Treatment-Residual Instruments for G-Estimation

Why Center Treatment Before Building a G-Estimation Moment?

In g-estimation, a candidate blip parameter is used to remove the modeled treatment effect from the observed outcome. At the true parameter, the resulting blipped-down outcome should have no remaining conditional mean association with treatment, given the measured pre-treatment history. A useful way to express that condition is to multiply the blipped-down outcome by treatment after subtracting its conditional mean given history.

This subtraction is not cosmetic. The residual has conditional mean zero, so it is orthogonal to every suitable function of history. At the same time, it retains the part of treatment variation that differs among people with the same history. That combination produces moments that can identify blip parameters while accounting for treatment assignment related to measured history.

Definition: Let $H$ denote pre-treatment history and $A$ treatment at the decision under study. The treatment-residual instrument is $$ R=A-E[A\mid H]. $$ For binary treatment, write $e(H)=P(A=1\mid H)$; then $R=A-e(H)$. An instrument formed by multiplying $R$ by a vector of history functions $q(H)$ is $q(H)R$.

From Conditional Mean Independence to a Moment

Work at one treatment decision. Assume consistency, and assume sequential exchangeability at this decision: conditional on $H$, treatment is independent of the relevant counterfactual outcome under no treatment. Suppose the structural nested mean model and its blip-down operation are correctly specified. Write $U_\gamma$ for the observed outcome transformed by candidate parameter $\gamma$ to remove the modeled treatment effect. At the true parameter $\gamma_0$, these conditions imply the conditional mean restriction $E[U_{\gamma_0}\mid A,H]=E[U_{\gamma_0}\mid H]$. This is a conditional-mean statement; a full conditional distributional independence is not required for the moment derivation.

The treatment model is used here to define $E[A\mid H]$, not to assert that treatment is randomized unconditionally. The derivation also requires integrability, such as $E[|R U_{\gamma_0}|]<\infty$, and for vector instruments, finite expectations of the corresponding components.

Theorem: Under the conditional mean restriction above, for any measurable $q(H)$ for which the moments exist, $$ E[q(H)\{A-E[A\mid H]\}U_{\gamma_0}]=0. $$

Proof. Set $e_A(H)=E[A\mid H]$. Conditional on $H$, the conditional mean restriction gives $E[U_{\gamma_0}\mid A,H]=m(H)$, where $m(H)=E[U_{\gamma_0}\mid H]$. Therefore $$ E[(A-e_A(H))U_{\gamma_0}\mid H] =E[(A-e_A(H))m(H)\mid H] =m(H)E[A-e_A(H)\mid H]=0. $$ Multiplying by $q(H)$, which is determined by $H$, and taking expectations yields the stated moment. The assumed integrability justifies these conditional expectations and the iterated-expectation step. $\square$

For binary treatment, the same moment has a more concrete interpretation. Let $\mu_a(H)=E[U_\gamma\mid A=a,H]$ and $e(H)=P(A=1\mid H)$. When $0<e(H)<1$, $$ E[(A-e(H))U_\gamma\mid H] =e(H)\{1-e(H)\}\{\mu_1(H)-\mu_0(H)\}. $$ Indeed, the $A=1$ contribution is $e(1-e)\mu_1$, and the $A=0$ contribution is $(1-e)(-e)\mu_0$. Their sum is the displayed expression. Thus, at the true parameter, the treatment-specific conditional means of the blipped-down outcome agree, and the residual moment is zero.

Linear Blips and Identification

A common single-decision additive model has blipped-down outcome $U_\gamma=Y-A\gamma^\top q(H)$, where $Y$ is the observed outcome and $q(H)$ is a $p$-component vector of specified history functions. The blip is $\gamma^\top q(H)$ for treated individuals. Using the same vector as the instrument gives the estimating equation $$ E[q(H)\{A-e(H)\}\{Y-A\gamma^\top q(H)\}]=0. $$ This equation is linear in $\gamma$. Under the conditional mean restriction for the true parameter, its coefficient matrix is $M=E[e(H)\{1-e(H)\}q(H)q(H)^\top]$ in the binary-treatment case.

Theorem: Suppose treatment is binary, the moments are finite, and the conditional mean restriction holds at $\gamma_0$. If $$ M=E[e(H)\{1-e(H)\}q(H)q(H)^\top] $$ is nonsingular, then the population estimating equation with instrument $q(H)\{A-e(H)\}$ has the unique solution $\gamma_0$.

Proof. Let $\Delta_Y(H)=E[Y\mid A=1,H]-E[Y\mid A=0,H]$. The conditional treatment contrast in $U_\gamma$ is $\Delta_Y(H)-\gamma^\top q(H)$, since subtracting $A\gamma^\top q(H)$ changes the treated mean by $\gamma^\top q(H)$ and leaves the untreated mean unchanged. Applying the binary residual identity componentwise, the population moment is $$ E[e(H)\{1-e(H)\}q(H)\{\Delta_Y(H)-\gamma^\top q(H)\}]. $$ At $\gamma_0$, the conditional mean restriction makes the bracket zero wherever treatment has positive probability in both arms, so $\gamma_0$ solves the equation. For any other $\gamma$, subtracting the moment at $\gamma_0$ from the moment at $\gamma$ gives $-M(\gamma-\gamma_0)$. If both parameters solve the equation, then $M(\gamma-\gamma_0)=0$. Nonsingularity implies $\gamma=\gamma_0$. $\square$

Worked Examples

Worked Example: Recovering a Constant Blip Across Two Histories

Suppose $H$ has values 0 and 1, each with probability $0.5$. The treatment probabilities are $e(0)=0.25$ and $e(1)=0.75$. In both strata, the conditional mean outcome contrast is 3: $E[Y\mid A=1,H]-E[Y\mid A=0,H]=3$. Use the constant blip $U_\gamma=Y-\gamma A$ and instrument $A-e(H)$.

In each stratum, $e(H)\{1-e(H)\}=0.25(0.75)=0.1875$. The conditional residual moment is therefore $0.1875(3-\gamma)$. Averaging over the two equally likely strata gives $$ E[(A-e(H))U_\gamma] =0.5(0.1875)(3-\gamma)+0.5(0.1875)(3-\gamma) =0.1875(3-\gamma). $$ Setting this to zero gives $\gamma=3$. The positive factor $0.1875$ confirms that the moment has a unique root. If treatment were deterministic in both strata, that factor would be zero and this moment would contain no information about the blip.

Worked Example: Two Blip Coefficients and the Correct Matrix

Let $H\in\{0,1\}$, with probability $0.5$ for each value, and let $e(0)=0.25$, $e(1)=0.75$. Use $q(H)=(1,H)^\top$ and $U_\gamma=Y-A(\gamma_0+\gamma_1H)$. Suppose the conditional outcome contrasts are 2 when $H=0$ and 5 when $H=1$. The corresponding true coefficients are $(\gamma_0,\gamma_1)=(2,3)$, since $2+3(0)=2$ and $2+3(1)=5$.

The matrix in the linear moment equation is $E[e(1-e)qq^\top]$. Each stratum contributes its probability, $0.5$, times $e(1-e)=0.1875$, for a weight of $0.09375$. At $H=0$, $q=(1,0)^\top$, so $qq^\top$ has a zero second row and column. At $H=1$, $q=(1,1)^\top$. Hence $$ M=0.09375 \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix} + 0.09375 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix} = \begin{pmatrix} 0.1875&0.09375\\ 0.09375&0.09375 \end{pmatrix}. $$

The right-hand side is $E[e(1-e)q\Delta_Y(H)]$. The $H=0$ contribution is $0.09375(1,0)^\top(2)=(0.1875,0)^\top$; the $H=1$ contribution is $0.09375(1,1)^\top(5)=(0.46875,0.46875)^\top$. Thus the right-hand side is $(0.65625,0.46875)^\top$. Checking the proposed coefficients, $$ M \begin{pmatrix} 2\\ 3 \end{pmatrix} = \begin{pmatrix} 0.1875(2)+0.09375(3)\\ 0.09375(2)+0.09375(3) \end{pmatrix} = \begin{pmatrix} 0.65625\\ 0.46875 \end{pmatrix}. $$ The determinant of $M$ is $0.1875(0.09375)-(0.09375)^2=0.0087890625$, which is positive, so the matrix is nonsingular and the coefficients are identified by these moments.

Worked Example: A Continuous-Treatment Residual

Suppose there is no $H$, $A\sim N(1,4)$, and $Y=2+1.5A+\varepsilon$, where $\varepsilon$ is independent of $A$ with mean zero and finite variance. For the blipped-down outcome $U_\gamma=Y-\gamma A$, the residual is $A-E[A]=A-1$.

Because $E[A-1]=0$, the intercept and the mean-zero error contribute nothing to the moment. Also, $E[(A-1)A]=E[(A-1)\{(A-1)+1\}]=\operatorname{Var}(A)=4$, and independence gives $E[(A-1)\varepsilon]=0$. Therefore $$ E[(A-1)U_\gamma] =1.5E[(A-1)A]-\gamma E[(A-1)A] =1.5(4)-\gamma(4). $$ The equation is zero at $\gamma=1.5$. The calculation uses treatment variation around its mean; if $A$ had zero variance, the moment would be zero for every $\gamma$, providing no identification.

Diagnostics, Alternatives, and Limitations

The residual construction clarifies the role of treatment assignment: only variation in $A$ left after conditioning on $H$ enters the instrument. In binary treatment, the conditional information factor is $e(H)\{1-e(H)\}$. It is small when treatment is nearly deterministic and exactly zero when one treatment arm is absent in a stratum. For multiple blip coefficients, a nearly singular matrix $M$ signals that the history functions are difficult to distinguish in the regions where both treatments occur. These are identification and numerical-stability warnings, not issues cured by changing the algebra.

The residual moment is an alternative to writing a conditional mean restriction directly or using a treatment-weighted g-method. It can yield a compact set of equations for blip parameters, while the parametric g-formula targets counterfactual means by modeling outcome and covariate distributions. The approaches rely on different modeling and estimation steps; the residual instrument does not remove the need for exchangeability, a suitable blip model, and treatment variation.

A common mistake is to reason that $E[A-e(H)\mid H]=0$ by itself makes any g-estimation equation valid. It only guarantees orthogonality to functions of $H$. The outcome must also satisfy the blipped-down conditional mean restriction at the true parameter. If the blip is misspecified, treatment remains associated with the transformed outcome given history, and the moment may select a parameter that is a weighted compromise rather than a scientifically correct effect. Likewise, a residual based on the wrong conditional treatment mean need not have the required orthogonality.

This derivation establishes population moments, not the finite-sample behavior of an estimator or the consequences of estimating the treatment mechanism. It also does not guarantee that the chosen instrument is efficient or that every component of a high-dimensional blip is identified. Those questions require additional analysis.

Specify the decision and history.
Use only covariates measured before the treatment decision in $H$, and define the candidate blipped-down outcome $U_\gamma$.

Center treatment conditionally.
Form $R=A-E[A\mid H]$, or $R=A-e(H)$ for binary treatment.

Choose history functions and check variation.
Construct $q(H)R$; in binary treatment, inspect overlap and whether the weighted matrix $M$ can distinguish the coefficients.

Set the moment to zero and assess its meaning.
Solve the estimating equation only under a defensible conditional mean restriction and correctly specified blip.

Check Your Understanding

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

For binary treatment, what is the treatment-residual instrument in terms of the propensity $e(H)$?
Why does the residual have zero conditional expectation given history, and what does that fact alone fail to establish?
If $e(H)=0.5$ in a stratum, what is the binary treatment information factor $e(H)\{1-e(H)\}$? If $e(H)=0$, what does the factor indicate?
In a two-coefficient additive blip model, what matrix condition gives uniqueness of the population moment solution?
For continuous treatment with $\operatorname{Var}(A)=0$, why can the treatment-residual moment not identify a blip coefficient?

Solutions. (1) It is $A-e(H)$. (2) By definition, $E[A-e(H)\mid H]=E[A\mid H]-e(H)=0$; this alone does not show that the blipped-down outcome has the required conditional mean independence from treatment. (3) At $e=0.5$, the factor is $0.5(0.5)=0.25$. At $e=0$, it is zero, indicating no treated observations in that stratum and no within-stratum treatment contrast information. (4) The matrix $E[e(H)\{1-e(H)\}q(H)q(H)^\top]$ must be nonsingular. (5) Zero treatment variance makes $A-E[A\mid H]$ zero almost surely when there is no history, so the moment is zero for every coefficient rather than selecting a unique value.