35.1 Introduction and Motivation
35.1.1 Why Match?
In many observational studies, researchers do not have the luxury of randomization. Subjects (people, firms, schools, etc.) typically select or are selected into treatment based on certain observed and/or unobserved characteristics. This can introduce systematic differences (selection bias) that confound causal inference. Matching attempts to approximate a randomized experiment by “balancing” these observed characteristics between treated and non-treated (control) units.
- Goal: Reduce model dependence and clarify causal effects by ensuring that treated and control subjects have sufficiently comparable covariates.
- Challenge: Even if matching achieves balance in observed covariates, any unobserved confounders remain a threat to identification (i.e., Matching is only a selection observables identification strategy). Matching does not magically fix bias from unobserved variables.
To understand why causal inference is difficult in observational studies, consider:
E(YTi|T)−E(YCi|C)=E(YTi−YCi|T)+[E(YCi|T)−E(YCi|C)]⏟Selection Bias
- The term E(YTi−YCi|T) is the causal effect (specifically the ATT).
- The term E(YCi|T)−E(YCi|C) reflects selection bias due to systematic differences in the untreated potential outcome across treated and control groups.
Random assignment ensures:
E(YCi|T)=E(YCi|C)
which eliminates selection bias. In observational data, however, this equality rarely holds.
Matching aims to mimic randomization by conditioning on covariates X:
E(YCi|X,T)=E(YCi|X,C)
For example, propensity score matching achieves this balance by conditioning on the propensity score P(X):
E(YCi|P(X),T)=E(YCi|P(X),C)
(See Propensity Scores for further discussion.)
The Average Treatment Effect (ATE) under matching is typically estimated as:
1NTNT∑i=1(YTi−1NCi∑j∈CiYCj)
where Ci denotes the matched controls for treated unit i.
Standard Errors in Matching
- Matching does not have a closed-form standard error for the ATE or ATT.
- Therefore, we rely on bootstrapping to estimate uncertainty.
Note: Matching tends to yield larger standard errors than OLS because it reduces the effective sample size by discarding unmatched observations.
35.1.2 Matching as “Pruning”
Matching can be thought of as “pruning” (a preprocessing step) (G. King, Lucas, and Nielsen 2017). The goal is to prune unmatched or poorly matched units before conducting analysis, reducing model dependence.
Without Matching:
- Imbalanced data → Model dependence → Researcher discretion → Biased estimates
With Matching:
- Balanced data → Reduces discretion → More credible causal inference
| Balance of Covariates | Complete Randomization | Fully Exact Matching |
|---|---|---|
| Observed | On average | Exact |
| Unobserved | On average | On average |
Fully blocked or exactly matched designs outperform randomized ones on:
- Imbalance
- Model dependence
- Efficiency and power
- Bias
- Robustness
- Research costs
35.1.3 Matching with DiD
Matching can be fruitfully combined with DiD when multiple pre-treatment periods are available. Such designs can help correct for selection bias under certain assumptions:
- When selection bias is symmetric around the treatment date, standard DID (implemented symmetrically around the treatment date) remains consistent (Chabé-Ferret 2015 ).
- If selection bias is asymmetric, simulations by Chabé-Ferret (2015) show that symmetric DID still outperforms matching alone, although having more pre-treatment observations can improve matching performance.
In short, matching is not a universal solution but often provides a helpful preprocessing step before conducting DiD or other causal estimation methods (J. A. Smith and Todd 2005).