Advanced Econometrics Takeshi Amemiya

Duration Models

11.1.3 Stationary Models—Basic Theory

We shall first explain a continuous-time Markov model as the limit of a discrete-time Markov model where the time distance between two adjacent time periods approaches 0. Paralleling (11.1.2), we define

Aj*(/) At = Prob[ith person is in state к at time t + At (11.2.1)

given that he or she was in state j at time /].

In Sections 11.2.1 through 11.2.4 we shall deal only with stationary (possibly heterogeneous) models so that we have Aj*(f) = Aj* for all t. •

Let us consider a particular individual and omit the subscript і to simplify the notation. Suppose this person stayed in state j in period (0, t) and then moved to state к in period (/, t + At). Then, assuming t/At is an integer, the probability of this event (called A) is

P( A) = (1 - Xj Aty/A, XJk At, (11.2.2)

where kj = (Sj^Ajfc) — Xjj determines the probability of exiting/ But using the well-known identity 1іти_00 (1 — и-1)" = e~ we obtain for small At

P(A) = exp (- Xjt)Ад At. (11.2.3)

Because At does not depend on unknown parameters, we can drop it and regard exp (—Xjt)Xjk as the contribution of this event to the likelihood func­tion. The complete likelihood function of the model is obtained by first taking the product of these terms over all the recorded events of an individual and then over all the individuals in the sample. Because of (11.2.3), a stationary model is also referred to as an exponential model.

Suppose M— 3 and a particular individual’s event history is as follows: This person stays in state 1 in period (0, tx), moves to state 2 at time tx and stays there until time tx + t2, moves to state 3 at time tx + t2 and stays there until tx + t2 + h, at which point this person is observed to move back to state 1. (The observation is terminated after we have seen him or her move to state 1.) Then this person’s likelihood function is given by

L = exp (— Xx tx )A12 exp (—A2f2)A23 exp (— A3/3)Я3j. (11.2.4)

If we change this scenario slightly and assume that we observe this person to leave state 3 at time tx + t2 + h but do not know where he or she went, we should change A31 to A3 in (11.2.4). Furthermore, if we terminate our observa­tion at time/, +12 +t3 without knowing whether he or she continues to stay in state 3 or not, we should drop A31 altogether from (11.2.4). In this last case we say “censoring (more exactly, right-censoring) occurs at time tx + t2 +t3."

Let us consider the simple case of M = 2. In this case we have A, = A12 and A2 = A21. (We are still considering a particular individual and therefore have suppressed the subscript /.) To have a concrete idea, let us suppose that state 1 signifies unemployment and state 2, employment. The event history of an individual may consist of unemployment spells and employment spells. (If the observation is censored from the right, the last spell is incomplete.) The individual’s likelihood function can be written as the product of two terms— the probability of unemployment spells and the probability of employment spells. We shall now concentrate on unemployment spells. Suppose our typi­cal individual experienced r completed unemployment spells of duration tx, t2,. . . , tT during the observation period. Then the contribution of these r spells to the likelihood function is given by

L — Xre~XT, (11.2.5)

where we have defined T = and have written A for Xx. The individual’s

complete likelihood function is (11.2.5) times the corresponding part for the employment spells.

We now wish to consider closely the likelihood function of one complete spell: e~**X. At the beginning of this section, we derived it by a limit operation, but, here, we shall give it a somewhat different (although essentially the same) interpretation. We can interpret e~u as P(T> t) where Гis a random variable that signifies the duration of an unemployment spell. Therefore the distribu­
tion function F( •) of T is given by

F(t) = 1 - P(T>t)= 1 - e~u. (11.2.6)

Differentiating (11.2.6) we obtain the density function/(•) of T:

fit) = Xe-*. (11.2.7)

Thus we have interpreted e~'uk as the density of the observed duration of an unemployment spell.

From (11.2.6) and (11.2.7) we have

№

1 - Fit)'

The meaning of Я becomes clear when we note

= Prob [leaves unemployment in it, t + ДО I has not left unemployment in (0, 0].

We call Я the hazard rate. The term originates in survival analysis, where the state in question is “life” rather than unemployment.

Still concentrating on the unemployment duration, let us suppose that we observe one completed unemployment spell of duration /,• for the і th individ­ual, {=1,2,. . . , N. Then, defining/'^) = k‘ exp (—Я'/,), we can write the likelihood function as

£ = П/^)’ (11.2.10)

(-і

which is a standard likelihood function of a model involving continuous random variables. Suppose, however, that individuals /=1,2,. . . ,n com­plete their unemployment spells of duration /, but individuals і = n+ 1,

. . . , N are right-censored at time /*. Then the likelihood function is given by

L*=if4h) П [1 - Ftf)], (11.2.11)

1-1 f-B+l

which is a mixture of densities and probabilities just like the likelihood func­tion of a standard Tobit model. Thus we see that a duration model with right-censoring is similar to a standard Tobit model.