A discretetime 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, t1).^{(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 discretetime 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) twoadult household becomes femaleheaded 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) femaleheaded household becomes twoadult 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 (t1). 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 (femaleheaded household, single maleheaded household, number of adults 1861, 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 femaleheaded household, are defined so that the event variable captures the full effect of the event. Using femaleheaded 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 t1 (lagged one period), and the third category captures the control (or level) variable: (1) femaleheaded household at time t and became femaleheaded at t (i.e., between t1 and t); (2) femaleheaded household at time t and became femaleheaded at t1 (i.e., between t2 and t1); and (3) femaleheaded household at time t and became femaleheaded prior to time t1. To capture all possible household combinations at time t, single maleheaded household at time t is included as a control variable, leaving twoadult household at time t as the omitted variable. In this example, the third variable (femaleheaded household at time t and became femaleheaded prior to time t1) provides information about how living in a femaleheaded household for two or more years affects the probability of entering and exiting poverty relative to living in a twoadult household. The following six control variables are defined with their interaction with the event variable in mind: (1) femaleheaded household for two or more years; (2) number of adults 1861 in the household, less the head and wife; (3) number of children in the household less those who enter at time t and t1; (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 twoadult to a femaleheaded 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.

Calculating the Likelihood an Event Occurs

The value of the estimated coefficients from the discretetime 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 twoadult to a femaleheaded 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 twoadult to a femaleheaded 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 twoadult to a femaleheaded household.


Left and Right Censoring

Our proposed discretetime logit hazard estimation approach takes account of rightcensored spells, while leftcensored spells are more problematic. Whether including or excluding leftcensored 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 LeftCensoring." He recommends that "when studying poverty transitions, using discretetime logistic regression, all observations from leftcensored spells should be included in [the] model to avoid selection bias." Iceland finds that omitting leftcensored cases potentially introduces greater bias in poverty transitions than including them because it would systematically exclude individuals in the midst of longterm poverty.^{(22)} Iceland (1997b) does not omit leftcensored 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 leftcensored spells. We do, however, identify leftcensored spells in the model using a dummy variable. With this design, the model of poverty entries that includes leftcensored 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 wellsuited to answering the research questions.

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