  In the comparative interrupted time series design (also called the method of
difference-in-differences), the change in outcome in a group exposed to
treatment in the periods before and after the exposure is compared to the
change in outcome in a control group not exposed to treatment in either period.
The standard difference-in-difference estimator for a comparative interrupted
time series design will be biased for estimating the causal effect of the
treatment if there is an interaction between history in the after period and
the groups; for example, there is a historical event besides the start of the
treatment in the after period that benefits the treated group more than the
control group. We present a bracketing method for bounding the effect of an
interaction between history and the groups that arises from a time-invariant
unmeasured confounder having a different effect in the after period than the
before period. The method is applied to a study of the effect of the repeal of
Missouri's permit-to-purchase handgun law on its firearm homicide rate. We
estimate that the effect of the permit-to-purchase repeal on Missouri's firearm
homicide rate is bracketed between 0.9 and 1.3 homicides per 100,000 people,
corresponding to a percentage increase of 17% to 27% (95% confidence interval:
[0.6,1.7] or [11%,35%]). A placebo study provides additional support for the
hypothesis that the repeal has a causal effect of increasing the rate of
state-wide firearm homicides.
