# Duration Variables

In this section we introduce basic concepts in duration analysis and present the commonly used duration distributions.

2.1 Survivor and hazard functions

Let us consider a continuous duration variable Y measuring the time spent in a given state, taking values in R+. The probabilistic properties of Y can be defined either by:

the probability density (pdf) function f(y), assumed strictly positive, or the cumulative distribution (cdf) function F( y) = Pn f (u)du, or the survivor function S( y) = 1 – F( y) = /Jf(u)du.

The survivor function gives the probability of survival to y, or otherwise, the chance of remaining in the present state for at least y time units. Essentially, the survivor function concerns the future.

In many applications the exit time has an economic meaning and may signify a transition into a desired or undesired state. Let us pursue the example of individual life expectancy. A related important indicator is the instantaneous mortality rate at age y. It is given by:

1

4y) = limnT P[y ^ Y < y + dy I Y > y]. (21Л)

dy^° dy

In this formula X(y) defines the probability per unit of time that a person dies within a short interval of dy (seconds) given that he/she is still alive at age y. It can easily be written in terms of the survivor function. Indeed we get:

= lim_LS( y) – S( y + dy)

dy^n dy S( y)

1 dS( y), S(y) dy ‘

S( y)‘

The hazard function is:

X(y) = ^4 = lim-1 P[ y < Y < y + dy |Y > y], Vy Є R+. (21.3)

S(y) dv^o dy

It gives the instantaneous exit rate per unit of time evaluated at y. Among often encountered exit rates are, besides the aforementioned mortality rate, the bankruptcy rate, and the failure rate of instruments.

The duration variable can equivalently be defined by S, f or X, in reason of the following relationship between the survivor function and the hazard function:

y

X(u)du.

о

This means that once we know the hazard function we can always find the survivor function.

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