14.4 Multiple Matching Variables

Distance Matching

\[d_{M}(\vec{x}, \vec{y}; Q) = \sqrt{(\vec{x} - \vec{y})^{T}\Sigma^{-1}(\vec{x} - \vec{y})}\]

Mahalanobis distance

• Image source: Lee, et al (2020)
• Curse of Dimensionality

Entropy Balancing

Entropy balancing enforces restrictions on moments

• means, variances, skewness, etc.

• Hainmueller, Jens. 2012. “Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies.” Political Analysis 20 (2): 25-46.

Choose weights \(w_{i}\) such that

\[\text{min} \sum_{\{i|D = 0\}} h(w_{i})\]

• balance constraint: \(\sum_{\{i|D = 0\}} w_{i}c_{ri}(X_{i}) = m_{r}\)
• normalization: \(\sum_{\{i|D = 0\}} w_{i} = 1\)