Maximum likelihood estimation with flow data

Maximum likelihood estimation with flow data is a parametric approach to deal with flow sampling data.

Assume that we have observations of a_i the time a person enters the state of interest, some observables x_i, and the censoring of the flow data takes on a particular form. In particular ${\displaystyle t_i=\min (t_i^U,L)}$, where t_i is the observed duration outcome, ${\displaystyle t_i^U}$ is the underlying continuous variable and L is the censoring threshold.^[1] For instance, when thinking about unemployment spells, a_i is the data of entering unemployment, x_i is a vector of worker characteristics, and t_i is the observed unemployment duration. If we only follow the workers for a certain period of time, this variable is necessarily a censored version of the true unemployment duration.

Two key assumptions allow for setting up the loglikelihood. First, a distributional form for the latent variable ${\displaystyle t_i^U}$ needs to be assumed. Second, independence between the true duration and the starting point of the spell is assumed, i.e.,

${\displaystyle F(t_i^U\vee x_i,a_i,L)=F(t_i^U\vee x_i)}$

where F is the conditional distribution of the underlying duration variable.^[2] This latter assumption allows us to model the probability that the variable is censored, i.e.,

${\displaystyle \mathrm{Pr}(t_i^U\ge L\vee x_i)=1-F(L\vee x_i)}$

which leads to the following log likelihood:

${\displaystyle {\displaystyle \sum _{i=1}^n}[d_i\log (f(t_i\vee x_i))+(1-d_i)\log (1-F(L\vee x_i))]}$

where f is the density associated with the distribution F and d_i is an indicator denoting whether t_i = L.^[3] Additionally, it is possible to have the threshold vary at the observational level, by replacing L by L_i in the formulas above.^[4]

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.^[5] Similarly, tests for censoring exist that compare the distribution of the generalized error under the censored and the uncensored assumption.^[6]

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