Advanced Econometrics Takeshi Amemiya
Nonstationary Models
So far we have assumed X)k(t) = Ajfc for all t (constant hazard rate). Now we shall remove this assumption. Such models are called nonstationary or semi - Markov.
Suppose a typical individual stayed in state yin period (0, t) and then moved to state к in period (t, t + AO - We call this event A and derive its probability P(A), generalizing (11.2.2) and (11.2.3). Defining m = t/At and using log (1 — є) = —є for small e, we obtain for sufficiently large m
(11.2.50)
The likelihood function of this individual is the last expression in (11.2.50) except for t/m.
Let us obtain the likelihood function of the same event history we considered in the discussion preceding (11.2.4). The likelihood function now becomes
 |
(11.2.51)
As in (11.2.4), A31 should be changed to A3 if the individual is only observed to leave state 3, and A31 should be dropped if right-censoring occurs at that time.
We shall concentrate on the transition from state 1 to state 2 and write A,2(t) simply as A(t), as we did earlier. The distribution function of duration under a nonstationary model is given by
(11.2.52)
which is reduced to (11.2.6) if A(r) = A. The density function is given by
(11.2.53)
The likelihood function again can be written in the form of (11.2.10) or
(11.2.11) , depending on whether right-censoring is absent or present.
Thus we see that there is no problem in writing down the likelihood function. The problem, of course, is how to estimate the parameters. Suppose we specify A'(0 generally as
To evaluate foA'(z) dz, which appears in the likelihood function, a researcher must specify x„ precisely as a continuous function of t—not an easy thing to do in practice. We shall discuss several empirical articles and shall see how these articles deal with the problem of nonstationarity, as well as with other problems that may arise in empirical work.
Model of Tuma. Tuma (1976) analyzed the duration of employment at a particular job. The rth individual’s hazard rate at time t is specified as
Xi{t) = P'xi + Oixt + ot2t2, (11.2.55)
where x, is a vector of socioeconomic characteristics of the rth individual.7 The parameter values are assumed to be such that the hazard rate is nonnegative over a relevant range of t. Because of the simple form of the hazard rate, it can easily be integrated to yield
J A‘(z) dz = fi'Xi t + ^-fi + ^-t3. (11.2.56)
Some people terminate their employment during the sample period, but some remain in their jobs at the end of the sample period (right-censoring). Therefore Tuma’s likelihood function is precisely in the form of (11.2.11).
Model of Tuma, Hannan, and Groeneveld. Tuma, Hannan, and Groene - veld (1979) studied the duration of marriage. They handled nonstationarity by dividing the sample period into four subperiods and assuming that the hazard rate remains constant within each subperiod but varies across different subperiods. More specifically, they specified
A'(0 = #x, for / Є Tp, p= 1,2,3,4, (11.2.57)
where Tp is the pth subperiod. This kind of a discrete change in the hazard rate creates no real problem. Suppose that the event history of an individual consists of a single completed spell of duration t and that during this period a constant hazard rate A(l) holds from time 0 to time т and another constant rate A(2) holds from time т to time t. Then this individual’s likelihood function is given by
L = е-*1*е-*2Х‘-Ч(2). (11.2.58)
Model of Lancaster. Lancaster (1979) was concerned with unemployment duration and questioned the assumption of a constant hazard rate. Although a simple search theory may indicate an increasing hazard rate, it is not clear from economic theory alone whether we should expect a constant, decreasing,
or increasing hazard rate for unemployment spells. Lancaster used a Weibull distribution, which leads to a nonconstant hazard rate. The Weibull distribution is defined by
F(t) = 1 - exp (—A/“). (11.2.59)
If a = 1,(11.2.59) is reduced to the exponential distribution considered in the preceding subsections. Its density is given by
fit) = Aat®-1 exp (—Яг®) (11.2.60)
and its hazard rate by
Aft) = { (11.2.61)
Whether a is greater than 1 or not determines whether the hazard rate is increasing or not, as is shown in the following correspondence:
(increasing hazard rate) (constant hazard rate) (decreasing hazard rate).
Lancaster specified the z'th person’s hazard rate as
A'(0 = a/®"1 exp ifi'xt), (11.2.63)
where x, is a vector of the z'th person’s characteristics. His ML estimate of a turned out to be 0.77, a result indicating a decreasing hazard rate. However, Lancaster reported an interesting finding: His estimate of a increases as he included more exogenous variables in the model. This result indicates that the decreasing hazard rate implied by his first estimate was at least partly due to the heterogeneity caused by the initially omitted exogenous variables rather than true duration dependence.
Because it may not be possible to include all the relevant exogenous variables, Lancaster considered an alternative specification for the hazard rate
nt) = vtm, (11.2.64)
where X‘(t) is as given in (11.2.63) and vt is an unobservable random variable independently and identically distributed as Gamma(l, <r2). The random
variable vt may be regarded as a proxy for all the unobservable exogenous variables. By (11.2.52) we have (suppressing i)
F{tv) = 1 - exp [-uA(r)], (11.2.65)
where A(/) = Io^(z) dz. Taking the expectation of (11.2.65) yields
F*(t) ■ EvF(tv) = 1 - [1 + (11.2.66)
Therefore, using (11.2.8), we obtain
A*(/) = A(r)[l + (11.2.67)
= A(1)[1-F*(r.
Because [1 — F*(t)]a2 is a decreasing function of t, (11.2.67) shows that the heterogeneity adds a tendency for a decreasing hazard rate. Under this new model (with a2 as an additional unknown parameter), Lancaster finds the MLE of a to be 0.9. Thus he argues that a decreasing hazard rate in his model is caused more by heterogeneity than true duration dependence.
Model of Heckman and Borjas. In their article, which is also concerned with unemployment duration, Heckman and Borjas (1980) introduced another source of variability of A in addition to the Weibull specification and heterogeneity. In their model, A also varies with spells. Let / denote the /th unemployment spell the /th individual experiences. Then Heckman and Borjas specify the hazard rate as
A<;(/) = a/"-1 exp ()S/X|7 + v^, (11.2.68)
where v, is unobservable and therefore should be integrated out to obtain the marginal distribution function of duration.8
Model of Flinn and Heckman. In a study of unemployment duration, Flinn and Heckman (1982) generalized (11.2.68) further as
A"(/) = exp |#Xtf(0 + cfii + у і 1 ^ 1 + y2 ** • (П -2.69)
The function (tk — 1 )/A is the Box-Cox transformation (see Section 8.1.2) and approaches log t as A approaches 0. Therefore putting A, = 0 and y2 = 0 in (11.2.69) reduces it to a Weibull model. Note that x(/is assumed to depend on t in (11.2.69). Flinn and Heckman assumed that changes in xu(t) occurred only at the beginning of a month and that the levels were constant throughout the month. The authors devised an efficient computation algorithm for handling
the heterogeneity correlated across spells and the exogenous variables varying with time.
