A Simple Approach to Treatment Effects on Durations When the Treatment Timing is Chosen

When a treatment unambiguously defines the treatment and control groups at a given time point, its effects are usually found by comparing the two groups' mean responses. But there are many cases where the treatment timing is chosen, for which the conventional approach fails. This paper sets up an ideal causal framework for such cases to propose a simple gamma-mixed proportional-hazard approach with three durations: the waiting time until treatment, the untreated duration from the baseline, and the treated duration from the treatment timing. To implement the proposal, we use semiparametric piecewise-constant hazards as well as Weibull hazards with a multiplicative gamma unobserved heterogeneity affecting all three durations. Despite the three durations interwoven in complex ways, surprisingly simple closed-form likelihoods are obtained whose maximization converges well. The estimators are applied to the same data as used by Fredriksson and Johansson (2008) for employment subsidy effects on unemployment duration to find about 11.1 month reduction.