  We introduce a new causal inference framework for time series data aimed at
assessing the effectiveness of heat alerts in reducing mortality and
hospitalization risks. We are interested in addressing the following question:
how many deaths and hospitalizations could be averted if we were to increase
the frequency of issuing heat alerts in a given location? In the context of
time series data, the overlap assumption - each unit must have a positive
probability of receiving the treatment - is often violated. This is because, in
a given location, issuing a heat alert is a rare event on an average
temperature day as heat alerts are almost always issued on extremely hot days.
To overcome this challenge, first we introduce a new class of causal estimands
under a stochastic intervention (i.e., increasing the odds of issuing a heat
alert) for a single time series corresponding to a given location. We develop
the theory to show that these causal estimands can be identified and estimated
under a weaker version of the overlap assumption. Second, we propose
nonparametric estimators based on time-varying propensity scores, and derive
point-wise confidence bands for these estimators. Third, we extend this
framework to multiple time series corresponding to multiple locations. Via
simulations, we show that the proposed estimator has good performance with
respect to bias and root mean squared error. We apply our proposed method to
estimate the causal effects of increasing the odds of issuing heat alerts in
reducing deaths and hospitalizations among Medicare enrollees in 2817 U.S.
counties. We found weak evidence of a causal link between increasing the odds
of issuing heat alerts during the warm seasons of 2006-2016 and a reduction in
deaths and cause-specific hospitalizations across the 2817 counties.
