[A4]
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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]
| 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 | ||||||
| Female-headed household (for more than 2 years) | 0.13 | 0.00 | 0.10 | 0.00 | 0.11 | 0.00 |
| Single male-headed 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 t-1) | 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 | ||||||
| Left-censored 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 | ||||||
| 1980-1989 | 0.56 | 0.00 | -- | -- | -- | -- |
| 1990-1996 | 0.18 | 0.00 | -- | -- | -- | -- |
| 1990 SIPP Panel (10/89-8/92) | -- | -- | 0.57 | 0.00 | -- | -- |
| 1997 | -- | -- | -- | -- | 0.26 | 0.00 |
| 1998 | -- | -- | -- | -- | 0.36 | 0.00 |
| 1999 | -- | -- | -- | -- | 0.38 | 0.00 |
| Number of person-years/months | 217,427 |
2,034,658 | 2,211,724 |
|||
| Notes: Table presents weighted means. Summary statistics based on person-years for the PSID and person-months for the SIPP. See Table 4 for weighted means of entry trigger events. | ||||||
| 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 | ||||||
| Female-headed household (for two or more years) | 0.47 | 0.00 | 0.35 | 0.00 | 0.34 | 0.00 |
| Single male-headed 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 t-1) | 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 | ||||||
| Left-censored 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 | ||||||
| 1980-1989 | 0.61 | 0.00 | -- | -- | -- | -- |
| 1990-1996 | 0.19 | 0.00 | -- | -- | -- | -- |
| 1990 SIPP Panel (10/89-8/92) | -- | -- | 0.57 | 0.00 | -- | -- |
| 1997 | -- | -- | -- | -- | 0.26 | 0.00 |
| 1998 | -- | -- | -- | -- | 0.25 | 0.00 |
| 1999 | -- | -- | -- | -- | 0.25 | 0.00 |
| Number of person-years/months | 35,445 | 272,639 | 517,902 | |||
| Notes: Table presents weighted means. Summary statistics based on person-years for the PSID and person-months for the SIPP. See Table 5 for weighted means of entry trigger events. | ||||||
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