Nonparametric Identification in Nonseparable Duration Models with Unobserved Heterogeneity

1 Introduction

A major strategy for identification in duration models is to impose that the hazard function of the duration variable is multiplicatively separable with respect to genuine duration dependence, observed and unobserved covariates. The mixed proportional hazard (MPH) model, which incorporates this assumption, is the most commonly used duration model in the econometric literature, Hausman and Woutersen (2008). Separability is a powerful source of identifying variation. It can be used to distinguish between genuine duration dependence and spurious duration dependence caused by dynamic selection. Furthermore, in models with competing risks, separability has been used to break the nonidentification result of Tsiatis (1975).

Despite these appealing properties, multiplicative separability has two major drawbacks. First, separating covariates from duration dependence is difficult to justify with economic theory. In job search models, for example, this is only justified with myopic agents and under very particular parametric assumptions, see Van den Berg (2001). These theoretical arguments are reinforced by recent experimental studies that find that duration dependence varies in a complex, nonproportional way with observed covariates, see Eriksson and Rooth (2014), Farber et al. (2016) and Kroft et al. (2013). Second, separating observed and unobserved characteristics implies that the effect of an observed covariate is the same for all individuals with identical observed characteristics. This assumption, however, is often too restrictive, Jun et al. (2016). While unobserved effect heterogeneity has been addressed by numerous studies in the context of regression models (see e.g. Matzkin 2007), it has not been considered in duration context.

The objective of this paper is to address the drawbacks of fully separable models. I study identification of hazard models of the type

where θ is the structural hazard function of a duration random variable and X and V represent observed and unobserved individual characteristics, respectively. These models are nonseparable in the sense that they allow for (i) arbitrary observed heterogeneity in the “true” duration dependence (through the function λ) and (ii) arbitrary interaction between observed and unobserved covariates (through the function r). The functions λ and r are unknown to the econometrician and are not assumed to belong to a parametric family.

This paper shows identification strategies for a comprehensive account of empirically relevant settings. In particular, identification is shown in single-spell models with and without time-varying covariates, in multiple-spell models with shared frailty and lagged duration dependence, in single-spell and multiple-spell competing risks models, as well as in treatment effects models where treatment is assigned during the individual spell in the state of interest.

Identification in this study relies on two main strategies. The first one is based on the assumption that there is one special regressor whose effect on the individual hazard is homogeneous w.r.t. unobserved characteristics. No assumption is imposed on the genesis of this regressor. Furthermore, this regressor is allowed to impact the weeding out process (i.e. the change of the conditional distribution of V given X over time) in an unrestricted way. This paper is the first to show identification in single-spell hazard models with interaction between observed and unobserved characteristics. Other single-spell nonparametric studies rely either on the fully separable structure of the MPH model, Elbers and Ridder (1982), Heckman and Singer (1984), Ridder (1986), Horowitz (1999) and Chiappori et al. (2015), or at least on the multiplicative separability of the unobserved heterogeneity as in the Mixed Hazard (MH) model, Brinch (2007) and McCall (1994).

The second strategy is based on multiple spells and a mild fixed-effects assumption that restricts unobserved heterogeneity to enter the hazard of at least two subsequent spells in the same way. The model is very general and does not require a finite mean of the unobserved heterogeneity. Moreover, previous spells are allowed to impact the hazard of subsequent spells (a so-called lagged duration dependence). Thus, these multiple-spell models can be interpreted as a hazard version of a dynamic panel model with fixed-effects. Importantly, the lagged duration effect is allowed to be arbitrary heterogeneous with respect to observed characteristics. This assumption is motivated by recent empirical findings in the unemployment duration literature, see e.g. Cockx and Picchio (2013). In the context of unemployment duration, lagged duration dependence, sometimes also referred to as scarring effects, is an object of interest on its own and its analysis has a long tradition, Heckman and Borjas (1980). The models in this paper also nest a shared frailty model, which is widely used in demographics and appropriate when multiple individuals from the same group share unobserved characteristics, Hougaard (2000). Existing multiple-spells studies either rely on fully separable variation in the observed heterogeneity, Honoré (1993), or at least impose separability of the unobservables, Frijters (2002) and Picchio (2012).^[1],[2] I show that the fixed-effects assumption is sufficient to avoid these restrictive assumptions. In particular, the fixed-effects assumption is used to derive a hazard analogue of a difference-in-differences approach.

Next, identification of nonseparable competing risks models is studied. Competing risks models arise naturally in unemployment duration context when more than one exit destination is possible, see e.g. Kyyrä and Ollikainen (2008) for an empirical example. Since only the first exit is observed, the marginal distributions are in general not identified. This paper shows that the nonseparable hazard structure (1) leads to identification when either there are multiple spells and the fixed-effects assumption is satisfied or when the researcher has access to a special regressor of the type discussed above. Previous studies have largely relied on the MPH structure, see Heckman and Honoré (1989), Abbring and Van Den Berg (2003a), Colby and Rilstone (2004) and Horny and Picchio (2010). One exception is the paper by Lee and Lewbel (2013) which nonparametrically identifies a bivariate accelerated failure time competing risk model.

This paper also contributes to the literature on treatment effects in the context of duration models. Much of this literature is discussed by Beyhum et al. (2024). In many studies, treatment is modeled as a binary variable, and is assumed that the treatment status realizes upon individuals entering into the state of interest (Blanco et al. 2020; Kastoryano and Beyhum 2020; Sant’Anna 2021). This assumption precludes many problems that are endemic to duration models. Most importantly, this assumptions states that the treatment status is observed. This assumption may be too restrictive for many cases such as unemployment search assistance, in which the treatment may realize at any time of the elapsed duration and thus will be censored if an individual exists unemployment prior to treatment, and, which makes identification more complicated, censoring is endogenous since the exit is co-determined by individual unobserved characteristics. Recent papers that deal with endogenous censoring of a dynamic treatment are for example Van den Berg et al. (2020a, 2020b) and Beyhum et al. (2024). A common assumption of these papers, as well as of the literature that assumes the treatment to be binary and realized upon inflow, is that the effect of the treatment does not depend on the duration of exposure to the treatment. In that sense, the effect is assumed to realize instantaneously. This assumption is technically important. Relaxing it requires identifying both the hazard function of the time to treatment from endogenously censored data and the duration dependence on the time spent in treatment. A binary treatment that realizes right at the beginning of a spell avoids these challenges. However, this assumption may also be restrictive in many important cases. The success of an Active Labor Market Policy such as training will depend on how long an individual is trained (Vooren et al. 2019). The success of many psychological therapies may depend on the length of exposure to the therapy (American Psychological Association 2017). The seminal paper by Abbring and Van Den Berg (2003b) allows the time to treatment to be endogenously censored and also allows that the treatment effect depends on the exposure to the treatment. The cost of doing so is that it is necessary to identify the full model, i.e. the underlying hazard functions of two duration variable (main duration and time to treatment). To that end, Abbring and Van Den Berg (2003b) have relied on the separability of the MPH model. This paper generalizes their results to a nonseparable context. Specifically, drawing on the results for nonseparable competing risks models shown here, the paper demonstrates that the treatment is identified if the time to treatment and the time to exit follow an augmented bivariate nonseparable duration model based on (1). The major cost for this generality is that the paper falls short of suggesting estimation procedures.

In all models discussed in this paper, identification of the function r draws on insights from the literature on nonadditive random functions, see Matzkin (2003) and Chesher (2007). In particular, r is assumed to be strictly increasing in the unobserved component. This is a natural generalization of the MPH assumption r(x, v) = id(v). Important MPH features such as spurious negative duration dependence in the empirical hazard translate to the general nonseparable setup in a straightforward way. Thus, this paper provides a link between the literature on identification in duration mixture models and the literature on identification in nonseparable regression models.

Nonparametric estimation of nonseparable duration models combines two challenges. First, nonparametric estimation of duration models, in particular the mixed proportional hazard model, is nontrivial. Second, nonseparable econometric models in general are hard to estimate (reference are given below). This paper restricts its focus to identification, which is also the major limitation of the paper.

The paper is structured as follows. In Section 2, the nonseparable model (1) is formally introduced and motivated. Identification results in single- and multiple spell models are presented in Section 3. Section 4 presents identification of competing risks models and Section 5 presents identification of duration treatment effect models. Section 6 concludes. All proofs and further supplementary results are in the Online Appendix.

2 Model and Motivation

For illustrative purposes, we build our exposition on a labor market example.^[3] Suppose that n unemployed individuals are searching for a job. Denote by T _i , i = 1, …, n the duration of unemployment of individual i, with T₁, …, T _n assumed to be independent and identically distributed. Let θ(t) be the unconditional hazard of T _i at elapsed length of unemployment t ≥ 0, θ ( t ) = lim h → 0 P { T ∈ [ t , t + h ) | T ≥ t } / h (the index i is omitted whenever this is possible). Let the random vector X _i represent observed characteristics of individual i that impact the duration T _i . X _i is assumed to have realized (just) prior entry into unemployment, so that its value is determined at t = 0. Typical examples are wage and experience in the preceding job spell, highest degree of education obtained until the moment of inflow into unemployment and gender. The realizations of X _i , denoted by small letters x, are assumed to be elements of a set X ⊂ R k , where k is a positive integer. By way of definition, X _i is time-constant. Time-varying covariates are considered in Section 3.2. Further, let V _i be a one-dimensional nonnegative unobserved random variable. V _i represents a single index of all unobserved individual characteristics that impact T _i . A typical example for factors contained in V _i is noncognitive skills. Analogously to the definition of X _i , V _i is required to be time-constant and determined before the start of the individual spell.

Henceforth, θ(t|X, V) denotes the individual hazard with notation in analogy to conditional probabilities. Conditionally on X and V, it is assumed fully specified. The empirical (or observed) hazard is denoted by θ(t|X). An important property of mixture hazard models (i.e. of hazard models allowing for unobserved heterogeneity V) is that T is a random variable even conditionally on realizations of X and V. The residual randomness, referred to as “the effect of luck” by Lancaster (1979), has been only scarcely discussed in the literature and has no established meaning. Lancaster (1990) interprets it as some intrinsic individual uncertainty, whereas Heckman (1991) distinguishes between characteristics known to the individual which are captured by V, and characteristics that are unknown to the individual and subsumed by the residual randomness. Decisions of agents are therefore based solely on the value of X, V. Heckman (1991) acknowledges, however, that this distinction has an arbitrary character. This paper follows both Lancaster (1979) and Heckman (1991) and assumes that the residual randomness is due to idiosyncratic factors that are independent of the factors determining the decisions of the agents. This interpretation is compatible with the well-known fact that the transformed duration Θ ( T | X , V ) ≔ ∫ 0 T θ ( t | X , V ) is independent of X and V, see e.g. Lancaster (1990).

With these preliminaries, consider the following model, which is referred here to as the generalized mixed hazard (GMH) model:

The functions λ and r are unknown to the econometrician. Two important special cases are the mixed proportional hazard (MPH) model,

and the mixed hazard (MH) model,

Both the MPH and MH models impose multiplicative separability of V. As observed by Lancaster (1985) and Chesher (2002), multiplicative separability of V effectively reduces the number of sources of stochastic variation from 2 to 1. The GMH model, on the contrary, allows X and V to interact through the function r in an arbitrary way. Furthermore, similarly to the MH model, the GMH model allows the duration dependence λ to depend on observed covariates.

Section A in the Appendix briefly discusses the relation of the GMH model to other existing models. In a nutshell, the accelerated failure time model and its generalizations the generalized AFT by Ridder (1990) and the extended generalized AFT by Brinch (2011) are not nested in the GMH model. Conversely, the GMH model is also not nested in those models.

The following examples motivate the GMH model.

Example 1: unobserved treatment heterogeneity. Van den Berg and Van der Klaauw (2006) evaluate the effect of counseling and monitoring on the re-employment chances of unemployed workers within a social experiment in the Netherlands. At inflow into unemployment, workers are either assigned at random to counseling and monitoring (Z = 1) or to a control group with no such services (Z = 0). The model estimated in their paper is the MPH model

where X denotes a list of controls. Separability of Z and V implies that the effect of the training does not depend on unobserved noncognitive abilities such as locus of control or on intrinsic motivation. However, recent empirical and theoretical evidence suggests that individuals with higher levels of locus of control tend to make higher effort when searching for a job Caliendo et al. (2015).^[4] This finding suggests that individuals with higher locus of control may benefit more from the counseling and monitoring process through more active participation. The simplest model that can capture this relationship is

This is a hazard version of the location-scale model in quantile regression, see He (1997). The coefficient γ captures the unobserved effect heterogeneity of Z.

Example 2: heterogeneous duration dependence. The baseline hazard function λ_MPH in model (3) represents the genuine duration dependence of the hazard. A commonly discussed reason for negative duration dependence of the unemployment hazard is stigma. Stigma occurs when the willingness of employers to hire unemployed individuals decreases with increasing spell of unemployment (Nüß 2018). For example, screening models predict that employers use the length of the current unemployment spell as a productivity signal, see e.g. Lockwood (1991). Recent experimental studies use fictitious job applications to actual job postings to analyze the effect of unemployment duration on the call-back rate. These studies provide evidence that the stigma-driven duration dependence might depend in a complex way on individual characteristics, in particular on the level of education and experience, as well on the type of occupation. As an example, Eriksson and Rooth (2014) find large drops of call-back rates over time for low- and medium-skilled occupations but little to no effect for high skilled jobs. Similar effects are found by Weber (2014). Farber et al. (2016) find that longer employment histories compensate for long ongoing unemployment spells, thus reducing or even eliminating stigma effects. Kroft et al. (2013) find significant heterogeneity of the duration dependence across levels of tightness of the local labor market. These empircial findings cannot be incorporated in a separable model. In particular, separability of λ and X implies that individual characteristics only shift the level of duration dependence.

The nonseparable duration dependence λ(t, X) in model (2), on the contrary, allows for a flexible interaction between time and observed characteristics. As an example, consider the Weibull specification

with a scale parameter λ₀ and a shape parameter λ₁ whose value is now allowed to depend on the value of the observed characteristics.^[5] The findings of Farber et al. (2016) can be modeled as λ₁ < 1 for shorter employment histories and λ₁ = 1 for long employment histories. Alternatively, flexibility can be achieved in a piecewise-constant bazeline hazard, in which the coefficients depend on x.

Example 3: heterogeneous measurement error. Consider a case in which the duration variable is measured with error, see e.g. Abrevaya and Hausman (1999). Common reasons for measurement errors in duration context are time aggregation, Bergström and Edin (1992), as well as under-reporting in retrospective surveys, Mathiowetz and Duncan (1988) and Aït-Sahalia (2007). Lancaster (1985) shows that a multiplicative measurement error in the duration variable, together with a Weibull specification of the latent baseline hazard, lead to a MPH model, with the multiplicative V resulting from a measurement error. His model imposes that the measurement error does not depend on observed covariates. In general, however, the measurement error will depend on demographic characteristics, Bound et al. (1989). Accounting for that possibility naturally leads to a nonseparable model, which is now demonstrated. Denote by T* the latent duration variable which might be measured imprecisely. Assume that the observed duration T satisfies

where η is a measurement error, k is some unknown function and V╨(X, T^*). Let the conditional distribution of the latent variable follow a Weibull specification, P{T* ≤ t∣X = x} = 1 − exp{−t ^α ψ(x)} with α being a (unknown) scalar and ψ an unknown function. Then the individual hazard of T satisfies θ(t∣x, v) = αt^α−1ψ(x)k(x,v) ^α , which is a special case of model (2) with λ(t, x) = αt^α−1ψ(x) and r(x, v) = k(x,v) ^α .

Roadmap. The intuition behind the choice of models/settings in this paper is the following. The ultimate goal of the paper is to show identification of nonseparable treatment effect models, in which a treatment S is administered at random points and maybe censored by the outcome T. It will be thus unobserved for those individuals who exit the state of interest before being treated. A major insight here is that before the treatment kicks in (i.e., for realizations t, s of T, S with t < s), the bivariate hazard model of T, S is a competing risk model. Thus, identification in competing risks models need to be studied before identification in treatment effects models. In turn, the building blocks of a competing risk model are hazard models for each of the competing risks. The paper thus presents first two different strategies for identification of hazard models of the type (2). The first strategy consists of a special regressor that is modelled separably in certain sense. The second strategy relies on multiple spells. Then the paper applies these strategies to competing risks models, and ultimately, applies the result for competing risks models to identify treatment effect models. This procedure is depicted in Figure 1.

Figure 1:

Overview of identification models and strategies.

3 Identification Under Random Right Censoring

4 Identification Under Endogenous Censoring: The Case of Competing Risks Models

An individual might consider several destinations of the transition out of unemployment. As an example, elderly unemployed might choose to search for a new job or to withdraw from the labor force, Kyyrä and Ollikainen (2008). In addition, the researcher might want to distinguish between full-time employment, part-time employment, employment on a short-term/long-term contract, and other forms of employment. These destinations are typically modeled as competing risks. The main feature of a competing-risks model is that the duration under each risk is latent and only the duration until the first cause of exit is observed. Put differently, the duration in each destination is potentially censored by the exit in a different destination. This censoring case is more involved than the case of random right censoring considered in the previous section. Specifically, since the distribution functions of each of the competing risks might depend on the same unobserved features of the individual, censoring is potentially endogenous. Without further structure, the “observed” joint (bivariate) survival function S(t₁, t₂) is nonparametrically nonidentified, Tsiatis (1975), which greatly complicates identification of the underlying structural model. This section studies identification of competing risks models when the hazard of each of the latent variables follows the GMH model (2).

Single-spell setting. Consider a setting with two risks, A and B. A generalization to a case with more than two risks is straightforward. Denote by T _i the latent duration under risk i = A, B. For each individual in the sample, the researcher observes T ˜ ≔ min { T A , T B } and the indicator function 1 { T A < T B } which informs about the cause of exit. Thus, instead of the joint survival function P{T _A > t _A , T _B > t _B }, the econometrician “knows” the crude survival functions P{T _i > t, T _j > T _i }, i, j = A, B, i ≠ j. Denote by V _A and V _B the risk-specific unobserved characteristics that impact T _A and T _B , respectively. We assume that (X, V _i ) fully determine the distribution of T _i and that T _A and T _B are conditionally independent given (X, V _A , V _B ). V _A and V _B are allowed to be dependent, while X╨(V _A , V _B ). This is the standard setup in competing risks models with covariates, see Heckman and Honoré (1989). The hazards θ _A and θ _B of T _A and T _B , respectively, are modeled as

Notably, r _i and λ _i are allowed to be risk-specific, i.e. it is not required that λ _A = λ _B or r _A = r _B .

The following result states that the GMH structure identifies the single-spell competing risk model when there is a special regressor (assumption A3) which is common for both risks A, B and when in addition this special regressor is separable from the duration dependence function λ (assumption A4).

Assumption A4 (ii)′ modifies assumption A4 (ii) to the competing risks setting. It can be relaxed along the lines of the discussion of assumption A4 (ii) above. Note that identification of ϕ _A , ϕ _B does not require the variation condition A4 (ii)′ and independence of X and V _A , V _B .

Discussion on nonidentification. Unlike assumption A4, assumption A4′ cannot be used to obtain identification of the competing risks model. The following discussion should provide the intuition. Due to the censoring problem, identification relies on the identified crude survival functions. These functions vary along a single time argument t. Identification with two risk-specific unobservables, however, requires either variation in two time arguments on an open subset of R + × R + or separable variation in x. Hence, the bivariate version of the MH model considered by Brinch (2007) (and in general, of any single-spell MH model with risk-specific unobservables but without separable x-variation) is not identified. Since the MH model is nested in the GMH model, the single-spell GMH model is not identified with competing risks under assumption A4′. This implies that when there is only one spell per individual, assumptions A3 and A4 are necessary for identification.

Multiple-spells setting. Consider now a setting with competing risks and multiple spells. For i = A, B, T_i,1, T_i,2 denote two consecutive spells of the latent variable under risk i. The hazards of the latent variables are modeled as

The researcher observes the tripple ( X , T ˜ k = min { T A k , T B k } , 1 { T A k < T B k } ) , k = 1,2 . (T_A1, T_A2) are assumed conditionally independent from (T_B1, T_B2) given X, V _A , V _B . However, conditionally on (X, V _A , V _B ), T_i1 may impact T_i2. Specification (21) and (22) contains a risk-specific fixed-effects assumption. In particular, for each risk i, the unobservables V _i and the generalized error function r _i are the same for both spells. However, both the unobservables and the function r are allowed to differ between risks. Furthermore, as in the single-spell competing-risks setup, V _A and V _B are allowed to be dependent.

The following proposition states that the fixed effects assumption used to identify the multiple-spells model in the previous section leads to identification in a competing risk setting.

Proposition 6 is based on similar assumptions and a similar difference-in-differences approach as Proposition 4 and so the intuition is common for both results. In particular, the use of the fixed-effects assumption is that the ratio of derivatives of crude survival functions is free of r and V. Furthermore, distinguishing between λ_i1 and ψ _i is possible, because the ratio of the subsurvival functions of the two spells depends on ψ _i in a way that varies with the elapsed duration in the second spell, which is not the case for λ_i1. Identification of r _i is achieved after the identification of λ_i1 and ψ _i .

The closest setup to the one considered in Proposition 6 is the one studied in Horny and Picchio (2010). They use multiple spells to prove identification in a competing risk model with a lagged duration dependence. Their result generalizes the single-risk results by Honoré (1993). In the models of Honoré (1993) and Horny and Picchio (2010), separable variation in X is crucial to identify the joint distribution of the unobservables. Identification of this distribution is necessary to distinguish between the lagged duration dependence ψ _i and the genuine duration dependence in the first spell λ_i1. The separable variation of X is provided by the structure of the MPH model. Model (21) and (22) does not require such variation.

A model that uses a fixed-effects assumption in a multiple-spell MPH context is developed in Abbring and Van Den Berg (2003a). The precise assumption there is V_i1 = V_i2. However, model (21) and (22) is substantially richer. In particular, X and V can interact nonseparably and a lagged duration dependence is allowed. Both these aspects are not allowed in Abbring and Van Den Berg (2003a).

5 Nonparametric Identification of Treatment Effects when the Treatment is Assigned During the Spell

Consider a setup, in which a treatment is assigned during the ongoing spell in the state of interest. As an example, ALMPs such as subsidized temporary work in the public sector are administered during unemployment. Related setups arise in epidemiology when a given medical treatment, such as kidney transplantation, is administered during an ongoing health condition.^[14] In such cases, the effect might vary with time to and duration of exposure.

To formalize this setup, let the random variable S denote time since entry into the state of interest (e.g. unemployment) until the start of treatment exposure. Contrary to X, V, which realize prior to or at begin of the spell in the state of interest, S realizes during the spell. As before, T denotes the duration of the underlying condition. In the example above, T is the duration of unemployment and S is the random time from the begin of unemployment until the individual starts temporary work as part of an ALMP. Define the vector V ≔ (V _T , V _S ), where V _T , V _S are two unobserved scalar random variables. Consider the following bivariate hazard model:

In equation (24), the hazard θ _S (t|X = x, V) of the treatment variable S is specified as the nonseparable model (2). θ _S depends on V only through the S-specific unobservables V _S . Equation (23) specifies the hazard of T. It is an augmented GMH model. Before the treatment is administered (t < s), this is simply the GMH model (2). After the begin of the exposure, the hazard function is modified by the component δ, which can be interpreted as a treatment effect. In particular, conditionally on X and V, S can impact T (only) through δ. δ is allowed to depend on time to exposure s on its duration. The latter is inferred from the difference between t and s. δ is also allowed to depend on observed heterogeneity x. V _S and V _T might be dependent, so that the model allows for endogenous selection into the treatment. Furthermore, the functions r _T , r _S can be different (and analogously for λ _T , λ _S ).

Model (23) and (24) is a generalization of the widely-used treatment effect model by Abbring and Van Den Berg (2003b). Identification in Abbring and Van Den Berg (2003b) relies on separable MPH structure of each of the two hazards θ _T , θ _S  – an assumption that is relaxed here.

For the further discussion, it is informative to review the differences between the treatment effect model (23), (24) and the lagged duration model (17) and (18). The latter imposes that the two durations T₁, T₂ realize sequentially. This implies that both durations are observed. In the treatment effect model, S is potentially censored by T. Moreover, in model (17) and (18), the hazard of T₂ depends on T₁ through ψ for potentially all values t₁, t₂ (i.e. ψ might be ≠1 for all t₁, t₂). In the treatment effect model, on the other hand, δ in (23) is equal to 1 when s > t. This condition is plausible when the treatment is not anticipated by the individuals. As an example, ALMPs often have random unanticipated variation that arises from the random order in which case workers treat the unemployed, see e.g. Sianesi (2004). Alternatively, individuals might be aware of the future time of the treatment but they might not be able to act upon this information. In the kidney example, patients might have little other choice but to wait for their turn to come.

Identification of the treatment effect δ proceeds in two steps. First, consider model (23) and (24) for values t < s. For such values, the model represents a GMH competing risks model. Under the assumptions from Section 4, λ _j , r _j , j = T, S and the joint distribution of r _T (X, V _T ), r _S (X, V _S ) can be identified. This is stated in the following proposition.

In a second step, knowledge of the components of the competing risks model is used to identify the treatment effect δ. The following result takes a general standpoint and assumes directly knowledge of these components (i.e. the source of this knowledge is ignored).

The proof of Proposition 8 is novel. Intuitively, when the competing risk model is identified, the derivative of the joint survival function of S and T with respect to t (which can be identified through its relationship to the crude survival functions) can be linked through a differential equation to the treatment effect δ. This differential equation is shown to have a unique solution, which leads to identification. Thus, it is the change in the marginal hazard ∂P(T > t, S > s)/∂t over elapsed durations t that identifies the treatment effect. On the contrary, Abbring and Van Den Berg (2003b) use variation of the marginal exit rate with respect to s, ∂P(T > t, S > s)/∂s. Their strategy requires smoothness of the treatment effect. This assumption is not necessary here.

6 Discussion

This paper has provided identification results for nonseparable duration models. These models are nonseparable in two ways. First, genuine duration dependence is allowed to depend on observed covariates. Second, observed and unobserved characteristics may interact in an arbitrary way. The models considered in the paper constitute a comprehensive set of settings considered in theoretical and applied duration studies. In particular, identification has been shown in single-spell models with and without time-varying covariates, in multiple models with shared frailty and lagged duration dependence, in single-spell and multiple-spell competing risks models, and in treatment effects models where treatment is assigned during the individual spell in the state of interest. The latter contribution is also the paper’s most important one. Proposition 8 and its Corollary 1 relax common assumptions on the values and timing on the treatment. Specifically, while most papers require the treatment to be binary and to realize at the beginning of the duration spell, this paper allows both the treatment to be continuous and to realize at later points in time. The former aspect is important when the duration of exposure to treatment matters. The latter aspect is relevant for most practical applications: hardly any treatment is applied right at the onset of a disease; no job searching is administered on day 1 of unemployment. Building on Abbring and Van den Berg (2003b), who allow for endogenous treatment timing and exposure-dependent effects under separability, this paper extends identification results to a nonseparable setting. Specifically, it shows that treatment effects can still be identified in a bivariate nonseparable duration model.

A natural follow-up of these results would be to develop estimation techniques and to additionally allow unobserved heterogeneity to interact with duration dependence. These topics remain for future research.where the last equality follows from the uniform distribution of F(T|X, V).