  Many empirical examples of regression discontinuity (RD) designs concern a
continuous treatment variable, but the theoretical aspects of such models are
less studied. This study examines the identification and estimation of the
structural function in fuzzy RD designs with a continuous treatment variable.
The structural function fully describes the causal impact of the treatment on
the outcome. We show that the nonlinear and nonseparable structural function
can be nonparametrically identified at the RD cutoff under shape restrictions,
including monotonicity and smoothness conditions. Based on the nonparametric
identification equation, we propose a three-step semiparametric estimation
procedure and establish the asymptotic normality of the estimator. The
semiparametric estimator achieves the same convergence rate as in the case of a
binary treatment variable. As an application of the method, we estimate the
causal effect of sleep time on health status by using the discontinuity in
natural light timing at time zone boundaries.
