C1: Simulation of the Likelihood Function.The high dimensional integrals in the likelihood
function are simulated using R draws from the joint distribution of and Q draws from the joint distribution of occupation and …rm speci…c human capital. The integral over the joint distribution of human capital is simulated using a modi…ed Geweke, e, and H jivassiliou (GH ) algorithm because the joint distribution of human capital is intractably complex. The type-speci…c simulated likelihood contribution is LSi ( i = k) = 1 R R X r=1 1 Q Q X q=1 e Ti Y t=1 Pr[Oitrq r i; o q it; f q it; ; Sit; i = k): (23) C2: Simulation of the Likelihood Function. With the exception of the integrals over the
distributions of …rm and occupation speci…c human capital, all integrals are simulated using simple
ncy simulators. This type of simulator is not practical in the case of the integral overfit and
oitbecause the distributions of these unobserved state variables are intractably complex. The integral that needs to be evaluated is the path probability over the sample period, denoted . The eon for this probability is
=
Z Z YTei
t=1
Pr[Oit ; Sit; i= k; oit; fit]dF(oi)dF(fi):
Note that the integral is over the joint distribution of fi and oi over the entire Tei years that person
are an enormous number of possible s ces of human capital that could occur. Calculating this distribution for each sample person is not practical. The solution is to use a modi…ed algorithm to simulate the integral. The intuition behind this method is the same as in Brien, Lillard, and Stern (2006). The complete algorithm is outlined below.
1. Draw orto r t 1 and ftrf r t 1: 2. Compute Pr[Oito r t; ftr]: 3. Compute r= r Pr[O ito r t; ftr]:
4. Ift=Tei, go to step 5. Otherwise, set t=t+ 1and go to step 1.
5. Repeat these steps for each of the R simulation draws. The simulated path probability is
= R1 PRr=1 r.
This algorithm simpli…es the problem because drawingfr
t andortconditional on the previous draw is very straightforward, while drawing from the complete distribution would be veryd cult.
C3: Outcome Probabilities. The most straightforward outcome probability found in the like-
lihood function is the probability of observing a person attending school or being unemployed. The likelihood contribution is simply the probability that the value of attending school exceeds the value of any other choice in the p s choice set, D
ne
t . Conditional on the unobserved state variables except", the choice probability is of the multinomial logit form,
Pr(dit =s ; oit; fit; Sit; i) = exp(Vt(s)) P k2Dne t exp(Vt(k)) . (24)
The probabilities for outcomes involving employment are similar to the non-employed outcome prob- abilities, except the choice probability is conditioned on the observed wage and multiplied by the wage density.
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