  We consider experimentation in the presence of non-stationarity, inter-unit
(spatial) interference, and carry-over effects (temporal interference), where
we wish to estimate the global average treatment effect (GATE), the difference
between average outcomes having exposed all units at all times to treatment or
to control. We suppose spatial interference is described by a graph, where a
unit's outcome depends on its neighborhood's treatment assignments, and that
temporal interference is described by a hidden Markov decision process, where
the transition kernel under either treatment (action) satisfies a rapid mixing
condition. We propose a clustered switchback design, where units are grouped
into clusters and time steps are grouped into blocks and each whole
cluster-block combination is assigned a single random treatment. Under this
design, we show that for graphs that admit good clustering, a truncated
exposure-mapping Horvitz-Thompson estimator achieves $\tilde O(1/NT)$
mean-squared error (MSE), matching an $\Omega(1/NT)$ lower bound up to
logarithmic terms. Our results simultaneously generalize the $N=1$ setting of
Hu, Wager 2022 (and improves on the MSE bound shown therein for
difference-in-means estimators) as well as the $T=1$ settings of Ugander et al
2013 and Leung 2022. Simulation studies validate the favorable performance of
our approach.
