  This paper considers identification and estimation of the causal effect of
the time Z until a subject is treated on a survival outcome T. The treatment is
not randomly assigned, T is randomly right censored by a random variable C and
the time to treatment Z is right censored by min(T,C). The endogeneity issue is
treated using an instrumental variable explaining Z and independent of the
error term of the model. We study identification in a fully nonparametric
framework. We show that our specification generates an integral equation, of
which the regression function of interest is a solution. We provide
identification conditions that rely on this identification equation. For
estimation purposes, we assume that the regression function follows a
parametric model. We propose an estimation procedure and give conditions under
which the estimator is asymptotically normal. The estimators exhibit good
finite sample properties in simulations. Our methodology is applied to find
evidence supporting the efficacy of a therapy for burn-out.
