  In the past decade, various exact balancing-based weighting methods were
introduced to the causal inference literature. Exact balancing alleviates the
extreme weight and model misspecification issues that may incur when one
implements inverse probability weighting. It eliminates covariate imbalance by
imposing balancing constraints in an optimization problem. The optimization
problem can nevertheless be infeasible when there is bad overlap between the
covariate distributions in the treated and control groups or when the
covariates are high-dimensional. Recently, approximate balancing was proposed
as an alternative balancing framework, which resolves the feasibility issue by
using inequality moment constraints instead. However, it can be difficult to
select the threshold parameters when the number of constraints is large.
Moreover, moment constraints may not fully capture the discrepancy of covariate
distributions. In this paper, we propose Mahalanobis balancing, which
approximately balances covariate distributions from a multivariate perspective.
We use a quadratic constraint to control overall imbalance with a single
threshold parameter, which can be tuned by a simple selection procedure. We
show that the dual problem of Mahalanobis balancing is an l_2 norm-based
regularized regression problem, and establish interesting connection to
propensity score models. We further generalize Mahalanobis balancing to the
high-dimensional scenario. We derive asymptotic properties and make extensive
comparisons with existing balancing methods in the numerical studies.
