AIPW and TMLE Under Nuisance-Model Misspecification

Why Misspecification Requires More Than a “Doubly Robust” Label

AIPW and TMLE combine an outcome regression with a propensity score to estimate a causal effect. Their double-robustness property is important: under standard conditions, either nuisance model can be misspecified while the average treatment effect (ATE) estimator remains consistent, provided the other nuisance model is correctly estimated. But this statement is not a guarantee that any combination of working models will succeed. If both nuisance limits are wrong, the estimators can converge to different, incorrect values. And consistency under one correct model does not, by itself, establish that an estimator is efficient.

The distinction is especially visible in TMLE. It updates an outcome regression so its predictions satisfy a targeting equation, then calculates the effect by substituting those updated predictions into the estimand. This substitution structure has useful properties, but the targeting equation is not a mechanism that automatically corrects two misspecified nuisance models.

Setup and the Two Estimators

Let \(A\) be a binary treatment, \(Y\) an outcome, and \(W\) baseline covariates. Write \(Y(1)\) and \(Y(0)\) for the potential outcomes under treatment and control. The target is the population ATE, \(\psi_0=E[Y(1)-Y(0)]\). For identification from observed data, assume consistency, conditional exchangeability \((Y(1),Y(0))\perp A\mid W\), and positivity: \(0<P(A=1\mid W)<1\) for covariate values in the target population. These assumptions identify the ATE as \(E[Q_0(1,W)-Q_0(0,W)]\), where \(Q_0(a,W)=E[Y\mid A=a,W]\).

For fitted nuisance estimates \(\hat g\) and \(\hat Q\), the AIPW estimator is

TMLE starts with an initial estimate of \(Q_0\), updates it using a fluctuation model involving the fitted propensity score, and reports the plug-in estimate

where \(\hat Q^*\) is the targeted outcome regression. In a standard ATE targeting step, the clever covariate is \(A/\hat g(W)-(1-A)/(1-\hat g(W))\). The update is chosen to make the empirical mean of this covariate times the outcome residual \(Y-\hat Q^*(A,W)\) equal to zero, or to solve the corresponding likelihood score equation. The precise fluctuation model depends on the outcome type.

What Double Robustness Says

Consider the probability limit of AIPW when the fitted nuisance functions converge to fixed limits \(g^*\) and \(Q^*\). Conditional on \(W\), the expectation of the AIPW expression is

If \(g^*=g_0\), the residual terms correct the difference between the fitted and true outcome regressions, and this conditional expectation reduces to \(Q_0(1,W)-Q_0(0,W)\). If \(Q^*=Q_0\), both residual terms have conditional expectation zero, regardless of whether \(g^*=g_0\). Taking expectations over \(W\) gives the true ATE in either case.

The same two cases explain TMLE’s double robustness. If the initial outcome regression is correct, its population residual has mean zero conditional on \(A,W\), so the targeting fluctuation has no population residual signal to correct; the regular targeting solution leaves the outcome regression at its truth. If the propensity score is correct, the population targeting equation has a different role: it makes the difference between the targeted plug-in contrast and the true ATE equal to zero. The calculation is given below.

Worked Example: Correct Propensity Score, Misspecified Outcome Regression

Why TMLE’s Targeting Equation Helps—and Where It Stops

For the ATE, let \(Q^*\) denote the population limit of the targeted outcome regression, and suppose the targeting step solves the population version of its score equation:

If \(g^*=g_0\), apply consistency and exchangeability to the two residual terms. Their expectations become \(E[Q_0(1,W)-Q^*(1,W)]\) and \(E[Q_0(0,W)-Q^*(0,W)]\), respectively. Therefore the targeting equation is

Rearranging shows that \(E[Q^*(1,W)-Q^*(0,W)]=E[Q_0(1,W)-Q_0(0,W)]\). Thus, with a correct propensity score, a targeted plug-in estimate is consistent even if the initial outcome regression was not correct, assuming the targeting procedure reaches a suitable solution.

If \(g^*\) is wrong as well as the initial outcome regression, the weighted residual equation no longer equals the difference in outcome-regression errors. It still imposes a score constraint, but that constraint need not make the plug-in contrast correct. This is the precise limit of the targeting argument: a solved equation is not the same as a correctly identified causal contrast.

Worked Example: Correct Outcome Regression, Misspecified Propensity Score

Worked Example: Both Nuisance Limits Are Wrong

Substitution Behavior Is Not the Same as Double Robustness

TMLE is called a substitution estimator because its final estimate substitutes fitted outcome predictions into the identified form of the parameter. For a binary outcome, a logistic fluctuation keeps predictions between 0 and 1, so the resulting ATE lies between -1 and 1. AIPW, by contrast, adds weighted residual corrections to an initial contrast and can produce a value outside this range. Respecting the parameter’s range can be valuable, but it does not guarantee that the estimate is unbiased, consistent, or efficient under misspecification.

There is also a useful algebraic connection. If a TMLE targeting step makes the empirical AIPW residual correction, evaluated at the targeted regression \(\hat Q^*\) and the same \(\hat g\), exactly zero, then the AIPW expression using \(\hat Q^*\) equals the TMLE plug-in estimate. This does not mean the AIPW estimator using the initial regression equals TMLE, nor does it make either estimate correct when both nuisance limits are wrong.

Consistency Is Not an Efficiency Guarantee

Under suitable regularity conditions, when both nuisance estimates converge to their true functions at appropriate rates, the AIPW and TMLE estimators can be asymptotically linear with the efficient influence function for the ATE. That supports efficient estimation and the corresponding influence-function-based variance calculation. When one nuisance limit is misspecified, double robustness may still give consistency, but the estimator’s first-order sampling variation need not be the efficient influence function’s variation. It is therefore not valid to infer efficiency merely from consistency.

In particular, an EIF-based standard error computed using a misspecified nuisance fit is not automatically justified by the double-robustness result. The variance procedure must be valid for the estimator’s actual large-sample behavior under the working models and estimation procedure. Cross-fitting and careful variance estimation address important estimation and inference issues, but they do not make a misspecified nuisance model correct.

Common Mistakes

• “Doubly robust means safe if both models are wrong.” It means consistency can hold when either one of the two nuisance functions is consistently estimated, under the stated conditions. It gives no general guarantee when both limits are wrong.
• “TMLE solves its score, so its answer must be right.” A score equation is a constraint. Its connection to the true ATE depends on conditions such as a correct propensity score or a correct outcome regression.
• “A bounded TMLE is necessarily better.” Substitution can respect parameter bounds, but a bounded estimate can still be biased or inconsistent.
• “Consistency implies efficiency.” Double robustness is a consistency property. Efficiency requires additional conditions, including suitable nuisance estimation and regularity.