Chapter 9 Doubly Robust Methods

Doubly Robust (DR) estimators are a class of causal inference methods that combine two modeling strategies—

1. Outcome regression models and

2. Treatment assignment (propensity score) models.

The key property is double robustness: the estimator of the causal effect remains consistent if either the outcome model or the treatment model is correctly specified (not necessarily both).

9.1 Why They Matter

• Problem: Using only outcome regression (risk of omitted confounders) or only propensity scores (risk of poor overlap/imbalance) can lead to bias.

• Solution: DR estimators merge both approaches, providing a safety net: if one model fails but the other is correct, the ATT/ATE estimate is still consistent.

• Applications: Widely used in observational studies, program evaluation, healthcare, education, and labor economics.

9.2 Core Ingredients

1. Outcome Model

   • Predicts the expected outcome given covariates and treatment:

     \[ \hat{m}(X,D) = \hat{E}[Y \mid X, D] \]

   • Example: linear regression, random forest, or other machine learning regressors.

2. Treatment Model (Propensity Score Model)

   • Models the probability of treatment given covariates:

     \[ \hat{p}(X) = P(D = 1 \mid X) \]

   • Example: logistic regression, boosted trees.

3. DR Estimator

   • Combines the two:

     \[ \hat{\theta}_{DR} = \frac{1}{n} \sum_{i=1}^n \Big[ \hat{m}(X_i, D_i) \;+\; \frac{D_i \cdot (Y_i - \hat{m}(X_i,1))}{\hat{p}(X_i)} \;-\; \frac{(1-D_i)\cdot (Y_i - \hat{m}(X_i,0))}{1-\hat{p}(X_i)} \Big] \]

   • Intuition: outcome model “predicts” \(Y\); propensity weighting corrects residual differences.

9.3 More on Outcome Models for Treated and Control

In the outcome regression step, the goal is to estimate what the outcome would be for each unit under both treatment states (\(D=1\) and \(D=0\)).

This means we need:

• \(\hat{m}_1(X) = \hat{E}[Y \mid D=1, X]\) (predicted outcome if treated)

• \(\hat{m}_0(X) = \hat{E}[Y \mid D=0, X]\) (predicted outcome if untreated)

There are two ways to get these:

1. Single pooled regression (with treatment indicator):

   • Fit one model:

     \[ Y \sim D + X + D \times X \]

   • This allows different intercepts/slopes by treatment status.

2. Separate regressions by group (common in practice):

   • Fit one regression for treated (\(D=1\)) and one for controls (\(D=0\)).

   • Then use each model to predict \(\hat{m}_1(X)\) and \(\hat{m}_0(X)\) for all units.

Both approaches are trying to achieve the same thing: predict counterfactual outcomes for each unit under both treatment states.

9.3.1 Why Some Articles Emphasize Separate Models

• It makes the intuition cleaner: you estimate the outcome model within each group, then apply it across the sample.

• It avoids strong functional form assumptions (a single pooled regression assumes the same structure across treated and controls, unless you include interactions).

• In machine learning implementations (e.g., random forests, boosting), it’s natural to just train two models — one per treatment status.

9.3.2 Key Point

👉 The distinction isn’t about the logic of DR, but about the implementation.

• Econometrics textbooks (like Wooldridge) often write it in pooled form for simplicity.

• Applied ML papers or software implementations often do separate regressions for treated vs. controls because it’s flexible.

9.3.3 Takeaway for Your Notebook

You can phrase it like this:

In practice, the outcome regression can be specified in two ways:

1. one pooled model with treatment indicators and interactions, or

2. separate models for treated and controls.

Both approaches aim to estimate the conditional expectation of outcomes under treatment and control, which are then combined with the propensity score model in the doubly robust estimator.

9.4 Advantages

1. Robustness: Consistency as long as one model is correctly specified.

2. Efficiency: Lower variance than using outcome regression or IPW alone.

3. Flexibility: Can use linear models or modern ML for either part.

9.5 Assumptions

1. Overlap: For all covariate values, both treated and untreated units exist.

2. No unmeasured confounding: All relevant confounders are observed.

3. Consistency/SUTVA: Treatment assignment is well-defined, no interference.

9.6 Example Applications

• Healthcare: Effect of new drug adoption on patient recovery, adjusting for patient characteristics.

• Labor Economics: Effect of training programs on wages when participants self-select.

• Education: Effect of tutoring on test scores, controlling for student demographics and school resources.

9.7 Summary

Doubly robust estimators are widely considered a best practice in causal inference:

• They provide a safeguard against misspecification.

• They unify regression and weighting methods.

• They extend naturally into advanced designs (e.g., DR DiD, Double Machine Learning).

They don’t remove the need for strong design and good covariates, but they improve the credibility and stability of causal effect estimates.

9.8 Extension: DR in Difference-in-Differences (DRDID)

• Extends DR logic to DiD setups.

• Estimates the ATT when parallel trends may hold only after conditioning on covariates.

• Uses both:

  • Outcome regression in pre/post periods, and

  • Propensity scores for treatment assignment.

• Still consistent if either the outcome regression or the propensity score model is correct.