Propensity Score Matching (PSM)

The propensity score is the conditional probability of receiving a treatment given a vector of observed covariates: e(X) = P(T=1 | X). It acts as a balancing score, meaning that at any given value of the propensity score, the distribution of covariates is expected to be identical between treated and control groups. This property, proven by Rosenbaum and Rubin in 1983, transforms a high-dimensional matching problem into a univariate one.

• Estimation: Typically estimated via logistic regression or non-parametric methods like gradient boosting.
• Dimensionality Reduction: Collapses dozens of covariates into a single scalar for matching.
• Diagnostic: The quality of the score is validated by checking covariate balance post-matching, not by model fit statistics like R-squared.

Once propensity scores are estimated, an algorithm pairs treated units with control units possessing similar scores. The choice of algorithm directly impacts the bias-variance trade-off of the resulting estimate.

• Nearest Neighbor Matching (NNM): Pairs each treated unit with the control unit having the closest score. Can be executed with or without a caliper—a maximum tolerated distance threshold to prevent poor matches.
• Kernel Matching: Uses a weighted average of all controls, where weights are inversely proportional to the distance in propensity score. This reduces variance but may increase bias.
• Stratification: Divides the sample into blocks based on score quantiles, then computes treatment effects within each block before averaging.

Valid causal inference via PSM requires the overlap assumption, also known as the common support condition. This mandates that for any value of the covariates X, there is a positive probability of being either treated or control: 0 < P(T=1 | X) < 1.

• Practical Enforcement: Researchers must visually inspect the histogram of propensity scores for both groups and trim observations that fall outside the region of common support.
• Minima-Maxima Rule: A conservative approach deletes all observations whose propensity score is less than the minimum or greater than the maximum of the opposite group.
• Violation Consequence: Extrapolating treatment effects over regions without overlap relies heavily on model specification rather than observed data, invalidating the non-parametric spirit of matching.

The success of PSM is not measured by the predictive accuracy of the propensity score model but by the degree to which it achieves covariate balance. After matching, the distribution of each covariate should be statistically indistinguishable between the treatment and control groups.

• Standardized Mean Difference (SMD): The difference in means divided by the pooled standard deviation. Values below 0.1 or 0.25 are generally considered adequately balanced.
• Variance Ratios: The ratio of variances between groups should be close to 1.
• Pseudo-R²: Re-running the propensity score model on the matched sample should yield a drastically lower pseudo-R², indicating that covariates no longer predict treatment assignment.

PSM only accounts for observed confounders. Sensitivity analysis quantifies how strongly an unmeasured confounder must influence both treatment assignment and the outcome to nullify the estimated treatment effect. This addresses the fundamental limitation of the unconfoundedness assumption.

• Rosenbaum Bounds: For binary outcomes, this method calculates how large the odds of differential assignment due to a hidden bias must be (Gamma, Γ) to render the result insignificant.
• Imbens' Sensitivity: For continuous outcomes, this framework parameterizes the partial correlation between the unobserved confounder and the treatment/outcome.
• Interpretation: A study is robust if the Gamma value required to overturn the result is large (e.g., Γ > 2), indicating the result is insensitive to moderate hidden biases.

PSM is naturally configured to estimate the Average Treatment Effect on the Treated (ATT), not the generic Average Treatment Effect (ATE). ATT answers the specific policy question: 'What was the effect of the intervention on those who actually received it?'

• ATT Definition: E[Y(1) - Y(0) | T=1]. The expected difference in outcomes for treated units relative to what they would have experienced had they not been treated.
• ATE Definition: E[Y(1) - Y(0)]. The expected effect for a randomly drawn unit from the entire population.
• Practical Implication: In PSM, control units are discarded if they are too dissimilar to treated units. This makes estimating ATE difficult without re-weighting schemes, as the matched control group represents the counterfactual for the treated, not the full population.

A re-weighting technique that corrects for selection bias by assigning each observation a weight equal to the inverse of its probability of receiving the treatment it actually received. Unlike PSM, which matches units, IPW creates a pseudo-population where treatment assignment is independent of observed covariates.

• Uses the entire sample without discarding unmatched units
• Can be combined with outcome regression in Doubly Robust Estimation
• Sensitive to extreme propensity scores near 0 or 1
• Stabilized weights reduce variance in practice

A causal inference method that combines a propensity score model and an outcome regression model. The key property: the estimator remains consistent if at least one of the two models is correctly specified, providing a safety net that PSM alone does not offer.

• Protects against model misspecification
• Uses both the treatment assignment mechanism and outcome structure
• Standard errors require special treatment (bootstrap or sandwich estimators)
• Implemented via Augmented Inverse Probability Weighting (AIPW)

The estimand that PSM is typically designed to recover. ATE measures the mean difference in outcomes between the scenario where all units receive treatment and the scenario where all units receive control. PSM estimates the Average Treatment Effect on the Treated (ATT) when matching treated units to controls.

• ATE = E[Y(1) - Y(0)] across the full population
• ATT = E[Y(1) - Y(0) | T=1] for the treated subpopulation
• ATE and ATT diverge when treatment effects are heterogeneous
• PSM with caliper matching typically targets ATT

An extraneous variable that influences both the treatment assignment and the outcome, creating a spurious association. PSM aims to balance these confounders across treatment and control groups. The validity of PSM rests on the conditional independence assumption (unconfoundedness): that all confounders are observed and included in the propensity score model.

• Unmeasured confounders violate the key identifying assumption
• Sensitivity analysis tests how strong an unmeasured confounder must be to nullify results
• Directed Acyclic Graphs (DAGs) help identify the minimal sufficient adjustment set

An adaptation of the random forest algorithm that estimates heterogeneous treatment effects by recursively partitioning data to find subgroups with distinct causal responses. While PSM estimates a single average effect, causal forests discover which units benefit most from treatment.

• Built on the Generalized Random Forest framework
• Provides valid confidence intervals via gradient-based inference
• Naturally handles high-dimensional covariate spaces
• Outperforms PSM when treatment effects vary substantially across subpopulations

A quasi-experimental technique that estimates treatment effects by comparing the change over time in outcomes for a treatment group versus a control group. Unlike PSM, which relies solely on cross-sectional covariate balancing, DiD exploits longitudinal variation and removes time-invariant unobserved confounders.

• Requires parallel trends assumption: outcomes would have evolved similarly absent treatment
• Often combined with PSM to create matched DiD designs
• Handles unobserved time-invariant confounders that PSM cannot address
• Event study plots visualize dynamic treatment effects over time