Matching for Causal Inference
Matching pairs or groups units with similar pretreatment covariates but different treatments, then compares their outcomes.
Nearest-neighbor match
For unit $i$, a nearest neighbor under covariate distance $d$ is
\[j(i)= \arg\min_{j:W_j\neq W_i}d(X_i,X_j).\]Using propensity-score distance gives
\[j(i)= \arg\min_{j:W_j\neq W_i} |\hat e(X_i)-\hat e(X_j)|.\]The opposite-treatment restriction is essential: a treated unit must be compared with a control unit, and vice versa.
Matched estimator for the treated
When each treated unit is matched to one control, the conceptual estimator
\[\hat\tau_{ATT}= \frac{1}{N_T} \sum_{i:W_i=1} (Y_i^{obs}-Y_{j(i)}^{obs})\]targets the average treatment effect on the treated, not automatically the ATE. The target changes with the matching scheme, target units, replacement rules, and weights.
Design choices
- With replacement: a strong control match may be reused; usually reduces bias but may increase dependence and reduce effective sample size.
- Without replacement: each control is used once; results can depend on match order.
- Caliper: rejects matches farther apart than a chosen score distance.
- Exact or coarsened exact matching: forces equality on selected covariates or categories.
- Mahalanobis matching: accounts for the covariance and scale of covariates.
Assessment after matching
Matching quality is judged by covariate balance and overlap, not by whether treatment-group outcome means become favorable. Examine standardized differences, variance ratios, covariate distributions, unmatched units, and sensitivity to design choices.
Discarding units changes the population represented by the estimate. State which units remain and which estimand the matched sample supports.
Matching versus randomized pairs
In Blocked and Paired Randomized Experiments, pairs are formed before random assignment and treatment is randomized within each pair. Observational matching builds comparisons after treatment already occurred and still depends on unconfoundedness.