Doubly Robust Estimation of Group-Time Effects

Why Combine Two Adjustment Strategies?

For a cohort first treated in period $g$, the effect at a later period $t$ depends on a counterfactual: what would that cohort’s outcome at $t$ have been without treatment? In a staggered-adoption setting, a natural comparison uses units still untreated at $t$. The earlier material on group-time effects establishes the target and the parallel-trends logic; the practical challenge is that treated and comparison units may differ in their pre-treatment characteristics.

Outcome regression addresses those differences by modeling changes among untreated units as a function of covariates. Propensity weighting instead reweights untreated units to resemble the treated cohort. A doubly robust estimator combines both adjustments. Under the identifying assumptions, it is consistent if either the outcome model or the propensity model is correctly specified—not necessarily both. This protection is valuable, but it does not repair failed parallel trends, lack of overlap, or a comparison group contaminated by treatment.

Target, Comparison Group, and Conditions

Definition: Let $G_i$ denote unit $i$’s first treatment period, with $G_i=\infty$ for a never-treated unit. For a cohort first treated in period $g$ and a post-treatment period $t\geq g$, the group-time average treatment effect is $$ ATT(g,t)=E\!\left[Y_{it}(g)-Y_{it}(0)\mid G_i=g\right]. $$ Here $Y_{it}(g)$ is the outcome at $t$ if unit $i$ first receives treatment at $g$, and $Y_{it}(0)$ is its outcome at $t$ if it remains untreated. Use period $g-1$ as the pre-treatment baseline and define $\Delta Y_i=Y_{it}-Y_{i,g-1}$. The comparison group consists of units untreated through $t$: $G_i>t$, including never-treated units.

Let $X_i$ contain covariates measured before treatment. In the comparison population for this particular $(g,t)$, define $D_i=1$ for members of cohort $g$, and $D_i=0$ for eligible controls. The relevant propensity score and untreated outcome regression are

e(X_i)=P(D_i=1\mid X_i,D_i\in\{0,1\}),\qquad m(X_i)=E[\Delta Y_i\mid D_i=0,X_i].

These are nuisance functions: they help estimate the effect but are not themselves the causal target. The following conditions give the estimator its causal interpretation.

No anticipation: treatment beginning in $g$ does not affect outcomes by $g-1$, so the baseline change is not already affected by treatment.
Conditional parallel trends: among units with the same $X_i$, the mean change that the treated cohort would have experienced without treatment equals the mean change among eligible controls: $$ E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i=g,X_i] = E[Y_{it}(0)-Y_{i,g-1}(0)\mid G_i>t,X_i]. $$
Overlap: treated units’ covariate values are represented among controls. In particular, $0<e(X_i)<1$ over the treated cohort’s covariate support, with probabilities sufficiently bounded away from 1 for stable weighting.
Pre-treatment covariates and valid sampling: $X_i$ is not affected by treatment, and the sampling design supports the population expectations and uncertainty calculations used.

Conditional parallel trends is an assumption about untreated outcome changes, not a consequence of fitting either nuisance model. Covariates should therefore be chosen and justified using subject-matter knowledge, not selected simply because they improve the estimated effect.

The Doubly Robust Estimator

The outcome-regression adjustment compares treated changes with the changes predicted for treated units by the untreated outcome model. Propensity weighting adds a correction based on control residuals, where each control residual is its observed change minus the model-predicted change. In a sample restricted to the cohort and its eligible controls, the estimator is

\widehat{ATT}(g,t) = \frac{1}{n_1}\sum_{i:D_i=1}\left(\Delta Y_i-\widehat m(X_i)\right) - \frac{1}{n_1}\sum_{i:D_i=0} \frac{\widehat e(X_i)}{1-\widehat e(X_i)} \left(\Delta Y_i-\widehat m(X_i)\right),

where $n_1=\sum_i D_i$ is the number of treated-cohort units in the restricted sample. The control weight is the estimated odds of cohort membership, $\widehat e(X_i)/(1-\widehat e(X_i))$. Dividing both sums by $n_1$ puts the weighted control correction on the treated-cohort scale. The formula targets an effect in the outcome’s units.

Theorem: Under no anticipation, conditional parallel trends, overlap, and regularity conditions for estimation, this estimator is consistent for $ATT(g,t)$ if either $m(X)$ is correctly specified or $e(X)$ is correctly specified. If both models are misspecified, consistency is not generally guaranteed.

Why the Two Routes to Consistency Work

The key identity from conditional parallel trends is that the treated cohort’s mean untreated change, conditional on $X$, is $m(X)$. Thus the effect can be expressed as the treated cohort’s mean observed change minus its mean predicted untreated change. The first part of the estimator estimates the difference between those two quantities, with the second part correcting for residual changes among controls.

If the outcome regression is correct, controls have mean residual zero at each $X$:

E[\Delta Y_i-m(X_i)\mid D_i=0,X_i]=0.

Consequently, weighting these residuals by any well-behaved function of $X_i$, including an estimated propensity odds that need not be correct, has expectation zero. The treated residual mean is then the treated cohort’s mean observed change minus its mean untreated change. Conditional parallel trends makes that difference $ATT(g,t)$.

Alternatively, suppose the propensity score is correct, while the outcome regression may be wrong. Correct odds reweight the control covariate distribution to match the treated covariate distribution. To see this, for any function $h(X)$,

E\!\left[(1-D)\frac{e(X)}{1-e(X)}h(X)\right] = E[D\,h(X)].

Both sides average over the restricted cohort-and-control population. The identity follows by conditioning on $X$: the left side has conditional factor $(1-e(X))e(X)/(1-e(X))=e(X)$, which is the treated probability on the right. Apply this identity to $h(X)=E[\Delta Y-m(X)\mid D=0,X]$. The weighted control residual removes the same covariate-specific residual component that appears in the treated residual mean. What remains is the treated-control difference in mean changes at the same $X$, which conditional parallel trends identifies as the treatment effect.

This argument explains the term doubly robust: there are two separate routes to consistency. It does not mean the estimator is protected against misspecification of both models, violations of the causal assumptions, or extreme weights that make estimation unstable.

Worked Example: Correct Outcome Regression, Misspecified Propensity

Suppose a hypothetical cohort and its eligible controls have a binary pre-treatment covariate $X$. The observed changes are:

Group	$X$	Observed $\Delta Y$ values
Treated cohort	High	10, 12
Treated cohort	Low	4, 6
Controls	High	5, 7
Controls	Low	1, 5

The control means give $\widehat m(\text{High})=6$ and $\widehat m(\text{Low})=3$. Treat this outcome regression as correct for the example. Suppose the propensity model is misspecified and assigns probabilities $0.75$ to High and $0.25$ to Low; the corresponding odds are 3 and $1/3$.

For the treated units, the outcome-regression residuals are $10-6=4$, $12-6=6$, $4-3=1$, and $6-3=3$. Their mean is

\frac{4+6+1+3}{4}=3.5.

The High control residuals are $5-6=-1$ and $7-6=1$; their weighted sum is $3(-1)+3(1)=0$. The Low control residuals are $1-3=-2$ and $5-3=2$; their weighted sum is $\frac{1}{3}(-2)+\frac{1}{3}(2)=0$. Therefore

\widehat{ATT}(g,t)=3.5-\frac{0+0}{4}=3.5.

The estimate is 3.5 outcome units. The propensity model was deliberately wrong, but the control residuals average to zero within each covariate group because the outcome regression is correct in this example. That illustrates the outcome-regression route to consistency; it is not evidence that arbitrary propensity misspecification is harmless in every finite sample.

Worked Example: Correct Propensity, Misspecified Outcome Regression

Use the same hypothetical changes. There are two treated units and two controls at each value of $X$, so the cohort-membership propensity is $0.5$ in both covariate groups. The correct odds are $0.5/(1-0.5)=1$. Now deliberately use a misspecified outcome model that predicts $\widehat m(X)=4$ for every unit.

The treated residuals sum to $(10-4)+(12-4)+(4-4)+(6-4)=16$, so their mean is $16/4=4$. The control residuals sum to $(5-4)+(7-4)+(1-4)+(5-4)=2$. Since every control weight is 1, the control correction is $2/4=0.5$. The estimate is

\widehat{ATT}(g,t)=4-0.5=3.5.

The covariate-specific treated-control differences in observed changes are $11-6=5$ for High and $5-3=2$ for Low. Averaging across the two equally sized treated-covariate groups gives $(5+2)/2=3.5$ outcome units. Correct propensity weighting balances those groups, so the misspecified outcome regression’s residual component cancels in the weighted comparison.

Cross-Fitting and Uncertainty

Flexible methods can estimate $m(X)$ and $e(X)$, but fitting and evaluating those models on the same observations can produce overfitting bias. Cross-fitting separates these tasks. Divide the independent sampling units into folds; for each fold, fit both nuisance models using the other folds, then predict $m(X_i)$ and $e(X_i)$ for units in the held-out fold. Assemble these out-of-fold predictions and calculate the doubly robust estimate. If outcomes are clustered—for example, people within a region—assign whole clusters to folds rather than splitting members of the same cluster across training and evaluation data.

Cross-fitting reduces the dependence between a unit’s residual and the nuisance-model fit used to predict it. For large-sample inference with flexible nuisance models, a standard sufficient rate condition is that the product of the nuisance estimation errors is small enough to be negligible at the $n^{-1/2}$ scale; under suitable norms and regularity conditions, having each error smaller than order $n^{-1/4}$ is sufficient. These rate statements do not replace overlap or causal identification assumptions. With small samples or weak overlap, cross-fitting cannot make unstable weights reliable.

An uncertainty estimate can be based on the estimator’s influence contributions, or on a bootstrap that resamples the independent sampling units and refits the nuisance models in each resample. If observations within clusters may be dependent, the independent units for uncertainty estimation are the clusters: use cluster-level influence contributions or resample whole clusters. A standard error describes sampling variability under the model and sampling assumptions; it does not account automatically for uncertainty about whether parallel trends holds.

Worked Example: Turning Influence Contributions Into a Standard Error

For illustration, suppose an analysis reports $\widehat{ATT}(g,t)=3.5$ outcome units and four independent clusters. After fitting the estimator, assume their estimated influence contributions, already centered to have mean zero, are $2,-1,3,-4$ outcome units. A simple cluster-based standard error uses the sample variance of those contributions divided by the number of clusters.

\bar{\phi}=0,\qquad s_{\phi}^{\,2} = \frac{2^2+(-1)^2+3^2+(-4)^2}{4-1} = \frac{30}{3}=10, \qquad SE(\widehat{ATT})=\sqrt{\frac{10}{4}}=1.581.

This illustrative standard error is 1.581 outcome units. A normal-approximation 95% interval is $3.5\pm1.96(1.581)$, or approximately $0.401$ to $6.599$ outcome units. Under repeated sampling and the conditions supporting the approximation, such intervals would cover the target effect about 95% of the time. Four clusters are generally too few for dependable large-sample cluster inference; the arithmetic illustrates the calculation, not a recommendation to rely on it with so few clusters.

Common Mistakes and Practical Checks

Calling the estimator robust to every problem. “Doubly robust” refers only to consistency when one of the two nuisance models is correct under the causal assumptions. It does not protect against failed parallel trends or anticipation.
Using treated controls. Controls must remain untreated through $t$. A unit already treated by $t$ does not provide the untreated outcome change required by this comparison.
Ignoring extreme propensity odds. If estimated probabilities are near 1, the odds can become very large. Inspect overlap and weight concentration; trimming or changing the target population requires explicit justification because it can change the estimand.
Using post-treatment covariates. A covariate affected by treatment is not a suitable pre-treatment adjustment variable for this construction.
Treating the estimate as an effect in every population. $ATT(g,t)$ is specific to cohort $g$ and period $t$. It is an average for that treated cohort, not automatically an effect for other cohorts or periods.

A careful analysis therefore identifies the cohort and eligible controls, defines the pre-treatment covariates, checks overlap, and states the conditional parallel-trends assumption. It then reports the cohort-time estimate with uncertainty appropriate to the sampling dependence, while describing which nuisance-model specification is being relied upon and why.

Check Your Understanding

Answer each question using the definitions and estimator above.

For a cohort first treated in period $g$, why must a comparison unit remain untreated through period $t$ when estimating $ATT(g,t)$?
State the conditional parallel-trends condition in words and explain why it is not guaranteed by fitting an outcome regression.
If the outcome regression is correct but the propensity model is wrong, what happens to the expected weighted control residual under the stated conditions?
In the estimator, why are control residuals weighted by $e(X)/(1-e(X))$ rather than by $e(X)$ alone?
What is cross-fitting intended to reduce, and how should folds be formed when observations are dependent within clusters?
Does a narrow confidence interval establish that conditional parallel trends is true? Explain.