  We propose a generalization of the synthetic control and synthetic
interventions methodology to the dynamic treatment regime. We consider the
estimation of unit-specific treatment effects from panel data collected via a
dynamic treatment regime and in the presence of unobserved confounding. That
is, each unit receives multiple treatments sequentially, based on an adaptive
policy, which depends on a latent endogenously time-varying confounding state
of the treated unit. Under a low-rank latent factor model assumption and a
technical overlap assumption we propose an identification strategy for any
unit-specific mean outcome under any sequence of interventions. The latent
factor model we propose admits linear time-varying and time-invariant dynamical
systems as special cases. Our approach can be seen as an identification
strategy for structural nested mean models under a low-rank latent factor
assumption on the blip effects. Our method, which we term "synthetic blip
effects", is a backwards induction process, where the blip effect of a
treatment at each period and for a target unit is recursively expressed as
linear combinations of blip effects of a carefully chosen group of other units
that received the designated treatment. Our work avoids the combinatorial
explosion in the number of units that would be required by a vanilla
application of prior synthetic control and synthetic intervention methods in
such dynamic treatment regime settings.
