
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, N_{r} 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 T_{t}.
Time 1 2 3 4 Number at Risk N_{r} N_{r}T_{1} N_{r}T_{1}T_{2} N_{r}T_{1}T_{2}T_{3} Number who Transition T_{1} T_{2} T_{3} T_{4} 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, P_{t}, is
P_{1} = Prob(exit poverty at t = 1, given one period in poverty) = T_{1}/N_{r}
P_{2} = Prob(exit poverty at t = 2, given not exit at t = 1) = T_{2}/(N_{r}T_{1}) [A1]
P_{3} = Prob(exit poverty at t = 3, given not exit at t = 1 or t = 2) = T_{3}/(N_{r}T_{1}T_{2})
P_{4} = Prob(exit poverty at t = 4, given not exit at t = 1, t = 2, or t = 3) = T_{4}/(N_{r}T_{1}T_{2}T_{3})
This is the KaplinMeier hazard estimator. The notation for the hazard rate for person i at time t, P_{it}, can be condensed and written as:
P_{it} = Prob(t = T_{i}  t d T_{i}). [A2]
This simply says that the hazard rate is the probability of exiting poverty (or entering poverty) at time t (T_{i} = t) given that the individual exits poverty (or enters poverty) at time t or later (T_{i} >= 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:
P_{it} = Prob(t = T_{i}  t <= T_{i}, 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]


Appendix B: Tables

Table B.1—Summary Statistics for Persons at Risk of Entering Poverty PSID SIPP Mean SD 1988 & 1990 1996 Mean SD Mean SD Change in Economic Status Change in state unemployment rate, t 0.09 0.00 0.03 0.00 0.03 0.00 Change in GDP, t (in billions) 164.86 0.29 24.02 0.04 95.95 0.04 Demographic Characteristics of Household Head Age: Less than 25 0.05 0.00 0.04 0.00 0.03 0.00 Greater than or equal to 55 0.25 0.00 0.27 0.00 0.26 0.00 Race/Ethnicity: Black 0.10 0.00 0.09 0.00 0.10 0.00 Hispanic   0.07 0.00 0.10 0.00 Educational attainment: Equal to high school 0.34 0.00 0.32 0.00 0.31 0.00 More than high school 0.46 0.00 0.47 0.00 0.56 0.00 Household Composition Femaleheaded household (for more than 2 years) 0.13 0.00 0.10 0.00 0.11 0.00 Single maleheaded household 0.06 0.00    >  Number of adults less head and wife 0.36 0.00 0.79 0.00 0.44 0.00 Number of children (less children that enter at t and t1) 1.10 0.00 1.02 0.00 1.07 0.00 Geographic Characteristics Region: Northeast 0.23 0.00 0.21 0.00 0.20 0.00 Midwest 0.29 0.00 0.26 0.00 0.25 0.00 West 0.18 0.00 0.20 0.00 0.21 0.00 Pacific 0.00 0.00     Urban area: MSA 0.58 0.00 0.74 0.00 0.81 0.00 Economic Characteristics State unemployment rate 6.98 0.01 6.02 0.00 4.47 0.00 GDP (in ten billions) 57.21 0.02 66.11 0.00 85.65 0.00 Spell Information Leftcensored spells (observed) 0.73 0.00 0.90 0.00 0.80 0.00 Number of previous spells (observed) 0.14 0.00 0.07 0.00 0.21 0.00 Time Period 19801989 0.56 0.00     19901996 0.18 0.00     1990 SIPP Panel (10/898/92)   0.57 0.00   1997     0.26 0.00 1998     0.36 0.00 1999     0.38 0.00 Number of personyears/months 217,427
2,034,658 2,211,724
Notes: Table presents weighted means. Summary statistics based on personyears for the PSID and personmonths for the SIPP. See Table 4 for weighted means of entry trigger events. Table B.2—Summary Statistics for Persons at Risk of Exiting Poverty PSID SIPP Mean SD 1988 & 1990 1996 Mean SD Mean SD Change in Economic Status Change in state unemployment rate, t 0.08 0.01 0.02 0.00 0.04 0.00 Change in GDP, t (in billions) 163.10 0.94 28.65 0.11 61.29 0.21 Demographic Characteristics of Household Head Age: Less than 25 0.13 0.00 0.09 0.00 0.08 0.00 Greater than or equal to 55 0.23 0.00 0.21 0.00 0.21 0.00 Race/Ethnicity: Black 0.43 0.00 0.27 0.00 0.24 0.00 Hispanic   0.17 0.00 0.22 0.00 Educational attainment: Graduate high school (two or more years ago) 0.32 0.00 0.30 0.00 0.31 0.00 Received an associate's degree or higher (two or more years ago) 0.17 0.00 0.22 0.00 0.26 0.00 Household Composition Femaleheaded household (for two or more years) 0.47 0.00 0.35 0.00 0.34 0.00 Single maleheaded household 0.09 0.00     Number of adults (less head and wife) 0.40 0.01 0.53 0.00 0.38 0.00 Number of children (less children that enter at t and t1) 1.71 0.01 1.89 0.00 1.85 0.00 Geographic Characteristics Region: Northeast 0.16 0.00 0.16 0.00 0.17 0.00 Midwest 0.29 0.00 0.23 0.00 0.20 0.00 West 0.10 0.00 0.17 0.00 0.22 0.00 Pacific 0.00 0.00     Urban area: MSA 0.52 0.00 0.66 0.00 0.75 0.00 Economic Characteristics State unemployment rate 7.34 0.02 6.37 0.00 4.92 0.00 GDP (in ten billions) 58.11 0.07 66.15 0.00 83.48 0.01 Spell Information Leftcensored spell 0.15 0.00 0.56 0.00 0.46 0.00 Number of previous spells (observed) 0.62 0.01 0.35 0.00 0.44 0.00 Time Period 19801989 0.61 0.00     19901996 0.19 0.00     1990 SIPP Panel (10/898/92)   0.57 0.00   1997     0.26 0.00 1998     0.25 0.00 1999     0.25 0.00 Number of personyears/months 35,445 272,639 517,902 Notes: Table presents weighted means. Summary statistics based on personyears for the PSID and personmonths for the SIPP. See Table 5 for weighted means of entry trigger events.

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