2. IDENTIFICATION WITH CONDITIONALLY DEPENDENT DURATION VARIABLES

The Heckman and Honoré (1989) competing risks framework can be generalized as follows. Let U₁ and U₂ be unobservables on (0,1) × (0,1) with joint distribution and density functions K and k and let x = (x₁,x₂,x₃,z) be a vector of covariates taking values on

. Recursively define two duration processes, T₁ and T₂:

where Φ_j, j = 1,2 are commonly referred to as the cumulative baseline hazard rates. As discussed in Heckman and Honoré (1989), special cases of this include the proportional and accelerated hazard rate models. This framework is also applicable to multiple spell models and various nonseparable hazard rate models. We index the x's to allow for exclusion restrictions such that x_j appears only in θ_j, j = 1,2. The term x₃ appears only in ρ, whereas z can be common to both hazard rates.¹

By the transformation theorem, the density and survivor functions of T₁,T₂ are derived as

where

and Φ_j′ denotes the derivative of Φ_j. We let

.

We formalize our assumptions as follows.

Assumption 1. The durations are generated as in equation (1) where (U₁,U₂) ∼ K : (0,1) × (0,1) → [0,1] , K(·,·) is strictly increasing and differentiable in its arguments.

Assumption 2. For j = 1,2, Φ_j(t) is differentiable and strictly increasing, Φ_j(0) = 0, Φ_j(1) = 1, and Φ_j(∞) = ∞.

Assumption 3. (θ₁(x₁,z),θ₂(x₂,z)) → R₊² is continuous. For j = 1,2, some

.

Assumption 4. ρ(t,x₃,z) → (a,b), a < 1 < b, is differentiable with respect to t, ρ(0,x₃,z) = 1, and for some

.

The statements regarding the domains, ranges, and inverse images of θ and ρ are made to ensure that, when evaluating one of the variates at a fixed point, this does not inadvertently impose restrictions on the values the other variates may take. The intuition of the results is very simple. In a competing risks framework, the observed duration variable is the minimum of the two duration processes. We find conditions under which the “observable” survivor function tends to the marginal survivor function of the processes. This kind of approach is sometimes referred to as “identification at infinity” as in Heckman (1990). Essentially, the assumption is made that, for some value of the conditioning variables (not necessarily infinity) or set of values, one of the dependent variables has a degenerate distribution. Given that, results like those of Elbers and Ridder (1982) can be applied directly. Moreover, because the parameters of interest are expressed directly in terms of the survivor function and its derivatives, these can be estimated directly by their sample analogs.

The partitioning of x = (x₁,x₂,x₃,z) allows for a subset of the observed covariates to enter into ρ. We show identification of this model under two different sets of assumptions. We either assume that x₃ is a subset of x with no elements in common with either x₁, x₂, or z and x₃ has a certain limiting impact on ρ, or we will show that certain shape restrictions on Φ₂ and ρ can identify the components of this model. Under the assumption that ρ(t,x₃¹,z) = 1, T₁ has no direct impact on T₂ and the duration processes become conditionally independent. In this situation identification essentially follows along the lines of Heckman and Honoré (1989, 1990). We impose the natural restriction that ρ(0,x₃,z) = 1 so that the marginal distributions with this model are also as in the conditionally independent model.

We let K₁(u₁) = K(u₁,1), K₂(u₂) = K(1,u₂), denote the marginal distribution functions and indicate their derivatives by K_j′, j = 1,2. The partial derivatives of K are indicated by K^(j), j = 1,2. Note that K_j(0) = 0 and K_j(1) = 1, j = 1,2.

Assumption 5. For j = 1,2 K_j′ is left continuous at 1 and K_j′(1) = L_j > 0.

Assumption 5 guarantees that a certain limiting ratio of the K_j′'s in the proofs that follow is well defined. Alternatively, one could allow for the K_j′'s to vanish, by, say, strengthening the smoothness assumption and using L'Hôpital's rule.

PROPOSITION 1. Let Assumptions 1–5 hold. Then, θ_j, Φ_j, j = 1,2, ρ, and K are identifiable.

Proof. The survivor function of the minimum of the two durations is given by

We can identify θ₂ by evaluating the ratio

and observing that

We can identify θ₁ symmetrically. To identify K note that

By varying θ₁ and θ₂ over

, we can trace out K. Because K is identifiable and increasing in its arguments, K₂ has a unique and identifiable inverse, H₂, such that

so that

and

The term Φ₁ is identified symmetrically. █

Remark 1. Without the identifying variable, x₃, it is still possible to identify the parameters using functional form restrictions. Note that in this case we can identify

In general, it is not possible to decompose the left-hand side into the two functions. However, for certain, fairly rich parametric models one can show that, if ρ(t) = ρ(t;a) and Φ(t) = Φ(t;b), where the true values are a₀ and b₀, say, then for any other values, a′ and b′, say, ρ(t;a₀) ≠ ρ(t;a′) and Φ(t;b₀) ≠ Φ(t;b′) on a set of positive measure. Consequently, it is possible to identify the two functions.

For example, suppose Φ(t) = t^a, ρ(t) = (1 + t)^b and say for two values a₀,b₀ and a′,b′ these are the same. Then

or

a constant, which can only hold on a set of measure zero. The rest of the identification follows as previously.

3. IDENTIFICATION WITHOUT EXCLUSION RESTRICTIONS

In many situations it may not be plausible to impose exclusion restrictions. For example, in bargaining situations, if both agents have access to the same information, there is no reason to expect that one agent will condition on less information than the other. However, it may be plausible to make an assumption as to how a covariate will impact on each agent's duration dependence, at least in a limiting sense. In this case the data are generated as

where x (a scalar) and z, taking values on

, are common to both θ₁ and θ₂. The durations respond asymmetrically to x such that for, say, large (small) values of x, we have θ₁ → 0 (θ₂ → 0) and we can assume that an observed exit is due to T₂ (T₁) being the minimum. It should be intuitive how the model is identified. The terms Φ₂ and θ₂ are identified using values of x such that, say, x ≥ x¹⁰. Conversely, Φ₁ and θ₁ are identified using values of x such that, say, x ≤ x²⁰ (x²⁰ < x¹⁰). We require that the relevant normalizations can be imposed within these ranges. The assumptions are as follows.

Assumption 6. The durations are generated as in equation (15) where (U₁,U₂) ∼ K : (0,1) × (0,1) → [0,1] , K(·,·) is strictly increasing and differentiable in its arguments.

Assumption 7. θ(x,z) = (θ₁(x,z),θ₂(x,z)) → R₊² is continuous;

.

Assumption 8. For all

such that x > x¹⁰, θ₁(x,z) = 0 and for all

such that x < x²⁰, θ₂(x,z) = 0, some x¹⁰ > x²⁰. For some

; for some

.

PROPOSITION 2. Let Assumptions 2 and 5–8 hold. Then Φ_j, j = 1,2 and K are identifiable, θ₁ is identifiable for x < x²⁰, and θ₂ is identifiable for x > x¹⁰.

Proof. Note first that for values of x such that x > x¹⁰

so that

We identify θ₁(x,z) symmetrically. Setting t = 1 we can identify K by varying θ₁ and θ₂ over

. As in the proof of Proposition 1, we can then identify Φ_j by inverting K_j, j = 1,2. █

Remark 2. Note that Proposition 2 only partially identifies the model (at least nonparametrically). Without imposing other restrictions we can identify θ₁(x,z)(θ₂(x,z)) only for those values of x,z such that x < x²⁰ (x > x¹⁰).

Remark 3. The presence of z in θ₁ and θ₂ allows us to vary these functions over

and identify K. Without z we would only be able to observe θ₁ and θ₂ over, say, (x²⁰,∞) × (x¹⁰,∞), so that K and Φ_j,j = 1,2, are only partially identifiable. This problem can be partially circumvented by assuming θ₁, θ₂ are differentiable. An explanation is in the working paper version of this paper (Paric and Rilstone, 1999).