AIPW Under Correct and Misspecified Nuisance Models

Why Nuisance-Model Specification Matters

The augmented inverse probability weighted (AIPW) estimator combines a propensity-score model with outcome-regression models. Its double-robustness property is a statement about consistency: under standard assumptions, the estimator can converge to the average treatment effect (ATE) if either nuisance model is correct. The property does not say that every combination of fitted models produces a reliable estimate, or that a correct model is guaranteed to make a finite-sample estimate stable.

To see exactly what the guarantee covers, it helps to work with the large-sample limits of the fitted models. When both fitted nuisance models are wrong, the AIPW estimator generally converges to a quantity different from the ATE. A bias identity makes that discrepancy explicit and shows why its size and direction depend on both model errors.

Setup and the AIPW Probability Limit

Let \(A\) be a binary treatment, with \(A=1\) for treated and \(A=0\) for untreated. Let \(X\) be measured pre-treatment covariates, and let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and control. The observed outcome is \(Y=AY(1)+(1-A)Y(0)\). The target is the ATE, \(\tau=E[Y(1)-Y(0)]\).

Write the true propensity score as \(e_0(X)=P(A=1\mid X)\), and the true conditional outcome means as \(m_{a0}(X)=E[Y\mid A=a,X]\) for \(a\in\{0,1\}\). Let \(e_*(X)\) and \(m_{a*}(X)\) denote the probability limits of the fitted propensity score and outcome regression. A star does not assert correctness: it allows a fitted model to converge to a misspecified limit.

Assumptions and definition: The causal interpretation here relies on consistency, conditional exchangeability \((Y(1),Y(0))\perp A\mid X\), and positivity \(0<e_0(X)<1\) on the target population. Also assume independent, identically distributed observations; finite expectations needed below; fitted propensity limits strictly between zero and one with integrable AIPW terms; and nuisance fits that converge sufficiently regularly for the sample average to have the stated probability limit. The true conditional mean \(m_{a0}(X)\) is identified with \(E[Y\mid A=a,X]\) under consistency and exchangeability.

For fitted functions \(\hat e\) and \(\hat m_a\), the AIPW score for one observation is

\hat\phi(O)=\hat m_1(X)-\hat m_0(X) +\frac{A}{\hat e(X)}\{Y-\hat m_1(X)\} -\frac{1-A}{1-\hat e(X)}\{Y-\hat m_0(X)\}.

The estimator is the sample average of these scores. Under the assumptions above, its probability limit is the expectation of the same score with the fitted functions replaced by their limits:

\tau_* = E\left[ m_{1*}(X)-m_{0*}(X) +\frac{A}{e_*(X)}\{Y-m_{1*}(X)\} -\frac{1-A}{1-e_*(X)}\{Y-m_{0*}(X)\} \right].

The target \(\tau\) equals \(E[m_{10}(X)-m_{00}(X)]\). Comparing it with \(\tau_*\) gives the central identity for this tutorial. Define the outcome-model errors \(\delta_a(X)=m_{a*}(X)-m_{a0}(X)\). Conditioning the score on \(X\), the treated residual has expectation \(e_0(X)\{m_{10}(X)-m_{1*}(X)\}\), while the control residual has expectation \(\{1-e_0(X)\}\{m_{00}(X)-m_{0*}(X)\}\). Substitution and collection of terms yield:

\tau_*-\tau = E\left[ \{e_*(X)-e_0(X)\} \left\{ \frac{\delta_1(X)}{e_*(X)} +\frac{\delta_0(X)}{1-e_*(X)} \right\} \right].

This identity states that the limiting bias depends on the product of propensity-model error and outcome-model error, with weights determined by the fitted propensity limit. It also fixes the signs: the errors are defined as fitted limit minus true conditional mean, not the reverse.

What Happens When the Models Are Correct?

If both nuisance models are correct, then \(e_*=e_0\) and \(\delta_1=\delta_0=0\), so the identity gives \(\tau_*-\tau=0\). Either condition on its own is enough for the same conclusion. If the propensity model is correct, the first factor is zero. If both outcome regressions are correct, both outcome errors are zero. This is the large-sample basis of double robustness.

Key result: Under the stated identification and regularity assumptions, AIPW is consistent for the ATE if the propensity-score model is correctly specified or if both treatment-specific outcome-regression models are correctly specified. If neither nuisance model is correct, the limiting bias is generally nonzero; it is zero only if the expectation in the bias identity happens to vanish.

Correct specification here refers to the relevant conditional function on the target population, not merely to a model fitting well in the observed sample. In particular, an outcome model must represent the appropriate conditional mean, and the propensity model must represent the treatment probability given the covariates. The identity also shows that the two errors can interact: a small error in one model does not, on its own, establish that the bias is small if the other error or the weighting factors are large.

Worked Example: Both Nuisance Models Are Correct

Consider a hypothetical population with two equally common covariate strata, \(L\) and \(H\). Suppose the true propensity scores are \(e_0(L)=0.25\) and \(e_0(H)=0.75\). The true conditional means are \(m_{10}(L)=8\), \(m_{00}(L)=6\), \(m_{10}(H)=12\), and \(m_{00}(H)=8\). Thus the conditional treatment effects are 2 and 4 outcome units, and the ATE is

\(\tau=0.5(8-6)+0.5(12-8)=0.5(2)+0.5(4)=3\) outcome units.

Suppose the fitted models converge to these true functions. Then \(e_*-e_0=0\) in each stratum, and \(\delta_1=\delta_0=0\). The bias identity gives zero bias in each stratum. Equivalently, the probability limit is \(0.5(2)+0.5(4)=3\) outcome units, the target ATE.

This conclusion is about the large-sample limit. It does not require a finite sample to have exactly balanced covariates or an estimate equal to 3; those are different questions.

Worked Example: Correct Propensity Model, Misspecified Outcome Models

Keep the same hypothetical population. Suppose the outcome-model errors are \(\delta_1(L)=\delta_0(L)=-0.5\) and \(\delta_1(H)=\delta_0(H)=-1\), so neither pair of outcome predictions is correct. But suppose the propensity model is correct: \(e_*=e_0\) in both strata.

In stratum \(L\), \(e_*-e_0=0.25-0.25=0\). In stratum \(H\), \(e_*-e_0=0.75-0.75=0\). Therefore each stratum's contribution to the bias identity is zero, regardless of the nonzero outcome errors. The limiting ATE estimate is \(3+0=3\) outcome units.

The fitted outcome regressions are misspecified, but the correct propensity model makes the residual corrections work in expectation. This is consistency, not a claim that poor outcome predictions are harmless for every finite-sample property or every inference procedure.

Worked Example: Correct Outcome Models, Misspecified Propensity Model

Again use the same population, but suppose the fitted propensity model converges to \(e_*(L)=e_*(H)=0.50\), rather than the true values 0.25 and 0.75. Let both outcome models be correct, so \(\delta_1(L)=\delta_0(L)=\delta_1(H)=\delta_0(H)=0\).

The bias expression in each stratum is zero because its bracketed outcome-error term is zero. Thus the limiting ATE estimate remains \(3\) outcome units, even though the propensity model is wrong in both strata. For example, in stratum \(L\), the propensity error is \(0.50-0.25=0.25\), but it multiplies a bracket equal to zero; the same reasoning applies in \(H\), where the propensity error is \(0.50-0.75=-0.25\).

This is the other half of double robustness. It depends on correct conditional outcome means for both treatment levels and does not turn an arbitrarily poor propensity model into a generally safe choice.

When Both Models Are Misspecified

When both nuisance models are wrong, the two factors in the bias identity are nonzero in general. Their product may be positive in some covariate regions and negative in others, so the overall bias depends on how those contributions average over the target population. A zero average can occur by cancellation, but that is not the double-robustness guarantee: small changes in the population or fitted limits can remove the cancellation.

Worked Example: Calculating Bias When Neither Model Is Correct

Use the same two-stratum population, whose true ATE is 3 outcome units. Let the misspecified propensity limit be \(e_*(L)=e_*(H)=0.50\). Choose outcome errors \(\delta_1(L)=\delta_0(L)=-0.5\) and \(\delta_1(H)=\delta_0(H)=-1\). Both nuisance models are now incorrect.

For stratum \(L\), the true propensity is 0.25. The conditional bias contribution is

\((0.50-0.25)\{(-0.5)/0.50+(-0.5)/(1-0.50)\}=0.25(-1-1)=-0.50\) outcome units.

For stratum \(H\), the true propensity is 0.75. The contribution is

\((0.50-0.75)\{(-1)/0.50+(-1)/(1-0.50)\}=(-0.25)(-2-2)=1.00\) outcome units.

Weighting by the stratum probabilities gives total bias \(0.5(-0.50)+0.5(1.00)=0.25\) outcome units. Therefore the AIPW probability limit is \(3+0.25=3.25\) outcome units, not the target 3. The opposite-signed stratum contributions partly cancel, but not completely.

For a sign check, the \(L\) contribution must be negative: the propensity error is positive, while the bracket is negative. Reversing either sign would give the wrong conditional bias and the wrong overall probability limit.

Diagnostics, Comparisons, and Limits

The double-robustness result is an asymptotic statement under identification, positivity, and regularity assumptions. It does not tell us which nuisance model is correct in a particular application. Nor can diagnostics certify that an unobserved confounder is absent or that conditional exchangeability holds. Model checks can reveal problems and guide revisions, but they cannot prove the assumptions required for causal identification.

Practical warning signs include a propensity fit that predicts treatment probabilities near zero or one, outcome-model predictions that are implausible in parts of the covariate space, and sensitivity of the estimate to reasonable model changes. Good balance on measured covariates is useful evidence about a fitted adjustment, but it does not establish correct outcome regressions or guarantee that the propensity model is correctly specified. Sparse overlap also makes the weighted residual corrections sensitive to individual observations; the large-sample identity should not be mistaken for a finite-sample stability guarantee.

Compared with outcome regression alone, AIPW adds weighted residual corrections and can remain consistent if the propensity model is correct when the outcome model is wrong. Compared with inverse probability weighting alone, it also uses outcome predictions and can remain consistent if the outcome regressions are correct when the propensity model is wrong. If both nuisance models are correct, AIPW has the standard efficient influence function under the relevant regularity conditions. If one or both limits are misspecified, consistency alone does not imply that an efficient influence-function standard error is valid; inference must reflect the fitted estimator and the conditions under which it is used.

The earlier material on influence-function standard errors explains how standard errors are formed when the relevant influence-function approximation applies. A misspecification-robustness argument for the point estimate is not, by itself, a proof that a particular variance estimate, confidence interval, or test is valid. Likewise, double robustness does not protect against confounding from omitted covariates, violations of positivity, incorrect treatment or outcome data, or arbitrary simultaneous model errors.

State the target and assumptions.
Identify the ATE, the measured covariates, and the identification and positivity assumptions needed for a causal interpretation.

Assess each nuisance model separately.
Ask whether the propensity limit equals \(e_0(X)\) and whether both outcome-regression limits equal their corresponding \(m_{a0}(X)\).

Apply the large-sample result narrowly.
If either model is correct, AIPW is consistent under the stated conditions. If neither is correct, use the bias identity to understand the possible limiting discrepancy rather than assuming it cancels.

Common pitfall: “AIPW is doubly robust, so it is consistent even if both models are wrong.” This is incorrect. The guarantee requires at least one correct nuisance component (the propensity model or both outcome regressions). With both wrong, the bias identity generally has a nonzero expectation; an accidental zero from cancellation is possible, but it is not guaranteed by double robustness.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

If \(e_*=e_0\) but the outcome models are wrong, what does the bias identity imply about the AIPW probability limit?
If both outcome regressions are correct but \(e_*\ne e_0\), which part of the bias identity ensures consistency?
Suppose a hypothetical population has two equally common strata. The ATE is 4 outcome units. Their conditional bias contributions are \(-0.2\) and \(0.6\). What is the overall limiting bias and AIPW probability limit?
Why does a zero average bias when both models are wrong not establish double robustness?

Solutions. (1) The propensity-error factor \(e_*-e_0\) is zero, so the expectation of the bias expression is zero; the probability limit is the ATE, subject to the stated assumptions. (2) Both outcome errors are zero, so the bracketed term is zero regardless of the propensity-model error; the estimator is consistent for the ATE. (3) The overall bias is \(0.5(-0.2)+0.5(0.6)=-0.1+0.3=0.2\) outcome units. The probability limit is \(4+0.2=4.2\) outcome units. (4) With both models wrong, the bias contributions may cancel for a particular population and pair of limits, but that cancellation is not guaranteed. Double robustness guarantees consistency when at least one nuisance component is correctly specified, not when both are incorrect.