Advanced Econometrics Takeshi Amemiya

Number of Completed Spells

The likelihood function (11.2.5) depends on the observed durations ti, t2, ■ ■ ■ , tr only through r and T. In other words, r and Tconstitute the sufficient statistics. This is a property of a stationary model. We shall show an alternative way of deriving the equivalent likelihood function.

We shall first derive the probability of observing two completed spells in total unemployment time T, denoted P(2, T). The assumption that there are two completed unemployment spells implies that the third spell is incomplete (its duration may be exactly 0). Denoting the duration of the three spells by t,, t2, and t3, we have

P(2, T) = P(0 ё t{ < T, 0 < t2 § T—tt, t3 Ш T-tx ~ h) (11.2.12)

к exp (—Az2){exp [-А(Г - z, - z2)]} dz2^j dzx

{kTfe~2T

2

It is easy to deduce from the derivation in (11.2.12) that the probability of observing r completed spells in total time T is given by

P(r, T) — - ^------- , (11.2.13)

which is a Poisson distribution. This is equivalent to (11.2.5) because Trand r! do not depend on the unknown parameters.

We can now put back the subscript і in the right-hand side of (11.2.5) and take the product over і to obtain the likelihood function of all the individuals:

L = flApexp(—A,7’I). (11.2.14)

/-і

Assuming that A, depends on a vector of the ith individual’s characteristics x„ we can specify

п Г Tt exp (fi'Xj) T' 1 U S^expOd'x,) Г Therefore, we have log Ly = X r, log Tt, + X і і - (x r/) lo8 X T‘ exP

(11.2.20)

(11.2.21)

Setting the derivative of (11.2.21) with respect to fi equal to 0 yields

d log Ly_ dfi - f ‘ '

X Tf exp (ft'x,)

which is identical to (11.2.19). Note that Lx is the likelihood function of a multinomial logit model (9.3.34). To see this, pretend that the / in (11.2.20) refers to the fth alternative and r, people chose the fth alternative. This is a model where the exogenous variables depend only on the characteristics of the alternatives and not on those of the individuals. Thus the maximization of L, and hence the solution of (11.2.19) can be accomplished by a standard multi­nomial logit routine.

We can write Lx as a part of the likelihood function L as follows:

where

Note that L2 is a Poisson distribution. Setting d log L2 /да = 0yields(l 1.2.17). We can describe the calculation of the MLE as follows: First, maximize Lx with respect to second, insert fi into L2 and maximize it with respect to a.