Doubly Robust Estimation

By the end of this lesson, you should be able to: write down a doubly robust estimator and say what each half is doing, name exactly which kind of error the second chance protects you from, and recognise the situation where having two models buys you nothing at all.

The promise

Lesson 5 modelled the outcome. Lessons 6 and 7 modelled the treatment. Both were guesses that could be wrong in their own way, so the obvious question is whether fitting both gives you anything.

It does, and the claim is specific. Fit an outcome model $\mu_0(X)$ for what untreated teams would have shipped, fit a propensity model $e(X)$, and combine them:

$\hat\tau_{ATT} = \frac{1}{n_1}\sum_i \left[ T_i\left(Y_i - \hat\mu_0(X_i)\right) - (1-T_i)\,\frac{\hat e(X_i)}{1-\hat e(X_i)}\left(Y_i - \hat\mu_0(X_i)\right) \right]$

Read it as the outcome model's answer, corrected by the residuals it left behind. If $\mu_0$ is right, those residuals average to zero whatever the weights are. If the weights are right, they reproduce the control mean whatever $\mu_0$ does. Get either one right and the bias dies.12

That's a real theorem, and it's why this estimator is everywhere. It's also routinely oversold, so the rest of this lesson is about what it actually promises.

What it does with the data you have

Start with the honest situation. Same 6,000 Alder Cloud teams, same four covariates, no ability column because there never is one. Fit both models on those four and combine.

Doubly robust bought nothing. It landed between the two estimators it was built from, three times the truth, exactly where each of them already was.

That isn't a failure of the estimator, it's the theorem working as stated. Two chances to be right is worth something only if one of them is live, and here neither is: both models are missing the same variable.