  Longitudinal Modified Treatment Policies (LMTPs) provide a framework for
defining a broad class of causal target parameters for continuous and
categorical exposures. We propose Local LMTPs, a generalization of LMTPs to
settings where the target parameter is conditional on subsets of units defined
by the treatment or exposure. Such parameters have wide scientific relevance,
with well-known parameters such as the Average Treatment Effect on the Treated
(ATT) falling within the class. We provide a formal causal identification
result that expresses the Local LMTP parameter in terms of sequential
regressions, and derive the efficient influence function of the parameter which
defines its semi-parametric and local asymptotic minimax efficiency bound.
Efficient semi-parametric inference of Local LMTP parameters requires
estimating the ratios of functions of complex conditional probabilities (or
densities). We propose an estimator for Local LMTP parameters that directly
estimates these required ratios via empirical loss minimization, drawing on the
theory of Riesz representers. The estimator is implemented using a combination
of ensemble machine learning algorithms and deep neural networks, and evaluated
via simulation studies. We illustrate in simulation that estimation of the
density ratios using Riesz representation might provide more stable estimators
in finite samples in the presence of empirical violations of the
overlap/positivity assumption.
