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 $$S(t)=1-F(t)=1-{\int }_0^{t}f(s)\mathrm{ds}$$ 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 $t$: $$\lambda (t)\equiv \underset{h\to 0}{\lim }\frac{\mathrm{Pr}(t\le T\le t+h|T>t)}{h}.$$ 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 $$\Lambda (t)\equiv {\int }_0^t\lambda (t)\mathrm{dt}.$$ $\Lambda (t)$ is also called the integrated hazard. It follows that $\Lambda (t)=-\ln S(t)$.

Common Distributions

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

• The exponential distribution:

$$F(t)=1-e^{-\gamma t}\text{or}\lambda (t)=\gamma .$$

• The Weibull distribution:

$$F(t)=1-e^{-\gamma t^{\alpha }}\text{or}\lambda (t)=\gamma \alpha t^{\alpha -1}.$$

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

Literature

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• Cox, D. R. (1962). Renewal Theory. London: Methuen.

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• Honoré, B. E. (1993). Identification results for duration models with multiple spells. Review of Economic Studies 60, 241–246.

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• 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.