Studies of Welfare Populations: Data Collection and Research Issues. Statistical Model

06/01/2002

The determinants of total-time-on, the number of spells, and the length of spells--as well as whether a recipient should be considered to be a short-termer, long-termer, or cycler--follows from the statistical features of her underlying time profile of participation. That time profile is generated mathematically by a discrete-time statistical process. The building blocks of any such process (in this case, moving on and off welfare) are a pair of hazard rates p[i|t,X(i),Z(i,t),H(i,t)] and q[i|t,X(i),Z(i,t),H(i,t)] for the probability that individual i moves onto the rolls at time t conditional on being off at t-1 and the probability that individual i moves off the rolls at time t conditional on being on at t-1, respectively. Here X(i) denotes time-invariant characteristics (such as family background at age 16 and race), Z(i,t) denotes the entire history of exogenous events that affect welfare transitions (such as business cycle and illnesses), and H(i,t) denotes the individual's entire history of welfare recipiency up through t-1. The variable t is taken literally to denote age, with t=0 at some initial age like 16. The probability functions p and q are taken over all unobservables in all time periods, consisting of all random events and shocks in the period prior to t. Thus we conceptualize all individuals as starting off at the same age, with certain fixed initial background characteristics, and then proceeding period by period through their lifetimes, moving on and off the rolls according to their individual-specific transition rates p and q . This constitutes a complete model of the process.

We will be interested in this chapter not in these structural transition rates, but rather in the distributions of welfare participation outcomes--that is, the types of patterns of participation that occur--that result from them over a particular calendar interval. Nevertheless, that different women have different patterns over such an interval necessarily arises from differences in the underlying hazards, and those hazards are a function of the variables denoted. A mutually exclusive categorization of all possible sources of heterogeneity in welfare patterns across women is the following: (1) heterogeneity in background characteristics, X(i) ; (2) heterogeneity in the vector of current and past time-varying exogenous events that differ across individuals, Z(i,t) ; and (3) heterogeneity in unobserved differences across individuals with the same X(i) and Z(i,t), both those which arise from different time-invariant unobserved characteristics (unobserved heterogeneity) as well as differences in current and past random shocks.(6) Thus any measure y of welfare participation patterns over a given calendar period from, say, t0 and t1 -such as total time on welfare, number of spells, average spell lengths--can be written as y (t 0,t 1 )=f(X,Z,e ), where X , Z , and e represent the three components just listed, over the interval from t0 to t1.(7) Given the function f, we can ask what types of mean characteristics are observed for women who have a particular welfare participation pattern y . Mathematically, we can write this as E[X | y(t 0 ,t1 )], and analogously for Z .

We will focus our empirical study below on X rather than Z . That is, we will examine the fixed, time-invariant characteristics (such as race, education, and average earnings and wages) of women with different welfare participation patterns. We will not examine time-varying characteristics, despite the fact that they presumably are important in explaining period-specific reasons for transitioning on and off the welfare rolls.

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