# Duration Models

Survival analysis concerns statistical models of the time until the first occurance of some event. In a biological setting, the event in question might be the death of an organism. In mechanics, it could be the failure of a machine. In economics, it could be entering the workforce.

The **survivor function** is the fundamental part of such models,
denoting the probability that the time of death is later than some
time $t$:
$$S(t)=\mathrm{Pr}(T>t)$$
where $T$ is a random variable denoting the time of death.
Let $f(t)$ denotes the pdf of $T$ and $F(t)$ denote the cdf of $T$.
We can see that $S$ must be non-increasing since
$$S(t)=1-F(t)=1-{\int}_{0}^{t}f(s)\phantom{\rule{thickmathspace}{0ex}}\mathrm{ds}$$
and $f$ is nonnegative.

Another fundamental object is the **hazard function** of $T$, $\lambda (t)$,
which gives the rate at which the probability of death is changing at some
time $t$:
$$\lambda (t)\equiv \underset{h\to 0}{\mathrm{lim}}\frac{\mathrm{Pr}(t\le T\le t+h|T>t)}{h}.$$
Using the definition of conditional probability and the cdf, we can write
$$\lambda (t)=\frac{f(t)}{1-F(t)}=\frac{f(t)}{S(t)}.$$

The **cumulative hazard function** of $T$, $\Lambda (t)$, is defined as
$$\Lambda (t)\equiv {\int}_{0}^{t}\lambda (t)\phantom{\rule{thickmathspace}{0ex}}\mathrm{dt}.$$
$\Lambda (t)$ is also called the **integrated hazard**. It follows
that $\Lambda (t)=-\mathrm{ln}S(t)$.

## Common Distributions

Some commonly used survivor functions (survival time distributions) include:

- The exponential distribution:

$$F(t)=1-{e}^{-\gamma t}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\lambda (t)=\gamma .$$

- The Weibull distribution:

$$F(t)=1-{e}^{-\gamma {t}^{\alpha}}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\lambda (t)=\gamma \alpha {t}^{\alpha -1}.$$

There is positive or negative duration dependence as $\alpha \gtrless 1$.

## Literature

Cox, D. R. (1959). The analysis of exponentially distributed life-times with two types of failure.

*Journal of the Royal Statistical Society: Series B*21, 411–421.Cox, D. R. (1962). Renewal Theory. London: Methuen.

Cox, D. R. (1972). Regression models and life-tables.

*Journal of the Royal Statistical Society: Series B*34, 187–202.Tsiatis, A. (1975). A Nonidentifiability Aspect of the Problem of Competing Risks.

*Proceedings of the National Academy of Sciences*72, 20–22.Peterson, A. V. (1976). Bounds for a joint distribution function with fixed sub-distribution functions: Application to competing risks.

*Proceedings of the National Academy of Science*73, 11–13.Heckman, J. J. and Singer (1984) A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data.

*Econometrica*52, 271–320.Kiefer N. M. (1988). Economic Duration Data and Hazard Functions.

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*Biometrika*76, 325–330.Han A. K. and J. A. Hausman (1990). Flexible Parametric Estimation of Duration and Competing Risk Models.

*Journal of Applied Econometrics*5, 1–28.Honoré, B. E. (1990). Simple Estimation of a Duration Model with Unobserved Heterogeneity.

*Econometrica*58, 453–473.Ridder, G. (1990). The non-parametric identification of generalized accelerated failure time models.

*Review of Economic Studies*57, 167–182.Honoré, B. E. (1993). Identification results for duration models with multiple spells.

*Review of Economic Studies*60, 241–246.Ferrall, C. (1997). Unemployment Insurance Eligibility and the School-to-Work Transition in Canada and the United States.

*Journal of Business & Economic Statistics*15, 115–129.Honoré, B. E. and E. Kyriazidou (2000). Panel data discrete choice models with lagged dependent variables.

*Econometrica*68, 839–874.Van den Berg, G. J. (2001). Duration Models: Specification, Identification, and Multiple Durations. J. J. Heckman and E. Leamer, editors, Handbook of Econometrics, Volume 5, 3381–3460. Amsterdam: North Holland.

Abbring, J. H., and G. J. Van den Berg (2003). The Identifiability of the Mixed Proportional Hazards Competing Risks Model.

*Journal of the Royal Statistical Society: Series B*65, 701–710.D’Addio, A. C. and B. E. Honoré (2006). Duration dependence and timevarying variables in discrete time duration models. Unpublished manuscript, Princeton University.

Honoré, B. E. and A. Lleras-Muney (2006). Bounds in Competing Risks Models and the War on Cancer.

*Econometrica*74, 1675–1698.Brinch, C. N. (2007). Nonparametric identification of the mixed hazards model with time-varying covariates.

*Econometric Theory*23, 349–354.Frederiksen, A., B. E. Honoré, and L. Hu (2007). Discrete time duration models with group-level heterogeneity.

*Journal of Econometrics*141, 1014–1043.Khan, S. and E. Tamer (2007). Partial rank estimation of duration models with general forms of censoring.

*Journal of Econometrics*136, 251–280.