  In estimating the causal effect of a continuous exposure or treatment, it is
important to control for all confounding factors. However, most existing
methods require parametric specification for how control variables influence
the outcome or generalized propensity score, and inference on treatment effects
is usually sensitive to this choice. Additionally, it is often the goal to
estimate how the treatment effect varies across observed units. To address this
gap, we propose a semiparametric model using Bayesian tree ensembles for
estimating the causal effect of a continuous treatment of exposure which (i)
does not require a priori parametric specification of the influence of control
variables, and (ii) allows for identification of effect modification by
pre-specified moderators. The main parametric assumption we make is that the
effect of the exposure on the outcome is linear, with the steepness of this
relationship determined by a nonparametric function of the moderators, and we
provide heuristics to diagnose the validity of this assumption. We apply our
methods to revisit a 2001 study of how abortion rates affect incidence of
crime.
