  Many causal processes have spatial and temporal dimensions. Yet the classic
causal inference framework is not directly applicable when the treatment and
outcome variables are generated by spatio-temporal point processes. We extend
the potential outcomes framework to these settings by formulating the treatment
point process as a stochastic intervention. Our causal estimands include the
expected number of outcome events in a specified area under a particular
stochastic treatment assignment strategy. Our methodology allows for arbitrary
patterns of spatial spillover and temporal carryover effects. Using martingale
theory, we show that the proposed estimator is consistent and asymptotically
normal as the number of time periods increases. We propose a sensitivity
analysis for the possible existence of unmeasured confounders, and extend it to
the Hajek estimator. Simulation studies are conducted to examine the
estimators' finite sample performance. Finally, we illustrate the proposed
methods by estimating the effects of American airstrikes on insurgent violence
in Iraq from February 2007 to July 2008. Our analysis suggests that increasing
the average number of daily airstrikes for up to one month may result in more
insurgent attacks. We also find some evidence that airstrikes can displace
attacks from Baghdad to new locations up to 400 kilometers away
