Doubly Robust Estimation

Two colleagues estimate the effect of a new margining policy on client attrition. One builds a regression model of attrition as a function of client covariates and policy exposure; the other builds a propensity-score model of who was exposed to the new policy and reweights (inverse probability weighting). Both approaches are correct in theory, but both rely on a model being right — the outcome regression's functional form, or the propensity model's specification — and there's no way to know in advance which, if either, is well specified. Doubly robust estimation asks: can we build one estimator that only needs one of these two models to be correct, without knowing in advance which one that will be?

An analogy: two independent locks on one door

Imagine a safe with two separate locks, either of which alone is enough to open it — a keypad code and a physical key. You don't need both to work; you need at least one. If the keypad's battery dies (the propensity model is wrong) but you still have the key (the outcome model is right), you get in fine, and vice versa. A doubly robust estimator is built the same way: it combines an outcome model and a propensity model such that if either one alone is correctly specified, the combined estimate is still consistent — you only fail if both locks fail simultaneously.

The estimator, one symbol at a time

Let $\hat{\mu}_1(X_i)$ and $\hat{\mu}_0(X_i)$ be predicted outcomes under treatment and control from an outcome regression model, and $\hat{e}(X_i)$ the estimated propensity score. The augmented inverse probability weighting (AIPW) estimator is:

$$\widehat{\text{ATE}} = \frac{1}{N}\sum_{i=1}^N \left[ \hat{\mu}_1(X_i) - \hat{\mu}_0(X_i) + \frac{D_i (Y_i - \hat{\mu}_1(X_i))}{\hat{e}(X_i)} - \frac{(1-D_i)(Y_i - \hat{\mu}_0(X_i))}{1-\hat{e}(X_i)} \right]$$

In plain English: start with the outcome model's prediction of the treatment effect for everyone, then add a correction term that uses the propensity weights to fix up any residual error the outcome model made — but only for the residual, not the whole prediction. If the outcome model is perfect, the residuals $Y_i - \hat{\mu}(X_i)$ are essentially noise and the correction term averages to roughly zero, leaving the (correct) outcome-model estimate. If instead the propensity model is perfect, the weighted-residual term exactly cancels the outcome model's bias, regardless of how wrong $\hat{\mu}$ was. Either model being right is enough.

Worked example 1: outcome model right, propensity model wrong

Suppose true attrition rates are 5% (treated) and 8% (control) for every client, and the outcome model correctly learns $\hat{\mu}_1 = 0.05$, $\hat{\mu}_0 = 0.08$ for everyone — no residuals, since the model is exactly right. Even if the propensity model is badly misspecified (say it estimates $\hat{e}(X_i) = 0.5$ for everyone when the truth varies wildly), the correction terms $D_i(Y_i - \hat\mu_1)/\hat e$ and $(1-D_i)(Y_i-\hat\mu_0)/(1-\hat e)$ are built from residuals $Y_i - \hat\mu(X_i)$ that are just mean-zero noise (since the outcome model is exactly right), so they average out regardless of the (wrong) weights multiplying them. The AIPW estimate collapses to $\hat\mu_1 - \hat\mu_0 = 0.05 - 0.08 = -0.03$, the correct ATE of −3 percentage points, unaffected by the broken propensity model.

Worked example 2: propensity model right, outcome model wrong

Now flip it: suppose the true propensity score is correctly estimated, $\hat{e}(X_i)$ exactly right for every client, but the outcome model is badly wrong — say it predicts $\hat\mu_1(X_i) = \hat\mu_0(X_i) = 0.065$ for everyone (a flat, uninformative guess, missing all real variation). The first two terms of the AIPW formula contribute a wrong, flat ATE estimate of 0. But the residual-weighted correction terms use $Y_i - \hat\mu(X_i)$, which is now the entire true effect (since $\hat\mu$ carries no signal) reweighted by the correct propensity score — exactly reconstructing standard IPW, which (with a correct propensity model) recovers the true −3 point ATE on its own. The flawed outcome model's bias gets fully corrected by the properly weighted residuals.

What this means in practice

Doubly robust estimation is the practical default when analysts are uncertain which model — the outcome regression or the propensity score — they trust more, which is nearly always. It's widely used in modern applied causal inference (and underlies methods like targeted maximum likelihood estimation), precisely because in real data neither a regression's functional form nor a propensity model's specification is ever provably correct; doubly robust methods hedge against getting either one wrong, at the modest cost of needing to fit two models instead of one. It does not protect against both models being wrong simultaneously, nor against violations of ignorability itself (see ignorability and unconfoundedness) — no amount of double robustness fixes an unmeasured confounder.