What is a propensity score?

hard

Answer

$e(x)\; = \;P(T\; = \;1\; \mid \;X\; = \;x)$
$(\mathrm{dimension}\;\mathrm{reduction}\;\mathrm{from}\; \mid X \mid \;\mathrm{to}\;1)$
$Uses: (1) matching on e(x), (2) IPW: weight treated by 1/e(x), control by 1/(1-e(x)) \to weighted average = ATE.$
$(3) Stratification by e(x) quintiles.$
$Estimated with logistic regression or ML.$
$Never use predicted probabilities of 0 or 1 (violates overlap \to wild variance).$

Check yourself — multiple choice

Random
$e(x)\; = \;P(T = 1 \mid X = x)$ ; Rosenbaum-Rubin: adjust on e(x) alone under unconfoundedness → matching / IPW / stratification; avoid e(x) near 0 or 1
Same as OLS
Not real

Propensity score: $e(x) = P(T = 1 \mid X)$ ; adjust on it via matching / IPW / stratification.

#causal-inference

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