# 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.