Correcting estimates of expected durations from discrete-time hazard models

Labour Economics 9 (2002) 303 – 339 www.elsevier.com/locate/econbase

Correcting estimates of expected durations from discrete-time hazard models Curtis Eberwein* Center for Human Resource Research, Ohio State University, 921 Chatham Lane, Suite 100, Columbus, OH 43221-2418, USA

Abstract This paper proposes corrections for estimating expected durations when ‘‘discrete-hazard’’ models are used to estimate the duration distribution in a two-state, event-history model. The corrections are based on the assumption that the continuous-time hazard rates are approximately constant within periods and make no parametric assumptions about the duration dependence of the continuous-time hazard rates. The corrections will be useful for applied work, especially when the time interval for the discrete-time model is quite long, as in the case of the SIPP data. The corrections are evaluated using simulations and are found to be robust for many types of hazard rates that display duration dependence. So long as the assumption that the hazards are constant within periods is not extremely violated, the corrected estimates of the expected durations are substantially better than the uncorrected estimates. Even if the hazards do change by extreme amounts within periods, the simulations indicate that two of the estimators still do very well as long as the volatility in the hazard is not early in new spells. An empirical application is explored and one of the estimators is found to be quite successful in correcting the bias. D 2002 Elsevier Science B.V. All rights reserved. JEL classification: C41; C15 Keywords: Duration models; Event histories; Monte Carlo Simulation

1. Introduction Empirical researchers are frequently interested in estimating duration distributions. For example, we might wish to estimate the expected duration of employment and unemployment spells or the expected time spent in or out of a welfare program. Frequently, researchers do not observe the exact length of durations in a state. Instead, they observe the state occupied at discrete moments in time. For example, one might observe whether an *

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individual was employed or unemployed at the beginning of each month. A common practice is to use such data to estimate ‘‘discrete hazards,’’ which are the probabilities that the state changes in the next observation. Since the parameters of hazard functions are hard to interpret economically, these hazards are then used to estimate the expected durations. Examples of this methodology include Ham and LaLonde (1996) and Eberwein et al. (1997). Because of this, the biases this paper focuses on are biases in expected durations, since these are the estimates that are comparable across different studies. These estimates are not consistent estimates of the expected durations of the spells. Rather, what is being estimated is the expected number of consecutive periods a state will be observed. This differs from the expected duration because it ignores the possibility of multiple changes of state within periods. For instance, a person might be employed on the first of the July, become unemployed on July 7th, become employed again on July 17th and remain employed on August 1st. A researcher observing only the state occupied on the first of each month could not distinguish this person from another person who was employed continuously throughout July. Because the usual estimator of the expected duration from these models ignores these short spells, it is an upward biased estimator of the expected duration. As shown below, the asymptotic bias increases as the time between observations increases, so estimates using different levels of time aggregation are not comparable. The bias will be most important when the time between observations is long, so that the probability of missing spells that occur entirely within periods is relatively large. An apparent way to resolve this problem is to ask the dates at which transitions occurred, not just the state occupied at the time of the interview. However, this can lead to other problems. For example, in the Survey of Income and Program Participation (SIPP), an important panel data set used to analyze poverty in the US, a ‘‘seam bias’’ has been noted in reported dates of transitions into or out of participation in welfare programs (Blank and Ruggles, 1994, 1996; Fitzgerald, 1991). The SIPP respondents are interviewed every 4 months and report, retrospectively, their participation in programs such as Food Stamps and Aid to Families with Dependent Children (AFDC). The seam bias arises because an unusually large number of reported transitions occur in the last month before the interview. As a result, analysts have estimated discrete-time hazard models based on 4-month intervals. However, using 4-month intervals raises two problems. First, the biases described above are likely to be quite important. Second, researchers cannot logically consider time intervals shorter than 4 months (Flinn and Heckman, 1982). The approach taken in this paper addresses both of these issues. Section 2 suggests three methods of correcting expected durations for a two-state discrete-hazard model based on the assumption that the continuous-time hazard rates are approximately constant within periods. None of these estimators makes any parametric assumptions about duration dependence. Section 3 applies the estimators to simulated data for several duration distributions. When the hazard rates in both states are constant, two of these estimators are consistent, while the third is shown to have at most a trivial bias when the hazard rates are not unreasonably large. Also, the loss of efficiency relative to the fullinformation, asymptotically efficient estimator (which uses the actual durations, not just the observed states at discrete times) is small for all three estimators. When the hazard rates are changing over time, the corrected estimators are biased but are typically substantially better than the uncorrected estimator. While the biases are always statistically significant, they are frequently of trivial magnitude, even when the hazards are

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changing by fairly large amounts within periods. Moreover, all three estimators are functions of only the discrete hazards, which are nonparametrically identified (i.e., one need not make any assumptions about the functional form of duration dependence in the continuous-time hazard). While it is possible to come up with counterexamples in which the corrections do poorly — or even fail to exist — these counterexamples involve very extreme changes in the hazard rates within periods. As long as researchers can reasonably rule out such extreme changes, the results indicate that the corrections represent a substantial improvement over the usual method of estimating expected durations with discrete hazards. The simulations also indicate that even very large changes in the hazard rate have little effect on two of the three estimators if the changes occur after the first two periods of a new spell. That is, for these two estimators, the assumption that the continuous-time hazards are approximately constant is only important for the hazard rates within the first few periods of a new spell. Section 4 applies the estimators to an employment/unemployment event history model using data on adult women who were part of a controlled experiment that studied the effects of government-sponsored classroom training on labor force outcomes. The data are monthly, but estimates of expected durations are obtained using both the full data and data in which the state is only observed every 3 months (i.e., quarterly data). It is shown that all three estimators improve the comparison between estimates from monthly and quarterly data when the corrections are applied to both levels of time aggregation. One of the three estimators is particularly successful. Section 5 provides concluding remarks.

2. The estimators 2.1. Preliminaries Before proceeding, a remark on the terminology used in this paper is in order. Hereinafter, the word spell will be used to refer to an interval of time spent in a particular state. When referring to a string of consecutive observations in the same state at discrete points in time, the words discrete spell will be used. Thus, the expected duration is the average length of a spell while the average discrete spell length would be the average number of consecutive observations of that state at discrete points in time. Most papers make no distinction between these, but the difference is important given the subject of this paper. Suppose we have a two-state event history and that the continuous-time hazard rates are constant. In particular, let ki be the hazard rate for state i. Define li to be the probability state j (not equal to i) is occupied at time t + 1, given state i was occupied at time t. Then a discrete-hazard model would be estimating the lis.1 The relationship between the discrete hazards and the continuous-time hazards is given by ki ¼ 

li lnð1  l1  l2 Þ: l1 þ l2

ð1Þ

This is obtained by inverting equation set 6.14 in Lancaster (1990, p. 112). Lancaster’s result implies that the sum of the discrete hazards must be less than one when the 1

Note that the length of time between observations of the state has been normalized to equal one.

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continuous hazards are finite and constant.2 The expected duration in each state is just the inverse of the continuous hazard rate. The expected number of consecutive observations in a particular state (i.e., the expected discrete spell length) is just the inverse of the discrete hazard. Thus, the ratio of the expected number of consecutive observations in a particular state to the expected duration is given by R¼

lnð1  l1  l2 Þ : l1 þ l 2

ð2Þ

Note that R depends only on the sum of the transition probabilities and not the individual values. It is easy to show that R is increasing in this sum and that it tends to one as the sum tends to zero and it tends to infinity as the sum tends to one. Thus, larger transition probabilities indicate greater inconsistency when the continuous hazard rates are constant. From Eq. (1) it is easy to show that: li ¼

ki ð1  ek Þ, k ¼ k1 þ k2 : k

ð3Þ

Suppose we were to change how often the states are observed such that the period of time between observations is now s times what it was before. Renormalizing so that the period between observations is one unit of time, the hazards would change to kiV = kis. Then qðsÞ ¼ li V  ki V ¼

ki ð1  eks Þ  ki s: k

ð4Þ

It is straightforward to show that q(0) = 0 and that qV(s) V 0, with strict inequality for s > 0. Thus, the inconsistency is increasing as the time between observations increases. Eq. (2) indicates that the asymptotic percentage bias, which is 100(R  1), is the same in both states when the hazards are constant. In fact, this result still holds when there is duration dependence in the hazard rates. In particular, assume the event history is an alternating renewal process. That is, the hazard rates can depend on time spent in the spell, but are independent of calendar time. Then we have the following result: Proposition 1. Assume the duration distributions both have positive, finite first and second moments in a two-state alternating renewal process. Let Di be the expected duration in state i and let Di* be the expected discrete spell length. Then Di*/Di is independent of i (The proof is in Appendix A). The intuition runs as follows. The long-run fraction of periods observed in state i is the same as the long-run fraction of time spent in state i. The first of these is the ratio of the expected discrete duration in state i to the sum of the expected discrete durations in both states. The second is the ratio of the expected duration in state i to the sum of the expected 2 This result requires that the hazards are constant. By allowing the hazards to change by an extreme amount within periods, it is possible to come up with an example in which the transition probabilities sum to a number greater than one. Such an example will be given in Section 3.

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durations, so these must be the same. If the bias were not the same in both states, the longrun fraction of time spent in each state and the long-run fraction of periods observed in each state could not both sum to one, which they must. It is tempting to think this result makes correcting the expected durations unimportant. If the spells are employment and unemployment, the estimates of the steady state unemployment rate are consistent even without correction for an alternating renewal process.3 Suppose a policy change could proportionately increase the expected duration of both employment and unemployment. While this does not affect the unemployment rate, it is not clear that we should be indifferent to this policy change. It would mean fewer unemployment spells and, thus, fewer times the typical worker needs to search for work. If there are any fixed costs to job search (i.e., costs that are occurred each time a worker searches for a job, but do not depend on how long a worker searches), this makes workers better off. It also means lower turnover for firms. In addition, workers would have, on average, longer tenure on the job. Thus, the policy change might well yield a positive benefit to society. On the other hand, if there is hysteresis for the long-term unemployed, the effect on welfare could be negative. Note also that the above result indicates that it might be better to use the expected discrete-spell lengths based only on the state occupied at the time of interview to estimate the average time spent in each spell when seam bias is present. In general, hazard rates will not be constant over time. First, the hazards might display duration dependence. That is, the hazard rates may depend on how long the state has been occupied. For example, unemployment hazards may fall as workers become stigmatized or discouraged. Second, hazards may depend on other variables that are changing over time. Employment hazards may rise, for instance, when macroeconomic indicators are bad. There is a third reason why hazards may change over time, namely unobserved heterogeneity. Some individuals may have higher hazard rates than others due to unobserved characteristics. As time passes, those with higher hazard rates are more likely to have exited a state, so that those remaining in the state are more likely to be those with low hazard rates. This introduces a change in the observed hazard rates similar to negative duration dependence (see Heckman and Singer, 1984). Unobserved heterogeneity is not relevant to the issue studied here. As Heckman and Singer (1984) showed, an appropriate way to handle it is to estimate a finite set of points of support for the heterogeneity (along with the probability distribution of these points). The appropriate way to estimate the expected duration is to estimate the expected duration for each point of support in the heterogeneity distribution and then integrate over all the points of support. The corrections proposed here could be applied to the expected durations for each point of support. 2.2. The estimators When the hazards are changing over time, it is not as simple to correct the estimates of the expected duration. However, suppose that the hazards are approximately constant 3 Equivalently, estimates of the ratio of the transition probabilities are consistent estimates of the ratio of the continuous-time hazard rates.

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within periods but change across periods.4 In this case, the ls will depend on the number of periods already occupied in the current state as well as (possibly) time changing regressors. If the continuous hazards were indeed constant within periods, but changing across periods, the expected duration in state i would be     T X 1 eki ðnÞ 1 ki ðnÞ  Di ¼ Si ðn  1Þ½1  e  n1þ þ SðT Þ T þ , ki ðnÞ 1  eki ðnÞ ki ðT Þ n¼1 ð5Þ  ki(n)

where Si(0) = 1, Si(n) = Si(n  1)e and ki (n) gives the hazard rate for state i given that state i has been occupied for n periods. The first term in the sum is the survivor function. The second term is the probability there is an exit from the current state between time n  1 and n given there was no exit before n  1. Thus, the product of the first two terms is the probability a spell ends between n  1 and n. The third term represents the expected duration in state i given the spell ends between n  1 and n. Eq. (5) assumes that the hazard is constant for n greater or equal to T. If this is not the case for any T, the expected duration is obtained by letting T tend to infinity. If the hazards are piecewise constant within periods but vary across periods, the empirical transition probabilities will be a mixture of transition probabilities. Suppose a discrete spell in state i has lasted from periods 1 to n. Let li(n) be the probability the observed state changes from i to j p i at the next observation. Let gi(n  m) be the probability the observed state changes from i to j p i in the next period given that the current spell in state i (which has been observed for n consecutive periods) actually began between m and m + 1, ma{0, 1, 2, . . ., n  1}. These differ because there is a possibility that an even number of transitions greater than zero occurred between earlier observations of state i.5 The relationship between these is given by li ðnÞ ¼

n1 X

Pi ðm,nÞgi ðn  mÞ,

ð6Þ

m¼0

where Pi(m, n) is the probability the current spell in state i began between date m and m + 1 given that state i has been observed in all periods from 1 to n. Note that Pi ðm, n þ 1Þ ¼

eki ðnmÞ Pi ðm, nÞ , maf0, 1, 2, . . . , n  1g: 1  li ðnÞ

ð7Þ

The numerator is the probability that state i is observed at time n + 1 and that the spell began between period m and m + 1 given that state i was observed up to period n. The

4

The hazard rates would have to be continuously differentiable for this to be a satisfactory approximation. In an earlier version of this paper, estimates of the hazards were obtained ignoring this distinction. An anonymous referee suggested a methodology to take this into account and provided invaluable assistance in working out the details. 5

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denominator is the probability state i is observed in period n + 1 given that state i was observed up to period n. The result follows from applying Bayes’ rule. It is now demonstrated that a closed-form, functional relationship exists between the reduced-form transition probabilities and the hazard rates when the hazards are piecewise constant within periods. First, if the initial observation of a state is in period 1, then the spell must have begun between time zero and time one (assuming no left censoring). Thus, li(1) = gi(1). For n > 1, using Eqs. (1), (6) and (7) we have li ðnÞ 

Pi ðm, nÞgi ðn  mÞ

m¼1

gi ðnÞ ¼

ki ðnÞ ¼ 

n1 X

Pi ð0, nÞ

,

gi ðnÞ lnð1  gi ðnÞ  gj ð1ÞÞ, jp i, gi ðnÞ þ gj ð1Þ

Pi ðm, n þ 1Þ ¼

eki ðnmÞ Pi ðm, nÞ , maf0, 1, 2, . . . , n  1g, 1  li ðnÞ

Pi ðn, n þ 1Þ ¼ 1 

n1 X

Pi ðm, n þ 1Þ:

ð8aÞ

ð8bÞ

ð8cÞ

ð8dÞ

m¼0

Given estimates of the li(n) one can then iterate on the above algorithm to obtain estimates of the gi(n) and the hazard rates.6 Note that if the estimates of the transition probabilities are maximum likelihood estimates (MLE) then, by the invariance principle, the estimates of the hazard rates will be MLE. Having obtained estimates of the hazard rates in this manner, one could then use the estimated hazards to replace the actual hazards in Eq. (5) in order to estimate the expected duration. In what follows, this estimator will be referred to as Estimator 1. A simpler correction can be obtained by directly estimating the percentage asymptotic bias, R, using Eq. (2). In particular, one could use the transition probabilities for the first observation of each state to calculate R, then divide the uncorrected estimates of the expected durations (i.e., the expected discrete spell length) by R to arrive at an estimate of

6 Due to sampling error, the estimates of the gi(n) and Pi(n, n + 1) could fall outside the unit interval. In the simulations of the Section 4, estimates of gi(n) were constrained to the interval [10  8,1 – 10  8], with the nearest boundary point assigned when estimates fell outside this interval. If Pi(n, n + 1) is estimated to be negative, it is assigned a value of zero with all other probabilities divided by their sum to ensure they sum to one.

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the expected duration. This is a somewhat naive correction, but it has the advantage of being computationally very simple. This simple correction will be referred to as Estimator 2 in this paper. Estimator 3 follows a somewhat different approach. First, the li are estimated, allowing these to change across periods, as with Estimator 1. Then the continuous hazards are estimated as above, once again assuming these are constant within periods. Then, for each period observed in the data, the probability of multiple transitions between dates is calculated. As this probability is generally quite complicated, three approximations were tried. Let Pk be the probability that exactly k changes of state occur between date n and date n + 1. In Appendix A, the first four of these probabilities are given (k = 0, 1, 2 and 3). The simplest approximation ignores the possibility of more than three transitions. Using this approximation, if the state is the same at time n and time n + 1 then either zero or two transitions occurred between n and n + 1. By Bayes’ rule the probability two transitions occurred would be P2/( P0 + P2). When the state changes between n and n + 1 either one or three transitions occurred. The probability three transitions occurred would be P3/ ( P1 + P3). For each observed period these probabilities are calculated and summed across periods to get an estimate of the number of ‘‘missed’’ spells. This is actually twice the sum, because transitions are missed only if there were two more transitions than observed in the data. Let N * be the number of observed discrete spells and N the estimate of the number of actual spells using this method (so N = N * + two times the sum of probabilities of two more transitions between dates). Then the uncorrected estimators are predicted to be biased by a factor of N/N *. The corrected estimator then divides the uncorrected estimator by N/N *.7 Proposition 2 in Appendix A shows that this is the appropriate correction. Proposition 2 suggests another possible correction when seam bias is present. First, estimate the discrete hazard based only on the states occupied at the time of the interview. Then, multiply the discrete expected duration by the number of observed discrete transitions divided by the number of total transitions reported retrospectively. This requires one to assume the seam bias arises because respondents mistakenly report the dates of transitions as being too close to the interview date, but that they accurately report the number of transitions. If, instead, seam bias arises because respondents are more likely to forget transitions that occurred more than a month before the interview, this method would not be appropriate. The correction used for Estimator 3 above ignores the possibility that more than three transitions could occur. When the state stays the same between n and n + 1 we know that an even number of transitions occurred. If the state changed we know an odd number of transitions occurred. Thus, calculating the exact probability that transitions were missed would involve an infinite sum of higher order probabilities. Since the Pk become complicated as k gets large, this problem becomes intractable, which is the reason for using the approximation above. But, by ignoring higher numbers of transitions, we will

7 The fraction of periods observed in state i is a consistent estimator of the long-run average state occupancy probability for state i (provided this exists). The bias arises because the number of discrete spells is less than the number of actual spells.

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underestimate the number of missed transitions. A second approximation assumes that the probability of j + 2 additional transitions in a period given that j > 1 transitions have already occurred is approximately Q, where Q is defined as follows. If the state did not change, Q is the probability of exactly two transitions in the current period given a new discrete spell in the current state. If the state changed, Q is the probability of exactly two transitions in the period if the period had started with a new spell in the other state.8 This overestimates the probability of k transitions for k > 3 since it assumes we have a full period for each two additional transitions to occur, when in fact there generally is less than a full period left. Using this approximation, the probability exactly 2k transitions (k > 0) occurred between n and n + 1 when the state is the same at n and n + 1 is approximated by P2k c P2 Q k  1, so that 1 X k¼0

P2k cP0 þ P2

1 X

Q k1 ¼ P0 þ

k¼1

P2 ¼ Z0 : 1Q

ð9Þ

Summing over the k and dividing by Z0 (to get the conditional probabilities) then yields a predicted number of missed spells of 1 X 1 1 2P2 2P2 kQ k1 ¼ : 2 Z0 Z 0 ð1  QÞ k¼1

ð10Þ

Similar reasoning establishes that the approximate probability of an odd number of transitions is 1 X k¼0

P2kþ1 cP1 þ P3

1 X

Q k1 ¼ P1 þ

k¼1

P3 ¼ Z1 : 1Q

ð11Þ

The approximate expected number of missed spells when the state changed between n and n + 1 is given by 1 X 1 1 2P3 P3 2kQ k1 ¼ : 2 Z1 Z 1 ð1  QÞ k¼1

ð12Þ

Since this second correction overestimates the probability of higher order transitions, a third correction was also tried. This method uses the same method as the second to calculate the

8

If the state stayed the same, two more transitions — say from two to four — would begin with the same state as the state occupied at the beginning of the period. Going from three to five transitions, however, would begin in the other state.

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probability that there were at least two hidden transitions, but counts all multiple transitions as being just two. Thus, the approximate number of expected missed transitions between n and n + 1 using this correction is

2

Z0  P0 Z1  P1 if no change of state; 2 otherwise: Z0 Z1

ð13Þ

It is not clear whether this will over or under estimate the number of missed spells. On the one hand, it overestimates the probability at least two transitions were missed — as with the second method. On the other hand, it underestimates the number of missed transitions by counting all missed transitions as two, rather than a higher possible even number – as with the first method. In the simulations presented in Section 2.3, the third method was used as it performed the best. We discuss this point in further detail below. All three proposed corrections to the expected duration in state i need only the firstperiod transition for state j. This is important because one might be primarily interested in the expected duration in only one state. For example, if one is interested in the average time spent on welfare, one needs only the transition probability back to welfare in the first period off of welfare. Thus, a sampling scheme that follows individuals on welfare until they are observed off welfare and then one more period would be sufficient to apply all the corrections proposed here. 2.3. Application of the estimators If expected durations are estimated holding all explanatory variables constant, as would be the case if one estimates the expected duration for a person with average characteristics, it is straightforward how to apply Estimators 1 and 2. Estimator 3 would not be appropriate in this case, as it requires going back to the original data. If characteristics differ across individuals, but are constant for each individual over time, it is straightforward to apply each of these corrections to each person. If there are time-changing explanatory variables, Estimator 2 is not appropriate. This is because the probability of multiple transitions given a first transition is changing over time, and Estimator 2 cannot account for this. Estimators 1 and 3 are still appropriate as one can still calculate the transition probability in the current state and what the transition probability would be if it were the first period in the other state conditional on the observed explanatory variables for each period. However, Estimator 3 assumes that all spells in the data are completed spells, so it cannot be applied when there is right censoring in the data. In Section 4, an adjustment to Estimator 3 that accounts for right censoring will be proposed. It will be shown that the adjustment works well for the particular empirical application considered there. As shown in Section 2.1, the application of these estimators will be most important when the time between observations is long. Examples include research on durations in welfare using the SIPP data, such as Blank and Ruggles (1994, 1996) and Fitzgerald (1991), where there are 4 months between observations.

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3. Monte Carlo simulations This section presents simulations to evaluate the proposed estimators. As a base, we first look at how the estimators perform when in fact the hazards are constant. The results in Table 1 are for the case in which the hazards are 0.1 and 0.04 in States 1 and 2, respectively (so that the true means are 10 and 25, respectively). The data were generated as follows. First, 5000 uniform (on (0,1)) pseudorandom numbers were generated. Recall that if u is a uniform random variable on (0,1), then  ln(1  u) is exponential with mean 1. Moreover, the integrated hazard — which is just the hazard times the duration, t, in this case — is also exponential with mean 1. Thus, the durations are recovered by setting the integrated hazard equal to the negative of the log of (1  u) and then solving for t. Having generated 5000 durations for each state in this manner, an event history is then simulated by assuming the first spell is in State 1 and starts at time zero, with states alternating until all 5000 spells in each state are used. Next, it is calculated which states are occupied at times n = 1, 2, 3, etc. until n is large enough to cover all the spells. Five estimates of the expected duration are then calculated. The first is the fullinformation estimate and uses the observed durations to calculate the sample mean. This is the full-information MLE, so it is both unbiased and asymptotically efficient. The second is the uncorrected limited-information estimate which uses only the states occupied at times 1, 2, 3, etc. and is the average discrete spell length. The next three are the corrected limited-information estimates, which are obtained by applying Estimators 1 through 3 discussed in Section 2. These estimates require estimates of the transition probabilities. Here, the empirical transition rates were used. That is, the probability a discrete spell that has lasted at least n periods ends between n and n + 1 is estimated to be the sample average number of times discrete spells lasting at least n periods ended between n and n + 1.9 This becomes problematic for large n because there will be few data points for spells that are among the longest observed. To handle this an ad hoc rule (similar to what many researchers use) was employed (see, e.g., Ham and Rea, 1987). Namely, if n is such that at least 5% of observed spells last n or more periods, the above estimate was used. For all longer spells, it is assumed that the transition probability is constant and the estimate is the average number of transitions for all such long spell observations. Put differently, long discrete spells are pooled.10 In practice, this implies that estimates of transition probabilities typically used at least 250 data points. Having calculated the estimates in this manner, the above procedure was repeated 1000 times to obtain 1000 realizations of each estimator. Reported in the tables are the means and standard deviations of the estimators in the sample of 1000. These standard deviations

9 When the hazards are constant, the transition probabilities are as well, but this information was not used in constructing the corrected estimates. Note also that these are the MLE estimates assuming piecewise constant hazards, so that by the invariance principle, Estimator 1 is the MLE of the expected duration under this assumption. 10 For example, suppose more than 5% of spells last 80 periods and less than 5% last 81 periods. Then for n < 81, the estimate is the fraction of all discrete spells lasting at least n periods that last exactly n periods. For n > 80, the estimate is the fraction of times a spell ends in any period greater than 80.

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Table 1 Constant hazards Estimator Full informationb Uncorrectedc Estimator 1 Estimator 2 Estimator 3

State 1 (hazard = 0.10)

State 2 (hazard = 0.04) a

Mean

S.D.

% Bias

Mean

S.D.

% Bias

10.01 10.73 10.01 10.01 10.00

0.130 0.145 0.146 0.150 0.150

NA 7.15 0.00  0.03  0.07

24.97 26.75 24.96 24.96 24.95

0.321 0.344 0.333 0.328 0.329

NA 7.13  0.04  0.05  0.09

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 realizations. a % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. b Average of actual durations. c Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

do not equal the standard errors of the point estimates. Rather, the standard errors equal the standard deviations divided by the square root of 1000. The results in Table 1 indicate that all three corrections do very well. There is no evidence of a bias for any of the three corrected estimators in the sample.11 The uncorrected estimators are biased by about 7.1%. It is not surprising that Estimators 1 and 2 are essentially unbiased since the assumptions they make are correct in this case and they are, therefore, consistent (though they could be biased in finite samples).12 The standard deviations for all three corrected estimators are similar and none are very large relative to the full-information MLE. All the implied efficiencies are between 86% and 98%, where an estimator’s efficiency is the ratio of the efficient estimator’s standard deviation to its own standard deviation. For Estimator 3, we also tried the other two approximations discussed in Section 2. The first approximation yields a small positive (and statistically significant) bias for the above distributions. The second gives results virtually indistinguishable from those in Table 1. To determine whether the second or third approximation should be used, simulations were run with higher (still constant) hazard rates (with higher hazards, the probability of more than three transitions increases, so the approximation gets worse). When the hazard rates are 0.5 and 0.4 (so the expected durations are 2 and 2.5) using the second approximation to calculate Estimator 3 results in a bias of  3.2% while the third approximation yields a bias of  0.9%, where the negative sign indicates that they underestimate the true mean. When the hazards are increased further to 1 and 0.5 (expected durations of 1 and 2) the biases are about  8.7% and  2.4% using the second and third approximations,

11 The biases are calculated relative to the full-information MLE. Of course, in this case, we know the true means, so these could have been used instead. Doing so does not change the results much. The bias of the uncorrected estimator is essentially the same using the true means and all other biases are not significantly different from zero. The results are presented in this way in order to be consistent with the other tables, where we cannot calculate the means analytically. 12 The estimates of the transition probabilities are consistent and thus so are the estimates of the hazard rates since the functional relationship is known and continuous.

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respectively. As the third approximation appears more robust, it is used to calculate all further realizations of Estimator 3. Moreover, such high transition rates would normally make analysis of durations unreliable: with these hazards many spells are taking place entirely between observations. For hazards of 0.4 and 0.5, the bias of the uncorrected estimator is about 51%. For hazards of 1 and 0.5, this rises to 93%. Thus, while some efficiency might be gained by improving the approximation, for reasonable hazards it would seem the gain would be small. The remainder of this section investigates how these estimators do when the hazards are not constant. With a few exceptions, the hazards will be assumed to have the form kðtÞ ¼ a 

b c þ , 1 þ h1 t 1 þ h2 t

so that Z 0

t

kðsÞds ¼ at 

b c lnð1 þ h1 tÞ þ lnð1 þ h2 tÞ, h1 h2

ð14Þ

where t is the length of time spent in a spell. All parameters are assumed to be nonnegative. They must, of course, be chosen such that the hazard never becomes negative for positive t. This functional form can generate a variety of shapes for the hazard. Figs. 1 – 12 show graphs of the hazards (against duration in the current spell) for Simulations 1 through 12. The method of generating samples is the same as above except that we can no longer solve for t analytically by setting the integrated hazard equal to  ln(1  u) where u is a uniform random variable. Instead, this equation is solved numerically using Newton’s method with an initial guess of zero. For hazard rates with the functional form given in Eq. (14), Newton’s method with an initial guess of zero was found to be computationally very efficient, with convergence almost always occurring after just a few iterations. The choice of functional form is somewhat arbitrary, as long as it is capable of generating a variety of shapes for the hazards. The reasons for choosing this functional

Fig. 1. Hazard functions for Simulation 1.

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Fig. 2. Hazard functions for Simulation 2.

Fig. 3. Hazard functions for Simulation 3.

Fig. 4. Hazard functions for Simulation 4.

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Fig. 5. Hazard functions for Simulation 5.

Fig. 6. Hazard functions for Simulation 6.

Fig. 7. Hazard functions for Simulation 7.

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Fig. 8. Hazard functions for Simulation 8.

Fig. 9. Hazard functions for Simulation 9.

Fig. 10. Hazard functions for Simulation 10.

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Fig. 11. Hazard functions for Simulation 11.

form are that it can generate many shapes for the hazard function — increasing, decreasing and nonmonotonic — and that numerically converting uniform pseudorandom numbers into durations takes very little computer time with this functional form. This last point is important because the simulation methodology requires generating a very large amount of data. Other functional forms would present a much higher computational burden, which would make simulating a large amount of data prohibitively costly. Tables 2– 9 below report the results of Simulations 2– 9.13 At the top of each table, the values of the parameters for the hazards in each state are reported. The rest of the tables are the same as Table 1. Inspection of Figs. 1 – 9 reveals a wide variety of different types of duration dependence. In Simulation 2, both hazards are decreasing. In this case, all three corrected estimators are now biased, but the bias is typically one-fourth to one-sixth the bias of the uncorrected estimator.14 Note that the assumption that the hazards are constant is seriously violated here. The hazard in State 1 falls from 0.2 at duration zero to 0.15 at duration one (a 25% drop). The hazard in State 2 falls from 0.1 to 0.075 from duration zero to duration one (also a 25% drop). Still, Estimators 2 and 3 both have biases under 2% in absolute value. Simulation 3 also has both hazards decreasing over time, but at a slower rate. Thus, the assumption that the hazards are constant within periods is a closer approximation in Simulation 3 than in Simulation 2. As expected, all three estimators perform better in this case. Indeed, Estimators 2 and 3 have a bias of less than 1% in Simulation 3 (compared to about 2% in Simulation 2). One reason the uncorrected estimates are upward biased is because the discrete hazard model does not allow for short spells. For example, if one period is a quarter, estimates of 13 We continue to use the sample average duration for the full-information estimator. This is no longer the MLE as it does not take into account duration dependence. Nonetheless, it is unbiased and the large samples imply that the standard errors are typically much less than one-tenth of 1% of the mean. 14 For the cases in which the hazards are not constant, the biases are always statistically significant. For example, in Simulation 7, Estimators 2 and 3 have biases of less than one-half of 1% in both states. A test for unbiasedness was performed as follows: for each realization, the realization of the unbiased estimator is subtracted from the realization of the corrected estimator. Under the null, this random variable has mean zero. The t statistics (with 999 degrees of freedom) were all bigger than eight in absolute value. Thus, even a bias that is fairly trivial in magnitude is highly statistically significant with these large sample sizes.

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Fig. 12. Hazard functions for Simulation 12.

expected durations in employment and unemployment will force a minimum duration of one quarter in either state. One possible remedy is to estimate the parameters of a continuous-time hazard model and then calculate the expected duration using the estimated continuous-time distribution. If this is done, even though no spells shorter than the period of observation appear in the data, short spells are allowed for at the stage of calculating expected durations given the parameter estimates. To investigate this possibility, suppose we know the hazard in Simulation 2 has the functional form given in Eq. (14). Further, suppose we know b = 0 and h2 = 1 so that only two parameters, a and c, need to be estimated (with b = 0, h1 is irrelevant). Note that the integrated hazard is always an exponential random variable with mean and variance both equal to one. A robust estimator is the Method of Moments Estimator (MME). In particular, one could set the sample mean and variance of the implied integrated hazard equal to one and solve these two equations for the two unknown parameters. Given estimates of the continuous-time hazard’s parameters, one can then estimate the expected duration given the estimated density function. Table 2 Decreasing hazards Estimator

State 1 (a = 0.10, b = 0, c = 0.10, h1 = 0, h2 = 1)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.05, b = 0, c = 0.05, h1 = 0, h2 = 1)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

8.22 9.37 8.53 8.39 8.39

0.118 0.135 0.135 0.139 0.142

NA 13.95 3.76 2.05 2.06

17.56 20.01 18.10 17.92 17.92

0.246 0.297 0.281 0.282 0.284

NA 13.92 3.07 2.02 2.03

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

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Table 3 Decreasing (at a slower rate) hazards Estimator

State 1 (a = 0.10, b = 0, c = 0.10, h1 = 0, h2 = 0.2)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.05, b = 0, c = 0.05, h1 = 0, h2 = 0.2)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

6.56 7.56 6.72 6.61 6.61

0.102 0.116 0.115 0.113 0.113

NA 15.27 2.53 0.78 0.80

14.56 16.78 14.87 14.67 14.68

0.254 0.280 0.264 0.268 0.271

NA 15.25 2.11 0.76 0.78

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

As a base case, the MME was applied to Simulation 2 using the actual durations. First, the spells were simulated. Next, the simulated spells were used to estimate the parameters. Finally, the expected durations were obtained by simulating 5000 durations using the estimated parameters as if they were the true parameters. This was repeated 100 times to obtain estimates of the mean and standard deviation of the estimated expected duration for each state. The results were an estimated expected duration of 8.23 in State 1 (standard deviation of 0.236) and 17.55 in State 2 (standard deviation of 0.493). The differences between these and the unbiased estimates in Table 2 are well within normal sampling error. Next, the parameters were estimated using the limited information given by the states occupied at discrete points in time. In particular, the procedure just described was repeated with the observed discrete spells replacing the actual durations in calculating the MME of the parameters and the implied expected durations. This yielded estimated

Table 4 Nonmonotonic hazards Estimator

State 1 (a = 0.1, b = 0.3, c = 0.4, h1 = 0.5, h2 = 0.2)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.05, b = 0.09, c = 0.05, h1 = 0.3, h2 = 0.1)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

4.72 5.29 4.71 4.67 4.66

0.072 0.080 0.080 0.078 0.088

NA 12.14  0.22  1.16  1.27

20.58 23.07 20.08 20.33 20.31

0.289 0.321 0.298 0.296 0.331

NA 12.12  2.39  1.18  1.29

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

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Table 5 Nonmonotonic hazards Estimator

State 1 (a = 0.1, b = 0.3, c = 0.4, h1 = 0.2, h2 = 0.5)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.05, b = 0.09, c = 0.05, h1 = 0.1, h2 = 0.3)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

12.52 13.64 13.09 12.81 12.82

0.174 0.185 0.186 0.191 0.194

NA 8.96 4.56 2.30 2.41

37.89 41.28 38.38 38.76 38.80

0.413 0.461 0.462 0.464 0.483

NA 8.92 1.27 2.27 2.38

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

expected durations of 9.38 for State 1 (standard deviation of 0.253) and 20.00 for State 2 (standard deviation of 0.554). These are essentially the same as the uncorrected estimators in Table 2.15 Thus, estimating the parameters of the continuous hazard by itself does not correct the problem, even when one has the advantage of knowing the functional form of the continuous-time hazard. The preceding discussion indicates that it is not that we fail to observe spells shorter than one period that is primarily responsible for the bias. Instead, the bias appears to be caused mainly by the fact that many of the longer discrete spells we observe actually contain more than one continuous-time spell. In all of Simulations 2 through 9, Estimators 2 and 3 perform fairly well. While the uncorrected estimator’s bias ranges from about 9% to 15%, Estimators 2 and 3 are never biased more than 2.5% and are frequently close to 1% or even less. The results for Estimator 1 are more mixed. It does worse than the other two in Simulations 2, 3 and 8. In Simulations 5, 6 and 9, it does better than the others in one state, but worse in the other. In one case (State 1 of Simulation 5), its bias is as high as 4.56% (compared to an 8.96% bias for the uncorrected estimator). In some simulations its bias differs considerably between the two states, indicating that it will alter estimates of the steady-state rate of being in each state. Simulation 10 investigates how increasing the time between observations affects the estimators. In particular, Simulation 10 is the same as Simulation 3, except the time between observations is three times as long. So, for example, if Simulation 3 were monthly data, Simulation 10 gives the results for the same hazards using quarterly data. Because the 15 In the data, we are still forcing a minimum of one period for the discrete spells in estimating the parameters. Two alternatives were explored: (i) subtracting .99 from each discrete spell and (ii) subtracting a uniform (on (0,1)) random variable from each discrete spell. The estimated expected durations for state 1 were 9.32 for (i) and 9.36 for (ii). For state 2, the estimates were 19.93 for (i) and 19.97 for (ii) (all standard deviations were essentially the same). In each case, the estimates are still essentially the same as the uncorrected estimators in Table 2.

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Table 6 Nonmonotonic hazards Estimator

State 1 (a = 0.1, b = 0.3, c = 0.4, h1 = 0.2, h2 = 0.5)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.05, b = 0.09, c = 0.09, h1 = 0.3, h2 = 0.1)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

12.52 13.95 13.04 12.75 12.75

0.174 0.185 0.180 0.185 0.185

NA 11.43 4.17 1.88 1.86

14.72 16.40 14.95 14.99 14.99

0.234 0.260 0.254 0.258 0.259

NA 11.41 1.58 1.87 1.84

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

time between observations is longer, the hazard changes more within periods. For example, for Simulation 3, both hazards fall by approximately 8.33% from the time a spell starts until it has lasted one period of time. For Simulation 10, with one period being three times as long, both hazards fall by 18.75% from the start of the spell until it is one period long. The results in Table 10 reveal that all three corrected estimators do not do as well in this case. The biases range from 4.77% to 7.95%, compared to 0.66% and 1.88% for Simulation 3. However, the improvement over the uncorrected estimator’s bias of about 47% remains substantial. The remaining simulations were designed to try and make the corrections look bad. Simulation 11 has extreme movements in the hazard rate. The hazard is constant at 0.05 in State 2. Initially, the hazard is very large in State 1. At t = 0, the hazard equals 10.2.

Table 7 One increasing hazard, one decreasing hazard Estimator

State 1 (a = 0.2, b = 0.1, c = 0, h1 = 0.2, h2 = 0)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.15, b = 0, c = 0.09, h1 = 0, h2 = 0.5)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

6.91 8.13 6.81 6.93 6.93

0.077 0.100 0.098 0.097 0.097

NA 17.70  1.28 0.36 0.28

5.39 6.33 5.49 5.40 5.40

0.074 0.084 0.082 0.084 0.085

NA 17.64 1.86 0.31 0.23

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

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Table 8 Increasing hazards Estimator

State 1 (a = 0.2, b = 0.1, c = 0, h1 = 0.2, h2 = 0)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.15, b = 0.09, c = 0, h1 = 0.3, h2 = 0)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

6.91 7.52 6.71 6.82 6.82

0.077 0.087 0.085 0.089 0.090

NA 8.96  2.71  1.16  1.30

8.92 9.72 8.66 8.81 8.80

0.098 0.111 0.109 0.112 0.113

NA 8.92  2.91  1.19  1.33

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

However, this hazard falls very rapidly and asymptotes toward 0.2. Indeed, by the time t = 0.1, the hazard has fallen all the way from 10.2 to 0.45. By t = 1, the hazard has fallen to 0.225 (this hazard uses a = 0.2, c = 10, h2 = 390 and all other parameters zero in Eq. (11)). For this duration distribution, there will be a fairly large number of very short spells. Even though the hazard falls so rapidly, it is initially so large that more than 10% of all spells end before t = 0.1. The rate at which spells end after this is much lower. Obviously, the assumption that the hazards are constant within periods is grossly violated early in spells in State 1. As Table 11 indicates, all three corrected estimators do much less well in this case, with biases in the 11 –13% range. This still represents a substantial improvement over the uncorrected estimator’s bias of about 27.9%. The assumption that the hazard asymptotes to a constant is correct in each of the above simulations. To investigate the sensitivity of the estimators to violations of this assump-

Table 9 Nonmonotonic hazards Estimator

State 1 (a = 0.2, b = 0.4, c = 0.4, h1 = 0.4, h2 = 0.1)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.3, b = 0.3, c = 0.2, h1 = 0.1, h2 = 0.5)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

3.37 4.11 3.27 3.33 3.32

0.040 0.052 0.051 0.052 0.052

NA 22.13  2.74  1.08  1.44

5.94 7.26 5.87 5.88 5.86

0.070 0.091 0.085 0.088 0.090

NA 22.12  1.31  1.09  1.45

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

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Table 10 Decreasing hazards Estimator

State 1 (a = 0.3, b = 0, c = 0.3, h1 = 0, h2 = 0.6)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.15, b = 0, c = 0.15, h1 = 0, h2 = 0.6)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

2.19 3.22 2.36 2.30 2.30

0.033 0.051 0.050 0.053 0.054

NA 47.04 7.95 4.81 4.83

4.84 7.12 5.20 5.07 5.07

0.069 0.105 0.097 0.102 0.108

NA 46.98 7.48 4.77 4.79

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

tion, a simulation was run with a hazard of 0.1 + 0.05 cos(2t) in State 1. The hazard in State 2 is constant and equal to 0.04. The results are in Table 12. All biases are estimated to be between 2% and 3%, compared to over 9% for the uncorrected estimator. Moreover, it appears that rapid change in the hazard in the first period causes the bias, rather than the fact that the hazard never approaches a constant in State 1. If the above simulation is repeated, but with a hazard of 0.1 + 0.05 cos (0.3t), so that the hazard still oscillates, but with a much longer period, the biases of the corrected estimates are all considerably less than one-tenth of 1%. It is possible to come up with even more extreme examples in which the estimators fail altogether. For example, suppose that both states have the same duration distribution and that the hazard is 0.02 for 0 < t < 0.95. For 0.95 < t < 1.05 the hazard is  10 ln (0.3) (approximately 12). For t > 1.05, the hazard is 0.05. One million durations for each state

Table 11 One constant hazard, one rapidly decreasing hazard Estimator

State 1 (a = 0.2, b = 0, c = 10, h1 = 0, h2 = 390)a c

Full Information Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 2 (a = 0.05, b = 0, c = 0, h1 = 0, h2 = 0)

Mean

S.D.

% Biasb

Mean

S.D.

% Bias

4.19 5.35 4.72 4.67 4.67

0.066 0.079 0.077 0.077 0.082

NA 27.91 12.85 11.60 11.57

19.95 25.52 22.27 22.26 22.26

0.277 0.408 0.378 0.386 0.414

NA 27.89 11.60 11.58 11.55

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a See Eq. (14) in the text. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

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Table 12 One constant hazard, one oscillating hazard Estimator Full Informationc Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 1a

State 2 b

Mean

S.D.

% Bias

Mean

S.D.

% Bias

9.98 10.90 10.18 10.25 10.25

0.131 0.138 0.138 0.138 0.140

NA 9.18 2.03 2.67 2.65

24.97 27.25 25.62 25.62 25.62

0.327 0.363 0.352 0.356 0.359

NA 9.13 2.65 2.62 2.61

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a The hazard in State 1 is 0.1 + 0.05 cos(2t). The hazard in State 2 is 0.04. b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

were calculated assuming both states have the same distribution and the transition probabilities for the first discrete period were estimated to be approximately 0.69 in both states. Thus, the first-period transition probabilities sum to a number greater than one. Since the log of a negative number does not exist, none of the corrected estimators exist. This distribution was rigged to get this result. The value of the hazard between 0.95 and 1.05, though cryptic-looking, was chosen to make it such that, among all spells lasting at least 0.95 units of time, 70% will end by at least a duration of 1.05. Thus, if the first observation of a state is at time n, as long as the duration of the spell in progress is at least 0.05 but not more than 0.95 at time n, it has more than a 70% chance of ending by time n + 1. Moreover, if the spell ends any time after n + 0.05, the hazard will remain low (0.02) until n + 1 in the other state so the probability of a transition back is actually quite low, despite the high transition probabilities. Because of this, the bias of the uncorrected estimator is actually low (the estimate of the full-information estimator indicated a mean duration of about 6.9 while the mean discrete spell length was 7.2). Thus, while Estimators 2 and 3 seem to do well with most time-changing hazards, empirical researchers are encouraged to think hard about whether extreme changes in the hazard such as this can be ruled out. Also, large values of the sum of the transition probabilities (close to or greater than one) might indicate that any estimates of the expected duration could be unreliable. In this case, the best remedy, if feasible, is to increase the period of observation. If, for this same event history, we make observations five times as often (so that now the jump occurs between time 4.75 and time 5.25 with the hazards reduced by a factor of five) the transition probabilities reach a maximum of about 0.36 in discrete period 4. Now all the corrections exist and the results are reported in Table 13. In this case, Estimator 1 does fairly poorly, with biases roughly four times that of the uncorrected estimator in absolute value. However, Estimators 2 and 3 do well, with all biases estimated to be considerably less than one-tenth of 1%. If one can rule out such large changes in the hazard rates within periods, it would seem that Estimators 2 and 3 do somewhat better than Estimator 1. Moreover, the results of Simulations 10 and 13 suggest that one only needs to rule out very sharp movements in the

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Table 13 One constant hazard, one discontinuous hazard Estimator Full Informationc Uncorrectedd Estimator 1 Estimator 2 Estimator 3

State 1a

State 2 b

Mean

S.D.

% Bias

Mean

S.D.

% Bias

34.30 34.44 33.82 34.30 34.29

0.970 0.974 0.988 0.975 0.976

NA 0.41  1.40 0.01  0.04

34.30 34.43 33.82 34.29 34.27

0.943 0.941 0.959 0.940 0.940

NA 0.38  1.40  0.02  0.06

Simulations use 5000 actual durations for each realization of the estimators. Results based on 1000 simulations. a For both states, the hazard is 0.004 for t < 4.75 and 0.01 for t > 5.25. For 4.75 < t < 5.25 the hazard is  2ln(0.3) (approximately 2.4). b % Bias is the percentage difference between the average of the estimator and the average of the unbiased, full-information estimator. c Average of actual durations. d Average number of consecutive periods observed in the state (uncorrected, limited-information estimator).

hazard within the first few periods of a discrete spell.16 In Simulation 13, there is still an extremely large jump in the hazard rate, but it takes place after the first period and both Estimators 2 and 3 are essentially unbiased. As Proposition 2 in Appendix A shows, the asymptotic bias depends only on the ratio of the number of observed spells to the number of actual spells. Thus, to empirically identify the bias, we need only identify the expected number of missed transitions. As multiple transitions imply new spells, what is most important is identifying the hazard rate within the first period of a spell. There does not seem to be a clear winner between Estimators 2 and 3. Estimator 2 might be preferred since it is much simpler. However, there are some situations where Estimator 2 cannot be applied. For example, suppose one has time-changing regressors in a hazard model. If, in calculating the expected duration, these are set to their average (or steady state) values, there is no problem. However, if they are allowed to change, Estimator 2 cannot deal with this, while Estimator 3 can, as it does not use only the first period transition probabilities to make the correction. Of course, normally one would want all variables at their steady state to calculate an expected duration. Still, some variables might change at a steady state. For example, seasonal dummies change within the steady state when it is given an annual interpretation. Age is another variable that one would normally have changing in calculating expected durations. Estimator 1 can also handle variables like seasonal dummies. While Estimator 3 appears to be the superior estimator, it is far more demanding computationally. In running the simulations described above, a single program simulated all of the data and calculated all five of the estimators. If this program is changed so that it does everything except calculate Estimator 3, the program runs in about one-third the time. In other words, in the simulations about two-thirds of the computer time is spent calculating Estimator 3.

16 Estimator 2, for example, uses only data from the first-period discrete transition probabilities, and thus is unaffected by duration dependence later in spells.

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Nonetheless, given the speed of computers these days it does not seem this should be a problem. In fact, all of the simulations reported above can be run in about ten hours on a notebook computer with a 333-mHz processor. Estimator 1 remains of interest even when Estimator 3 is feasible because it can be easily used when there are censored spells in the data. Nonetheless, Estimator 3 may still be useful, with suitable modification, when there are censored spells. This will be discussed further in Section 4. In an earlier version of this paper, the above simulations were run making no distinction between the observed transition probabilities and the transition probability conditional on how long the true continuous duration is (the l and g of Section 2.2). The results for all the estimators were at least as good as those reported here, so it appears that nothing is lost — and some simplicity gained — by just using the observed transition probabilities. As a final note on the simulations, the above models were also simulated for ‘‘small’’ and ‘‘medium’’ sample sizes. More precisely, all the above models were run with only 150 durations for each state (the small sample size) and 800 durations for each state (the medium sample size). The results were virtually the same as for the larger sample case, but with larger standard errors.

4. Empirical application In this section, the corrections proposed in this paper are used to estimate expected durations in employment and unemployment using data from the classroom training component of the National JTPA Study, an experimental evaluation of the programs financed under the Job Training Partnership Act. For this application, unemployment includes being out of work and actively looking for work (the usual definition) as well as being out of the labor force. The individuals in the sample are all adult women who were disadvantaged workers eligible for JTPA classroom training (JTPA-CT) in the late 1980s. In the experiment, some of these women were randomly assigned to the treatment group, which received access to JTPA-CT. The rest were assigned to the control group, which did not receive access to JTPA training. For a detailed description of the experiment and JTPA, see Orr et al. (1996) and Bloom et al. (1997). The estimates here are for expository purposes; for a detailed evaluation of the effects of being offered or receiving JTPA-CT, see Eberwein et al. (1997). The JTPA-CT data are monthly. To evaluate the corrections, we constructed quarterly data (i.e., data in which we only observe the state occupied every 3 months). Discrete hazard models were estimated for employment and unemployment spells using a simple logit model.17 These hazards were then used to estimate expected durations. Table 14 below gives the resulting estimates of the expected durations — corrected and uncorrected — for both quarterly and monthly data. Table 15 gives the percentage difference

17 Only spells beginning after the start of the sample (fresh spells) were used. Because a woman’s past employment history affects her eligibility for JTPA-CT, spells in progress at the beginning of the sample are not comparable to fresh spells.

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Table 14 Estimated employment and unemployment expected durations for disadvantaged women (JTPA classroom training sample, adult women) Expected duration estimates Controls

Treatments

Overall

(i) Quarterly data (uncorrected) Unemployment Employment

22.62 31.09

17.11 32.64

18.86 32.15

(ii) Quarterly data (corrected) Unemployment Employment

19.08 25.64

14.05 26.14

15.65 25.98

(iii) Monthly data (uncorrected) Unemployment Employment

19.34 26.66

15.93 29.08

17.01 28.31

(iv) Monthly data (corrected) Unemployment Employment

18.48 24.77

15.15 26.84

16.21 26.18

All estimated expected durations are in months. Estimated expected durations are based on logit discrete hazards estimated using the sample of 1940 women in the JTPA classroom training data. Corrected estimates use Estimator 1 to calculate expected durations from the discrete transition probabilities. Treatments (1324 women) are those randomly selected to receive the offer of access to JTPA classroom training while controls (616 women) are those who were not offered training. The overall column refers to the average for the full sample of both treatments and controls.

between the estimates using quarterly and monthly data when (1) none of the estimates are corrected; (2) the quarterly estimates are corrected, but the monthly estimates are not; and (3) all estimates are corrected. Tables 17 and 18 in Appendix B give the estimated logit Table 15 Percentage difference between expected duration estimates using quarterly or monthly data for the disadvantaged women in the JTPA-CT sample % Difference between quarterly and monthly estimates Controls (1) Quarterly data uncorrected, monthly data uncorrected Unemployment 16.9 Employment 16.6 (2) Quarterly data corrected, monthly data uncorrected Unemployment  1.4 Employment  4.8 (3) Quarterly data corrected, monthly data corrected Unemployment 3.2 Employment 3.5

Treatments

Overall

7.4 12.2

10.9 13.6

 11.8  10.1

 8.0  8.2

 7.3  2.6

 3.5  1.8

No. (1) compares (i) and (iii) from Table 14. No. (2) compares (ii) and (iii) from Table 14. No. (3) compares (ii) and (iv) from Table 14.

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parameters for the monthly and quarterly data, respectively. To model duration dependence, a polynomial in (t  1)/t, where t is the duration in the spell, was used.18 The correction used in Table 14 is Estimator 1. We obtain the expected durations reported in Table 14 by first estimating an expected duration for each individual and then averaging the estimates over all the individuals in the sample. As Table 15, part (1) shows, there is a considerable difference between the uncorrected estimates from monthly and quarterly data. For the overall sample, the quarterly estimates of the expected durations are 10.9% larger for unemployment and 13.6% larger for employment. If only the quarterly estimates are corrected (Table 15, part 2), the quarterly estimates become smaller by 8.0% and 8.2% for unemployment and employment, respectively. Part 3 of Table 15 gives the percentage difference when both the monthly and quarterly estimates are corrected. In this case, the quarterly estimates are smaller than the monthly estimates by 3.5% and 1.8% for unemployment and employment, respectively. Thus, the estimates are three to six times closer when the correction is applied to both, indicating that the correction can improve the comparability of studies using different levels of time aggregation when the correction is applied to each level of aggregation. Estimator 2 does not apply in this case because there are explanatory variables (age and the local unemployment rate) that change over time. Nonetheless, it is of interest to see how applying this correction using the first period transition probabilities for an individual with characteristics evaluated at the means affects the estimates. Application of Estimator 2 implies that the upward bias is 7.8% for the monthly data and 28.6% for the quarterly data. Correcting the expected durations yields estimated expected durations in unemployment of 14.67 months for the quarterly data and 15.78 months for the monthly data, for a difference of  7.0%. For employment, the estimated expected durations would be 25.00 and 26.26 for quarterly and monthly data, a difference of  4.8%. While this is an improvement over the uncorrected estimates, it is less than the improvement obtained with Estimator 1. The likely reason is that this correction ignores both heterogeneity across individuals and the time-changing nature of some of the variables. There is also a problem with applying Estimator 3. As is typical in data sets, many of the spells observed in the JTPA-CT data are right-censored. Recall that Estimator 3 relies on estimating the number of continuous spells that were missed in the data to estimate the bias. This is done by estimating the probability a spell was missed for each data point and then summing all these probabilities. The problem is, with censored data we are not observing all the periods in some discrete spells, so we underestimate the number of missed spells relative to observed spells. Three methods were attempted in applying Estimator 3 in the presence of rightcensored spells. Table 16 presents the results from all three methods. In (i), censoring

18 To avoid problems associated with extrapolation out of sample — such as those described in Eberwein et al. (2002) — the duration dependence terms were frozen at their values at 24 months (eight quarters) for unemployment spells and 30 months (10 quarters) for employment spells. These were chosen so that approximately 5% of the observed spells last this long. For simplicity, no unobserved heterogeneity was estimated.

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Table 16 Estimated expected durations using Estimator 3 Data set

Estimated expected duration Quarterly

Percent difference Monthly

(i) No extrapolation out of sample Unemployment 16.22 Employment 27.64

16.03 26.68

1.2 3.6

(i) 24-month extrapolation out of sample Unemployment 15.08 Employment 25.71

15.45 25.71

 2.4 0.0

(i) 60-month extrapolation out of sample Unemployment 14.28 Employment 24.35

15.10 25.13

 5.4  3.1

is ignored and the estimates are what they would be if all the observed spells were completed. As discussed above, this likely underestimates the bias, but it does so for both levels of time aggregation, and so could yield comparable estimates. In (ii) and (iii), simulated ‘‘data’’ are used. In particular, when a spell is censored, the computer simulates additional periods for that spell using a random number and the estimated transition probability to determine whether the spell ends. Additional periods are added to the censored spell until it either ends or reaches a fixed time limit. For (ii), the time limit is 24 months (eight quarters), while in (iii), it is 60 months (20 quarters). The results in (i) indicate that Estimator 3 is about as successful as Estimator 1 in making the estimates from quarterly and monthly data comparable. The percentage difference is only 1.2% for unemployment and 3.6% for employment. The results in (ii) are somewhat better, with  2.4% for unemployment and 0.0% for employment (the difference was in the fourth decimal place). Simulating further out of sample did not help. In (iii), the differences of  5.4% and  3.1% are worse than the estimates obtained by ignoring censoring. Theoretically, simulating further out of sample should improve the estimates. The likely reason this did not work with the JTPA-CT data is that there are no observations as long as 60 months in the data — the maximum is 30 months — so that (iii) uses estimates of the transition probabilities that are identified only by extrapolation, while the 24-month simulations in (ii) do not. In fact, 24 months was chosen so that a reasonable amount of data (about 5% of the spells) was available to estimate the hazard rates being used. Thus, using simulated data when there are censored spells to calculate Estimator 3 seems to work well as long as one does not extrapolate to durations not observed in the data. It is somewhat surprising that the estimators performed as well as they did for these data. Inspection of the estimated monthly discrete transition probabilities for several individuals in the data revealed that the discrete hazard falls gradually for unemployment spells. For employment spells, however, the discrete hazard rises rapidly in the first few periods, with the discrete hazard in the third month roughly twice the discrete hazard in

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the first month. Thus, the assumption that the hazard is constant within periods is seriously violated for the quarterly data. Despite this, Estimators 1 and 3 do a reasonable job of correcting the quarterly estimates so as to make them comparable to the monthly estimates.

5. Conclusions This paper has proposed some corrections for the bias in estimating expected durations that results from using discrete hazard models. An advantage of these estimators is that none of them make any parametric assumptions about the way in which the hazards are changing across periods and each depends only on the discrete transition probabilities, which are empirically identified. The corrections were evaluated using Monte Carlo simulations and an empirical application. The results indicate that two of the estimators perform fairly well so long as the hazard rates do not change by an extreme amount between observations of the state. Even when the hazards do change quite a bit between observations, the corrections can still give a substantial improvement over uncorrected estimates, though it is possible to come up with extreme examples for which the estimators fail to exist. As long as researchers can rule out extreme changes in the within-period hazards, Estimators 2 and 3 appear to give a substantial improvement over uncorrected estimates without having to make any parametric assumptions about the functional form of the continuous-time hazard. The results also indicate that only large changes in the hazard rate within the first few periods of a new spell can cause a nontrivial bias in Estimators 2 and 3. Moreover, the examples in which the estimators did poorly or failed to exist involved such extreme changes in the hazard rates that it is hard to imagine that many conclusions could be drawn about the underlying duration distribution without increasing the frequency of observations. The example in which the corrections do not exist involves the hazard rates in both states jumping at a single point in time from only 0.02 to approximately 12, then remaining at this value for only one tenth of a period before dropping to 0.05. Even with this extreme change in the hazard rate, if observations are made five times as often, so that the jump in the hazard is one-half the length of a period, Estimators 2 and 3 do very well. In an empirical application using data from the National JTPA Study, we show that applying the corrections developed here has a nontrivial impact on estimated expected durations. We also show how application of these corrections helps reconcile estimates obtained using data with differing levels of time aggregation.

Acknowledgements The author thanks John Ham, Robert LaLonde, Jeffrey Smith, John Galbraith, Ngo Van Long, the seminar participants at Ohio State University and an anonymous referee for useful comments on earlier versions of this paper. Any remaining errors are the responsibility of the author.

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Appendix A A.1. Proofs Proof of Proposition 1. Define P(i j t) to be the probability that state i is occupied at time t. Lancaster (1990, Theorem 2, p. 97) shows that regardless of the initial state, lim Pði j tÞ ¼

t!1

Di uQi : D1 þ D2

This is the long-run state occupancy probability. That is, it is the long-run fraction of time expected to be spent in state i. Suppose we observe the state for some arbitrarily large number of discrete periods, T. Define the following random variables: Ni(T ) is the number of discrete periods observed in state i and Mi(T ) is the number of discrete spells in state i. An asymptotically unbiased estimator of Q1 is the fraction of periods spent in State 1, given by q1 ðT Þ ¼

N1 ðT Þ N1 ðT Þ : ¼ N1 ðT Þ þ N2 ðT Þ T

To see this, define the function f (t) = P(1j t)  Q1. By construction, f (t) tends to zero as t tends to infinity. For any T, let t*(T ) be the largest integer that is less than or equal to the square root of T. Define eðT Þ ¼ sup j f ðtÞ j , t>t *ðTÞ

and note that, by definition, e(T) tends to zero as T (and, therefore, t*(T)) tends to infinity. Then

E½q1 ðT Þ ¼

t *ðT Þ T T 1 1X 1X 1 X E½N1 ðT Þ ¼ ½Q1 þ f ðtÞ ¼ f ðtÞ þ Q1 þ f ðtÞ: T T t¼1 T t¼1 T t¼t*ðT Þþ1

By construction, jf (t)j < 1 for all t so that   * ðT Þ  t *ðT Þ  1 tX 1   V pffiffiffiffi , f ðT Þ <    T t¼1 T T which tends to zero as T tends to infinity. Also,       T T  1 X 1 X t *ðT Þ   f ðT Þ < eðT Þ ¼ 1  eðT Þ, T T  T t¼t*ðT Þþ1  t¼t*ðT Þþ1

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which also tends to zero as T tends to infinity because e(T ) and t*(T )/T both tend to zero as T tends to infinity. Thus,

lim E½q1 ðT Þ ¼ Q1 ¼

T !1

D1 : D1 þ D2

Note that: q1 ðT Þ ¼

N1 ðT Þ=M1 ðT Þ : ½N1 ðT Þ=M1 ðT Þ þ ½N2 ðT Þ=M2 ðT Þ½M2 ðT Þ=M1 ðT Þ

As the sample average discrete spell is a consistent estimator of the expected discrete spell length, we have that as T tends to infinity, with probability one: Ni ðT Þ ! Di*: Mi ðT Þ Moreover, because neither duration distribution is defective, as T tends to infinity the number of discrete spells in each state must tend to infinity with probability one. Since this is an alternating renewal process, the number of discrete spells in State 1 can differ from the number of discrete spells in State 2 by at most one. Thus, M2 ðT Þ !1 M1 ðT Þ with probability one. Using these results, it is straightforward to show by continuity that q1 ðT Þ !

D1* D1 þ D2* *

with probability one. Note that, because q1(T ) is always in [0,1] for all T its asymptotic expectation must tend to the same limit (i.e., it cannot have any extreme values tending to infinity that occur with a probability that tends to zero). Thus, D1* D1 ¼ : D1 þ D2* D1 þ D2 *

Symmetric reasoning establishes the analogous result for State 2. Thus, Di* D1* þ D2* ¼ Di D1 þ D2 for i = 1, 2 . Since the right-hand side does not depend on i, this establishes the result.5

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Proposition 2. Maintain the assumptions and definitions of Proposition 1 and define Ri(T) to be the total time spent in state i between 0 and T and Ki(T) to be the actual number of continuous-time spells in state i between time 0 and time T. Then Mi ðT Þ Ni ðT Þ Ki ðT Þ Mi ðT Þ is a consistent estimator of Di. Proof of Proposition 2. By construction, Ri(T )/T is the sample fraction of time spent in state i and, therefore, must converge to Qi with probability one as T tends to infinity. Define o(T ) implicitly as Ri ðT Þ Ni ðT Þ ¼ þ oðT Þ, T T and note that the proof of Proposition 1 implies that o(T ) tends to zero with probability one as T tends to infinity. Rearranging, we have Ri ðT Þ Mi ðT Þ Ni ðT Þ R1 ðT Þ þ R2 ðT Þ ¼ þ oðT Þ, Ki ðT Þ Ki ðT Þ Mi ðT Þ Ki ðT Þ where we use the fact that T = R1(T ) + R2(T ) by definition. As censoring becomes unimportant as T tends to infinity, the term on the left-hand side is a consistent estimator of Di. Since o(T ) tends to zero with probability one, to complete the proof it suffices to show the term multiplying o(T ) is bounded above with probability one. But this term is the time spent in both states divided by the number of times the process has renewed in state i, so it is obvious this must tend toward the sum of expected durations, D1 + D2, which is finite by assumption. 5 A.2. Transition probabilities Next, we give the probabilities of exactly k transitions occurring in one period assuming the current state is State 1. The relevant probabilities for State 2 are symmetric. Here, k1(n) is the hazard rate for State 1 assuming State 1 has been occupied in the previous n discrete observations. k1 and k2 are the hazards in States 1 and 2 when the discrete spell is new. Given this notation, the probability of zero transitions is P0 ¼ 1 

Z

1

k1ðnÞ ek1 ðnÞt dt ¼ ek1 ðnÞ :

0

The probability of exactly one transition is:

P1 ¼

Z 0

1

k1 ðnÞek1 ðnÞt ek2 ð1tÞ dt ¼

k1 ðnÞ ek2 ð1  eðk1 ðnÞk2 Þ Þ: k1 ðnÞ  k2

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This has the indeterminant form 0/0 if the hazards are equal. Evaluating the limit gives P1 ¼ kek if k1(n) = k2 = k. The probability of exactly 2 transitions is given by P2 ¼

Z

1

k1 ðnÞek1 ðnÞt

0

k2 ek1 ð1tÞ ð1  eðk2 k1 Þð1tÞ Þdt: k2  k1

Integrating yields:   ðk1 ðnÞk2 Þ k1 ðnÞk2 k1 1  eðk1 ðnÞk1 Þ ðk2 k1 Þ 1  e e e P2 ¼ : k2  k1 k1 ðnÞ  k1 k1 ðnÞ  k2 This has the form 0/0 if any two hazards are equal. Evaluating these limits gives P2 ¼

k1 k2 ðk1  k2 Þ2

ek1 ½ek1 k2  1  ðk1  k2 Þ

if k1(n) = k1, P2 ¼

  k22 1  eðk2 k1 Þ ek1  ek1 k2 k2  k1 k2  k1

if k1(n) = k2, P2 ¼

k1 ðnÞk2 ðk1 ðnÞ  k2 Þ2

ek2 ½k1 ðnÞ  k2  1 þ eðk1 ðnÞk2 Þ 

if k1 = k2, and P2 ¼

1 2 k k e 2

if k1(n) = k1 = k2 = k. The probability of exactly three transitions is: Z 1 k1 k2 P3 ¼ k1 ðnÞek1 ðnÞt ek2 ð1tÞ ½eðk1 k2 Þð1tÞ  1 þ ðk1  k2 Þð1  tÞdt: ðk1  k2 Þ2 0 Integrating yields P3 ¼

 1  eðk1 ðnÞk1 Þ k1  k2 ðk1 ðnÞk2 Þ k2 e eðk1 k2 Þ þ e 2 k1 ðnÞ  k1 k1 ðnÞ  k2 ðk1  k2 Þ k1 ðnÞk1 k2

) # 1  ðk1  k2 Þ k1  k2 ðk1 ðnÞk2 Þ  þ  ½1  e k1 ðnÞ  k2 ðk1 ðnÞ  k2 Þ2 "

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Again, this has the form 0/0 if any two hazards are equal. Taking the limits gives   k21 k2 2  ðk1  k2 Þ k2 ðk1 k2 Þ ðk1 k2 Þ P3 ¼ e 2e  ½1  e  k1  k2 ðk1  k2 Þ2 if k1(n) = k1, P3 ¼

k1 k22 ðk1  k2 Þ

ek2 2



1  eðk1 k2 Þ 1  1  ðk1  k2 Þ 2 k1  k2



if k1(n) = k2, P3 ¼

k1 ðnÞk22 ðk1 ðnÞ  k2 Þ3

e

k2



1 ðk1 ðnÞ  k2 Þ2  ðk1 ðnÞ  k2 Þ þ 1  eðk1 ðnÞk2 Þ 2



if k1 = k2, and P3 ¼

1 3 k ke 6

if k1(n) = k1 = k2 = k.

Appendix B. Parameter estimates of hazards for the JTPA-CT data Please see Tables 17 and 18.

Table 17 Parameter estimates for the discrete hazards out of employment and unemployment, adult women JTPA-CT (monthly data) Variable

Coefficient

Employment hazard (logit coefficients) Constant  2.117 Experimental status  0.050 Highest grade completed  0.077 No high school degree 0.165 Kids under four 0.052 Single 0.057 Married  0.002 Black 0.160 Hispanic  0.014 Age  0.010 Age2/100  0.008 Local unemployment rate  0.021 (t  1)/t  2.900 [(t  1)/t]2 17.038

S.E.

t Statistic

Mean

0.678 0.069 0.025 0.086 0.077 0.084 0.090 0.077 0.103 0.032 0.044 0.018 8.295 35.220

 3.122  0.729  3.046 1.929 0.670 0.674  0.017 2.067  0.140  0.310  0.178  1.136  0.350 0.484

1.000 0.701 11.464 0.469 0.314 0.342 0.212 0.325 0.124 32.928 11.509 5.494 0.742 0.626

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Table 17 (continued ) Variable

Coefficient

S.E.

t Statistic

Mean

Employment hazard (logit coefficients) [(t  1)/t]3  20.217 [(t  1)/t]4 5.759

48.640 21.920

 0.416 0.263

0.538 0.470

Unemployment hazard (logit coefficients) Constant  0.898 Experimental status 0.163 Highest grade completed 0.060 No high school degree  0.018 Kids under four  0.218 Single  0.080 Married 0.104 Black  0.108 Hispanic 0.064 Age  0.077 Age2/100 0.085 Local unemployment rate  0.054 (t  1)/t  5.137 [(t  1)/t]2 21.557 [(t  1)/t]3  29.131 11.627 [(t  1)/t]4

0.835 0.089 0.030 0.106 0.099 0.112 0.110 0.105 0.129 0.040 0.053 0.023 11.250 48.590 68.100 31.100

 1.075 1.830 1.990  0.166  2.188  0.714 0.946  1.022 0.495  1.903 1.583  2.299  0.457 0.444  0.428 0.374

1.000 0.671 11.125 0.549 0.343 0.339 0.208 0.312 0.134 32.346 11.035 5.636 0.670 0.545 0.454 0.386

t denotes the number of periods in the spell. Experimental status, No high school degree, Kids under four, Single, Married, Black and Hispanic are all dummy variables (divorced and white are default groups). Kids under four is equal to one if there are children four years old or younger present in the household. The omitted group consists of control group members with a high school degree and no kids under four who are divorced, widowed or separated and who are white. Estimates obtained from a sample of 1940 women.

Table 18 Parameter estimates for the discrete hazards out of employment and unemployment, adult women JTPA-CT (Quarterly Data) Variable

Coefficient

Employment hazard (logit coefficients) Constant  0.290 Experimental status  0.026 Highest grade completed  0.090 No high school degree 0.196 Kids under four 0.118 Single 0.023 Married 0.039 Black 0.170 Hispanic  0.124 Age  0.019 Age2/100 0.006 Local unemployment rate  0.006 (t  1)/t 4.166 [(t  1)/t]2  17.855 [(t  1)/t]3 24.447 [(t  1)/t]4  11.950

S.E.

t Statistic

Mean

0.770 0.081 0.029 0.099 0.090 0.099 0.104 0.090 0.122 0.037 0.049 0.022 18.260 82.990 122.80 59.390

 0.377  0.322  3.095 1.971 1.322 0.236 0.380 1.887  1.013  0.523 0.130  0.286 0.228  0.215 0.199  0.201

1.000 0.700 11.457 0.471 0.312 0.346 0.210 0.333 0.124 32.764 11.402 5.455 0.496 0.351 0.257 0.193

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Table 18 (continued ) Variable

Coefficient

Unemployment hazard (logit coefficients) Constant 0.936 Experimental status 0.241 Highest grade completed 0.061 No high school degree 0.026 Kids under four  0.298 Single  0.156 Married 0.161 Black 0.057 Hispanic  0.112 Age  0.136 Age2/100 0.163 Local unemployment rate  0.018 (t  1)/t 57.452 [(t  1)/t]2  268.8 [(t  1)/t]3 402.8  196.8 [(t  1)/t]4

S.E.

t Statistic

Mean

1.028 0.113 0.038 0.133 0.124 0.144 0.136 0.133 0.169 0.050 0.066 0.030 33.65 154.6 231.4 113.4

0.910 2.141 1.600 0.199  2.395  1.090 1.185 0.425  0.662  2.734 2.485  0.606 1.707  1.739 1.741  1.736

1.000 0.675 11.147 0.544 0.346 0.326 0.212 0.293 0.133 32.167 10.919 5.607 0.393 0.264 0.184 0.132

t denotes the number of periods in the spell. Experimental status, No high school degree, Kids under four, Single, Married, Black and Hispanic are all dummy variables (divorced and white are default groups). Kids under four is equal to one if there are children four years old or younger present in the household. The omitted group consists of control group members with a high school degree and no kids under four who are divorced, widowed or separated and who are white. Estimates obtained from a sample of 1940 women.

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