WORKING PAPER 2002:20
Dynamically assigned
treatments: duration models,
binary treatment models,
and panel data models
Jaap H. Abbring
Gerard J. van den Berg
wp 2002-20.qxd 02-11-11 14.10 Sida 1
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Duration Models, Binary Treatment Models, and Panel Data Models
Jaap H. Abbring Gerard J. van den Berg
y
August 30, 2002
Abstract
Often, the moment of a treatment and the moment at which the out-comeofinterestoccursarerealizationsofstochasticprocesseswith depen-dent unobserveddeterminants. Notably,bothtreatment and outcome are characterizedbythemomenttheyoccur.Wecomparedi erentmethodsof inferenceofthetreatmente ect,andwearguethatthetimingofthe treat-ment relative totheoutcomeconveys usefulinformationon thetreatment e ect,whichis discardedinbinary treatment frameworks.
Department of Economics, Free University Amsterdam, and Department of Economics, UniversityCollegeLondon
y
Departmentof Economics,Free UniversityAmsterdam, DeBoelelaan1105, NL{1081HV Amsterdam,TheNetherlands,TinbergenInstitute,IFAU-Uppsala,andCEPR.
Keywords: program evaluation, treatment e ects, bivariate duration analysis, selection bias,hazardrate,unobservedheterogeneity, xede ects,randome ects.
JELcodes:C14,C31,C41.
Thanks to participants at the Tenth Panel Data Conference in Berlin, 2002, in particu-lartoourdiscussantBoHonore,forhelpful comments.
In virtually every real-life treatment situation,both the treatment and the out-comeof interestare realizedatspeci cpointsin time. Typicalexamples include the e ect of trainingprograms orpunitive bene tsreductions on unemployment durations,thee ect ofthehiringofreplacementworkers onstrikedurations,and the e ect of promotions on tenure. An extensive literature on the evaluation of social programs and treatment e ects exists, but in general this literature does not address nor exploit the speci c informationon the timing of events (see, for instance,Heckman, LaLondeandSmith,1999,foranoverview ofthisliterature). Abbringand Vanden Berg(2002) (henceforth AVdB) advanceonthis litera-ture by explicitly considering the evaluation of treatment e ects in a \duration variables"context.Considerasubjectinacertainstate.Afteracertainstochastic amount of time, the subject leaves this state. The subject may receive a treat-ment at some stochastic moment before it leaves the state. The parameter of interest is the e ect of the treatment on the exit rate out of the state. AVdB adopt an explicit general model framework for the distribution of the durations untiltreatment and outcome. (In fact,they present the modelinterms of coun-terfactual variables, but in the present paper we suppress this for expositional convenience.) It allows the duration variablesto be dependent by way of depen-dent unobserved determinants, with each single duration having its own Mixed Proportional Hazard (MPH) model speci cation. In addition to this, a causal e ect of the realized treatment works on the exit rate out of the current state from the moment the treatment is realized onwards. The MPH model is by far themost populardurationmodel(see Vanden Berg,2001,for asurvey).Models tting into the AVdB model framework have been estimated by, for example, Cardand Sullivan(1988),Gritz (1993),Lillard(1993), Lillardand Panis (1996), Bonnal, Fougere and Serandon (1997), Abbring, Van den Berg and Van Ours (1997),and Vanden Berg,Vander Klaauwand VanOurs (2004).
AVdB demonstrate thattheir baselinemodel, includingthe causal treatment e ect, is non-parametrically identi ed from single-spell duration data, that is, from a random sample of subjects owing into the state of interest and having subsequently been followed over time. This result has a number of notable as-pects. First, it does not require exclusion restrictions on observed covariates, so the data need not contain a variable that a ects the treatment assignment but doesnot a ect the outcome of interest other than by way of the treatment. Ex-clusion restrictions are often diÆcult to justify. If a variable is observed by the analyst then itis oftenalsoobservable tothe individuals underconsideration. If
maybe subjecttotreatment,thenhe takeshisvalue ofthe variableintoaccount todeterminehis optimalstrategy, andthis strategy a ects the rate atwhich the individual leaves the state of interest. Indeed, if the individual knows that the variable is an important determinant of the treatment assignment process then he may have a strong incentive to inquire the actual value of the variable. Sec-ondly,theresult doesnotrequire parametricfunctionalformassumptionsonthe bivariateprobabilitydistributionoftheunobserved heterogeneitytermsoronthe durationdependence, the covariate e ects, and the treatment e ect. Obviously, thisisadesirableproperty. Thirdly,asmentionedabove,the AVdBmodelallows for selectione ects by way of unobserved determinants a ectingboth the treat-ment assignment and the outcome. In other words, it is not necessary to make aconditionalindependence assumptionstating thatthe data are able tocapture all systematic determinants of the process of treatment assignment so that the remaining observed variation inthe treatment assignment is independent of the determinantsoftheoutcome ofinterest.Inapplications,suchanassumptionmay be diÆcult to justify, for example if the treatment assignment is carried out by caseworkers whousediscretionarypower,taking individualcharacteristicsofthe subject intoaccount that are unobserved to the analyst.
Standard methods of treatment evaluation often rely on exclusion restric-tions, parametric functional form assumptions on the joint distribution of the \errorterms"inthe model,orconditionalindependenceassumptions,toidentify the treatment e ect. In this sense, treatment evaluation with the AVdB model comparesfavorably tothosemethods. Theaimof the presentpaperistoprovide abetter understanding of this. More precisely,we examinewhich informationor variation in the data enables identi cation of the treatment e ect in the AVdB model framework, by comparing the model speci cation and the data to those usedinstandard methodsof treatment evaluationinthe presence ofselection ef-fects. Speci cally,we makea comparisonto latent variablemethodswith binary treatment indicators and to panel data methods (see e.g. Wooldridge, 2002, for atextbook overview).
The paper is organized as follows. In Section 2 we present the AVdB model framework. Section 3 makes the comparison to latent variablemethods. Section 4makes the comparison topanel data methods. Section5 concludes.
cally assigned treatments
Considerasubjectinacertainstate.Afteracertainamountoftime,the subject leaves this state. The subject may receive a treatment at some moment before itleaves the state. We are interested in the determinantsof the event of leaving the state, so the latter event is the event of interest, and the durationuntil this event is the variable of interest. To x thoughts, consider an individual who is unemployed and who moves to employment ata certain point of time, and who may receive a training at some point during his spell of unemployment. For the sakeofconvenience,weusetheterm\individual"ingeneraltodenotethesubject. We normalize the point of time at which the individual enters the state to zero. The durations T
m
and T p
measure the duration until the event of interest and the duration until treatment, respectively. The populationthat we consider concernsthein owintothestate,andtheunconditionalprobabilitydistributions thatarede ned belowaredistributionsinthein owintothe state.Whetherthis is the in ow at a xed point of calendar time or the total in ow over time (or the in owatanother rangeof in owdates) depends onthe application athand.
The two durations are random variables. We use t m
and t p
to denote their realizations. We assume that, for a given individual in the population, the du-rationvariablesare absolutelycontinuous random variables.We assume that all individual di erences in the joint distribution of T
m ;T
p
can be characterized by explanatoryvariablesX;V,whereX isobserved andV isunobserved tothe ana-lyst.Ofcourse,the jointdistributionof T
m ;T
p
jX;V canbeexpressed intermsof the distributions of T p jX =x;V and T m jT p = t p
;X =x;V. The latter distribu-tionsare inturncharacterizedby theirhazardrates
p (tjx;V)and m (tjt p ;x;V), respectively. 1
Asnotedintheintroduction,weareinterestedinthecausale ectoftreatment on the exit out of the current state. The treatment and the exit are character-ized by the moments at which they occur, and we are interested in the e ect of the realization of T
p
on the distribution of T m
. To proceed, we assume that, 1
For a nonnegative random (duration) variable T, the hazard rate is de ned as (t) = lim
dt#0
Pr(T 2 [t;t+dt)jT t)=dt. Somewhat loosely, this is the rate at which the spell is completedattgiventhatithasnotbeencompletedbefore,asafunctionoft.Itprovidesafull characterizationofthedistributionofT(seeLancaster,1990,andVandenBerg,2001).Consider thedistributionof aduration variableconditionalonsomeother variables.It iscustomaryto useavertical\conditioningline"within theargumentofahazardratein ordertodistinguish between(ontheleft-handside)thevalueofthedurationvariable atwhich thehazardrateis evaluated, and(ontheright-hand side)thevariablesthatareconditionedupon.
m p the realization of T
p
a ects the shape of the hazard of T m
from t p
onwards, in a deterministicway. This implies that the causal e ect is captured by the e ect of t p on m (tjt p ;x;V) for t > t p
. It is ruled out that t p a ects m (tjt p ;x;V) on t2 [0;t p
]. The latter could be called a \no anticipation" assumption, because it basically means that the individual's behavior before the moment of treatment does not depend on the future realization of the moment of treatment. AVdB show that such anassumption is crucialfor identi cation.
2 LetV :=(V m ;V p ) 0
bea(21)-vectorofunobservedcovariates.LetT p ?? V m jx;V p , implyingthat p (tjx;V)= p (tjx;V p ).Furthermore, letT m ?? V p jt p ;x;V m , so that m (tjt p ;x;V) = m (tjt p ;x;V m
). Somewhat loosely, one may say that V p
(V m
) captures the unobserved determinants ofT
p (T
m
).Nowletusturn tothe speci -cationsofthe hazardrates
m (tjt p ;x;V m )and p (tjx;V p
).Weadoptthe following modelframework, Model 1. p (tjx;V p ) = p (t) p (x)V p (1) m (tjt p ;x;V m ) = m (t) m (x)Æ(tjt p ;x) I(t>tp) V m (2) 2
Inreality,thereisoftennoinformationavailableonthedegreetowhichanactualtreatment is anticipated. Even if some anticipation cannot be ruled out, there is virtually never any information on the moment at which the individual receivesinformation on the moment of treatmentunless themomentoftreatmentis fullypredictableat anindividuallevel.Thefact thatarealizationoftheeventofinterestcouldbeduetotheanticipationofafuturetreatment hashauntedtheempiricalliteratureontreatmente ects.Manystandardtreatmentevaluation studies su er from a potential bias due to anticipatory e ects. This includes studies using \di erence-in-di erences"methodswhereone\di erence"concernsacomparisonbetween pre-andpost-treatmentcircumstances(seeHeckman,LaLondeandSmith,1999,foranoverview). Nowsupposethatthedeterminantsofthestochastic processoftreatmentassignmenta ect theindividual'sexit rateoutofthestateof interestbefore theactualrealizationof the treat-ment.Then thetreatment programis said to havean ex ante e ect on exit out of the state of interest. Such an e ect is to beexpected in well-establishedprograms. The ex ante e ect shouldnotbeconfusedwithanticipationoftherealization oftheprocessoftreatment assign-ment,becausein thelattercasethe individualknowsthestochasticoutcome ratherthan the determinantsoftheprocess.Theexante e ectcanbecontrastedtotheexposte ectof treat-ment,whichisthee ectofarealizedtreatmentontheindividualexitrate{thisisofcoursethe e ectwefocusoninthispaper.Wedonotdealwithexantee ectsoftreatment.Identi cation wouldrequireadditionalinformation,suchasstrongfunctional-formassumptions,instruments foracomparisonofaworldwithatreatmentprogramto aworldwithouttheprogram,orthe impositionofaneconomic-theoreticstructureonthemodel.
otherwise.
Apart fromthe term involving Æ(tjt p
;x), the above hazard rates have Mixed ProportionalHazard(MPH)speci cations.Thefunction
i
(t)iscalledthe \base-linehazard"sinceitgivestheshapeofthehazardrate
i
foranygivenindividual. Thehazardrate issaid tobedurationdependent ifitsvaluechanges overt. Pos-itive(negative)durationdependence means that
i
(t)increases (decreases). The term
i
(x) is called the \systematic part" of the hazard. In applied duration analysis, it is common to specify
i
(x)=exp (x 0
i
), so that the hazard function is multiplicative in all separate elements of x. In biostatistics, is often called the treatment e ect if x captures whether the subject has received a treatment at the beginning of the spell, but we avoid such confusing terminology. Finally, the term V
i
is called the \unobserved heterogeneity term". (AVdB also analyze alternativemodelspeci cations,notablywith Ædependingont;x;and onathird unobserved heterogeneity term, say V
Æ .) ThetermÆ(tjt p ;x) I(t>tp)
capturesthetreatmente ect.Thenotationusedhere requiressomediscussion.First,notethattherearealternativebutobservationally equivalent ways of capturingthis e ect. Forexample,one may suppress I(t>t
p ) and rede ne Æ(:) such that it is zero if t t
p
. This is however less attractive from an expositional point of view. Forexample, in our setup a constant treat-ment e ect is relatively easy to capture. The function Æ(tjt
p
;x) is not identi ed on t 2 [0;t
p
], so by not restricting its values on this interval we create an un-interesting identi cation problem. In the sequel, it is silently understood that all statements concerning Æ(tjt
p
;x), including identi cation statements, concern Æ(tjt p ;x) onf(t;t p ;x) 2[0;1) 2 X :t >t p
g. It is useful to separate t from the other arguments of Æ(:) by way of avertical \conditioningline", because we will occasionallyintegrate overt.
Clearly, treatment is ine ective if and only if Æ(tjt p
;x) 1. Now suppose Æ(tjt
p
;x) is equal to a constant larger than one. If T p
is realized then the level of the individual exit rate out of the current state increases by a xed amount. This stochastically reduces the remaining duration in that state, in comparison tothe casewherethe treatmentisgivenatalaterpointoftime.More ingeneral, we allow the treatment e ect to vary with the moment of treatment t
p
, with x, andwith theelapsedtimet inthecurrent state.Asaresult,the individuale ect may alsovary with the time t t
p
since (the onset of) treatment.
Forconvenience, wemakeanumberofnormalizationsandregularity assump-tions onthe determinants of the model(see AVdB). Notably, the individual val-ues of x are taken to be time-invariant. For our results it is useful to point out
atesaway,by assumingthat thereisonlyasinglepointoftime atwhichpossible treatment and outcome may occur. Time-varying covariates are potentiallyvery useful for the identi cationof durationmodels (see Honore,1991, and Heckman and Taber, 1994).
AmoresubstantiveassumptionisthatX isindependentfromV m
;V p
.Wealso assumethat E(V
i
)<1. This is acommon assumption in the analysis of single-spellduration data with MPH-type models (see Heckman and Taber, 1994, and Van den Berg, 2001,for surveys).
It isuseful tophrase the problemof the identi cationof the treatment e ect inthepresenceof\selectivity",inthecontextofourModel1.First,notethatthe datatypicallyprovideobservationsonrealizationsof T
m
andx. Inaddition,if T p iscompletedbeforetherealizationt
m
thenwealsoobservetherealizationt p
, oth-erwise we merely observe that T
p
exceeds t m
. Now consider the (sub)population of individuals with a given value of x. The individuals who are observed to re-ceive a treatment at a date t
p
are a non-random subset from this population. The most important reason for this is that the distribution of V
p
among them doesnot equal the corresponding populationdistribution,becausemost individ-uals with high values of V
p
have already had the treatment before. IfV p
and V m are dependent, then by implicationthe distribution of V
m
amongthem does not equalthe corresponding populationdistribution either.A second reasonfor why the individuals who are observed to receive a treatment at a date t
p
are a non-random subset is that, in order to observe the fact that treatment occurs at t
p , the individual should not have left the state of interest before t
p
. Because of all this,the treatmente ect cannotbe inferredfromadirect comparisonof realized durations t
m
of these individuals to the realized durations of other individuals. If the individuals with a treatment at t
p
have relatively short durations then this can befor two reasons: (1) the individualtreatment e ect ispositive,or(2) these individuals have relatively high values of V
m
and would have found a job relatively fast anyway. The second relation is called a spurious relation as it is merely due to the limited observability of the set of explanatory variables. This relationisreferredtoas\selectivity".IfV
m andV
p
are independentthenI(t>t p
) isan\ordinary"exogenous time-varyingcovariatefor T
m
,and one may inferthe treatment e ect froma univariatedurationanalysis based onthe distribution of T m jt p ;x;V m
mixed over the distribution of V m
. However, in general there is no reason toassume independence of V
m
and V p
, and if this possible dependence is ignoredthen the estimate of the treatmente ect may be inconsistent.
restric-p p m p m
properties) depend on their determinants. More speci cally, the model speci -cationis nonparametric in the sense that we donot make parametric functional formassumptionsontheprobabilitydistributionoftheunobserved heterogeneity terms,the baselinehazards, the systematichazards, and the treatmente ect.
Wedonot imposethatthereareobservedexplanatoryvariablesthatdoa ect T
p
butdonota ect T m
otherthanbywayoft p
.Inotherwords,wedonotexclude elementsof x from
m
(x)that are included in p
(x).
We now examineModel1 froma number of di erent angles.First, the prob-abilityPr (T m >t m ;T p >t p jx) can beexpressed as Pr(T m >t;T p >t p jx)= Z 1 0 Z 1 0 exp( m (x)v m [ m (minft;t p g)+I(t >t p )(tjt p ;x)] p (x)v p p (t p ))dG(v m ;v p ) with i (t):= Z t 0 i ( )d (tjt p ;x):= Z t t p m ( )Æ( jt p ;x)d The joint density of T
m ;T
p
jx follows from di erentiation with respect tot m
and t
p
.Notethat Model1and the aboveincludeaspeci cation ofthe distributionof T p for T p >T m
, but this speci cation is immaterial,as it does not play any role inthe paper orindeed inany empiricalanalysis.
To make comparisons to other models, it is useful to rewrite Model 1 as a regression-typemodel.It iswell known that the integrated hazardof a duration distribution has an exponential distribution with parameter 1 (see e.g. Ridder, 1990, Horowitz,1999). From this it follows that we can write
log p (T p )= log p (x) logV p +" p (3) log[ m (minfT m ;t p g)+I(T m >t p )(T m jt p ;x)]= log m (x) logV m +" m (4) where" p and" m
haveanExtreme Value{TypeI(EV1)probabilitydistribution. This distributiondoes not have any unknown parameters; itsdensity equals
f(" i )=e " i e e " i ; 1<" i <1
p termsontheright-handside,whereV
p and "
p
arerandomvariables.Equation(4) states that the random variable T
m jt
p
;x is distributed as the sum of the terms onthe right-handside, where V
m and "
m
are randomvariables.Thereholds that " m ?? " p ; and (" m ;" p ) ?? (X;V m ;V p ) but V p and V m
are allowed to be dependent. It is important to stress that the fact that "
p
and " m
have a fully speci ed distribution does not mean that we make a parametric functional form assumption on the distributions of T
p jx;V p and T m jt p ;x;V m
. This is because the left-hand sides of the above regression-type equations specify unknown transformations of the dependent variables: the integratedbaselinehazard
p forT
p
,andageneralizedintegratedbaselinehazard (including treatment e ects depending ont
p
) for T m
. A regression equation for, say,T p jx;V p statesthatT p equals 1 p (exp ( log p (x) logV p +" p )),where 1 p (:) isthe inverse of p (:).
The fact that we specify the assignment of treatment by way of specifying the hazard rate of a durationdistribution (rather than by way of specifying the individual realization of the duration variable) implies that there is a random component inthe assignment that is by de nition independent of all other vari-ables. This randomcomponent isrepresented by the term "
p
inthe \regression" equation (3) for T
p
. One may interpret this random component in terms of the randomnessof the outcome of T
p at tjT
p
t that remainsif the hazardrate at t isspeci ed.Consider anindividual who has not yet been given atreatment and who has not yet left the state of interest, at time t. Basically, in a small time interval[t;t+dt), the probability oftreatment is
p (tjx;V
p
)dt, and the probabil-ity of no treatment is 1
p (tjx;V
p
)dt. This is aBernoulli trial. Given the value of
p (tjx;V
p
)dt, itsoutcome iscompletely random.In practice, such randomness may re ect behavior of the institution that supplies or imposes the treatments, orit may re ectpurely randomexternal shocks.
Recallthatthedataideallyprovidei.i.d.observationsonrealizationsofT m ;I(T m > T p );andT p I(T m >T p
),givenx.ThelattertermindicatesthatifT p
iscompleted before the realization of T
m
then we also observe the realization of T p
, whereas otherwise we merely observe that the realization of T
p
exceeds the realization of T
m
.Toanalyze theidenti cationweassumethatthe data actuallyprovideexact knowledgeof the whole jointdistribution of T
m ;I(T m >T p ); andT p I(T m >T p ), given x.
Identi ability is a property of the mapping from the model determinants
i ;
i
m m p p m p
unique mapping from the domain to the data. The modelis non-parametrically identi ed if this mapping has an inverse, i.e. if for given data there is a unique set of functions
i ;
i
(i=m;p),Æ and Gin the domain that isable to generate these data (ofcourse, these data must bein the image of the mapping).
Notethatthis approachtreatsthe individual-speci cunobserved heterogene-itytermsasrealizationsofrandomvariables.Intermsofpaneldataanalysis,this means that V
m ;V
p
are treated as \random e ects" when estimating the model with, for example, Maximum Likelihood.An alternative approach treats the in-dividual values of V
m ;V
p
as unknown individual-speci c parameters (or, equiva-lently, as\incidental"parameters or \ xed e ects").
AVdB demonstrate that Model 1 (as characterized by the functions m , p , m , p
, Æ, and G) is non-parametrically identi ed from single-spell data. The result implies that the treatment e ect is identi ed without exclusion restric-tionsorparametricfunctionalformrestrictionsonthedistributionofunobserved heterogeneity.
We nowturn tothecase wherethedata covermultiplespellsofanindividual inthestate ofinterest,i.e.if the dataare \multiple-spell"data.Weassumethat anindividualhasa xedvalueofV
m ;V
p
.Foragivenindividual,thedi erentspells provide independent drawings from the jointdistribution of the T
m ;I(T m >T p ); and T p I(T m >T p ), given x;V m and V p
. The extension to more than two spells is trivial. Also note that the setup includes cases in which physically di erent individualsshare the same value of V
m ;V
p
and weobserve one duration for each of these individuals.Such a group of individualsis usually called astratum.
Since V m
and V p
are unobserved, the durationvariables given x are not inde-pendent across spells. In fact, any stochastic dependence across spells can only be due to the presence of heterogeneity. It is ruled out by assumption that real-izations of T
m or T
p
in one spell a ect the distributions of durations in another spell.This may be a strong assumption in some applications. For example, par-ticipation in a training program for the unemployed may have an e ect on the durations of future unemployment spells.
Compared to Model 1, the AVdB model for multiple spells is much more exible. It allows for interaction between t and x in the individualhazard rates, sothat the MPHstructure isrelaxed substantially.It alsoallowsfor dependence of V
m ;V
p
in the in ow on x, and it does not require E(V) < 1 anymore. It is allowed that the individual hazard rates in the second spell depend on t;x in a di erent way than they do in the rst spell. The size of the treatment e ect may also be di erent across the two spells. The individual values of x in
still requires that the treatment e ect and the unobservables a ect the hazard rates multiplicatively. AVdB demonstrate that this modelis non-parametrically identi ed from multiple-spelldata.
3 Comparison to latent variable models with bi-nary treatment assignment
We start this section by considering identi cation of a treatment e ect in stan-dard latent variable models with binary treatment assignment (see Maddala, 1983,andWooldridge,2002,foroverviews).Werestrictattentiontothe relations between the mean ofthe endogenousvariablesontheone hand and the explana-toryvariableson the other. This is inline with the spirit inwhich these models are interpreted (see e.g. Heckman (1990) for identi cation results based on full information). Consider Y =x 0 Y Y +x 0 Y Æ 0 I(Z >0)+" Y Z =x 0 Z Z +" Z (5) where Y, Z, " Y and " Z
are continuous random variables,Z > 0 indicates treat-ment, I(Z >0)isa binarytreatmentindicator, and x
0 Y
Æ 0
is the treatment e ect. Weassumethat inferenceisbasedonarandomsampleof subjectswith informa-tion on Y, x
Y , x
Z
and I(Z >0) for each subject. Furthermore, x :=(x Y ;x Z ) is independent of " Y ;" Z
. We take the parameter Z
to be identi ed from the data on I(Z > 0) and x
Z
. It is generally acknowledged that either exclusion restric-tions(statingthat somecovariatesinx
Z
withnon-zero parametersin Z
are not includedin x
Y
) orparametricfunctional-formassumptionsonthe joint distribu-tion of ("
Y ;"
Z
) are required for identi cation of the treatment-e ect parameter Æ
0
.Toillustratethis,suppose one aimstoidentifyÆ 0
fromthe di erencebetween the means of [YjZ > 0;x] and [YjZ 0;x], as a function of x := x
Y ;x Z . We have that E(YjZ >0;x) E(YjZ 0;x)= x 0 Y Æ 0 +E(" Y j" Z > x 0 Z Z ;x Z ) E(" Y j" Z x 0 Z Z ;x Z ) (6) Ifwe donot makeparametric functionalformassumptions onthe joint distribu-tionof"
Y ;"
Z
(6) can be linearin x Z
Z
. Ifinadditionwedonot makeanexclusion restriction then obviouslyÆ
0
is not identi ed from this linear expression.
Nowsuppose thatwedonotobserveY butonlyI(Y >0).Thisisthe\binary treatment { binary outcome" speci cation. For example, Z > 0 may indicate whether an unemployed individual has participated in a training program, and Y > 0 may indicate whether he has found ajob. The binary speci cation for Y e ectivelyreduces theinformationin the data,soidenti cationneedsat least as many assumptions as above(see Cameron and Heckman, 1998, for results).
Before we contrast the identi cation results, it should be noted that using latent variable models with binary treatment assignment or discrete-time panel data models when Y is in fact a (function of a) duration variable leads to a numberofintractablepracticalproblems.First,itrequiresaggregationovertime. Often, it is recorded whether a treatment occurs in a baseline period, and it is recorded whether the outcome (i.e. the duration variable) is realized in the subsequent period. Then, the question arises what to do when the treatment and the outcome occur in the same period. In addition, it is not clear how to deal with observations that are right-censored before the end of the observation window. It is common that a spell in a state of interest can end in di erent ways. Usually, only a subset of these are deemed interesting. For example, an unemployment spellcan end becauseof atransitiontowork, but alsobecauseof a transition into education, militaryservice, etcetera. A study of the transition rate to work may treat transitions toother destinations as independently right-censoring the durationuntil work. The treatment e ect estimate may be biased ifsuchobservationsare discarded orifthey are treatedasobservations ofexit to work.
Now letus consider the similaritiesand di erences between the above model and the duration model of Section 2 in its regression representation (3-4), and thecorrespondingidenti cationresults.Toshapethoughts,onemay,intheabove latent variable model, interpret Y as logT
m and Z > 0 as T p < T m . Of course, alternativeinterpretationsarepossible,andeachof themisimperfect.Moreover, ifModel1is thetrue modelthen ingeneraltheparameter Æ
0
inthe abovemodel speci cation isa complicatedfunction of all parameters of Model1.
The mostnotablesimilaritybetween the modelsconcerns the proportionality assumptions in Model 1 and the additivity assumption in regression equations. Both impose some \smoothness" by excluding certain interactions of time and explanatoryvariablesattheindividuallevel.TheMPHspeci cationdoesimpose more structure than a regression speci cation, because the former decomposes the \error term"into two terms.
modelandthedatausedforModel1concerns thefactthatthelatterincorporate thetimingofthetreatmentwhereastheformerdonot.Atthesametime,themost fundamental di erence between the identi cation result for the latent variable model and the identi cation result for Model 1 concerns the fact that the latter does not need exclusion restrictions while the latter even allows the treatment e ect tohave atime dimension.This suggests that the timingof events provides potentially very useful informationonthe treatment e ect.
We shed more light on this by returning to Model 1. A single observation is equivalent to an observation of a realization of minfT
m ;T p g;I(T m > T p ) and T m I(T m > T p
) given x. The pair minfT m ;T p g;I(T m > T p
) is called the identi- ed minimum of T
m ;T
p
. The data and the model can be decomposed into two parts:the identi ed minimumgiven x,and the durationfromT
p untilT m if T p is the rst to be realized. Let us examine the part of the modelthat speci es the distribution of this identi ed minimum given x. This is in fact the well-known competing risksmodelwith dependent risks, wherethe dependence runs by way ofrelatedunobserved heterogeneityterms (seeHeckmanand Honore, 1989, Lan-caster, 1990). Notethat the speci cation of the competing risksmodeldoes not depend onÆ,whichisintuitivelyobvious. Heckmanand Honore(1989)showthat under anumberof assumptions, the dependent competing risks modelis identi- edfromsingle-spelldataontheidenti edminimumandx.Soifthedataprovide exact knowledgeof the jointdistribution ofthe identi ed minimum given xthen m ; p ; m ; p
;andGcanbededuced. Ofcourse,weobservemore thanthe iden-ti ed minimum and x, namely T
m I(T
m > T
p
). The latter, which is intimately linked to the time distance between the moment of outcome and the moment of treatment, can be used to identify Æ(tjt
p
;x). Intuitively, one can compare the actualdistribution of this time distance tothe distributionthat would prevail if Æ=1.The latter distribution isidenti ed fromthe competing risks model.
To clarify this further, we compare the observable exit rates m
att of those who are treated exactlyatt tothose who are not yet treatedat t:
log m (tjT p =t;x) log m (tjT p >t;x) Thereholds that
m (tjT p =t;x)= m (t) m (x) Æ(tjt;x) E(V m jT m t;T p =t;x) m (tjT p >t;x)= m (t) m (x) E(V m jT m t;T p >t;x)
log m (tjT p =t;x) log m (tjT p >t;x)= (7) logÆ(tjt;x)+logE(V
m jT m t;T p =t;x) logE(V m jT m t;T p >t;x) The selection e ect logE(V
m jT m t;T p = t;x) logE(V m jT m t;T p > t;x) is identi edfromthecompetingriskspartofthemodel,thatis,frominformationon events in [0;t). In words, we know the average \quality" (V
m
) among treatment andcontrolgroupsatt fromthecompetingrisks partof themodel. This enables identi cation of Æ(tjt;x). A given treatment works and onlyworksfrom the day the subject is exposed to its possible e ects, whereas unobserved heterogeneity a ects the observed exit rate out of the current state fromday 1.
3
The above results based on the partitioning of model and data reinforce the conclusionsfromthecomparisonbetweenthebinarytreatmentapproachandthe durationmodel approach. First of all, the timing of events conveys useful infor-mation on the treatment e ect. Secondly, the MPH speci cation is important, because this is assumed for the identi cationof the competing risks model (see alsoAbbring and Vanden Berg, 2001a,for intuition behind the identi cationof the competing risks model).
4 Comparison to panel data models Consider the dynamic panel data model
W t =x 0 W;t W +x 0 W;t Æ 0 I(Z t >0)+V W + W;t Z t =x 0 Z ;t Z +V Z + Z ;t ; (8)
where the index t denotes time. We deliberately introduce new notation W t
for theoutcome becausebelowwe adopttwodi erentinterpretationsofW
t
interms of duration outcomes. Inference is based on a random sample of subjects with informationonW t , W t 1 , x W;t , x W;t 1 ,x Z ;t , x Z ;t 1 ,I(Z t >0)and I(Z t 1 >0)for each subject. We take
j;t
to be i.i.d. across time and across j = W;Z, so that 3
In most econometric evaluation approaches, a treatment e ect can only be identi ed if someofthevariationin theassignmentoftreatmentisnotfullycollinearwiththevariationin theother determinants ofthe outcomeof interest.In ourframework,thisis taken careofby therandomcomponent"
p
in the\regression"equation(3) forT p . Thevariation in " p a ects T m
runs by way of the relation between V W and V Z . We take E[ j;t jx j ] = 0, where forsake ofbrevity weuse x
j :=(x j;t ;x j;t 1 ).Wecan treatV W as a xed e ect by taking rst di erences of W t across time, E[W t W t 1 jZ t 1 0;Z t >0;x Y ;x Z ]=(x W;t x W;t 1 ) 0 W + x 0 W;t Æ 0 : (9) Clearly, Æ 0 isidenti ed. Nowsupposethat
W
variesovertime.Theequivalentoftheright-handsideof equation(9)equalsx 0 W;t W;t x 0 W;t 1 W;t 1 +x 0 W;t Æ 0 .Asaresult,Æ 0 isunidenti ed fromE[W t W t 1 jZ t 1 0;Z t >0;x W ;x Z ].A\di erence-in-di erences"however gives E[W t W t 1 jZ t 1 0;Z t >0;x W ;x Z ] E[W t W t 1 jZ t 1 0;Z t 0;x W ;x Z ]=x 0 W;t Æ 0 : (10) so Æ 0
is again identi ed. Note that this does not require the speci cation of a modelequationfor the treatmentassignment Z
t . Moreover, Z t and X W;t may be dependent, and the method even works in the absence of observed covariates. Also,clearly, Æ
0
can beallowed todepend on t.
We can translate the dynamic panel data model towards our model in two ways.Wecaneither(i)letW
t
beasurvivalindicatorforasingleoutcomeduration T m and set W t = 1 () T m
<t, or (ii) relate each W t
to a separate outcome duration, say T
m;t .
Option(i)suggestsalinkbetweenoursingle-spelldurationmodelanda multi-period panel data model. In fact, because a duration variable has a time unit, single-spelldurationmodels areoftenthoughttobesimilartomulti-periodpanel datamodels.However, this apparentsimilarityisdeceptive.Inthe caseof (i),we onlyobserveonedurationoutcomeperindividual,soW
t 1
=1=)W t
=1,which isnotnecessarilysointhegeneralcaseofspeci cation(8).Thelatterspeci cation allows for more variation in W
t and W t 1 given Z t 1 ;Z t ;x W ;x Z . This variation distinguishespaneldatamodelsfromour durationmodel. Asaresult,we cannot apply (9)and, in particular, wecannot dodi erence-in-di erences likein(10).
Nevertheless, the resultsat the end ofthe previous section(see equation (7)) suggest that an approach in the spirit of the di erence-in-di erences approach might be possible, provided that one can usefully compare di erent sets of in-dividuals across time, and provided that the treatment assignment process is speci ed. In Abbring and Van den Berg (2001b) we develop such an approach.
di erences.
The point ofdeparture inAbbring and Vanden Berg(2001b) isthe observa-tion that di erence-in-di erences amounts to an examinationof a speci c inter-action term: the extent to which the outcome over time di ers between treated andcontrols.AbbringandVandenBerg(2001b)focus onthe lograteatwhicha treatmentisgiven conditionalonthe momentof exit(log
p (tjx;t m )witht<t m ) andexaminewhether this behaves di erently asafunctionoft whent
m
is di er-ent.Intuitively,if Æ>1thenmanyofthose who\die"att
m received atreatment shortlybeforet m ,so p (tjx;t m
)willtend toincrease shortly beforet m ,relativeto p (tjx;t m )forlargert m
.Thisamountstoexaminingtheinteractiontermbetween themomentoftreatmenttandtheoutcomet
m
inlog p
(tjx;t m
).Itturnsoutthat thisapproachallowsone todistinguishbetween acausaltreatmente ect and se-lectivity.Intuitively, iftreatmentand outcome aretypicallyrealizedvery quickly after each other, no matter what the values of the other outcome determinants are,thenthisisevidenceofapositivecausaltreatmente ect.The selectione ect doesnot giverise to thesame typeof quick succession ofevents.The xvariables play norole here. However, Æ is assumed to beconstant over t and t
p .
This result again illustrates the usefulness of the information in the timing of events to assess the treatment e ect. Both in the panel data approach and in ourdurationmodelapproach,the treatmente ect works fromaspeci c pointof time onwards, whereas the selection e ect works at all points of time in a more permanentway. Inboth approaches, separabilityassumptions,ruling outcertain interaction e ects of the determinants of the individual outcome of interest, are needed.Inthepaneldataapproach,additivityoftreatmente ectandunobserved heterogeneityintheoutcome equation(8) iscrucial.The previoussectionargued that in the duration model approach the additivity of the determinants of the individual log outcome hazard rate log
m (tjt p ;x;V m ) (equation (2)) is crucial. The results in this section so far emphasize that what is particularly crucial is theadditivityoftreatmente ectandunobserved heterogeneityinthisloghazard rate(althoughthisbyitselfseemstobeinsuÆcienttoidentifythewholeduration model includingthe way the treatmente ects varies with t and t
p ).
Theseseparabilityassumptionsattheindividuallevelenableanempirical dis-tinction between the treatment e ect, that works from a speci c point of time, andtheselectione ectthatworksatallpointsoftime.Thedurationapproachis muchmoreinvolvedbecauseofthedynamicnatureofselectivityinduration anal-ysis:as timeproceeds, the compositionof the survivors changes,so the selection e ect changes.
towards our duration model framework. Let Z t
indicate whether a treatment is given during the t-th spell. This interpretation allows for a straightforward comparison with our multiple-spell duration model. In both cases we observe multiple outcomes for an individual with the same unobserved covariates. This suggeststhatwecanremovetheroleofunobservedheterogeneityinmultiple-spell data by some sort of conditional likelihood or rst-di erencing approach. AVdB develop ananalogueof the xed-e ectpanel data estimatorbased on(10).Both estimators do not require any observed explanatory variables or a speci cation of the treatmentassignment process.
5 Conclusions
Variation in the durationuntil treatment relative to the durationuntil the out-comeof interest conveys useful informationon the causaltreatmente ect inthe presence of selectione ects. This informationis discarded ina binary treatment framework.Analysisofthedurationvariablesallowsforinferenceoncausal treat-mente ects if novalidinstrumentsareavailableand ifconditionalindependence assumptions can not be justi ed. With single-spell duration data, this works as follows. If treatment and outcome are typically realized very quickly after each other, no matter what the values of the other outcome determinants are, then thisistaken asevidenceofapositivecausaltreatmente ect. Theselectione ect doesnot give rise tothe same type of quick succession ofevents.
Wemakeanumberofquali cations.First,itispivotalthatindividualsdonot anticipate the realization of the moment of treatment, because then the treat-ment works from a moment in time that precedes the actual participation. It is obvious that this would lead to incorrect inference. See AVdB for an extensive methodologicalanalysis of this issue. Secondly, the information in the timing of events isuseful forinference onhowthe causaltreatment e ect varies with time andwith the time since treatment. This provides potentiallyvery useful insights into the workings of the treatment, and this is important from a policy point of view. Thirdly, it is an important topic for further research to investigate to what extent the identi cation results for the single-spell duration model frame-workare robustwithrespect toseparabilityassumptionsembeddedinthe model framework.
The resultslead tosome suggestionsfor empiricalwork. First,itisuseful not to discard information on the timing of the treatment, and, in particular, not to round-o such information into a binary treatment indicator. Secondly, it is
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