# Duration dependence

The duration dependence describes the relationship between the exit rate and the time already spent in the state. Technically it is determined by the hazard func­tion, which may be a decreasing, increasing, or constant function of y. Accord­ingly, we distinguish (i) negative duration dependence; (ii) positive duration dependence; and (iii) absence of duration dependence.

Negative duration dependence

The longer the time spent in a given state, the lower the probability of leaving it soon. This negative relationship is found for example in the job search analysis. The longer the job search lasts, the less chance an unemployed person has of finding a job.

Positive duration dependence

The longer the time spent in a given state, the higher the probability of leaving it soon. Positive duration dependence is observed in the failure rate of instruments which are getting used up in time, or depreciate gradually. For example, the longer a lightbulb works, the higher the probability that it fails within the next hour (say).

Absence of duration dependence

The hazard function is constant. In this case there is no relationship between the duration spent in the state and the probability of exit.

The absence of duration dependence is often imposed as a simplifying al­though very restrictive assumption. It implies that items, such as instruments or

machines, do not deteriorate: for example, an item, which has been in use for ten hours, is as good as a new item with regard to the amount of time remaining until the item fails. This effect is usually not supported by the data and durations observed in empirical research usually belong to the first or second category or else display a nonmonotone hazard function. For this reason empirical hazards often need to be studied case by case. Let us consider, for instance, a typical hazard function representing the rate of bankruptcies. A newly created firm has a low probability of failure; however six months to two years later the failure rate increases sharply. The bankruptcy rate usually shows a tendency to diminish for companies which operate for a fairly long time, acquire more experience, and become better known to their customers and suppliers.