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This section describes the methods used to analyze our three research questions: (1) What are the dynamics behind changes in the poverty rate over time? (2) What events increase individuals' likelihood of entering and exiting poverty? and (3) What is the likelihood of entering and exiting poverty given these different events? We answer the first research question using the count method and answer the second and third research questions using the multivariate hazard method. While some researchers have used the count method to examine the relationship between events and transitions, using this descriptive approach is problematic because individuals can experience more than one event at a time, thereby making it impossible to identify the relative importance of the different events in the individuals' transitions.
The count method is used to examine both the absolute number of individuals entering and exiting poverty, as well as the probability of entering and exiting poverty at a point in time. The number of people entering and exiting poverty is obtained by calculating changes in individuals’ poverty statuses across two years. The number of people who enter poverty in year t is defined as the number of persons not poor last year, at t-1, who are poor this year, at t. Similarly, the number of people who exit poverty in year t is defined as the number of persons poor last year, at t-1, who are not poor this year, at t. For our notation, let ENt represent the number of individuals who enter poverty in year t and EXt represent the number of persons who exit poverty in year t. Equation 1 (presented in the previous section) shows that these are two of the components needed to decompose the poverty rate.
Looking at entries and exits in the context of the poverty rate equation (Equation 1) provides answers to one of the primary questions: What are the dynamics behind changes in the poverty rate over time? This descriptive analysis provides information about the relative importance of poverty entries and poverty exits in defining the overall poverty rate. For example, we can examine whether poverty rates remained high in some years because the number of entries and exits were low or because both entries and exits were high. A simple table, like the one shown below, is used to identify whether there are any patterns in poverty entry and/or poverty exits between the mid-1970s and mid-1990s.
| Year | Number Poor | Number Enter Poverty | Number Exit Poverty | Net Change in Number Poor | Population | Poverty Rate |
|---|---|---|---|---|---|---|
| 1974 | NP,74 | |||||
| 1975 | NP,75 | EN75 | EX75 | EN75 – EX75 | N75 |  |
| 1976 | NP,76 | EN76 | EX76 | EN76 – EX76 | N76 |  |
| 1977 | NP,77 | EN77 | EX77 | EN77 – EX77 | N77 |  |
| 1978 | NP,78 | EN78 | EX78 | EN78 – EX78 | N78 |  |
The number of entries and exits are used to calculate the probability of entering or exiting poverty at a point in time. The probability of entering poverty is defined as the ratio of the number of people who enter poverty in year t (ENt) and the number of people not poor in year t-1 (Nnp,t-1), or
[2]
Similarly, the probability of exiting poverty is defined as the ratio of the number of people who exit poverty in year t (EXt) and the number of people poor in year t-1 (Np,t-1), or
[3]
Note that the sum of Nnp,t-1 and Np,t-1 is the total population in year t-1.
The definitions above highlight, for example, that for an individual to enter poverty in year t, that individual cannot be poor in year t-1. While this appears obvious, it is very important to keep in mind when examining poverty entry and exit rates. The percentage of individuals entering poverty is calculated from the population of individuals not poor, which is the majority of the U.S. population, while the percentage of individuals exiting poverty is calculated from the population of individuals who are poor, which is small fraction of the U.S. population. So, even if the same number of individuals enter and exit poverty in a year, the poverty entry rate will be substantially lower than the poverty exit rate. Eller (1996), for example, finds a 3.0 percent poverty entry rate in 1993 and a 21.6 poverty exit rate in 1993. These percentages provide no information about whether more people entered or exited poverty in 1993. The absolute numbers of entries and exits, defined as ENt and EXt above, do provide this information.
A discrete-time multivariate hazard model is used to analyze events that trigger individuals’ entries into and exits from poverty. A hazard model simply provides information about the likelihood (i.e., probability) of experiencing an event at time t (e.g., exiting poverty) given that the event has not occurred prior to time t (e.g., the person is in poverty in the period prior to t, t-1).(15) Our multivariate hazard model allows the probability of experiencing an event at time t (e.g., exiting poverty) to depend on a set of explanatory variables, which includes among other characteristics, age, race, gender, and educational attainment, as well as the trigger events. This multivariate framework allows us to determine the relative importance of multiple events in poverty transitions, something that cannot be learned from a descriptive analysis. Separate poverty entry and exit equations are estimated.
Our discrete-time hazard model assumes that the probability of entering (or exiting) poverty in a given period (e.g., year) is represented by a logit specification.(16) The logit specification is popular as it is very tractable and restricts the transition probabilities to lie between zero and one (Allison 1984). Several studies of poverty dynamics have used the logit specification (Stevens 1994 and 1999, Iceland 1997b). With this assumption, the probability of entering (or exiting) poverty for person i at time t can be written as:
[4]
where
[5]
In this model, the vector T represents transition events, the primary focus of this analysis, and the vector X represents control variables.(17) The transition and control variables are based on our conceptual model. Our model of poverty entries includes the following transition events: (1) child under age six enters household, (2) two-adult household becomes female-headed household,(18) (3) young adult (under age 25) sets up own household, (4) loss of employment (of head, spouse, and other household members)—measured as a change from positive to zero hours work (PSID) and from with job to no job (SIPP), (5) nondisabled household head becomes disabled, and (6) weakening economy (change in state unemployment rate and change in GDP).
Our model of poverty exits include similar, although slightly different transition events: (1) female-headed household becomes two-adult household, (2) gain in employment (of head, spouse, and other household members)—measured as a change from zero to positive hours work (PSID) and from no job to with job (SIPP), (3) disabled household head becomes nondisabled, (4) household head receives high school degree, (5) household head receives advanced degree (associates degree or higher), and (6) strengthening economy (change in state unemployment rate and change in GDP). Because some of these events are choice variables (and thus potentially endogenous), this model does not necessarily identify causal relationships. Instead, it measures conditional relationships—the relationship after controlling for other events and characteristics.
An important issue is the extent to which events that occur in earlier periods are allowed to affect transitions in the current period. That is, to what extent lags enter the model. An immediate fall in income, say due to the loss of a job, may not cause a household to instantly fall below the poverty threshold if it is eligible for unemployment insurance. A household may fall below the poverty threshold only when unemployment insurance benefits run out. Similarly, a young adult who sets up her/his own household may only fall into poverty after private transfers from parents stop; and a change in educational attainment may only help an individual out of poverty after she/he obtains a higher paying job. Based on this theory of the timing between events and a poverty transition, we allow lags to enter the model for up to one year. In the yearly PSID data, we include a measure of the event at time t and a one year lag (t-1). In the monthly SIPP data, we include the event at time t and four quarterly lags.
Control variables include characteristics of the household head (age, race, and educational attainment), household (female-headed household, single male-headed household, number of adults 18-61, number of children), geographic characteristics (region and MSA), economic indicators (state unemployment rate and GDP), poverty spell information (observed duration of current spell at time t, observed number of prior spells, left censored spell identifier), and year identifiers.
Control variables that are tied to the event variables, such as female-headed household, are defined so that the event variable captures the full effect of the event. Using female-headed household as an example, three categories are created such that the first category captures the event at time t, the second category captures the event at time t-1 (lagged one period), and the third category captures the control (or level) variable: (1) female-headed household at time t and became female-headed at t (i.e., between t-1 and t); (2) female-headed household at time t and became female-headed at t-1 (i.e., between t-2 and t-1); and (3) female-headed household at time t and became female-headed prior to time t-1. To capture all possible household combinations at time t, single male-headed household at time t is included as a control variable, leaving two-adult household at time t as the omitted variable. In this example, the third variable (female-headed household at time t and became female-headed prior to time t-1) provides information about how living in a female-headed household for two or more years affects the probability of entering and exiting poverty relative to living in a two-adult household. The following six control variables are defined with their interaction with the event variable in mind: (1) female-headed household for two or more years; (2) number of adults 18-61 in the household, less the head and wife; (3) number of children in the household less those who enter at time t and t-1; (4) graduated from high school two or more years ago; and (5) received an associates degree or higher two or more years ago.(19)
Our analysis with PSID data further examines whether the events that trigger entries and exits differ for persons in long versus short poverty spells. It may be the case that changes in household composition, such as a shift from a two-adult to a female-headed household, result in long spells of poverty, whereas changes in employment cause only short poverty spells. We define a "long" poverty spell as one that has lasted four or more years and a "short" poverty spell as one that has lasted less than four years. We estimate separate models for short and long poverty spells.
The value of the estimated coefficients from the discrete-time multivariate hazard models do not have a straightforward interpretation. We can use these coefficients to determine whether an event increases or decreases an individuals' likelihood of experiencing a poverty transition, but alone, they do not provide information about the degree to which individuals are more or less likely to transition. We can, however, use these estimated coefficients and individuals' own characteristics to calculate the likelihood of entering poverty (or exiting poverty) when an event occurs. To calculate the likelihood of entering poverty with a shift from a two-adult to a female-headed household, for example, we (1) calculate each individual's estimated probability (i.e., likelihood) of entering poverty when the event is assumed to occur(20) and (2) average these estimated probabilities (i.e., likelihoods) across individuals. The average of these estimated probabilities gives the average likelihood of entering poverty when the event occurs.
We also calculate how the likelihood of entering/exiting poverty changes when the event occurs. To do this we first calculate (1) the average likelihood of entering poverty when the event occurs and (2) the average likelihood of entering poverty when the event does not occur.(21) Next, we calculate the difference between these two likelihoods, where this difference provides an estimate of how the likelihood of entering/exiting poverty changes when an event occurs. To quantify, for example, how a shift from a two-adult to a female-headed household affects poverty entries, we calculate the difference in the probability of entering poverty when the household structure shift does occur versus the probability of entering poverty when the household structure shift does not occur. This difference in the probabilities provides an estimate of how the likelihood of entering poverty changes with a shift from a two-adult to a female-headed household.
Our proposed discrete-time logit hazard estimation approach takes account of right-censored spells, while left-censored spells are more problematic. Whether including or excluding left-censored spells in an analysis produces misleading results depends on whether the analysis is trying to answer questions regarding poverty transitions or poverty duration. Iceland (1997a) looks at this exact topic in his paper "The Dynamics of Poverty Spells and Issues of Left-Censoring." He recommends that "when studying poverty transitions, using discrete-time logistic regression, all observations from left-censored spells should be included in [the] model to avoid selection bias." Iceland finds that omitting left-censored cases potentially introduces greater bias in poverty transitions than including them because it would systematically exclude individuals in the midst of long-term poverty.(22) Iceland (1997b) does not omit left-censored cases from his model because his focus is on how urban labor market characteristics affect transitions out of poverty, not the precise duration of poverty.(23) As our analysis focuses on poverty transitions, we incorporate left-censored spells. We do, however, identify left-censored spells in the model using a dummy variable. With this design, the model of poverty entries that includes left-censored spells, for example, examines "first observed poverty entry," not "first entry."
Summary: To summarize, we use the count method and the multivariate hazard model to answer our three research questions on the dynamics of poverty. We use the count method to examine the dynamics behind the poverty rate and the multivariate hazard model to examine events associated with poverty entries and exits. These methods are chosen because they are well-suited to answering the research questions.
15. The basic hazard model is defined in detail in Appendix A. The basic hazard model can be used to measure individuals’ likelihood of exiting poverty, but this more basic form of the model does not provide information about how different factors (i.e., transition events) affect the likelihood of exiting poverty.
16. We use a discrete-time, not a continuous-time, multivariate hazard model because poverty transitions are observed in large discrete time periods—a month or a year.
17. Some individuals enter (or exit) poverty more than once, so are included in the model more than once. Our standard errors are adjusted for this.
18. See discussion of control variables that are tied to event variables below for additional information on how this event is measured relative to other household combinations.
19. Changes in educational attainment are events only in the poverty exit model, so these last two variables pertain only to the exit model.
20. For the poverty entry models, the probability
individual i enters poverty at time t is expressed as
,
where
and Xi and Ti represent individual i's own
characteristics (see Equations 4 and 5). When calculating the estimated
probability of entering poverty when an event is assumed to occur, individual's
own characteristics are used except for the one transition event that is
assumed to occur (i.e., the event indicator variable is set to one).
21. These average likelihood values are calculated as described above.
22. Stevens (1999) is also concerned about bias from omitting left-censored spells from her examination of demographic characteristics (i.e., not transition events) associated with poverty exit and reentry. She finds the bias from omitting left-censored spells from her exit and reentry probabilities is extremely small (p. 572).
23. Stevens (1999) is also concerned about bias from omitting left-censored spells from her models that estimate exit and reentry rates. She similarly argues that omitting left-censored spells may over-estimate poverty exit rates at long durations. Stevens (1999) estimates her models both with and without left-censored spells. She finds the bias from omitting left-censored spells from her exit and reentry probabilities is extremely small (p. 572).
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