Maximum likelihood estimation with flow data is a parametric approach to deal with flow sampling data. Assume that we have observations of ai the time a person enters the state of interest, some observables xi, and the censoring of the flow data takes on a particular form. In particular
Two key assumptions allow for setting up the loglikelihood. First, a distributional form for the latent variable
where F is the conditional distribution of the underlying duration variable. This latter assumption allows us to model the probability that the variable is censored, i.e.,
which leads to the following log likelihood:
where f is the density associated with the distribution F and di is an indicator denoting whether ti = L. Additionally, it is possible to have the threshold vary at the observational level, by replacing L by Li in the formulas above.
Tests of specification in duration models encompass testing for the validity of the imposed functional form. Tests of restrictions on the functional form are similar to those testing for unobserved heterogeneity, where the restriction imposes no such heterogeneity. Nevertheless, it is often desirable to test for such heterogeneity, as this can bias the estimation of the hazard rate. Similarly, tests for censoring exist that compare the distribution of the generalized error under the censored and the uncensored assumption.