Orthogonal Pseudo-Outcomes for CATE Learning

From Treatment-Effect Targets to Pseudo-Outcomes

A conditional average treatment effect is a function of observed covariates: it describes how the average effect varies across covariate profiles. Learning this function requires a way to connect observed outcomes, which reveal only one treatment condition for each unit, to the unobserved contrast of interest. An orthogonal pseudo-outcome provides that connection: under appropriate conditions, its conditional mean given the covariates equals the conditional treatment effect.

This construction combines an outcome model with inverse-probability weighting. The outcome model predicts the outcome under each treatment condition; the weights use the treatment propensity to correct residuals. This augmentation gives the pseudo-outcome a useful robustness property: its conditional expectation remains the target if either the propensity model is correct or both outcome models are correct.

For a causal interpretation, assume consistency, so \(Y=WY(1)+(1-W)Y(0)\); conditional exchangeability, so \((Y(1),Y(0))\) is independent of \(W\) given \(X\); and positivity, so \(0<e_0(x)<1\) for covariate values in the target population. These assumptions identify the CATE as \(\tau(x)=\mu_{1,0}(x)-\mu_{0,0}(x)\). They do not ensure that a particular fitted model will estimate this function well; that also depends on the data and learning method.

The Augmented Inverse-Probability-Weighted Pseudo-Outcome

Let \(q_1(x)\) and \(q_0(x)\) be candidate outcome models, and let \(g(x)\) be a candidate propensity model. For an observation \(O=(Y,W,X)\), define the augmented inverse-probability-weighted (AIPW) pseudo-outcome as follows.

The leading difference \(q_1(X)-q_0(X)\) supplies an initial estimate of the treatment contrast. For a treated observation, the treated residual \(Y-q_1(X)\) adjusts that initial estimate; for a control observation, the control residual enters with a minus sign. The inverse propensity factors adjust for the fact that treatment assignment probabilities may differ across covariate profiles. The result is an outcome-like quantity, but it is not an individual treatment effect: even under the assumptions above, its conditional mean—not each realized value—is the CATE.

Conditional Mean, Robustness, and Orthogonality

The key property follows by taking a conditional expectation given \(X=x\). Since \(E[W\mid X=x]=e_0(x)\), and the observed outcome has conditional mean \(\mu_{w,0}(x)\) in treatment arm \(w\), the pseudo-outcome has conditional expectation

To express its deviation from \(\tau(x)=\mu_{1,0}(x)-\mu_{0,0}(x)\), define outcome-model errors \(\delta_1(x)=q_1(x)-\mu_{1,0}(x)\) and \(\delta_0(x)=q_0(x)-\mu_{0,0}(x)\). Subtracting the true contrast gives

In the last equality, the coefficient of \(\delta_1\) is \(1-e_0/g=(g-e_0)/g\). The coefficient of \(\delta_0\) is \(-1+(1-e_0)/(1-g)=(g-e_0)/(1-g)\), which explains the plus sign between the two terms in braces.

Double robustness is a statement about the conditional expectation, not a guarantee that an estimate is accurate under arbitrary misspecification. If both nuisance models are wrong, the displayed bias formula shows how their errors combine. It also shows local orthogonality: around the truth, the leading bias is a product of a propensity error and an outcome-model error, rather than a first-order error in either one alone.

The orthogonality implication can be seen by writing \(g(x)=e_0(x)+\eta(x)\), \(q_1(x)=\mu_{1,0}(x)+\delta_1(x)\), and \(q_0(x)=\mu_{0,0}(x)+\delta_0(x)\). Where the denominators are bounded away from zero, the bias is

For small errors, this is of product order: to first order, perturbing only \(g\) while holding the outcome models correct gives no bias, and perturbing only the outcome models while holding \(g=e_0\) gives no bias. This is the sense in which the score is locally orthogonal to nuisance-model errors. It does not mean errors cancel in every finite sample, nor does it remove the need for adequate overlap or sufficiently accurate nuisance estimates.

Learning a CATE by Regression

Suppose nuisance estimates \(\widehat q_1(x)\), \(\widehat q_0(x)\), and \(\widehat g(x)\) have been obtained. Compute one pseudo-outcome for each observation:

Then fit a regression or other conditional-mean learner with \(\widehat\Gamma_i\) as the response and \(X_i\) as predictors. The population target of squared-error regression is \(E[\Gamma\mid X=x]\): for any fixed \(x\), the value \(m\) minimizing \(E[(\Gamma-m)^2\mid X=x]\) is the conditional mean. Under the identification assumptions and either nuisance-model correctness condition, that conditional mean is \(\tau(x)\). Thus regression on the pseudo-outcome turns CATE learning into a conditional-mean prediction problem.

Practical Checks and Common Pitfalls

Inverse-probability factors can make pseudo-outcomes highly variable when fitted propensities are near zero or one. This is a practical consequence of weak overlap: some covariate profiles provide little information about one treatment condition. Inspect propensity estimates and the distribution of the pseudo-outcomes, and be explicit about any stabilization or trimming choices, since such choices can change the analysis and sometimes the target population.

A second issue is that nuisance models and the final regression are estimated rather than known. Flexible models can overfit, and using the same observations at every fitting stage can complicate finite-sample behavior. The orthogonality calculation explains why certain nuisance errors have reduced first-order impact; it does not by itself guarantee good performance with overfitting, poor overlap, or a final regression that cannot represent the CATE pattern.

• Mistake: treating \(\widehat\Gamma_i\) as an individual treatment effect. A pseudo-outcome is a noisy constructed response. Its conditional mean, not its observation-specific value, targets the CATE.
• Mistake: claiming robustness when both nuisance components are wrong. Double robustness requires a correct propensity model or both correct outcome models. If both are wrong, the product-bias formula generally does not vanish.
• Mistake: dropping the minus sign before the control residual. The definition uses \(-(1-W)\{Y-q_0(X)\}/\{1-g(X)\}\). With errors written as \(q_0-\mu_{0,0}\), the resulting control contribution to the bias has a plus sign.
• Mistake: interpreting orthogonality as immunity to error. Orthogonality removes first-order sensitivity at the truth under suitable conditions; it does not remove finite-sample bias, variance, or overlap problems.