  Time plays a fundamental role in causal analyses, where the goal is to
quantify the effect of a specific treatment on future outcomes. In a randomized
experiment, times of treatment, and when outcomes are observed, are typically
well defined. In an observational study, treatment time marks the point from
which pre-treatment variables must be regarded as outcomes, and it is often
straightforward to establish. Motivated by a natural experiment in online
marketing, we consider a situation where useful conceptualizations of the
experiment behind an observational study of interest lead to uncertainty in the
determination of times at which individual treatments take place. Of interest
is the causal effect of heavy snowfall in several parts of the country on daily
measures of online searches for batteries, and then purchases. The data
available give information on actual snowfall, whereas the natural treatment is
the anticipation of heavy snowfall, which is not observed. In this article, we
introduce formal assumptions and inference methodology centered around a novel
notion of plausible time of treatment. These methods allow us to explicitly
bound the last plausible time of treatment in observational studies with
unknown times of treatment, and ultimately yield valid causal estimates in such
situations.
