Imputation Estimation for Staggered Difference-in-Differences

Why Impute Untreated Outcomes?

With staggered adoption, different units begin treatment in different periods. Once a unit is treated, its observed outcome no longer reveals what would have happened to it untreated. Imputation estimation addresses this missing counterfactual directly: fit a model for untreated outcomes using observations that are untreated, predict the untreated outcomes for treated unit-periods, and compare those predictions with the observed outcomes.

The approach makes the source of identification visible. The causal comparison depends on the untreated-outcome model being appropriate for the treated observations, not on a single regression coefficient whose weighting may be difficult to interpret. It also permits treatment effects to vary across units and over time. The cost is that predictions rely on a specified model and on enough untreated data to estimate and apply it.

The Untreated-Outcome Model and Identification

A common imputation specification models the conditional mean of the untreated potential outcome as an additive unit and time component, optionally with covariates:

Here \(\alpha_i\) captures persistent differences across units, \(\lambda_t\) captures common period changes, \(X_{it}\) is a vector of covariates, and \(\mathcal{F}\) denotes the conditioning information used to specify the model, such as treatment timing and observed covariates. The fixed effects require a normalization for estimation, but predicted values do not depend on which equivalent normalization is chosen. A model may omit \(X_{it}\) when no covariates are included.

The equation is a restriction on untreated potential outcomes, not on observed outcomes after treatment. Estimate its parameters only from untreated observations: never-treated units and periods before a unit first receives treatment. Do not use treated outcomes to fit the untreated-outcome model, because treatment effects would then distort the predictions intended to represent no treatment.

For a causal interpretation, the untreated-outcome model must correctly describe the relevant conditional mean, and the untreated observations must provide support for estimating it and predicting the treated cells. In particular, a unit fixed effect cannot be estimated for a unit with no observed untreated period unless some additional structure supplies that information. There must also be no anticipation: before its treatment begins, a unit’s observed outcome follows its untreated potential outcome. Consistency and no interference are needed so observed outcomes correspond to the relevant potential outcomes and one unit’s treatment does not change another unit’s outcome. These conditions do not make treatment adoption random; they state what must hold for the model-based counterfactual comparison to be credible.

The causal assumption is stronger than merely fitting a regression well to the untreated sample. A good fit among untreated observations does not, by itself, establish that the model correctly predicts the untreated trajectory of treated units. That extrapolation is the key identifying step.

Worked Example: Predicting One Treated Unit’s Counterfactual

Aggregating Imputed Effects

An imputation procedure first produces treated-cell estimates. A single summary requires weights, and those weights determine the estimand. For a collection \(\mathcal{T}\) of treated unit-periods with nonnegative weights \(w_{it}\) that sum to 1, define

Equal weights across treated observations estimate an average over those observations, so units observed treated for more periods contribute more. Equal weights across units instead give each unit the same total weight, which changes the target when treatment duration differs. Weights can also select a particular cohort or event time. Always state which cells are included and how they are weighted; an aggregate is not defined by the imputation step alone.

Why Counterfactual Estimation Uncertainty Matters

An imputed effect contains uncertainty from two sources: the treated outcome and the fitted model used to predict its untreated counterfactual. Treating \(\widehat{Y}_{it}(0)\) as known would omit the second source and generally understate uncertainty. A useful derivation makes this explicit for a linear model with fixed design matrices and independent, homoskedastic errors. Let \(Z_0\) be the design matrix for untreated observations, \(Y_0\) their outcomes, and \(Z_1\) the design matrix for the treated cells being averaged. Let \(w\) contain the aggregation weights, and define \(c=Z_1'w\). With ordinary least squares,

The second expression follows because the weighted average of the fitted counterfactuals is \(w'Z_1\widehat{\beta}=c'\widehat{\beta}\). If the treated-cell errors are independent of the errors used to fit the untreated model, then the two random components in this difference have zero covariance. When each error has variance \(\sigma^2\), the variance is

The first term is variation in the treated outcomes; the second is uncertainty from estimating the counterfactual model. This formula is a justification under the stated simplified assumptions, not a universal standard-error recipe. Panel outcomes often have serial dependence within units, and treated outcomes can be correlated with untreated observations from the same units. In those settings, the covariance between the two components also matters.

In practice, use an inference procedure that matches the sampling and treatment-assignment structure. A common approach is a cluster-robust variance estimator at the level at which treatment is assigned, or a cluster bootstrap that resamples those units and refits the untreated-outcome model in every replication. The entire imputation-and-aggregation procedure must be repeated so that uncertainty in the fitted counterfactuals is included. With few treated clusters, conventional cluster-robust approximations can be unreliable; the number and distribution of treated clusters need to inform the choice and interpretation of inference.

Practical Checks and Common Pitfalls

Before interpreting an imputation estimate, inspect which observations actually identify the untreated-outcome model. A model can be estimable overall yet fail to predict particular treated cells: for example, a treated unit with no untreated observations offers no direct information about its unit fixed effect. Covariate values or time patterns outside the untreated sample’s support also require extrapolation. Such cells should not be treated as automatically identified merely because software produces a fitted value.

A frequent error is to fit the untreated-outcome model using all observations, including treated outcomes, then describe the resulting fitted values as untreated counterfactuals. This can feed treatment effects into the prediction model and pull estimated effects toward zero or otherwise distort them. The correction is to fit the specified model on untreated observations only, predict counterfactuals for treated cells, and report the model and support assumptions that justify those predictions.

A second pitfall is to report an aggregate without defining its weights. Equal weight per treated observation, per unit, per cohort, or per event time can answer different questions. A third is to calculate uncertainty as though the fitted counterfactuals were fixed. Inference must reflect estimation of the untreated model and the dependence structure of the panel.