24 Estimating QTEs and DTEs Under Unconfoundedness
24.1 Treatment Support and Estimation Strategy
Before translating identifying equations into sample estimators, we must consider the support of the treatment \(X_i\). That affects how we implement the regression specifications and whether we can avoid parametric assumptions.
- If \(X_i\) is continuously distributed, we treat \(X_i\) as a covariate in our regression specifications (e.g., as part of the index \(\psi(X_i, W_i)\)). The causal parameters at specific values of interest, \(x_1\) and \(x_2\), are recovered from the estimated regression functions.
- If \(X_i\) has finite support, we have more flexibility regarding how treatments interact with other covariates. Typically, one either splits the sample and estimates separate models within each treatment subsample \(\{i : X_i = x\}\) or includes treatment dummies in the regression specifications.
24.2 Estimation under Unconditional Unconfoundedness
Under unconditional unconfoundedness (as in a randomized controlled trial), potential outcomes are independent of treatment assignment. Recall that causal parameters are identified by the observed outcome distributions within each treatment group (section 19). Estimation is therefore straightforward.
24.2.1 Discrete Treatment
When the treatment \(X_i\) is discrete, estimation requires no modeling assumptions. The unconditional potential outcome CDF \(F_{Y^x}(y)\) is estimated nonparametrically using the empirical CDF of the outcome \(Y_i\) within the subsample assigned to treatment level \(x\): \[ \hat{F}_{Y^x}(y) = \frac{1}{\sum_{i=1}^N \mathbb{I}\{X_i = x\}} \sum_{i=1}^N \mathbb{I}\{Y_i \leq y\} \mathbb{I}\{X_i = x\}. \tag{24.1}\]
Similarly, the unconditional potential outcome quantile \(Q_{Y^x}(\tau)\) is estimated by the sample \(\tau\)th quantile of the subsample with \(X_i = x\): \[ \hat{Q}_{Y^x}(\tau) = \inf \left\{ q : \hat{F}_{Y^x}(q) \geq \tau \right\}. \tag{24.2}\]
In this setting, the estimated QTEs and DTEs are differences in subsample quantiles and empirical CDFs.
24.2.2 Continuous Treatment
If the treatment \(X_i\) is continuous, we cannot construct subsamples for specific values of \(x\). Instead, we pool the data and rely on the smoothing properties of distribution or quantile regression to interpolate across treatment levels:
- Estimating the CDF: We estimate \(F_{Y^x}(y)\) by running a distribution regression of \(Y_i\) on a vector of treatment transformations \(\psi(X_i)\). This yields the fitted conditional CDF \(\hat{F}_{Y|X}(y|x) = \Lambda(\psi(x)'\hat{\beta}(y))\). The estimator for the potential outcome CDF is then obtained by evaluating the fitted model at the specific treatment value of interest: \[ \hat{F}_{Y^x}(y) = \Lambda\left(\psi(x)'\hat{\beta}(y)\right). \]
- Estimating the Quantiles: We estimate \(Q_{Y^x}(\tau)\) by running a quantile regression of \(Y_i\) on \(\psi(X_i)\), which yields the linear index evaluator: \[ \hat{Q}_{Y^x}(\tau) = \psi(x)'\hat{\beta}(\tau). \]
24.3 Estimation under Conditional Unconfoundedness
In case unconfoundedness only holds conditional on covariates \(W_i\), we must adjust for these covariates in our regression specifications. The estimation strategy depends on whether we are interested in conditional or unconditional causal parameters and is somewhat more involved. For simplicity, we assume the treatment \(X_i\) is discrete for the remainder of this section.
24.3.1 Conditional Parameters
If we are interested in the conditional parameters \(F_{Y^x|W}(y|w)\) and \(Q_{Y^x|W}(\tau|w)\), adjustment is straightforward. We apply distribution or quantile regression of \(Y_i\) on \(\psi(W_i)\) within the subsample \(\{i : X_i = x\}\). Since conditioning on \(W_i\) is sufficient to restore unconfoundedness, the regression specifications estimated within this subsample identify the causal parameters: \[ \begin{aligned} \hat{F}_{Y^x|W}(y|w) & = \Lambda\left(\psi(w)'\hat{\beta}(y)\right), \\ \hat{Q}_{Y^x|W}(\tau|w) & = \psi(w)'\hat{\beta}(\tau), \end{aligned} \] where the parameters \(\hat{\beta}(y)\) and \(\hat{\beta}(\tau)\) are estimated solely using observations for which \(X_i = x\).
24.3.2 Unconditional CDFs via Regression G-Computation
When interest lies in the unconditional potential outcome distribution \(F_{Y^x}(y)\) in the population, we must integrate out the controls via the G-computation (standardization) formula: \[ F_{Y^x}(y) = \E\left[ F_{Y|X, W}(y|x, W_i) \right]. \] This structure suggests a simple two-step estimator:
- Step 1 (Conditional Estimation): Run a distribution regression of \(Y_i\) on \(\psi(W_i)\) using only the subsample with \(X_i = x\). This yields the coefficient estimates \(\hat{\beta}(y)\) and the conditional CDF estimator \(\hat{F}_{Y|X, W}(y|x, w) = \Lambda(\psi(w)'\hat{\beta}(y))\).
- Step 2 (Averaging/Standardization): Average these predicted probabilities over the entire sample’s covariate distribution (including units that received a different treatment): \[ \hat{F}_{Y^x}(y) = \frac{1}{N} \sum_{i=1}^N \Lambda\left(\psi(W_i)'\hat{\beta}(y)\right). \tag{24.3}\]
By averaging over the entire sample’s covariates, we reconstruct the counterfactual distribution as if everyone in the population had been assigned to treatment \(x\). This G-computation estimator consistently estimates \(F_{Y^x}(y)\) under standard regularity conditions, and differences in these CDFs directly yield the DTE.
24.3.3 Unconditional Quantiles via Inverse Probability Weighting
Estimating unconditional quantiles \(Q_{Y^x}(\tau)\) requires a different approach: because the quantile operator is non-linear, we cannot average the conditional quantiles \(\hat{Q}_{Y^x|W}(\tau|W_i)\) to obtain the unconditional quantile.
To overcome this, we exploit the propensity-score weighted moment condition (21.4): \[ \E\left[ \frac{\mathbb{I}\{X_i = x\}}{P(X_i = x | W_i)} \mathbb{I}\{Y_i \leq Q_{Y^x}(\tau)\} \right] = \tau. \tag{24.4}\]
Firpo (2007) shows that we can estimate \(Q_{Y^x}(\tau)\) via a two-step optimization procedure:
- Step 1 (Propensity Score): Estimate the propensity score \(p(w) \equiv P(X_i = x | W_i = w)\) using a standard binary response model (such as a logit or probit) on the full sample, which yields the fitted probabilities \(\hat{p}(W_i)\).
- Step 2 (Weighted Quantile Regression): Solve the weighted check-loss minimization problem: \[ \hat{Q}_{Y^x}(\tau) = \argmin_{q} \sum_{i=1}^N \frac{\mathbb{I}\{X_i = x\}}{\hat{p}(W_i)} \rho_\tau(Y_i - q). \tag{24.5}\]
Because there are no covariates in the check-loss minimization problem other than the constant threshold \(q\), Equation 24.5 is a weighted sample quantile. This is computationally trivial to solve and avoids modeling the full conditional quantile process.
24.4 Asymptotic Properties
Despite relying on first-step semiparametric or nonparametric estimators (either the distribution regression coefficients \(\hat{\beta}(y)\) or the propensity scores \(\hat{p}(W_i)\)), both the G-computation estimator \(\hat{F}_{Y^x}(y)\) and the IPW estimator \(\hat{Q}_{Y^x}(\tau)\) achieve the parametric rate of convergence.
Under standard regularity conditions, they are \(\sqrt{N}\)-consistent and asymptotically normal: \[ \sqrt{N}\left(\hat{Q}_{Y^x}(\tau) - Q_{Y^x}(\tau)\right) \xrightarrow{d} \mathcal{N}\left(0, \sigma^2(\tau)\right). \]
The averaging step in G-computation and the integration in the IPW check-loss objective function act as “smoothing” operators, filtering out the first-step estimation noise to allow convergence at the \(\sqrt{N}\) parametric rate.
For detailed derivations of the semiparametric efficiency bounds and the exact sandwich formulas for the asymptotic variances, see Firpo (2007) for QTEs and Donald and Hsu (2014) for DTEs.