  In recent years, there is a growing body of causal inference literature
focusing on covariate balancing methods. These methods eliminate observed
confounding by equalizing covariate moments between the treated and control
groups. The validity of covariate balancing relies on an implicit assumption
that all covariates are accurately measured, which is frequently violated in
observational studies. Nevertheless, the impact of measurement error on
covariate balancing is unclear, and there is no existing work on balancing
mismeasured covariates adequately. In this article, we show that naively
ignoring measurement error reversely increases the magnitude of covariate
imbalance and induces bias to treatment effect estimation. We then propose a
class of measurement error correction strategies for the existing covariate
balancing methods. Theoretically, we show that these strategies successfully
recover balance for all covariates, and eliminate bias of treatment effect
estimation. We assess the proposed correction methods in simulation studies and
real data analysis.
