Duration models with time varying data

Many duration models use regressors that do not vary over time. However, both continuous and discrete-time duration models may exhibit time-varying regressors. For instance, medical trials can use different dosage levels over the trial, or during a single spell, the marital status or unemployment rate can change. Time-varying covariates are generally more easily handled in discrete-time problems, as covariates are rarely continuously observed. There are two key issues that arise when treating the time-varying covariates as fixed. First, a misspecification treat arises if the entire history of the variable matters rather than the specific values it takes on at certain time periods. Second, feedback treats arise when the variables are not strictly exogenous . These issues can be related to strict and weak exogeneity, however, subtle differences between the two cases have prompted researchers to define external and internal regressors for hazard functions with time-varying covariates.

For the case of discrete-time duration data, setting up the likelihood function when the regressors vary over time takes more work than in the case with time-invariant covariates, but one can assume conditional independence such that

D( T∨T ≥ a_m-1, x_m, c_m ) = D( T∨T ≥ a_m-1, x_m ) for m=1,...,M

This condition states that the observed covariates x_m sufficiently capture the censoring variable c_m. If conditioning on lagged variables is necessary, one can just include the relevant lagged variable in the conditioning set.

One can show that P (y_m=1 ∨ y_m-1= 0, x_m, c_m = 0) = P(a_m-1 ≤ T < a_m ∨ T ≥ a__m-1 , x_m) = 1 - exp[- ∫ λ (s ; x_m , θ)ds]

along with P(y_m = 0 ∨ y_m-1 = 0, x_m, c_m= 0) and use this to set up the partial likelihood for person i. So, essentially, the partial likelihood with time-varying covariates resembles the partial likelihood with time-invariant covariates where the time-invariant covariates are replaced by the time-variant covariates. This likelihood, however, only holds under the assumptions stated above.

The hazard specification λ(t; x_m, θ)= κ(x_m; β) λ_m for a_m-1≤ t<a_m can still be used for time-varying regressors. This specification has been used to estimate the effect of unemployment insurance on unemployment spells, while certain parameterizations also allow for time-specific coeffients β_m. For the specific case of predetermined regressors, discrete-time duration models are available.