  In an observational study, it is common to leverage known null effect to
detect bias. One such strategy is to set aside a placebo sample -- a subset of
data immune from the hypothesized cause-and-effect relationship. Existence of
an effect in the placebo sample raises concern of unmeasured confounding bias
while absence of it corroborates the causal conclusion. This paper establishes
a formal framework for using a placebo sample to detect and remove bias. We
state identification assumption, and develop estimation and inference methods
based on outcome regression, inverse probability weighting, and doubly-robust
approaches. Simulation studies and an empirical application illustrate the
finite-sample performance of the proposed methods.
