IN OUR STRUCTURAL ESTIMATION, we assume throughout that ε t is distributed as a Type 1 extreme value. The computational advantages of parameterizing G(ε) this way are most evident from Lemma A.1 below, where we provide formulas for A t (h) and B t (h), the value of human capital on and off the equilibrium path, and also an expression for marginal disturbances, q jk [p t (h)].
LEMMA A.1: If ε jkt is independently and identically distributed as a Type I extreme value with location and scale parameterswhere p 0t (h) is the probability that the optimal choice is retirement,The IIA property of Type 1 extreme values implies that the marginal idiosyncratic shock for a manager who is indifferent between the best job match (j k) and retiring is the log-odds ratio of the probability that a manager with characteristics (t h) who accepts employment in (j k) versus retiring. The logodds ratio does not depend on the other components of the conditional-choice probability vector. The greater the probability of retirement observed in equilibrium, the less important is the human-capital component, and the higher is the unobserved shock for the marginal person.PROOF OF LEMMA A.1: The formula for q jk [p t (h)] given by (S-1) is well known (e.g., Hotz and Miller (1993)). Denoting the probability density function of ε * jkt ≡ d jk ε jkt by dG(ε * jkt ), we first derive an expression for E[exp(−ε * jkt /b t )] and then use it in our derivation of the formula for A t (h t ):