The second function on the right is called the likelihood function
Because we know that the actual length of the career is longer than we observe then the contribution of these observations to the likelihood function
is just the survivor function, S([t.
Based on Likelihood Ratio Test statistics (LRT) the student-t log-likelihood function
s strongly preferred to a normal likelihood function
10 Notice that in the special case where these constants are all set to zero, the functional form suggested by Hamilton (1991) collapses to the traditional maximum likelihood function
The likelihood function
now depends on Pr(x [where] [x.
To see this, note that p(v |is greater than~ - |Beta~Z |where~ u |is greater than~ - |Alpha~X) does not appear in the likelihood function
for the Heckman model nor is the expectation of Y*, denoted by g, conditional upon u |is greater than~ - |Alpha~X.
less than or equal to] n - k and choosing the one that maximizes the (logarithm of the) likelihood function
The log likelihood function
1(m,k) (equation 3) can be rewritten as:
2] are presented in non-normalized form, which makes them suitable for a likelihood function
, but not suitable as probability densities.
Because we know that the actual length of the career is longer than we observe, then the contribution of these observations to the likelihood function
is just the survivor function (S).
Estimators produced by maximizing the likelihood function
in a Poisson regression are generally preferred over those produced by least squares based on their statistical properties, and the data are automatically weighted based on the expected number of deaths (highly correlated with population size).
Assuming a Gaussian error distribution, the likelihood function