Answer-first: Propensity scores balance covariates by reducing a high-dimensional selection problem into a single number: the probability of receiving treatment given observed characteristics. When treated and control units are matched or weighted on this score, their covariate distributions become similar, making treatment assignment “as good as random” conditional on observables.
Warm intro (and where to find the “why” pages)
If you’re staring at a propensity score question at 11:47pm, feeling stuck, behind, or low-key panicking because the matching logic feels abstract, you’re not alone. Propensity score problems look straightforward—“estimate a probability and match”—but under exam pressure, students freeze when asked to explain why this single score balances covariates and how it reduces selection bias.
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Answer first
Propensity scores balance covariates because units with the same propensity score have the same distribution of observed characteristics. By matching, weighting, or subclassifying on the propensity score, we replicate the covariate balance of a randomized experiment—but only with respect to observed variables.
Problem setup
Let \(D_i\) be a binary treatment and \(X_i\) a vector of observed covariates. The propensity score is:
\[ e(X_i) = \Pr(D_i = 1 \mid X_i). \]
The key identification assumption is selection on observables:
\[ (Y_i(0), Y_i(1)) \perp D_i \mid X_i. \]
This means that once we control for \(X_i\), treatment assignment is as good as random.
Step-by-step solution (WordPress-safe MathJax)
Step 1: Reduce high-dimensional covariates to a single score
Instead of matching on all components of \(X_i\), we match on the scalar \(e(X_i)\). Rosenbaum and Rubin (1983) showed that:
\[ D_i \perp X_i \mid e(X_i). \]
This is the balancing property.
Step 2: Balance covariates using matching or weighting
Once we estimate \(e(X_i)\), we can:
- match treated and control units with similar scores,
- weight observations by inverse probability of treatment, or
- subclassify units into propensity score bins.
All three methods create covariate balance.
Step 3: Estimate the treatment effect
After balancing, we estimate:
\[ \tau = \mathbb{E}[Y_i(1) – Y_i(0)]. \]
Because treated and control units now look similar on observables, differences in outcomes can be attributed to treatment.
Step 4: State the identifying assumption clearly
Propensity scores only work if:
\[ (Y_i(0), Y_i(1)) \perp D_i \mid X_i. \]
This is also called unconfoundedness or conditional independence.
Importantly: propensity scores do not fix unobserved confounding.
Intuition
Propensity scores mimic random assignment by ensuring that treated and control units have similar observed characteristics