# Transition Events in the Dynamics of Poverty. Appendix A: Hazard Rate model

The hazard rate is the probability of experiencing an event at time t (i.e., making a transition) given that the event has not occurred prior to time t. The table below provides an example of the information necessary to calculate the hazard rate over four periods (e.g., years). Time t goes from 1 to 4, Nr people are initially at risk of an event (e.g., exiting poverty), and the last row represents the number of individuals who experience an event (e.g., transition out of poverty) at each time t, which is represented by Tt.

Time 1 2 3 4
Number at Risk Nr Nr-T1 Nr-T1-T2 Nr-T1-T2-T3
Number who Transition T1 T2 T3 T4

In each period, the hazard rate is simply the number who experience an event over the number at risk. So, the hazard rate at time t equal one to four, Pt, is

P1 = Prob(exit poverty at t = 1, given one period in poverty) = T1/Nr

P2 = Prob(exit poverty at t = 2, given not exit at t = 1) = T2/(Nr-T1)                              [A1]

P3 = Prob(exit poverty at t = 3, given not exit at t = 1 or t = 2) = T3/(Nr-T1-T2)

P4 = Prob(exit poverty at t = 4, given not exit at t = 1, t = 2, or t = 3) = T4/(Nr-T1-T2-T3)

This is the Kaplin-Meier hazard estimator. The notation for the hazard rate for person i at time t, Pit, can be condensed and written as:

Pit = Prob(t = Ti | t d Ti).         [A2]

This simply says that the hazard rate is the probability of exiting poverty (or entering poverty) at time t (Ti = t) given that the individual exits poverty (or enters poverty) at time t or later (Ti >= t).

Moving to a multivariate hazard framework allows the hazard rate to depend not only on time, but also on a set of explanatory variables, call them X. The hazard rate in the multivariate framework can be simply modified from the above equation to include these explanatory variables, X, and be written as:

Pit = Prob(t = Ti | t <= Ti, X).             [A3]

Moving from this form of the hazard rate to the estimating equation requires an assumption about how the hazard rate depends on the explanatory variables. With this assumption, the hazard rate for person i at time t can be written as:

[A4]

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