  $2^K$ factorial designs are widely adopted by statisticians and the broader
scientific community. In this short note, under the potential outcomes
framework (Neyman, 1923; Rubin, 1974), we adopt the partial identification
approach and derive the sharp lower bound of the sampling variance of the
estimated factorial effects, which leads to an "improved" Neymanian variance
estimator that mitigates the over-estimation issue suffered by the classic
Neymanian variance estimator by Dasgupta et al. (2015).
