  We consider missingness in the context of causal inference when the outcome
of interest may be missing. If the outcome directly affects its own missingness
status, i.e., it is "self-censoring", this may lead to severely biased causal
effect estimates. Miao et al. [2015] proposed the shadow variable method to
correct for bias due to self-censoring; however, verifying the required model
assumptions can be difficult. Here, we propose a test based on a randomized
incentive variable offered to encourage reporting of the outcome that can be
used to verify identification assumptions that are sufficient to correct for
both self-censoring and confounding bias. Concretely, the test confirms whether
a given set of pre-treatment covariates is sufficient to block all backdoor
paths between the treatment and outcome as well as all paths between the
treatment and missingness indicator after conditioning on the outcome. We show
that under these conditions, the causal effect is identified by using the
treatment as a shadow variable, and it leads to an intuitive inverse
probability weighting estimator that uses a product of the treatment and
response weights. We evaluate the efficacy of our test and downstream estimator
via simulations.
