved heterogeneity leads to more optimistic
conclusions about monetary incentives than Meyer (1996).
7
Appendices
Appendix 1
Rc
We define zis,max = 0 is θ(s, x)ds;
Edis
Z
= Pr{tis ≤ cis } = Pr{
tis
θ(s, x)ds ≤
0
= 1 − e−zis,max ;
Ezis =
Z
zis,max
Z
cis
0
θ(s, x)ds} = Pr{zis ≤
Z
cis
θ(s, x)ds}
0
zis ezis dzis + zis,max e−zis,max
0
= 1 − e−zis,max = Edis .
Q.E.D.
Appendix 2,
In this appendix we are concerned with the estimation of two hazard models. The first
model is the exponential model with a exponential mixing distribution.
θ = η where η ∼ Gamma(α, β).
(9)
This yields the following density function for t given η
f (t|η) = ηe−ηt .
Let M (η) denote the mixing distribution; the density of a observed realization has the following form.
Z
g(t) =
ηe−ηt dM (η)
Z α α −η(β+t)
αβ α
β η e
dη =
=
(10)
Γ(α)
(β + t)α
However, the density of equation (??) could also be generated by the second model.
θ=
α
.
(β + t)
23
(11)
It follows from ER that we need a regressor to distinguish between the models of equation
(9) and (11). Suppose this regressor has two values and that we normalize φ(x0 ) to be one
and denote φ(x1 ) by γ. We first estimate γ.
1 P
x=x si (ε)
N0
γ̂ = 1 P 0
.
x=x1 si (ε)
N1
In large samples we have the following for equation (9).
1 P
γε
x=x Esi
N0
.
=
γ̂ ≈ 1 P 0
ε
x=x1 Esi
N
1
Appendix 3A
We first censor the treatment and control group at c and calculate d¯0 (c) =
d¯1 (c) =
P
x=x1
NP
1
di
P
x=x0
N0
di
and
. Assume that d¯0 (c) < d¯1 (c), otherwise relabel. Using the moment function
di
P
di
1
− x=x
, or, equivalently, g1 (c1 (θ)) = d¯0 (c)− d¯1 (c1 ) yields a censoring
g1 (c1 (θ)) =
N0
N1
point c1 and d¯0 (c) < d¯1 (c) implies c1 < c0 . Note that
Z
1 c0
θi du = φ(x0 ) ∗ c0
s0,max =
η 0
Z c1
1
s1,max =
θi du = φ(x1 ) ∗ c1 .
η 0
x=x0
In large samples, the moment function g1 (c1 (θ)) yields s0,max = s1,max . This suggest the
1)
following estimator for φ(x
φ(x0 ) .
c1
φ(x1 )
= .
(12)
φ(x0 )
c
Appendix 4
With full compliance and time-invariant treatment effect, theorem 2 would just be corollary of theorem 1 and X would always have the same value as R. With partial compliance, X
is a probabilistic function of R and η. For example, in our application, R = 1 is a necessary
condition for treatment, X = 1 but the people with a low values of η refuse more often to
participate in the re-employment bonus experiment. As a result, X is correlated with η and
X is not randomly assigned. In general, we do not need {X = 1 ⇒ R = 1}; it suffices for our
estimator that R and X are correlated (for example, path-dependent preferences will do).
The artificial censoring point is a function of the regressor X, i.e. we censor at c0 if
X = 0 and at c1 if X = 1. Consider moment function g1 (c1 |c0 ) and let N0 and N1 denote
the number of individuals for which R = 0 and R = 1 respectively.
1 X
1 X
di −
di
g1 (c1 |c0 ) =
N0
N1
R=0
R=1
X
X
X
X
1
1
1
1
=
di +
di −
di −
di
N0
N0
N1
N1
R=0,X=0
R=0,X=1
24
R=1,X=0
R=1,X=1
Artificial censoring ensures that
Eg1 (c1 |c0 ) = E{
1
N0
X
di +
R=0,X=0
1
N0
X
R=0,X=1
di −
1
N1
X
R=1,X=0
di −
1
N1
X
R=1,X=1
di } = 0
while correlation between R and X ensures that Eg1 (c1 |c0 ) is monotone in c1. Choosing
c0 = ε, 2ε, 3ε, ... and then determining c1 gives enough moments to identify the baseline
hazard in the setting of theorem 1. We now add moments by choosing c1 = ε, 2ε, 3ε, ... to
identify the treatment effect as a piecewise-constant function of time. For efficiency, we also
use the moments of g2 .
Appendix 5
Variances of moment functions
g1 = d¯1 − d¯0
Eg1 (γ 0 , λ0 ) = 0
X
1
Eg1 g10 = E(d¯1 − d¯0 )2 = 2 E{ (di1 − di0 )}2
N
X
1
1
i
=
E{ (d1 − di0 )2 } = E{(di1 − di0 )2 }
N2
N
(13)
An estimator for the expression in (13):
Estimator(Eg1 g10 ) =
1 X i
{ (d1 − di0 )2 }.
N2
g1 = d¯1 − d¯0
Eg1 (γ 0 , λ0 ) = 0
X
1
Eg1 g10 = E(d¯1 − d¯0 )2 = 2 E{ (di1 − di0 )}2
N
X
1
1
i
=
E{ (d1 − di0 )2 } = E{(di1 − di0 )2 }
2
N
N
8
(14)
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