Retirement and Economic Behavior

By Henry J. Aaron; Gary Burtless | Go to book overview

over time as the probability of retirement increases. Changes in the z2 variables are more difficult to interpret, but equation 12 gives the change in the average of z2: multiplied by a variable that is positive and decreasing over time.


Appendix C: The Competing-Risks Model

The competing-risks model can be cast into a latent variable framework very similar to the probit model. Let nk for k = 1, . . ., K denote K competing risks. Introduce the latent random variable

for k = 1, . . ., K, which would be the length of period if the particular risk were the only risk present. Denote the distribution function of by . However, is a latent variable in general since it cannot necessarily be observed. Instead, only the minimum of the theoretical lifetimes is observed:

(13)

.

The information we have here, including the actual outcomes, includes the fact that if Y is greater than x, then the survivor distribution function, F + ̅y(x) = 1 - Fy(x), along with the survivor function, ry(x) = fy(x)/F + ̄y(x), gives the conditional failure rate function for the observable variable Y. If gk(x) denotes the probability of failure from nk in the interval (x, x + dx), then in the presence of all k risks we find

(14)

,

so long as the probability of more than one simultaneous failure is negligible.

The competing-risks model bears a close relationship to the sample selection models in econometrics. Here we know the value of the left- hand-side variables in both cases (for K = 2) while in the sample selection models, we only know the value of the left-hand-side variables for one of the equations since the other equation is a probit model. Nonetheless, the competing-risks model has had a somewhat dubious reputation,

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