Durations as Dependent Variables of a Regression Equation

Suppose that each individual experiences one complete spell. Then the likeli­hood function is

N

L = J"JA, exp (—A^j). (11.2.25)

/-і

The case of a person having more than one complete spell can be handled by behaving as if these spells belonged to different individuals. Assume as before

Аі = exp (0′ xt). (11.2.26)

But, here, we have absorbed the constant term a into fi as there is no need to separate it out.

We shall derive the asymptotic covariance matrix of the MLE fi. We have

which is the continuous-time analog of (11.1.65). Therefore we have r^log L

Подпись: (11.2.29)

image854 Подпись: (11.2.30)
Подпись: exp (—A,-z) dz = y, л,-
Подпись: (11.2.28)

Therefore, by Theorem 4.2.4, fi is asymptotically normal with asymptotic covariance matrix

Now, suppose we use log /,• as the dependent variable of a linear regression equation. For this purpose we need the mean and variance of log t(. We have5

Подпись:E log tj = Xj J (log z) exp (— kjZ) dz

= — c — log A,,

image859 Подпись: (11.2.32)

where c = 0.577 is Euler’s constant, and

= j + (c + logA,)2.

Therefore we have

Подпись: (11.2.33)V]Qgt, = J.

We can write (11.2.31) and (11.2.33) as a linear regression

log ti + c = – fi’Xi + U(, (11.2.34)

where Eiij = 0 and Км, = л2/6. Because {и,} are independent, (11.2.34) de­fines a classical regression model, which we called Model 1 in Chapter 1. Therefore the exact covariance matrix of the LS estimator is given by

Comparing (11.2.35) with (11.2.30), we see exactly how much efficiency we lose by this method.

Alternatively, we can define a nonlinear regression model using tt itself as the dependent variable. From (11.2.28) we have

U = exp (—P’Xj) + vt, (11.2.36)

where EVj = 0. We have by integration by parts

£/? = I r2A, exp(-A, z)fife = -|. (11.2.37)

Jo

Therefore we have

VVj = exp (— 20’x(). (11.2.38)

This shows that (11.2.26) defines a heteroscedastic nonlinear regression model. The asymptotic normality of the NLWLS estimator /Jwls can be proved under general assumptions using the results of Sections 4.3.3 and

6.5.3. Its asymptotic covariance matrix can be deduced from (4.3.21) and

(6.1.4) and is given by

^Avls = exp (20’xt) M 1 = ^ x, x’) ‘. (11.2.39)

Therefore the NLWLS estimator is asymptotically efficient. In practice Vv, will be estimated by replacing 0 by some consistent estimator of 0, such as 0^ defined earlier, which converges to 0 at the speed of 4n. The NLWLS estima­tor using the estimated Vvt has the same asymptotic distribution.

The foregoing analysis was based on the assumption that tt is the duration of a completed spell of the ith individual. If some of the spells are right-censored and not completed, the regression method cannot easily handle them. How­ever, maximum likelihood estimation can take account of right-censoring, as we indicated in Section 11.2.1.

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