11 Doubly Robust Estimation
How to read this chapter. Section 11.2–Section 11.4 (the AIPW construction, double robustness, the exact remainder, and the optimal-augmentation class with its perfect-square variance argument), Section 11.7 (the efficient influence function and the efficiency bound), Section 11.10 (asymptotic inference with estimated nuisances), and the simulation study of Section 11.9 form a complete first-reading path. The starred sections deepen the picture and may be deferred: Section 11.5–Section 11.6 recast the optimal augmentation as a Hilbert-space projection (the efficiency statement of Section 11.7 can be read without them and gains resonance on a second pass), and Section 11.8 is enrichment on building double robustness into the regression fit itself. The learning objectives above are tiered the same way.
11.1 Why Combine Outcome Regression and Weighting?
Under consistency, conditional exchangeability, and positivity (Chapter 6), the average treatment effect (ATE) \(\tau = \E\{Y(1) - Y(0)\}\) is identified by either of the following equivalent expressions: \[\tau = \E\{\mu_1(X) - \mu_0(X)\}, \qquad \mu_t(x) = \E(Y \mid T=t,\, X=x),\] or \[\tau = \E\!\left\{\frac{TY}{\pi(X)} - \frac{(1-T)Y}{1-\pi(X)}\right\},\] where \(\pi(x) = P(T=1 \mid X=x)\) is the propensity score. Without the causal assumptions, these expressions still define functionals of the observed-data distribution, but they do not in general equal \(\tau\).
These formulas suggest two basic estimation strategies.
- Outcome regression (prediction estimator): estimate \(\mu_1(x)\) and \(\mu_0(x)\), then average \(\hat\mu_1(X_i)-\hat\mu_0(X_i)\) over the sample.
- Inverse probability weighting (IPW): estimate \(\pi(x)\) and reweight observed outcomes to reconstruct the potential-outcome means.
Each strategy has strengths and weaknesses. Regression-based estimators can be statistically efficient when the outcome model is well specified, but may be biased under misspecification. IPW estimators are often conceptually appealing because they directly use the treatment model, but they can be unstable when estimated propensity scores are close to \(0\) or \(1\) (Rosenbaum and Rubin 1983).
This chapter develops a third strategy: combine both models to construct an estimator that is consistent when either model is correct, and achieves the semiparametric efficiency bound when both are correct (under regularity conditions developed in Section 11.10). We derive the same estimator in three complementary ways: first as a bias-corrected prediction estimator (Section 11.2), then as the optimal member of a class of augmented estimators (Section 11.3–Section 11.6), and finally as the efficient influence function in a semiparametric model (Section 11.7).
11.2 The Prediction Estimator and Its Bias
To understand the source of bias it is useful to work first with generic working regression functions \(m_t(x)\), not necessarily equal to the true conditional means. The prediction estimator (also called the regression estimator) based on \(m_t\) is \[\hat\tau_{\mathrm{pred}}(m) = \frac{1}{n}\sum_{i=1}^n \bigl\{m_1(X_i) - m_0(X_i)\bigr\}.\] Define the corresponding population prediction estimand \(\tau_{\mathrm{pred}}(m) = \E\{m_1(X)-m_0(X)\}\). Under ignorability, the target parameter is \(\tau = \E\{\mu_1(X)-\mu_0(X)\}\), where \(\mu_t(x)=\E\{Y(t)\mid X=x\}=\E(Y\mid T=t,X=x)\). Hence \[\tau_{\mathrm{pred}}(m) - \tau = \E\bigl[(m_1(X)-\mu_1(X)) - (m_0(X)-\mu_0(X))\bigr]. \tag{11.1}\] Equation Equation 11.1 shows that \(\tau_{\mathrm{pred}}(m) = \tau\) if and only if \(\E[\{m_1(X)-\mu_1(X)\} - \{m_0(X)-\mu_0(X)\}] = 0\). Correct specification of both arm-specific working regressions is a simple sufficient condition, but it is not necessary: the two integrated misspecification errors may cancel. In practice we substitute fitted models \(m_t = \hat\mu_t\); if \(\hat\mu_t\) converges to a deterministic limit \(\mu_t^\dagger\), the same decomposition gives the probability-limit bias \(\E[\{\mu_1^\dagger(X)-\mu_1(X)\} - \{\mu_0^\dagger(X)-\mu_0(X)\}]\) of the prediction estimator. (Exact finite-sample unbiasedness is not claimed when the regressions are estimated from the same data.)
If a propensity score estimator \(\hat\pi\) is available, we can estimate the two bias terms in Equation 11.1 from the observed data. Define residuals \(e_i(t) = Y_i(t) - \hat\mu_t(X_i)\). Since \(Y_i(1)\) is observed only for treated units and \(Y_i(0)\) only for control units, \[\widehat{\mathrm{Bias}}(\hat\tau_{\mathrm{pred}}) = -\frac{1}{n}\sum_{i=1}^n \frac{T_i}{\hat\pi(X_i)}\,e_i(1) + \frac{1}{n}\sum_{i=1}^n \frac{1-T_i}{1-\hat\pi(X_i)}\,e_i(0).\] The bias-corrected prediction estimator is therefore \[\hat\tau_{\mathrm{AIPW}} = \hat\tau_{\mathrm{pred}} - \widehat{\mathrm{Bias}}(\hat\tau_{\mathrm{pred}}) = \hat\mu_{1,\mathrm{dr}} - \hat\mu_{0,\mathrm{dr}}, \tag{11.2}\] where \[\hat\mu_{1,\mathrm{dr}} = \frac{1}{n}\sum_{i=1}^n \frac{T_i}{\hat\pi(X_i)}\bigl\{Y_i - \hat\mu_1(X_i)\bigr\} + \frac{1}{n}\sum_{i=1}^n \hat\mu_1(X_i), \tag{11.3}\] \[\hat\mu_{0,\mathrm{dr}} = \frac{1}{n}\sum_{i=1}^n \frac{1-T_i}{1-\hat\pi(X_i)}\bigl\{Y_i - \hat\mu_0(X_i)\bigr\} + \frac{1}{n}\sum_{i=1}^n \hat\mu_0(X_i). \tag{11.4}\]
11.3 The Augmented IPW Estimator
Notation for the rest of the chapter. From this point on, \(\mu_t(x)\) inside estimating functions and population identities denotes a generic (or working) outcome regression, not necessarily equal to the truth; the truth is written \(\mu_t^*(x) = \E(Y\mid T=t,X=x)\), and the fitted version that replaces \(\mu_t(x)\) in practice is \(\hat\mu_t(x)\). Similarly, \(\pi(x)\) inside estimating functions denotes a working propensity model; the true propensity score is written \(\pi^*(x) = P(T=1\mid X=x)\), and the fitted version used in practice is \(\hat\pi(x)\). Whether a particular theorem assumes \(\pi = \pi^*\) or allows an arbitrary working \(\pi\) will be stated explicitly. We also use the unsubscripted scalar \(\mu_t = \E\{Y(t)\}\) for the marginal counterfactual mean; the convention is that \(\mu_t\) without an argument is the scalar mean and \(\mu_t(x)\) with an argument is the regression function.
Collecting the terms in Equation 11.2 gives the estimating-equation form. The AIPW estimator solves \(\mathbb{P}_n\{\phi(O;\tau,\hat\eta)\} = 0\) where \[\phi(O;\tau,\eta) = \left[\frac{T}{\pi(X)}\{Y-\mu_1(X)\}+\mu_1(X)\right] - \left[\frac{1-T}{1-\pi(X)}\{Y-\mu_0(X)\}+\mu_0(X)\right] - \tau, \tag{11.5}\] and \(O=(X,T,Y)\) denotes one observation. Solving explicitly, \[\hat\tau_{\mathrm{AIPW}} = \frac{1}{n}\sum_{i=1}^n \Bigg[\hat\mu_1(X_i) - \hat\mu_0(X_i) + \frac{T_i}{\hat\pi(X_i)}\{Y_i - \hat\mu_1(X_i)\} - \frac{1-T_i}{1-\hat\pi(X_i)}\{Y_i - \hat\mu_0(X_i)\}\Bigg]. \tag{11.6}\]
The AIPW estimator enjoys two important properties: double robustness and optimality within the augmented estimator class. We establish double robustness here; the optimality property is developed in Section 11.4.
Theorem 11.1 (Double Robustness) Let \(\phi(O;\tau,\eta)\) be the estimating function in Equation 11.5. Suppose consistency, conditional exchangeability, and positivity hold, so that \(\tau = \E\{\mu_1^*(X)-\mu_0^*(X)\}\). Assume the working functions are admissible: \(0 < \pi(x) < 1\) on the support of \(X\), and the expectations displayed in Equation 11.5 exist. Then \(\E\{\phi(O;\tau,\eta)\} = 0\) if either:
- \(\mu_t(x) = \mu_t^*(x)\) for \(t=0,1\), regardless of whether \(\pi(x)\) is correctly specified; or
- \(\pi(x) = \pi^*(x)\), regardless of whether \(\mu_0\) and \(\mu_1\) are correctly specified.
Proof. Case 1: outcome regression correct. If \(\mu_t(X) = \E(Y\mid T=t,X)\), then conditional on \(X\), \[\E\!\left[\frac{T}{\pi(X)}\{Y-\mu_1(X)\}\,\middle|\,X\right] = \frac{P(T=1\mid X)}{\pi(X)}\,\E\{Y-\mu_1(X)\mid T=1,X\} = 0,\] and similarly the untreated augmentation term vanishes. Therefore \(\E\{\phi\} = \E\{\mu_1(X)-\mu_0(X)\} - \tau = 0\).
Case 2: propensity score correct. Suppose \(\pi(X) = \pi^*(X)\) but \(\mu_0,\mu_1\) are arbitrary (possibly misspecified). We compute the conditional expectation of the first bracket of \(\phi\) given \(X\): \[\begin{aligned} \E\!\left[\frac{T}{\pi(X)}\{Y-\mu_1(X)\} + \mu_1(X)\,\Big|\, X\right] &= \frac{\E[TY\mid X]}{\pi(X)} - \mu_1(X)\,\frac{\E[T\mid X]}{\pi(X)} + \mu_1(X) \\ &= \mu_1^*(X) - \mu_1(X) + \mu_1(X) = \mu_1^*(X), \end{aligned}\] where the second line uses \(\E[T\mid X] = \pi^*(X) = \pi(X)\) (so the \(\mu_1\) terms cancel) and \(\E[TY\mid X] = \pi(X)\mu_1^*(X)\). By symmetry, the control bracket has conditional expectation \(\mu_0^*(X)\). Hence \(\E\{\phi\mid X\} = \mu_1^*(X) - \mu_0^*(X) - \tau\), and taking expectations over \(X\) and invoking the identification formula gives \(\E\{\phi\} = 0\). \(\square\)
Theorem 11.2 (The Exact Double-Robustness Remainder) Under the assumptions of Theorem 11.1, for any admissible working pair \((\pi, \mu_0, \mu_1)\), \[\E\{\phi(O;\tau,\eta)\} = \E\!\left[\{\pi^*(X)-\pi(X)\}\left\{\frac{\mu_1^*(X)-\mu_1(X)}{\pi(X)} + \frac{\mu_0^*(X)-\mu_0(X)}{1-\pi(X)}\right\}\right]. \tag{11.7}\]
Proof. Conditioning on \(X\), \(\E[\,T\{Y-\mu_1(X)\}/\pi(X)\mid X\,] = \{\pi^*(X)/\pi(X)\}\{\mu_1^*(X)-\mu_1(X)\}\), hence the treated bracket of Equation 11.5 satisfies \[\E\!\left[\frac{T}{\pi(X)}\{Y-\mu_1(X)\}+\mu_1(X)\,\middle|\, X\right] - \mu_1^*(X) = \{\mu_1^*(X)-\mu_1(X)\}\,\frac{\pi^*(X)-\pi(X)}{\pi(X)},\] and symmetrically the control bracket satisfies \[\E\!\left[\frac{1-T}{1-\pi(X)}\{Y-\mu_0(X)\}+\mu_0(X)\,\middle|\, X\right] - \mu_0^*(X) = -\{\mu_0^*(X)-\mu_0(X)\}\,\frac{\pi^*(X)-\pi(X)}{1-\pi(X)}.\] Subtracting the two brackets, using \(\tau = \E\{\mu_1^*(X)-\mu_0^*(X)\}\), and taking expectations over \(X\) gives Equation 11.7. \(\square\)
Identity Equation 11.7 packages the main structural facts of this chapter into a single display.
Double robustness: the integrand is a product of a propensity-score error and an outcome-regression error, so it vanishes when either factor is identically zero — Cases 1 and 2 of Theorem 11.1 are both immediate.
Orthogonality at the truth: the remainder is bilinear in the two errors, so its Gateaux derivative with respect to either nuisance, evaluated at \((\pi^*, \mu^*)\), is zero — the property developed as Neyman orthogonality in Section 11.10.
The product rate: if the working functions are replaced by estimates \((\hat\pi, \hat\mu_t)\) fitted on an independent sample (so that Equation 11.7 applies conditionally on that sample) and the fitted propensity score satisfies \(\varepsilon \leq \hat\pi \leq 1-\varepsilon\), the Cauchy–Schwarz inequality bounds the conditional remainder by \[|R_2| \;\leq\; \varepsilon^{-1}\,\|\hat\pi-\pi^*\|\bigl(\|\hat\mu_1-\mu_1^*\| + \|\hat\mu_0-\mu_0^*\|\bigr), \tag{11.8}\] with \(\|\cdot\|\) the \(L^2(P)\) norm — exactly the product structure behind the rate condition Equation 11.25 of Section 11.10.
Theorem 11.3 (Double-Robust Consistency) Write \(\hat\tau_{\mathrm{AIPW}} = \mathbb{P}_n\,\psi(O;\hat\eta)\) for the sample average in Equation 11.6, where \(\psi(O;\eta) = \phi(O;\tau,\eta) + \tau\) does not involve \(\tau\). Suppose (i) \(\hat\mu_t \to \mu_t^\dagger\) and \(\hat\pi \to \pi^\dagger\) in \(L^2(P_X)\)-probability, with \(\varepsilon \leq \hat\pi(x),\, \pi^\dagger(x) \leq 1-\varepsilon\) for some \(\varepsilon > 0\) and \(\E(Y^2) < \infty\); and (ii) \((\mathbb{P}_n - P)\,\psi(O;\hat\eta) = o_p(1)\). If either \(\mu_t^\dagger = \mu_t^*\) for both \(t\) or \(\pi^\dagger = \pi^*\), then \(\hat\tau_{\mathrm{AIPW}} \overset{p}{\to} \tau\).
Proof. Decompose \[\hat\tau_{\mathrm{AIPW}} - \tau = (\mathbb{P}_n - P)\,\psi(O;\hat\eta) + P\{\psi(O;\hat\eta) - \psi(O;\eta^\dagger)\} + \E\{\phi(O;\tau,\eta^\dagger)\}.\] The first term is \(o_p(1)\) by (ii). The second tends to zero in probability by (i), the bounded denominators, and the Cauchy–Schwarz inequality. The third is zero by Theorem 11.1 — equivalently, by Equation 11.7 — under either correctness condition. \(\square\)
The importance of the AIPW estimator goes beyond bias correction. In Section 11.4 we show that it is the variance-optimal member of a natural class of augmented estimators indexed by arbitrary control functions \(b_t(x)\). Later, in Section 11.7, we will see that the same estimating function \(\phi\) is exactly the efficient influence function for the ATE under the nonparametric model. Thus the same object arises from three perspectives: bias correction, optimal augmentation, and semiparametric efficiency.
11.4 A Class of Augmented Estimators
The AIPW estimator has a second important property beyond double robustness: it is the variance-minimizing estimator within a broad class of augmented IPW estimators. We now define this class and establish the optimality result.
For any functions \(b_1(x)\) and \(b_0(x)\) that are square-integrable with respect to the distribution of \(X\) — we write \(b_t \in \mathcal{L}^2 \equiv L^2(P_X)\) — define \[\hat\mu_{1,b} = \frac{1}{n}\sum_{i=1}^n \frac{T_i}{\pi(X_i)}\,Y_i - \frac{1}{n}\sum_{i=1}^n \left\{\frac{T_i}{\pi(X_i)}-1\right\} b_1(X_i), \tag{11.9}\] \[\hat\mu_{0,b} = \frac{1}{n}\sum_{i=1}^n \frac{1-T_i}{1-\pi(X_i)}\,Y_i - \frac{1}{n}\sum_{i=1}^n \left\{\frac{1-T_i}{1-\pi(X_i)}-1\right\} b_0(X_i), \tag{11.10}\] and let \(\hat\tau_b = \hat\mu_{1,b} - \hat\mu_{0,b}\). Think of \(b_t(x)\) as any working approximation to \(\E\{Y(t)\mid X=x\}\): the first term of \(\hat\mu_{t,b}\) is an IPW estimator of \(\mu_t\), and the second subtracts a mean-zero correction indexed by \(b_t\). The correction is mean-zero under the true \(\pi\) regardless of \(b_t\), so every member of the family is unbiased; only the variance depends on the choice of \(b_t\), which is what makes this a meaningful optimization problem.
The following theorem shows that every estimator in this class is unbiased for \(\tau\) under a correctly specified propensity score, regardless of the choice of \(b_t\), and provides an explicit formula for its total variance.
Theorem 11.4 (Unbiasedness and Variance of the AIPW Class) Under joint conditional exchangeability \(\{Y(0), Y(1)\} \indep T \mid X\) (the form used in the variance calculation below; Chapter 4 discusses its relation to the weaker per-arm form), SUTVA, and positivity strengthened to strict overlap (\(\varepsilon \leq \pi^*(x) \leq 1-\varepsilon\) for some \(\varepsilon > 0\)), with the true propensity score used as \(\pi(x)\) in the estimator, for any \(b_0, b_1 \in \mathcal{L}^2\) and \(\E\{Y(t)^2\} < \infty\):
- Unbiasedness: \(\E(\hat\tau_b) = \tau\).
- Total variance: \[\begin{aligned} \mathrm{Var}(\hat\tau_b) &= \frac{1}{n}\,\E\!\left[\left(\frac{1}{\pi(X)}-1\right)\{Y(1)-b_1(X)\}^2 + 2\{Y(1)-b_1(X)\}\{Y(0)-b_0(X)\}\right. \\ &\hspace{5em}\left. + \left(\frac{1}{1-\pi(X)}-1\right)\{Y(0)-b_0(X)\}^2\right] + \frac{1}{n}\,\mathrm{Var}\{Y(1)-Y(0)\}. \end{aligned} \tag{11.11}\]
Proof. Let \(\xi = \frac{T}{\pi(X)}\{Y-b_1(X)\} + b_1(X) - \frac{1-T}{1-\pi(X)}\{Y-b_0(X)\} - b_0(X)\), so that \(\hat\tau_b = n^{-1}\sum_{i=1}^n \xi_i\) and, for i.i.d. observations, \(\mathrm{Var}(\hat\tau_b) = n^{-1}\mathrm{Var}(\xi)\). By SUTVA, \(Y = TY(1)+(1-T)Y(0)\), so \[\xi = \frac{T}{\pi(X)}\{Y(1)-b_1(X)\} + b_1(X) - \frac{1-T}{1-\pi(X)}\{Y(0)-b_0(X)\} - b_0(X).\]
Part (i). Since \(\pi(X) = \pi^*(X)\), we have \(\E[T/\pi(X)\mid X] = 1\), so \(\E[(T/\pi(X)-1)b_1(X)] = 0\). Under conditional exchangeability, \(T \indep Y(1)\mid X\), so \(\E[(T/\pi(X))\,Y(1)\mid X] = \E[Y(1)\mid X] = \mu_1^*(X)\). Hence \(\E[\hat\mu_{1,b}] = \mu_1\) and, by symmetry, \(\E[\hat\mu_{0,b}] = \mu_0\), so \(\E(\hat\tau_b) = \tau\).
Part (ii). We apply the law of total variance, conditioning on \((X, Y(1), Y(0))\). By joint conditional exchangeability, the only remaining randomness is \(T \mid \{X, Y(1), Y(0)\} \sim \mathrm{Bernoulli}(\pi(X))\).
Conditional mean. Since \(\E[T/\pi(X)\mid X]=1\) and \(\E[(1-T)/(1-\pi(X))\mid X]=1\), we get \(\E[\xi \mid X, Y(1), Y(0)] = Y(1) - Y(0)\).
Conditional variance. \(\xi\) is linear in \(T\): \[\xi = T\!\left[\frac{Y(1)-b_1(X)}{\pi(X)} + \frac{Y(0)-b_0(X)}{1-\pi(X)}\right] - \frac{Y(0)-b_0(X)}{1-\pi(X)} + b_1(X) - b_0(X).\] Using \(\mathrm{Var}(T\mid X) = \pi(X)(1-\pi(X))\), \[\begin{aligned} \mathrm{Var}(\xi \mid X, Y(1), Y(0)) &= \pi(X)(1-\pi(X))\left[\frac{Y(1)-b_1(X)}{\pi(X)}+\frac{Y(0)-b_0(X)}{1-\pi(X)}\right]^{2} \\ &= \left(\frac{1}{\pi(X)}-1\right)\{Y(1)-b_1(X)\}^2 + 2\{Y(1)-b_1(X)\}\{Y(0)-b_0(X)\} \\ &\quad + \left(\frac{1}{1-\pi(X)}-1\right)\{Y(0)-b_0(X)\}^2. \end{aligned}\] Taking expectations over \((X, Y(1), Y(0))\) and adding the between-group term \(\mathrm{Var}(\E[\xi\mid X,Y(1),Y(0)]) = \mathrm{Var}(Y(1)-Y(0))\) gives Equation 11.11. \(\square\)
Every member of the class is unbiased for \(\tau\), for any choice of \(b_t\). The first term of Equation 11.11 depends on \(b_t\); the second, \(n^{-1}\mathrm{Var}\{Y(1)-Y(0)\}\), does not.
Theorem 11.5 (Optimal Control Functions) The control functions \(b_1^*(X) = \E\{Y(1)\mid X\}\) and \(b_0^*(X) = \E\{Y(0)\mid X\}\) form a minimizing pair for the total variance in Equation 11.11; the full set of minimizing pairs, and the uniqueness of the resulting estimator, are described in the remark below. The resulting AIPW estimator coincides with the bias-corrected prediction estimator in Equation 11.2.
Proof. Write \(\mu_t^*(x) = \E\{Y(t)\mid X=x\}\) and set \(u(X) = \mu_1^*(X)-b_1(X)\) and \(v(X)=\mu_0^*(X)-b_0(X)\). Decompose \[Y(t) - b_t(X) = (\mu_t^*(X)-b_t(X)) + (Y(t)-\mu_t^*(X)) = d_t(X) + \varepsilon_t,\] where \(d_1=u\), \(d_0=v\), and \(\E[\varepsilon_t\mid X]=0\). Under strong ignorability, \(T\indep (Y(1),Y(0))\mid X\), so \(\varepsilon_t\) is also independent of \(T\) given \(X\).
Substituting into the \(b_t\)-dependent term of Equation 11.11 and using \(\E[\varepsilon_t\mid X]=0\), the law of iterated expectations eliminates every cross term between \((u,v)\) and \((\varepsilon_1,\varepsilon_0)\), leaving \[\underbrace{\E\!\left[\left(\frac{1}{\pi(X)}-1\right)u(X)^2 + 2\,u(X)v(X) + \left(\frac{1}{1-\pi(X)}-1\right)v(X)^2\right]}_{=:\,Q(b_0,b_1)} + C, \tag{11.12}\] where \(C = \E[(1/\pi-1)\mathrm{Var}\{Y(1)\mid X\} + (1/(1-\pi)-1)\mathrm{Var}\{Y(0)\mid X\}] + 2\E[\varepsilon_1\varepsilon_0]\) does not depend on \(b_0\) or \(b_1\).
It remains to show \(Q(b_0,b_1)\geq 0\) with equality at \(u\equiv 0\), \(v\equiv 0\). Write \[\left(\frac{1}{\pi}-1\right)u^2+2uv+\left(\frac{1}{1-\pi}-1\right)v^2 = \frac{1-\pi}{\pi}\,u^2 + 2uv + \frac{\pi}{1-\pi}\,v^2 = \left(\sqrt{\frac{1-\pi}{\pi}}\,u+\sqrt{\frac{\pi}{1-\pi}}\,v\right)^{2},\] a perfect square, so \(Q(b_0,b_1)\geq 0\). The choice \(b_t^*(X)=\mu_t^*(X)\), i.e. \(u\equiv 0\) and \(v\equiv 0\), achieves \(Q=0\) and hence \(\mathrm{Var}(\hat\tau_b) \geq n^{-1}[C + \mathrm{Var}\{Y(1)-Y(0)\}] =: \mathrm{Var}(\hat\tau_{b^*})\). \(\square\)
11.5 The Projection Interpretation \(\star\)
The class-specific optimality of AIPW within the augmented IPW family (Theorem 11.5) has an elegant interpretation in terms of projections in a Hilbert space of estimators. This framing is a working analogy: the collection of square-integrable random variables, quotiented by a.s. equality, is a genuine Hilbert space \(L^2(P)\) under the inner product \(\langle X, Y\rangle = \E(XY)\), which coincides with \(\mathrm{Cov}(X, Y)\) whenever either argument has mean zero. Theorem 11.6 is an instance of the \(L^2\) projection theorem in that space, applied to the closed subspace \(\Lambda\) of mean-zero elements; see Tsiatis (2006) (Chapter 2) for the careful construction.
This projection view does not define a new estimator; rather, it explains why the augmentation used in Section 11.4 is variance reducing and why the optimal correction must be orthogonal to the augmentation space.
11.5.1 Optimal Estimation via Projection
Let \(\hat\theta_0\) be an unbiased estimator of \(\theta\). Define the augmentation space \(\Lambda\) as a pre-specified closed linear subspace of mean-zero square-integrable random variables that are computable from the observed data without knowledge of \(\theta\). Elements of \(\Lambda\) are called admissible augmentation terms. The observability restriction is essential: if \(\Lambda\) were taken to be all mean-zero square-integrable random variables, the true parameter error \(\hat\theta_0-\theta\) itself would be in \(\Lambda\), and the “projection” would yield the degenerate estimator \(\theta\) with zero variance — a construction that requires knowing the very quantity we are trying to estimate. Restricting \(\Lambda\) to observable corrections removes this degeneracy. For any \(\hat b \in \Lambda\) the estimator \(\hat\theta_b = \hat\theta_0 - \hat b\) remains unbiased for \(\theta\). We seek the \(\hat b^* \in \Lambda\) that minimizes \(\mathrm{Var}(\hat\theta_b)\).
Theorem 11.6 (Optimal Projection) The optimal correction is \(\hat b^* = \Pi(\hat\theta_0\mid\Lambda)\), the \(L^2\) projection of \(\hat\theta_0\) onto \(\Lambda\). It is characterized by: (1) \(\hat b^* \in \Lambda\); (2) \(\mathrm{Cov}(\hat\theta_0-\hat b^*,\;\hat b) = 0\) for all \(\hat b \in \Lambda\). The optimal estimator is \[\hat\theta_{\mathrm{opt}} = \hat\theta_0 - \hat b^* = \Pi(\hat\theta_0 \mid \Lambda^\perp),\] the projection of \(\hat\theta_0\) onto the orthogonal complement \(\Lambda^\perp\). It satisfies the Pythagorean identity \[\mathrm{Var}(\hat\theta_0) = \mathrm{Var}(\hat\theta_{\mathrm{opt}}) + \mathrm{Var}(\hat b^*), \tag{11.15}\] so in particular \(\mathrm{Var}(\hat\theta_{\mathrm{opt}}) \leq \mathrm{Var}(\hat\theta_0)\).
Proof sketch. \(\Lambda\) is a closed linear subspace of the Hilbert space \(L^2(P)\) of square-integrable random variables (its elements additionally have mean zero), so the \(L^2\) projection theorem gives a unique \(\hat b^* \in \Lambda\) satisfying the orthogonality condition \(\mathrm{Cov}(\hat\theta_0 - \hat b^*,\, \hat b) = 0\) for all \(\hat b \in \Lambda\). Any other choice \(\hat b \in \Lambda\) yields \[\mathrm{Var}(\hat\theta_0 - \hat b) = \mathrm{Var}(\hat\theta_{\mathrm{opt}} + (\hat b^* - \hat b)) = \mathrm{Var}(\hat\theta_{\mathrm{opt}}) + \mathrm{Var}(\hat b^* - \hat b) \geq \mathrm{Var}(\hat\theta_{\mathrm{opt}}),\] where the middle equality uses \(\mathrm{Cov}(\hat\theta_{\mathrm{opt}},\, \hat b^* - \hat b) = 0\). Specializing to \(\hat b = 0\) gives Equation 11.15. \(\square\)
The Pythagorean identity shows that the variance of the initial estimator decomposes orthogonally: the optimal estimator retains only the component in \(\Lambda^\perp\), and the discarded component \(\hat b^*\) accounts for the remainder. Because every element of \(\Lambda\) has mean zero, constants are orthogonal to \(\Lambda\), so \(\Pi(\hat\theta_0 \mid \Lambda) = \Pi(\hat\theta_0 - \theta \mid \Lambda)\): projecting the estimator and projecting its centered estimation error yield the same correction \(\hat b^*\). The centered form is the one used in the semiparametric theory of Appendix C, where the projected objects — scores and influence functions — are mean zero by construction. Tsiatis (2006) calls \(\Lambda\) the augmentation space precisely because its elements are the corrections that reduce variance.
11.6 Projection in the Causal Inference Setting \(\star\)
We now specialize the projection framework to the ATE, assuming \(\pi(X)\) is known. To align with the arm-by-arm structure of Section 11.4, we decompose the Horvitz–Thompson estimator as \(\hat\tau_{\mathrm{HT}} = \hat\mu_{1,\mathrm{HT}} - \hat\mu_{0,\mathrm{HT}}\), where \[\hat\mu_{1,\mathrm{HT}} = \frac{1}{n}\sum_{i=1}^n \frac{T_i Y_i}{\pi_i}, \qquad \hat\mu_{0,\mathrm{HT}} = \frac{1}{n}\sum_{i=1}^n \frac{(1-T_i)Y_i}{1-\pi_i},\] and \(\pi_i = \pi(X_i)\).
11.6.1 Arm-wise Augmentation Spaces
Define separate augmentation spaces for the treated and control arms: \[\Lambda_1 = \left\{n^{-1}\sum_{i=1}^n\!\left(\frac{T_i}{\pi_i}-1\right) b_1(X_i) : b_1 \in \mathcal{L}^2\right\}, \tag{11.16}\] \[\Lambda_0 = \left\{n^{-1}\sum_{i=1}^n\!\left(\frac{1-T_i}{1-\pi_i}-1\right) b_0(X_i) : b_0 \in \mathcal{L}^2\right\}. \tag{11.17}\] Because \(\E[(T_i/\pi_i-1)\mid X_i]=0\) and \(\E[(1-T_i)/(1-\pi_i)-1\mid X_i]=0\), every element of \(\Lambda_1\) (resp. \(\Lambda_0\)) has expectation zero, so \(\hat\mu_{1,\mathrm{HT}} - \hat b_1\) (resp. \(\hat\mu_{0,\mathrm{HT}} - \hat b_0\)) remains unbiased for \(\mu_1\) (resp. \(\mu_0\)) for any \(b_t\). The combined augmentation space \(\Lambda = \Lambda_1 + \Lambda_0\) is the arm-wise sum, whose elements are corrections that leave \(\hat\tau_{\mathrm{HT}}\) unbiased for \(\tau\).
The sum adds nothing, however: \(\Lambda_1\) and \(\Lambda_0\) are the same subspace, parameterized in two different ways. Writing \(T_i/\pi_i - 1 = (T_i - \pi_i)/\pi_i\) and \((1-T_i)/(1-\pi_i) - 1 = -(T_i - \pi_i)/(1-\pi_i)\), and noting that when \(\pi\) is bounded away from \(0\) and \(1\) the maps \(b_1 \mapsto b_1/\pi\) and \(b_0 \mapsto -b_0/(1-\pi)\) are bijections of \(\mathcal{L}^2\), \[\Lambda = \Lambda_1 = \Lambda_0 = \left\{n^{-1}\sum_{i=1}^n (T_i - \pi_i)\, a(X_i) \;:\; a \in \mathcal{L}^2\right\}. \tag{11.18}\] This is the geometric face of the reparameterization identity Equation 11.13: the pair \((b_0, b_1)\) supplies two functions to sweep out a space indexed by the single function \(a_b\), which is why the joint indexing in Theorem 11.5 is redundant (the remark on uniqueness of the minimizing estimator). We nevertheless retain the arm-wise labels \(\Lambda_1\) and \(\Lambda_0\): they are the convenient coordinates for the arm-wise projections of Theorem 11.7 below.
11.6.2 Computing the Projections
Theorem 11.7 (Arm-wise Projections onto the Augmentation Spaces) Assume the setting of Theorem 11.4: \(\pi\) known and bounded away from \(0\) and \(1\), and \(\E\{Y(t)^2\} < \infty\). The optimal corrections are \[\Pi(\hat\mu_{1,\mathrm{HT}} \mid \Lambda_1) = n^{-1}\sum_{i=1}^n\!\left(\frac{T_i}{\pi_i}-1\right) b_1^*(X_i), \qquad b_1^*(x) = \E\{Y(1)\mid x\},\] \[\Pi(\hat\mu_{0,\mathrm{HT}} \mid \Lambda_0) = n^{-1}\sum_{i=1}^n\!\left(\frac{1-T_i}{1-\pi_i}-1\right) b_0^*(X_i), \qquad b_0^*(x) = \E\{Y(0)\mid x\}.\]
Proof. We verify the treated-arm projection; the control arm follows by symmetry. Denote the residual \(R_1 = \hat\mu_{1,\mathrm{HT}} - n^{-1}\sum_i (T_i/\pi_i - 1)\,b_1^*(X_i)\), which simplifies to \[R_1 = \frac{1}{n}\sum_{i=1}^n \left[\frac{T_i}{\pi_i}\{Y_i - b_1^*(X_i)\} + b_1^*(X_i)\right] =: \frac{1}{n}\sum_{i=1}^n r_{1i}.\] By Theorem 11.6, it suffices to show \(\mathrm{Cov}(R_1,\, \hat b) = 0\) for every \(\hat b = n^{-1}\sum_j (T_j/\pi_j - 1)\,b_1(X_j) \in \Lambda_1\). Writing \(\hat b\) as \(n^{-1}\sum_j b_j\) and using independence across units, cross terms with \(i \neq j\) vanish, so \(\mathrm{Cov}(R_1,\hat b) = n^{-1}\,\mathrm{Cov}(r_{1i},\, b_i)\), and since \(\E(b_i) = 0\) this reduces to showing \(\E[r_{1i}\, b_i] = 0\).
Condition on \((X_i, Y_i(1), Y_i(0))\); the only remaining randomness is \(T_i \mid X_i \sim \mathrm{Bernoulli}(\pi_i)\). Using \(T_i Y_i = T_i Y_i(1)\) and \(b_1^*(X_i) = \E\{Y_i(1)\mid X_i\}\), \[\E\!\left[\frac{T_i}{\pi_i}\{Y_i(1) - b_1^*(X_i)\}\left(\frac{T_i}{\pi_i}-1\right)\,\middle|\, X_i, Y_i(1), Y_i(0)\right] = \{Y_i(1)-b_1^*(X_i)\}\,\E\!\left[\frac{T_i}{\pi_i}\!\left(\frac{T_i}{\pi_i}-1\right)\,\middle|\, X_i\right].\] Since \(T_i^2 = T_i\), we have \(\E[T_i^2/\pi_i^2 \mid X_i] = 1/\pi_i\) and \(\E[T_i/\pi_i \mid X_i] = 1\), so the inner expectation is \(1/\pi_i - 1\). Taking the outer expectation and using \(\E\{Y_i(1)-b_1^*(X_i)\mid X_i\} = 0\), this contribution to \(\E[r_{1i}\,b_i]\) vanishes. The constant term \(b_1^*(X_i)\) in \(r_{1i}\) also contributes zero because \(\E[(T_i/\pi_i - 1)\mid X_i] = 0\). \(\square\)
11.6.3 Recovering the AIPW Estimator
By Theorem 11.6, the optimal estimators of \(\mu_1\) and \(\mu_0\) are the projections onto \(\Lambda_1^\perp\) and \(\Lambda_0^\perp\), respectively: \[\begin{aligned} \hat\mu_{1,\mathrm{opt}} &= \hat\mu_{1,\mathrm{HT}} - n^{-1}\!\sum_{i=1}^n\!\left(\frac{T_i}{\pi_i}-1\right)\mu_1^*(X_i) = \frac{1}{n}\sum_{i=1}^n \left[\frac{T_i}{\pi_i}\{Y_i-\mu_1^*(X_i)\}+\mu_1^*(X_i)\right], \\ \hat\mu_{0,\mathrm{opt}} &= \hat\mu_{0,\mathrm{HT}} - n^{-1}\!\sum_{i=1}^n\!\left(\frac{1-T_i}{1-\pi_i}-1\right)\mu_0^*(X_i) = \frac{1}{n}\sum_{i=1}^n \left[\frac{1-T_i}{1-\pi_i}\{Y_i-\mu_0^*(X_i)\}+\mu_0^*(X_i)\right]. \end{aligned}\] Subtracting, \(\hat\tau_{\mathrm{opt}} = \hat\mu_{1,\mathrm{opt}} - \hat\mu_{0,\mathrm{opt}}\), which is precisely the AIPW estimator Equation 11.6 with \(b_t^*(X) = \E\{Y(t)\mid X\}\), confirming Theorem 11.5 by a different route.
11.7 The Efficient Influence Function and Semiparametric Efficiency
This section states the semiparametric efficiency result and interprets it in light of the estimator development above; no new derivation is given. A complete proof requires explicit characterization of the nuisance tangent space and is beyond the scope of these notes; see Tsiatis (2006) for a rigorous development.
We now return to the efficient influence function introduced in Chapter 10 and show that it is exactly the estimating function underlying AIPW. This identification is crucial: it explains why AIPW is not only doubly robust, but also semiparametrically efficient when both nuisance functions are estimated consistently at suitable rates.
Under the nonparametric model for \(O=(X,T,Y)\), the efficient influence function for the ATE is \[\varphi_{\mathrm{eff}}(O) = \frac{T}{\pi^*(X)}\{Y-\mu_1^*(X)\} - \frac{1-T}{1-\pi^*(X)}\{Y-\mu_0^*(X)\} + \mu_1^*(X) - \mu_0^*(X) - \tau, \tag{11.19}\] which by definition is a function of the true nuisance components \(\pi^*\) and \(\mu_t^*\). The AIPW estimating function \(\phi(O;\tau,\eta)\) in Equation 11.5 has the same algebraic form as \(\varphi_{\mathrm{eff}}\), but with working nuisances substituted in place of the truths; the implementable AIPW estimator is obtained by further substituting the fitted versions. The identification that matters for efficiency is at the population level: when the working nuisances coincide with the truths, \(\phi(O;\tau,\eta^*) = \varphi_{\mathrm{eff}}(O)\), and this is the efficient influence function \(\varphi^*(O)\) recorded in Chapter 10.
Chapter 10 also flagged two central properties of \(\varphi^*\) whose proofs were deferred to this chapter: double robustness and semiparametric efficiency. Both are now in hand: double robustness is Theorem 11.1, and semiparametric efficiency follows from the identification \(\varphi_{\mathrm{eff}} = \varphi^*\) together with the convolution theorem below. The remark on which parts of the variance formula are identified has already verified the variance side by direct computation: the minimized class variance Equation 11.14 equals \(\E\{\varphi_{\mathrm{eff}}(O)^2\}\).
The natural decomposition noted in Chapter 10 — an IPW-style residual correction plus a regression-centered term — is exactly the structure that emerged from both the bias-correction derivation of Section 11.2 and the optimal-augmentation derivation of Section 11.4. The augmentation space \(\Lambda\) corresponds to a specific component of the nuisance structure: it is the finite-sample analogue of the treatment-mechanism score subspace \[\mathcal{H}_T = \bigl\{a(X)\{T-\pi(X)\} : a \in L^2(P_X)\bigr\},\] as the common form Equation 11.18 makes explicit. This parametrization presumes strong overlap, \(\epsilon \leq \pi(X) \leq 1-\epsilon\) almost surely. Appendix C defines the treatment-score space instead by the conditional mean-zero condition \(\E\{h(T,X) \mid X\} = 0\), which under weak positivity is strictly larger; the two agree under strong overlap. Orthogonalizing the Horvitz–Thompson estimating function against \(\mathcal{H}_T\) yields the AIPW form; this removes a nuisance-score component, but it is not the general definition of the efficient influence function, which is the canonical gradient (see also Appendix C).
To state the bound precisely, define the observed-data functional \[\Psi(P) = \E_P\{\mu_{1,P}(X) - \mu_{0,P}(X)\}, \qquad \mu_{t,P}(x) = \E_P(Y \mid T=t, X=x),\] on the nonparametric model for \(O=(X,T,Y)\) satisfying positivity and \(\E_P\{\varphi_{\mathrm{eff}}(O)^2\} < \infty\). The functional \(\Psi(P)\) is defined for every such \(P\) without causal assumptions; under consistency, conditional exchangeability, and positivity (Chapters 4–6), it equals the causal ATE \(\tau\). The efficiency theory below concerns estimation of \(\Psi(P)\); the causal interpretation rides on the identification assumptions.
Theorem 11.8 (Semiparametric Efficiency Bound for the ATE) Under the nonparametric model for \(O=(X,T,Y)\), every regular asymptotically linear estimator \(\hat\tau\) of \(\Psi(P)\) (equivalently, of the ATE under the identification assumptions) satisfies \[\mathrm{Avar}(\sqrt{n}(\hat\tau - \tau)) \;\geq\; \E\{\varphi_{\mathrm{eff}}(O)^2\},\] and the bound is attained by estimators whose influence function equals \(\varphi_{\mathrm{eff}}\) in Equation 11.19.
11.8 Doubly Robust Regression: Weighted and Augmented Approaches \(\star\)
The AIPW estimator achieves double robustness by adding an explicit bias-correction term to the prediction estimator. An equally important question is how to build double robustness directly into the outcome model fit, so that the prediction estimator is itself doubly robust without any separate augmentation step. The unifying concept behind this section is the internal bias calibration (IBC) condition: a simple algebraic identity that, when enforced by the fitting procedure, forces the prediction estimator to coincide with AIPW. The two constructions below — IPW-weighted least squares and regression on a clever covariate — are distinct ways of ensuring IBC holds, and the second is the targeting step that underlies TMLE.
11.8.1 The Internal Bias Calibration Conditions
Let \(\hat\mu_t(x)\) denote any fitted outcome model for arm \(t\), and let \(\hat\pi_i = \hat\pi(X_i)\). The prediction estimator \(\hat\tau_{\mathrm{pred}} = n^{-1}\sum_i\{\hat\mu_1(X_i)-\hat\mu_0(X_i)\}\) requires no augmentation if the IPW-weighted residuals vanish: \[\sum_{i=1}^n \frac{T_i}{\hat\pi_i}\bigl\{Y_i - \hat\mu_1(X_i)\bigr\} = 0, \tag{11.20}\] \[\sum_{i=1}^n \frac{1-T_i}{1-\hat\pi_i}\bigl\{Y_i - \hat\mu_0(X_i)\bigr\} = 0. \tag{11.21}\] We call Equation 11.20–Equation 11.21 the internal bias calibration (IBC) conditions (Firth and Bennett 1998), following the terminology introduced in the context of prediction estimation in survey sampling. The word internal reflects the fact that these constraints are imposed within the outcome-model fitting step itself: once they hold, no separate bias-correction term is needed afterward. When both IBC conditions hold, the augmentation terms in Equation 11.6 are zero by construction, so \(\hat\tau_{\mathrm{pred}} = \hat\tau_{\mathrm{AIPW}}\) and the prediction estimator is itself doubly robust. The two approaches below are two ways of fitting \(\hat\mu_t\) so that the IBC conditions are satisfied by construction.
11.8.2 Weighted Regression Approach
Suppose the outcome model for arm \(t\) is parameterized as \(\mu_t(X;\theta_t)\) for a finite-dimensional parameter \(\theta_t\), with the model including a constant term. Instead of fitting \(\theta_t\) by ordinary least squares among the units with \(T_i=t\), use IPW-weighted least squares: minimize \[\sum_{i=1}^n \frac{T_i}{\hat\pi_i}\bigl\{Y_i - \mu_1(X_i;\theta_1)\bigr\}^2 \tag{11.22}\] for \(t=1\), and analogously with \((1-T_i)/(1-\hat\pi_i)\) for \(t=0\). This is essentially IPW-weighted estimation of the regression coefficients: each unit’s contribution to the least-squares criterion is upweighted by the inverse of its probability of being in that treatment arm. The normal equation of Equation 11.22 with respect to the intercept component of \(\theta_1\) is \[\sum_{i=1}^n \frac{T_i}{\hat\pi_i}\bigl\{Y_i - \mu_1(X_i;\hat\theta_1)\bigr\} = 0,\] which is exactly the IBC condition Equation 11.20. Hence the IPW-weighted fitted values automatically satisfy the IBC condition for any model containing a constant, and the resulting prediction estimator is doubly robust; see Robins et al. (1994) and Bang and Robins (2005) for a general treatment. The key point is not that weighted least squares creates a fundamentally different doubly robust estimator; rather, it constructs fitted values for which the prediction estimator algebraically equals the AIPW estimator.
11.8.3 Augmented Model Approach and the Clever Covariate
The augmented approach achieves the same calibration by modifying the covariate set rather than the fitting weights. Let \(\hat\mu_1^{(0)}(X_i)\) be any initial fit of the treated outcome model (e.g., from OLS or a flexible machine-learning method). Augment it by including the clever covariate \(\hat\pi_i^{-1}\) (Laan and Rubin 2006; Laan and Rose 2011) and run OLS of \(Y_i\) on \(\hat\mu_1^{(0)}(X_i)\) and \(\hat\pi_i^{-1}\) among treated units, fitting the model \[Y_i = \alpha_1 + \beta_1\hat\mu_1^{(0)}(X_i) + \gamma_1\hat\pi_i^{-1} + e_i(1), \qquad T_i = 1. \tag{11.23}\] Denote the OLS estimates by \(\hat\alpha_1\), \(\hat\beta_1\), \(\hat\gamma_1\). The fitted outcome model evaluated at all units is \(\hat\mu_1(X_i) = \hat\alpha_1 + \hat\beta_1\hat\mu_1^{(0)}(X_i) + \hat\gamma_1\hat\pi_i^{-1}\), and the prediction estimator of \(\mu_1\) is \(\hat\mu_1^{\mathrm{pred}} = n^{-1}\sum_{i=1}^n \hat\mu_1(X_i)\).
The normal equation of Equation 11.23 for \(\hat\gamma_1\) (the coefficient on \(\hat\pi_i^{-1}\)) is \[\sum_{i=1}^n \frac{T_i}{\hat\pi_i}\bigl\{Y_i - \hat\mu_1(X_i)\bigr\} = 0, \tag{11.24}\] which is exactly the IBC condition Equation 11.20. Since Equation 11.24 equals zero, we can add it to \(\hat\mu_1^{\mathrm{pred}}\) without changing its value, giving the AIPW representation \[\hat\mu_1^{\mathrm{pred}} = \frac{1}{n}\sum_{i=1}^n \hat\mu_1(X_i) + \frac{1}{n}\sum_{i=1}^n \frac{T_i}{\hat\pi_i}\bigl\{Y_i - \hat\mu_1(X_i)\bigr\}.\] The IPW-weighted residual sum is zero by the IBC condition, so the prediction estimator coincides with the AIPW estimator.
The model Equation 11.23 offers two additional degrees of freedom beyond the fixed-offset augmentation \(\hat\mu_1^{(0)}(X_i) + \hat\gamma_1\hat\pi_i^{-1}\). First, the free coefficient \(\hat\beta_1\) allows the scale of the initial fit to be recalibrated: if \(\hat\mu_1^{(0)}\) systematically over- or under-predicts, \(\hat\beta_1 \neq 1\) absorbs this scaling error. Second, the intercept \(\hat\alpha_1\) absorbs any residual mean shift. The covariate \(\hat\pi_i^{-1}\) is called clever because it is chosen so that the fitted regression satisfies the same score equation Equation 11.24 that appears in the AIPW bias correction. When the initial outcome model is misspecified, including this covariate lets the regression absorb the component of the model error that is correlated with \(\hat\pi_i^{-1}\), so the fitted prediction estimator becomes algebraically equivalent to a debiased estimator. Whether the additional parameters \((\hat\alpha_1,\hat\beta_1)\) yield a measurable efficiency gain over the fixed-offset form depends on the data-generating process: the improvement is largest when the residual misspecification of \(\hat\mu_1^{(0)}\) has substantial linear projection onto \(\mathrm{span}\{1,\hat\mu_1^{(0)}(X),\hat\pi^{-1}(X)\}\) and negligible otherwise. Section 11.9 provides a simulation in which the gain is substantial.
11.9 Lab: Simulation Study
This lab compares four estimators of the ATE: the IPW estimator \(\hat\tau_{\mathrm{HT}}\), the AIPW estimator \(\hat\tau_{\mathrm{AIPW}}\), the fixed-offset augmented-model estimator \(\hat\tau_{\mathrm{aug}}\), and the improved augmented-model estimator \(\hat\tau_{\mathrm{aug}^+}\) of Equation 11.23. A \(2\times 2\) design over nuisance-model correctness — outcome regression (OR) correct or misspecified, crossed with propensity score (PS) correct or misspecified — allows the lab to demonstrate double robustness directly: the augmented estimators remain consistent whenever at least one nuisance model is correct, and fail only when both are misspecified. The lab also reports the empirical coverage of the 95% Wald confidence interval based on the influence-function variance estimator of Section 11.10.2, illustrating the distinction between double-robust consistency and valid asymptotic inference. Full R code is provided in the supplementary file chapter11_lab.R.
11.9.1 Simulation Setup
Generate \(n = 1000\) i.i.d. observations as follows. Draw \(X_i \sim N(0,1)\), then \(T_i \mid X_i \sim \mathrm{Bernoulli}(\pi^*(X_i))\) with true propensity score \[\pi^*(x) = \expit\{0.2\,x + 0.2\,(x^2 - 1)\}.\] The quadratic term is centered at \(\E(X^2) = 1\) so that the marginal treatment probability is roughly balanced. The potential outcomes are \[Y_i(1) = 1 + X_i + 0.5 X_i^2 + \varepsilon_i(1), \qquad Y_i(0) = X_i + 0.5 X_i^2 + \varepsilon_i(0),\] with \(\varepsilon_i(t) \overset{\mathrm{i.i.d.}}{\sim} N(0,1)\) independent of \((X_i, T_i)\), giving true ATE \(\tau = 1\). The observed outcome is \(Y_i = T_i Y_i(1) + (1-T_i) Y_i(0)\). The quadratic term enters both the true OR and the true PS; a linear-only fit omits it in either or both places, yielding the four scenarios below.
| Scenario | Outcome regression fit | Propensity score fit |
|---|---|---|
| S1 | correct: \(Y \sim (1, X, X^2)\) | correct: \(T \sim (1, X, X^2)\) |
| S2 | correct: \(Y \sim (1, X, X^2)\) | misspecified: \(T \sim (1, X)\) |
| S3 | misspecified: \(Y \sim (1, X)\) | correct: \(T \sim (1, X, X^2)\) |
| S4 | misspecified: \(Y \sim (1, X)\) | misspecified: \(T \sim (1, X)\) |
In each scenario the OR is fitted arm-wise by linear regression and the PS by logistic regression, with the \(X^2\) term either included or omitted as indicated. The fitted \(\hat\pi\) is clipped to \([10^{-3}, 1-10^{-3}]\) for numerical stability. Clipping is a finite-sample regularization, not part of the theoretical estimator: in the runs reported below it is never active (zero of the four million fitted propensity scores per specification fall outside the clipping interval), so it leaves the probability limits of \(\hat\pi\) — and hence the “PS correct” labels — untouched. Double robustness predicts that \(\hat\tau_{\mathrm{AIPW}}\) is consistent in Scenarios S1–S3 and loses its consistency guarantee in S4.
11.9.2 Estimator Construction
Given \(\hat\pi_i = \hat\pi(X_i)\) and initial fitted values \(\hat\mu_t(X_i) \equiv \hat\mu_t^{(0)}(X_i)\), let \(c_{1i} = \hat\pi_i^{-1}\) and \(c_{0i} = (1-\hat\pi_i)^{-1}\).
IPW estimator. \(\hat\tau_{\mathrm{HT}} = n^{-1}\sum_i \{T_i Y_i/\hat\pi_i - (1-T_i)Y_i/(1-\hat\pi_i)\}\).
AIPW estimator. \(\hat\tau_{\mathrm{AIPW}}\) as in Equation 11.6 with fitted nuisances.
Fixed-offset augmented-model estimator. Within each arm, regress \(Y_i - \hat\mu_t(X_i)\) on \(c_{ti}\) without an intercept to obtain \(\hat\gamma_t\), set \(\hat\mu_t^{\mathrm{aug}}(X_i) = \hat\mu_t(X_i) + \hat\gamma_t c_{ti}\), and compute \(\hat\tau_{\mathrm{aug}} = n^{-1}\sum_i\{\hat\mu_1^{\mathrm{aug}}(X_i) - \hat\mu_0^{\mathrm{aug}}(X_i)\}\).
Improved augmented-model estimator. Within each arm, regress \(Y_i\) on \((1, \hat\mu_t^{(0)}(X_i), c_{ti})\) by OLS to obtain \((\hat\alpha_t, \hat\beta_t, \hat\gamma_t)\), set \(\hat\mu_t^{\mathrm{aug}^+}(X_i) = \hat\alpha_t + \hat\beta_t\hat\mu_t(X_i) + \hat\gamma_t c_{ti}\), and compute \(\hat\tau_{\mathrm{aug}^+}\) analogously.
Variance estimators and Wald intervals. For each estimator \(\hat\tau_\bullet\), the asymptotic variance is estimated by the empirical variance of the plug-in influence values, \(\hat V_\bullet = \{n(n-1)\}^{-1}\sum_i (\hat\varphi_{\bullet,i} - \bar\varphi_\bullet)^2\), with \(\hat\varphi_{\bullet,i}\) the influence-function contribution Equation 11.26 evaluated at the estimator-specific fitted values (for HT, the influence value is simply \(T_i Y_i/\hat\pi_i - (1-T_i)Y_i/(1-\hat\pi_i) - \hat\tau_{\mathrm{HT}}\)). The 95% Wald interval is \(\hat\tau_\bullet \pm 1.96\sqrt{\hat V_\bullet}\).
11.9.3 Results
Repeat the simulation \(B = 2000\) times and record bias, variance, MSE, and empirical 95% Wald coverage for each estimator.
| Scenario | Metric | \(\hat\tau_{\mathrm{HT}}\) | \(\hat\tau_{\mathrm{AIPW}}\) | \(\hat\tau_{\mathrm{aug}}\) | \(\hat\tau_{\mathrm{aug}^+}\) |
|---|---|---|---|---|---|
| S1: OR \(\checkmark\), PS \(\checkmark\) | Bias \(\times 10^3\) | \(0.55\) | \(-0.44\) | \(-0.45\) | \(-0.45\) |
| Var \(\times 10^3\) | \(6.82\) | \(4.25\) | \(4.25\) | \(4.30\) | |
| MSE \(\times 10^3\) | \(6.82\) | \(4.25\) | \(4.25\) | \(4.30\) | |
| Cov (%) | \(99.9\) | \(95.0\) | \(94.9\) | \(94.5\) | |
| S2: OR \(\checkmark\), PS \(\times\) | Bias \(\times 10^3\) | \(188.56\) | \(0.62\) | \(0.62\) | \(0.62\) |
| Var \(\times 10^3\) | \(6.32\) | \(4.33\) | \(4.33\) | \(4.33\) | |
| MSE \(\times 10^3\) | \(41.87\) | \(4.32\) | \(4.32\) | \(4.32\) | |
| Cov (%) | \(67.4\) | \(94.2\) | \(94.2\) | \(94.2\) | |
| S3: OR \(\times\), PS \(\checkmark\) | Bias \(\times 10^3\) | \(1.46\) | \(2.02\) | \(2.25\) | \(-25.26\) |
| Var \(\times 10^3\) | \(5.74\) | \(4.71\) | \(4.41\) | \(8.02\) | |
| MSE \(\times 10^3\) | \(5.74\) | \(4.71\) | \(4.41\) | \(8.66\) | |
| Cov (%) | \(99.9\) | \(98.4\) | \(98.2\) | \(93.2\) | |
| S4: OR \(\times\), PS \(\times\) | Bias \(\times 10^3\) | \(188.42\) | \(188.77\) | \(188.62\) | \(-5.42\) |
| Var \(\times 10^3\) | \(6.28\) | \(6.30\) | \(6.29\) | \(4.26\) | |
| MSE \(\times 10^3\) | \(41.78\) | \(41.93\) | \(41.87\) | \(4.29\) | |
| Cov (%) | \(67.5\) | \(33.1\) | \(33.1\) | \(94.7\) |
| Scenario | Metric | \(\hat\tau_{\mathrm{HT}}\) | \(\hat\tau_{\mathrm{AIPW}}\) | \(\hat\tau_{\mathrm{aug}}\) | \(\hat\tau_{\mathrm{aug}^+}\) |
|---|---|---|---|---|---|
| S1 | aveSE \(\times 10^3\) | \(128.83\) | \(64.23\) | \(64.20\) | \(64.12\) |
| empSD \(\times 10^3\) | \(82.58\) | \(65.20\) | \(65.20\) | \(65.56\) | |
| ratio | \(1.56\) | \(0.99\) | \(0.98\) | \(0.98\) | |
| S2 | aveSE \(\times 10^3\) | \(114.68\) | \(63.43\) | \(63.43\) | \(63.43\) |
| empSD \(\times 10^3\) | \(79.50\) | \(65.77\) | \(65.77\) | \(65.77\) | |
| ratio | \(1.44\) | \(0.96\) | \(0.96\) | \(0.96\) | |
| S3 | aveSE \(\times 10^3\) | \(128.91\) | \(84.88\) | \(81.75\) | \(74.28\) |
| empSD \(\times 10^3\) | \(75.74\) | \(68.60\) | \(66.39\) | \(89.57\) | |
| ratio | \(1.70\) | \(1.24\) | \(1.23\) | \(0.83\) | |
| S4 | aveSE \(\times 10^3\) | \(114.84\) | \(78.78\) | \(78.74\) | \(63.55\) |
| empSD \(\times 10^3\) | \(79.27\) | \(79.37\) | \(79.34\) | \(65.27\) | |
| ratio | \(1.45\) | \(0.99\) | \(0.99\) | \(0.97\) |
The results illustrate double robustness directly.
S1 (both correct). All four estimators are approximately unbiased. \(\hat\tau_{\mathrm{HT}}\) has about 60% higher MSE (\(6.82\) vs. \(4.25 \times 10^{-3}\)) because it discards the outcome model. \(\hat\tau_{\mathrm{AIPW}}\), \(\hat\tau_{\mathrm{aug}}\), and \(\hat\tau_{\mathrm{aug}^+}\) are essentially equivalent, confirming that the augmentation parameters converge to \((0, 1, 0)\) when the OR is correct; with both nuisance models correct, Neyman orthogonality implies that the recalibration carries no first-order efficiency cost (the remark on behavior under a correct outcome model). Wald coverage is close to nominal for the three PS-and-OR estimators. HT’s coverage is far above nominal for a more subtle reason. The plug-in influence value treats the propensity score as known, so \(\hat V_{\mathrm{HT}}\) targets the known-\(\pi\) asymptotic variance of the IPW estimator. The lab, however, estimates \(\pi\) by correctly specified logistic maximum likelihood, and Chapter 10 showed that ML estimation of the propensity parameters can only reduce the asymptotic variance of \(\hat\tau_{\mathrm{HT}}\) relative to plugging in the true \(\pi^*\). In this DGP the reduction is large: the average estimated standard error exceeds the empirical standard deviation by more than 50%, and the mismatch does not shrink as \(n\) grows, because it reflects the omitted first-order nuisance correction of Chapter 10, not a finite-sample effect. Re-running the lab with the true \(\pi^*\) plugged in restores HT coverage to the nominal level.
S2 (OR correct, PS misspecified). HT is badly biased (bias \(\approx 0.19\), MSE \(\approx 42 \times 10^{-3}\)) and its Wald interval undercovers (67%). The three augmented estimators are essentially indistinguishable from one another: when the OR is correct, they converge asymptotically to the same limit regardless of PS specification. Their bias is under \(1 \times 10^{-3}\), MSE is about \(4.32 \times 10^{-3}\), and Wald coverage is \(\approx 94.2\%\). This is the first direct demonstration of double robustness: PS misspecification is absorbed by the correctly specified OR model.
S3 (OR misspecified, PS correct). HT, AIPW, and \(\hat\tau_{\mathrm{aug}}\) remain consistent through the correct PS model, with biases under \(3 \times 10^{-3}\). HT again over-covers for the reason given under S1; AIPW and \(\hat\tau_{\mathrm{aug}}\), whose intervals were close to nominal in S1, now also over-cover (\(98.4\%\) and \(98.2\%\)) — the Takeaways below explain why. \(\hat\tau_{\mathrm{aug}}\) achieves the lowest MSE (\(4.41 \times 10^{-3}\)): the targeted regression reduces residual variance without disturbing the IBC constraint. \(\hat\tau_{\mathrm{aug}^+}\) shows a modest finite-sample bias (\(-25 \times 10^{-3}\)) induced by extrapolating the three-term arm-wise regression through high-leverage clever-covariate values; its Wald coverage is \(93.2\%\), still close to nominal. This is the second double-robustness demonstration: OR misspecification is absorbed by the correctly specified PS model.
S4 (both misspecified). Double robustness provides no guarantee, and this is what the simulation shows. HT, AIPW, and \(\hat\tau_{\mathrm{aug}}\) are all badly biased, and Wald coverage is catastrophic for AIPW and aug (\(\approx 33\%\)). The variance estimator still produces a number, but that number targets the spread of a sampling distribution that is not centered on \(\tau\): the Wald interval is concentrated around the wrong value. The diagnostics confirm the mechanism: the aveSE/empSD ratio is \(\approx 0.99\), so the interval has nearly the right width in the wrong place.
Takeaways. Comparing Scenarios S2–S3 against S4 is the crisp illustration of Theorem 11.1: AIPW is consistent whenever at least one nuisance model is correct, and only S4 breaks the guarantee. Wald coverage is more subtle, and the table repays a careful reading. AIPW coverage is close to nominal in S1 and S2 and collapses to \(33\%\) in S4 — the collapse is precisely the behavior anticipated by the remark on double robustness versus efficient inference, since the plug-in variance targets the spread around a misspecified limit, not around \(\tau\).
Scenario S3, however, shows persistent over-coverage (\(98.4\%\) for AIPW), and this is not a finite-sample accident: S3 is exactly the \((\pi^*, \mu^\dagger)\) configuration analyzed in the remark on double robustness versus efficient inference. Because \(\mu_t^\dagger \neq \mu_t^*\), the AIPW moment is not orthogonal in the \(\pi\)-direction, so estimating \(\pi\) affects the sampling distribution at first order; as in the discussion of HT under S1, ML estimation of a correctly specified propensity model reduces the true sampling variance below the quantity targeted by the plug-in influence values, so the Wald interval is conservative. The gap between average standard error and empirical standard deviation does not close as \(n\) grows — it is an asymptotic phenomenon — and re-running S3 with the true \(\pi^*\) plugged in restores nominal coverage.
The mirror-image configuration is S2, where the analogous first-order contribution comes from estimating the correct outcome model under a misspecified \(\pi\); in this DGP that contribution happens to be negligible, which is why S2 coverage looks nominal. Neither the size nor the direction of these omitted contributions is guaranteed in general: under flexible machine-learning nuisances or in other designs, one-correct-model coverage can be conservative or anticonservative. The remedy for the omitted first-order term is a variance estimator that accounts for it — the stacked sandwich of the remark on double robustness versus efficient inference; cross-fitting, by contrast, addresses the same-sample empirical-process problem of flexible nuisance estimation (Chapter 12) and does not by itself restore orthogonality at a misspecified limit.
The lab also isolates a secondary phenomenon visible in S3: when the PS is correct but the OR is misspecified, the fixed-offset \(\hat\tau_{\mathrm{aug}}\) can reduce MSE relative to the plug-in AIPW estimator by shrinking residual variance through the IBC-enforced regression, without disturbing consistency. Efficiency gains of this type are the practical motivation for targeting methods such as TMLE (the remark on TMLE).
11.10 Asymptotic Inference with Estimated Nuisance Functions
A fundamental question in the practical use of the AIPW estimator is: under what conditions is the effect of estimating the nuisance functions \(\pi^*\) and \(\mu_t^*\) asymptotically negligible, so that \(\hat\tau_{\mathrm{AIPW}}\) behaves as if the nuisance functions were known? Answering this question determines when plug-in nuisance estimators, including flexible machine-learning methods, can be used without invalidating large-sample inference on \(\tau\).
11.10.1 Asymptotic Normality
Under suitable regularity conditions, \[\sqrt{n}(\hat\tau_{\mathrm{AIPW}}-\tau) = \frac{1}{\sqrt{n}}\sum_{i=1}^n\varphi_{\mathrm{eff}}(O_i) + o_p(1) \overset{d}{\longrightarrow} N\!\bigl(0,\;\E\{\varphi_{\mathrm{eff}}(O)^2\}\bigr),\] where \(\varphi_{\mathrm{eff}}\) in Equation 11.19 is evaluated at the true nuisances. The \(o_p(1)\) remainder captures the effect of nuisance estimation error. Neyman orthogonality implies that this error enters the AIPW estimator only through a second-order remainder: the Gateaux derivative of the estimating equation with respect to each nuisance function, evaluated at the truth, is zero. A sufficient condition for the remainder to be negligible is the product-rate condition \[\|\hat\pi-\pi^*\|\cdot\|\hat\mu_t-\mu_t^*\| = o_p(n^{-1/2}), \qquad t=0,1, \tag{11.25}\] where \(\|\cdot\|\) denotes the \(L_2(P)\) norm. The drift part of the remainder is exactly the identity Equation 11.7 evaluated at the fitted nuisances, and Equation 11.25 is the condition that the Cauchy–Schwarz bound Equation 11.8 converts into \(o_p(n^{-1/2})\).
A symmetric sufficient condition for Equation 11.25 is that each nuisance estimator converge slightly faster than \(n^{-1/4}\), namely \(\|\hat\pi-\pi^*\| = o_p(n^{-1/4})\) and \(\|\hat\mu_t-\mu_t^*\| = o_p(n^{-1/4})\). (Exact \(n^{-1/4}\) rates would yield a product of order \(O_p(n^{-1/2})\), not \(o_p(n^{-1/2})\), and are therefore borderline.) This is much weaker than the parametric rate \(n^{-1/2}\) required of each individually. In modern applications with flexible machine-learning methods, this statement is usually paired with sample splitting or cross-fitting; without such devices, additional empirical-process conditions are needed to justify the asymptotic expansion. Chapter 12 develops the cross-fitting approach in detail.
11.10.2 Variance Estimation and Confidence Intervals
Because \(\hat\tau_{\mathrm{AIPW}}\) is asymptotically linear with influence function \(\varphi_{\mathrm{eff}}\), its asymptotic variance can be estimated consistently by the empirical variance of the plug-in influence values. Compute \[\hat\varphi_i = \frac{T_i}{\hat\pi(X_i)}\{Y_i-\hat\mu_1(X_i)\} - \frac{1-T_i}{1-\hat\pi(X_i)}\{Y_i-\hat\mu_0(X_i)\} + \hat\mu_1(X_i)-\hat\mu_0(X_i) - \hat\tau_{\mathrm{AIPW}}, \tag{11.26}\] and set \[\hat V = \frac{1}{n(n-1)}\sum_{i=1}^n(\hat\varphi_i-\bar\varphi)^2, \qquad \bar\varphi = \frac{1}{n}\sum_{i=1}^n\hat\varphi_i. \tag{11.27}\] The Wald confidence interval is \(\hat\tau_{\mathrm{AIPW}} \pm z_{1-a/2}\sqrt{\hat V}\).
Theorem 11.9 (Consistency of the Variance Estimator) Suppose \(\sqrt{n}(\hat\tau_{\mathrm{AIPW}}-\tau) = n^{-1/2}\sum_i\varphi_{\mathrm{eff}}(O_i)+o_p(1)\) with \(\E\{\varphi_{\mathrm{eff}}(O)^2\} < \infty\), and that the estimated influence values Equation 11.26 satisfy \[\frac{1}{n}\sum_{i=1}^n \bigl\{\hat\varphi_i - \varphi_{\mathrm{eff}}(O_i)\bigr\}^2 = o_p(1).\] Then \(n\hat V \overset{p}{\to} \E\{\varphi_{\mathrm{eff}}(O)^2\}\); that is, \(\hat V\) is consistent for the asymptotic variance of \(\hat\tau_{\mathrm{AIPW}}\).
The second condition is not implied by asymptotic linearity, which controls the average of the true influence values but not the empirical second moment of the estimation error \(\hat\varphi_i - \varphi_{\mathrm{eff}}(O_i)\). It holds, for example, when both nuisance estimators are \(L^2\)-consistent for the truth, the fitted propensity scores are bounded away from \(0\) and \(1\), and standard moment conditions hold.
11.10.3 Looking Ahead: When Plug-In Fails
For the fitted AIPW estimator, the root-\(n\) expansion and the plug-in influence-function variance estimator rest on the product-rate condition Equation 11.25 and an unstated empirical-process condition on the nuisance estimators; the finite-sample oracle projection identity of Section 11.5 involves no nuisance estimation and does not depend on them. For correctly specified parametric nuisance models of fixed dimension (e.g., logistic regression for \(\pi\) and linear regression for \(\mu_t\)), under the usual identifiability, overlap, and moment conditions, both requirements are routinely verified: the parametric rate \(n^{-1/2}\) is more than enough for the product-rate bound, and standard Donsker arguments dispose of the empirical-process remainder. (A misspecified parametric model converges to the wrong limit, and the product-rate condition to the truth fails entirely — the situation of the remark on double robustness versus efficient inference.)
The situation changes qualitatively when the nuisances are estimated with flexible machine-learning methods — random forests, gradient boosting, neural networks, regularized regressions in high dimensions, and the like. Two difficulties arise. First, the individual nuisance convergence rate may be slower than \(n^{-1/4}\), or hard to characterize explicitly, so the product-rate condition becomes uncomfortably close to binding. Second, the machine-learning function classes are typically not Donsker, so the empirical-process remainder need not vanish when the same data are used for both nuisance estimation and score evaluation — plug-in inference can fail even when the nuisance estimates are predictively accurate.
Chapter 12 addresses both difficulties by (a) formalizing Neyman orthogonality as the first-order insensitivity property invoked above and exhibiting the AIPW score as an orthogonal score, and (b) decoupling nuisance estimation from score evaluation via cross-fitting. Under these devices, the same-data empirical-process problem is resolved: the product-rate condition suffices for the nuisance remainder to be \(o_p(n^{-1/2})\), without a Donsker requirement on the nuisance function classes, provided that nuisance consistency, bounded moments, overlap, and score regularity conditions also hold. Each nuisance estimator is then allowed to converge slightly faster than \(n^{-1/4}\). The variance estimator Equation 11.27 carries over essentially unchanged, using out-of-fold influence values in place of in-sample ones. The resulting double/debiased machine learning (DML) estimator (Chernozhukov et al. 2018) is thus a direct extension of the AIPW development here and the efficiency argument of Section 11.7.
11.11 Comparison of Regression, IPW, and AIPW
All consistency statements in the table below are conditional on the causal identification assumptions of Chapter 6: consistency, conditional exchangeability, and positivity. Without these, model correctness identifies a regression functional but not the causal estimand \(\tau\).
| Estimator | Uses \(\mu_t\) | Uses \(\pi\) | Consistent under regularity if | Main weakness |
|---|---|---|---|---|
| Regression (prediction) | Yes | No | outcome model correct | Sensitive to OR misspecification |
| IPW (Horvitz–Thompson) | No | Yes | propensity model correct | Unstable under weak overlap |
| AIPW | Yes | Yes | either model correct | Requires estimating both nuisances and careful inference (cross-fitting or regularity conditions) |
When overlap is weak, the IBC-based approaches of Section 11.8 may improve numerical stability and finite-sample behavior by folding the propensity score into the outcome-model fit rather than relying on a separate multiplicative correction. They do not, however, eliminate the underlying weak-overlap problem: where covariate support is lacking for one of the treatment arms, no nonparametric estimator under the observed-data model can recover the unsupported portion of the ATE without additional extrapolation assumptions. In such cases the practical recommendation is to change the target estimand — through trimming, overlap weighting, or restriction to a subgroup with adequate support — rather than to expect any algebraic refinement of AIPW to fully repair it.
11.12 Chapter Summary
The prediction estimator is biased when the outcome model is misspecified; the bias can be estimated with the propensity score and subtracted to yield the AIPW estimator (Section 11.2).
The AIPW estimator has two key properties. First, double robustness (Theorem 11.1): it is consistent whenever either the outcome model or the propensity score model is correctly specified. Second, within the class of augmented IPW estimators indexed by \(b_t(x)\), the choice \(b_t^*(x)=\E\{Y(t)\mid X\}\) minimizes variance when both nuisance components are correctly specified (Theorem 11.4–Theorem 11.5). This is a class-specific optimality result.
The same class-specific optimum can be derived from a Hilbert-space projection: the optimal estimator is the projection of the HT estimator onto \(\Lambda^\perp\), and the Pythagorean identity governs the variance reduction (Theorem 11.6).
Under the nonparametric observed-data model and regularity conditions, the AIPW estimating function is the efficient influence function for the ATE, and its variance gives the semiparametric efficiency bound (Section 11.7). The class-specific optimum therefore coincides with the global semiparametric efficiency bound: no regular asymptotically linear estimator, inside or outside the augmented IPW class, can improve upon AIPW’s asymptotic variance. The bias-correction, optimal-augmentation, and semiparametric-efficiency derivations all pick out the same object.
Double robustness can also be enforced within the outcome-regression fitting step, either by IPW-weighted regression or by an augmented regression model that includes the inverse propensity score as a clever covariate. The latter is closely related to the targeting idea used in TMLE (Laan and Rubin 2006) (Section 11.8).
The simulation study (Section 11.9) shows that when the propensity score model is correct, IPW, AIPW, and the augmented-model estimators are all consistent. When the outcome regression is misspecified but the propensity score is correct, the fixed-offset augmented-model estimator can outperform the plug-in AIPW estimator in MSE because the targeted regression fit reduces residual variance while preserving the IBC condition. The improved augmented-model estimator adds additional recalibration but can introduce finite-sample bias (S3) and offers no general guarantee when both nuisance models are misspecified (S4).
For asymptotic inference, the central issue is whether the effect of estimating nuisance functions is asymptotically negligible. Neyman orthogonality yields a second-order remainder, and the product-rate condition Equation 11.25 is sufficient for root-\(n\) inference. In modern machine-learning settings, cross-fitting is often used to make these conditions more plausible; Chapter 12 develops this approach.
11.13 Exercises
1. Bias of the prediction estimator.
- Verify the bias formula Equation 11.1 by writing \(\tau_{\mathrm{pred}}(m)-\tau\) as a difference of working-model errors and simplifying.
- Assume the working propensity score equals the true propensity score, and treat the working regressions \(m_t\) as fixed functions. Show, using iterated expectations, that \(\E\{\widehat{\mathrm{Bias}}(\hat\tau_{\mathrm{pred}})\} = \tau_{\mathrm{pred}}(m) - \tau\), i.e., the weighted-residual expression is unbiased for the actual prediction bias, which is generally nonzero. Under what condition on the working outcome regressions is this expectation zero?
- Construct a simple example (binary \(X\), binary \(T\), binary \(Y\)) in which both the estimated bias is nonzero and the outcome model is misspecified, but the AIPW estimator is still consistent.
2. The AIPW class and optimal control functions.
- Verify that \(\hat\tau_b\) with \(b_t=\mu_t\) equals the AIPW estimator Equation 11.6.
- Using Equation 11.11, show that \(b_t^*(x)=\E\{Y(t)\mid x\}\) minimizes conditional variance by completing the square in \(b_t\). Then use the reparameterization Equation 11.13 to characterize the full set of minimizing pairs (the remark on uniqueness of the minimizing estimator).
- Suppose \(\pi(x)=1/2\) for all \(x\). Simplify Equation 11.11 and interpret the result.
3. Double robustness.
- Verify Case 1 of Theorem 11.1: with \(\mu_t=\mu_t^*\) and arbitrary \(\pi\), show \(\E\{\phi\}=0\).
- Verify Case 2: with \(\pi=\pi^*\) and arbitrary \(\mu_t\), show \(\E\{\phi\}=0\).
- Provide a counterexample showing \(\E\{\phi\}\neq 0\) when both models are misspecified.
4. Projection and the Pythagorean identity.
- Verify \(\mathrm{Cov}(\hat\theta_{\mathrm{opt}},\hat b^*)=0\) from the definition of \(\hat b^*\).
- Deduce Equation 11.15 from the decomposition \(\hat\theta_0 = \hat\theta_{\mathrm{opt}} + \hat b^*\).
- Explain why \(\hat\theta_{\mathrm{opt}}\) is still unbiased for \(\theta\) even though \(\hat b^* \in \Lambda\) is subtracted.
5. Semiparametric efficiency. Let \(\varphi_{\mathrm{IPW}}(O) = TY/\pi^*(X) - (1-T)Y/\{1-\pi^*(X)\} - \tau\) denote the influence function of the IPW estimator with known propensity score.
- Show that \[\varphi_{\mathrm{IPW}}(O) - \varphi_{\mathrm{eff}}(O) = \{T - \pi^*(X)\}\, a^*(X), \qquad a^*(x) = \frac{\mu_1^*(x)}{\pi^*(x)} + \frac{\mu_0^*(x)}{1-\pi^*(x)},\] and conclude \(\E\{\varphi_{\mathrm{eff}}(O)^2\} \leq \E\{\varphi_{\mathrm{IPW}}(O)^2\}\) by verifying that the cross term \(\E(\varphi_{\mathrm{eff}}\,\Delta)\) with \(\Delta = \varphi_{\mathrm{IPW}} - \varphi_{\mathrm{eff}}\) vanishes. (Hint: condition on \(X\) and use \(T^2 = T\).) Observe that the difference has exactly the form of the augmentation elements in Equation 11.18.
- Conclude from part (a) that the efficiency gain is \[\E\{\varphi_{\mathrm{IPW}}^2\} - \E\{\varphi_{\mathrm{eff}}^2\} = \E\bigl[\pi^*(X)\{1-\pi^*(X)\}\, a^*(X)^2\bigr],\] which depends on the levels of the regression functions \(\mu_t^*\), not only on the conditional variances of the potential outcomes. In particular, show that adding the same constant \(c\) to both potential outcomes leaves \(\tau\), \(\varphi_{\mathrm{eff}}\), and every conditional variance unchanged, yet the gain depends on \(c\). What does this suggest about centering outcomes before IPW estimation? (Relate your answer to the augmented class Equation 11.9–Equation 11.10 with constant \(b_t\).)
- Give one reason why an efficient estimator based on \(\varphi_{\mathrm{eff}}\) may still be disfavored in practice relative to a simpler, less efficient alternative.
6. Augmented model and the clever covariate.
- Let \(\hat\mu_1^{(0)}(X_i)\) be any initial outcome model fit. Show that the normal equation for \(\hat\gamma_1\) in the augmented model Equation 11.23 is exactly the IBC condition Equation 11.20, and conclude that the prediction estimator can always be written in AIPW form.
- Explain why \(\hat\gamma_1 \overset{p}{\to} 0\) when the initial model \(\hat\mu_1^{(0)}\) is correctly specified, stating the full-rank condition you need. Does coefficient convergence alone imply that including the clever covariate carries no first-order efficiency cost? Explain what additional property delivers that conclusion when both nuisance models are correct (the remark on behavior under a correct outcome model).
- Explain in words why \(\hat\pi_i^{-1}\) is called the clever covariate: what bias does it absorb, and why does this make the prediction estimator a debiased estimator even when \(\hat\mu_1^{(0)}\) is misspecified?
7. Coverage of the Wald confidence interval (computational). Use the data-generating process of Section 11.9 with the propensity score generalized to \(\pi^*(x) = \expit\{0.2\, x + \gamma(x^2 - 1)\}\), so that the baseline (\(\gamma = 0.2\)) coincides with the lab and larger \(\gamma\) degrades overlap. Implement the AIPW estimator and the influence-function variance estimator Equation 11.27. Following Scenario S3 of the lab, use logistic regression of \(T\) on \((1, X, X^2)\) for \(\hat\pi\) (correct PS) and OLS of \(Y\) on \((1, X)\) separately in each arm for \(\hat\mu_t\) (misspecified OR); truncate \(\hat\pi\) to \([10^{-3}, 1-10^{-3}]\).
- With \(\gamma = 0.2\), \(n = 500\), \(B = 2000\) replications, compute the empirical coverage of the 95% Wald interval. Compare the average standard error with the empirical standard deviation of \(\hat\tau_{\mathrm{AIPW}}\) across replications.
- Repeat part (a) with \(\gamma \in \{0.35,\, 0.45\}\), which produces increasingly weak overlap at large \(|X|\). Report coverage, average SE, and empirical SD. What do you observe? (The grid stops below \(1/2\) deliberately: for \(X \sim N(0,1)\), \(\E\{1/(1-\pi^*(X))\} = 1 + e^{-\gamma}\,\E\{e^{\gamma X^2 + 0.2 X}\}\) is finite only for \(\gamma < 1/2\), so at \(\gamma \geq 1/2\) the efficient influence function has infinite variance and the root-\(n\) Wald theory under study no longer applies. Verify the threshold with the Gaussian moment-generating function.)
- Which assumption of the asymptotic theory (Section 11.10) is stressed as \(\gamma\) grows, and which of the remedies in the Extreme Propensity Scores warning box would you try first? (No simulation required; a paragraph suffices.)
8. The exact double-robustness remainder. Let \(R(\pi, \mu_0, \mu_1)\) denote the right-hand side of Equation 11.7.
- Evaluate \(R\) at the probability limits of the four lab scenarios: argue that \(R = 0\) in S1–S3 while \(R \neq 0\) in general in S4, and reconcile this with the bias pattern in the results table.
- Prove the bound Equation 11.8 from Equation 11.7 under \(\varepsilon \leq \hat\pi \leq 1-\varepsilon\), and explain why the product structure — rather than a sum of the two nuisance errors — is what makes the rate condition Equation 11.25 attainable with flexible machine-learning nuisances.
- Show directly from Equation 11.7 that the Gateaux derivatives of the population moment with respect to each nuisance vanish at \((\pi^*, \mu^*)\), and relate this to Neyman orthogonality.
9. Cross-fitting does not restore orthogonality. Consider the setting of the remark on double robustness versus efficient inference: the propensity model is correctly specified and fitted by maximum likelihood at the parametric rate, so \(\hat\pi \to \pi^*\), while the outcome model is misspecified, so \(\hat\mu_t \to \mu_t^\dagger \neq \mu_t^*\). Suppose the nuisances are fitted on one half of the sample and the AIPW score is averaged over the other half (a single sample split).
- Expand \(\sqrt{n}(\hat\tau_{\mathrm{AIPW}} - \tau)\) around \((\pi^*, \mu^\dagger)\) and identify the term \(\sqrt{n}\,D_\pi[\hat\pi - \pi^*]\), where \(D_\pi\) is the Gateaux derivative displayed in the remark on double robustness versus efficient inference.
- Explain why independence of the two half-samples removes the empirical-process remainder but leaves \(\sqrt{n}\,D_\pi[\hat\pi - \pi^*] = O_p(1)\) intact, so the plug-in variance Equation 11.27 remains invalid.
- Which of the remedies in the remark on double robustness versus efficient inference addresses this term, and what does it require in practice?