Estimating the Derivative Function and Counterfactuals in Duration Models with Heterogeneity

Abstract

This paper presents a new estimator for counterfactuals in duration models. The counterfactual in a duration model is the length of the spell in case the regressor would have been different. We introduce the structural duration function, which gives these counterfactuals. The advantage of focusing on counterfactuals is that one does not need to identify the mixed proportional hazard model. In particular, we present examples in which the mixed proportional hazard model is unidentified or has a singular information matrix but our estimator for counterfactuals still converges at rate N ^1/2, where N is the number of observations. We apply the structural duration function to simulate important policy effects, including a change in welfare benefits.

Keywords:

• Heterogeneity
• Mixed proportional hazard model
• Transformation model

JEL Classification:

• C41
• C24
• J64

ACKNOWLEDGMENT

We thank Matthew Harding and Ketan Patel for excellent research assistance. We thank Andrew Chesher, Nicole Lott, Pierangelo de Pace, and seminar participants at UCL for helpful comments.

Notes

See Lancaster (1990) and Van den Berg (2001) for overviews and Han and Hausman (1990) and Meyer (1996) for applications.

Ridder (1990) gives identification proofs for the closely linked generalized accelerated failure time model and Hausman and Woutersen (2005) estimate a duration model with time-varying regressors; Honoré and Hu (2010) give a recent review of the transformation model.

Van den Berg (2001) argues against relying on very short spells for estimating a duration model.

In the mixed proportional hazard model, one can derive the following equality, ln −Xβ −ln (v) +ln {−ln (U)}, where U is uniformly distributed on [0,1]; we condition on v and U in this case, and we condition on ϵ when we consider the transformation model of Eq. (2).

See also Vytlacil (2002).

We often surpress the arguments and write T* rather than T*(X, X*, T).

We thank an anonymous referee for stressing the importance of a stochastic

A set of functions ℋ is closed under the power transformation if f(t) ∈ ℋ implies {f(t)}^α ∈ ℋ for every α > 0. Here, however, we have e ^H(t) ∈ ℋ implies {e ^H(t)}^α ∉ ℋ for every α > 0, α ≠ 1.

Gørgens and Horowitz (1996) also derive an estimator for H(.) for this case of censored data.

See U.S. Department of Health and Human Services (HHS), “National Evaluation of Welfare-to-Work Strategies: How Effective are Different Welfare-to-Work Approaches?” (2000) for more information about the study's design. See this document for a bibliography of previous research using these data.

These strategies were also tested in Atlanta, GA and Grand Rapids, MI.

US HHS op. cit.

Sanctions were implemented by individual welfare offices, and not all subjects who failed to attend an orientation were sanctioned. The NEWWS study found no clear relationship between sanctioning rates within a city and participation rates in welfare-to-work programs.

Brock and Harknett (1998) found that about 66% of the people directed to attend an orientation in Riverside actually attended.

We group Asians with Whites because their employment and earnings characteristics are more similar to that of Whites than Blacks or Hispanics.

For the censored case, we need assumptions A1–A7 plus Chen's (2002) A8–A9.