Summary Notes for DML

Apr 24, 2024 1 min read causal inference, econometrics

Statistical Setting

We observe data $(X_i, Y_i, W_i) \in \mathcal{X} \times \mathbb{R} \times {0,1}$ according to the potential outcomes model and we assume the following:

(SUTVA)
${Y_i(0), Y_i(1)} \perp W_i | X_i \quad$ (Unconfoundedness)
$0 < e(X) < 1$, where $e(X)$ is the propensity score $\quad$ (Overlap)

We can use regression adjustment, IPW and AIPW to estimate $\tau = \mathbb{E}[Y_i(1) - Y_i(0)]$. Here, we define $\mu(w_i, X_i) := \mathbb{E}(Y_i(w_i)|X_i)$, $$ \begin{aligned} \tau= & \mathbb{E}\left[\mu\left(1, X_i\right)-\mu\left(0, X_i\right)\right] \ = & \mathbb{E}\left[\frac{Y_i W_i}{e\left(X_i\right)}-\frac{Y_i\left(1-W_i\right)}{1-e\left(X_i\right)}\right] \ = & \mathbb{E}\left{\frac{\left[Y_i-\mu\left(1, X_i\right)\right] W_i}{e\left(X_i\right)}-\frac{\left[Y_i-\mu\left(0, X_i\right)\right]\left(1-W_i\right)}{1-e\left(X_i\right)}\right} \ & + \mathbb{E}\left[\mu\left(1, X_i\right)-\mu\left(0, X_i\right)\right] \end{aligned} $$

Unfortunately, neither regression adjustment (first equation) nor IPW estimator (second equation) is Neyman orthogonal, making them unsuitiable for plugging-in machine learning estimators. In sharp contrast, AIPW estimator (third equation) is both Neyman orthogonal and “double robust”.