Covariate Adjustment In Regression Discontinuity

Why Adjust for Covariates Near a Cutoff?

In a regression discontinuity (RD) design, treatment assignment changes at a known cutoff in a running variable. The estimated treatment effect is the discontinuity in the outcome at that cutoff, under the design's identification conditions. A predetermined covariate—measured before assignment and not caused by treatment—can help predict the outcome near the cutoff. Accounting for such a covariate may reduce unexplained variation and improve precision.

Adjustment is not what identifies the RD effect. Identification comes from the cutoff design and the continuity conditions for potential outcomes; adjustment cannot make those conditions true. The purpose of covariates is narrower: use relevant, pre-assignment information while preserving the local comparison and its interpretation. The earlier material on robust bias-corrected RD inference addresses inference for local polynomial estimates; covariate adjustment must be incorporated consistently into the estimator and its inference procedure.

Definition: A predetermined covariate $X_i$ is a variable measured before treatment assignment, whose value is not caused by the treatment or by the assignment decision at the cutoff. In covariate-adjusted RD, $X_i$ is included alongside the running variable's local polynomial terms to account for outcome variation that it predicts.

Let $R_i$ be the running variable, $c$ the cutoff, $D_i=1\{R_i\ge c\}$ the treatment-assignment indicator in a sharp RD design, and $Y_i$ the outcome. A typical local linear regression, fitted within a bandwidth around $c$, is

Y_i=\alpha+\tau D_i+\beta_-(R_i-c)+\beta_+D_i(R_i-c)+\gamma'X_i+\varepsilon_i.

The observations receive kernel weights that typically decrease with their distance from $c$. The slopes on the running variable can differ on the two sides; $\tau$ is the fitted difference between the right- and left-side limits at the cutoff, holding the covariates fixed. Centering $X_i$ at any fixed reference value changes the intercepts but not this difference when the same covariate coefficient applies on both sides.

This specification is an illustration, not a prescription to use one covariate coefficient in every application. The local polynomial order, bandwidth, kernel, weights, and any side-specific terms must be chosen as part of a coherent analysis. Covariate adjustment should not replace the local polynomial in the running variable or turn the analysis into a global regression.

Why Continuity Matters for Adjustment

The key point can be seen by comparing the adjusted and unadjusted local fits. Suppose the conditional outcome relation near the cutoff includes a common linear covariate effect $\gamma'X_i$. Write $\Delta_X$ for the right-side limit of the mean covariate minus its left-side limit. At the cutoff, the unadjusted outcome difference includes both the adjusted outcome difference and the part associated with the covariate difference:

\tau_{\mathrm{unadjusted}}=\tau_{\mathrm{adjusted}}+\gamma'\Delta_X, \qquad \text{so}\qquad \tau_{\mathrm{adjusted}}=\tau_{\mathrm{unadjusted}}-\gamma'\Delta_X.

This is a local relationship under the stated common-effect specification; it is not an identity that holds regardless of model form. If the covariate's conditional mean is continuous at $c$, then $\Delta_X=0$, and adjustment does not change the population cutoff discontinuity in this specification. It can still improve precision by explaining outcome variation. If the covariate itself jumps at the cutoff, the two estimates can differ. That difference should not be treated as an automatic correction: it may signal a problem for the RD comparison, and the adjusted coefficient then depends on how the model handles that covariate.

Key principle: Use predetermined covariates to explain outcome variation, not to create the cutoff contrast. If adjustment changes the estimated discontinuity substantially, investigate why; do not assume that the adjusted estimate is automatically more credible.

Worked Examples

Worked Example: A Small Precision Gain Without a Material Change in the Estimate

Suppose a hypothetical sharp RD analysis estimates the effect of a training offer on a test score measured in points. The running variable is a qualifying score, and $X_i$ is a pre-assignment baseline test score. The same bandwidth and kernel weights are used in the unadjusted and adjusted local linear fits. The estimated baseline-score difference at the cutoff is $0.03$ points, and the fitted outcome association is $1.30$ outcome points per baseline-score point.

The fitted adjustment implied by the local relationship is

\gamma'\Delta_X=1.30(0.03)=0.039\text{ outcome points}.

If the unadjusted estimated cutoff difference is $1.80$ points, the corresponding adjusted difference is approximately

1.80-0.039=1.761\text{ points}.

Suppose the adjusted estimate is reported as $1.76$ points with $SE=0.44$ points, compared with $SE=0.62$ points without adjustment. The adjusted estimate is very close to the unadjusted one, while its reported standard error is smaller. For illustration, a normal-approximation 95% interval using the adjusted standard error is

1.76\pm1.96(0.44)=1.76\pm0.8624=[0.898,2.622]\text{ points}.

Thus, under this approximation, the interval ranges from about $0.90$ to $2.62$ points. It is not a 95% probability statement about the fixed effect. In an actual RD analysis, the reported uncertainty must come from an inference procedure appropriate to the local estimator, including any bias correction used; this example assumes the displayed standard error is appropriate for that purpose. The example illustrates a possible precision gain, not evidence that adjustment always reduces the standard error.

The baseline score is measured before assignment, and its estimated cutoff difference is small, so the example is consistent with adjustment preserving the cutoff comparison. Those facts alone do not establish all RD identification assumptions.

Worked Example: A Covariate Jump Changes the Adjusted Estimate

Consider a hypothetical local analysis of a program's effect on monthly earnings, measured in hundreds of dollars. A predetermined covariate, prior earnings, predicts current earnings with a fitted coefficient of $2.0$ hundreds of dollars per prior-earnings unit. Suppose the local difference in mean prior earnings at the cutoff is $0.40$ units, and the unadjusted estimated earnings jump is $3.0$ hundreds of dollars.

The fitted part of the unadjusted jump associated with the covariate difference is

\gamma'\Delta_X=2.0(0.40)=0.80\text{ hundreds of dollars}.

Under the common-effect local model, the adjusted estimate is therefore

\tau_{\mathrm{adjusted}}=3.0-0.80=2.20\text{ hundreds of dollars}.

The adjusted estimate is $2.20$ hundreds of dollars per month, or $\$220$ per month; the unadjusted estimate is $\$300$ per month. This arithmetic does not show that the adjusted estimate is the correct causal effect. A discontinuity in a predetermined covariate deserves investigation because it means the covariate distributions do not have matching limits at the cutoff. Adjustment can change the target comparison by subtracting the fitted contribution of that discontinuity, but it cannot by itself establish why the covariate jumps or restore the design's identifying conditions.

Worked Example: Why a Post-Treatment Variable Is Not a Safe Adjustment

Suppose, hypothetically, that a treatment increases a measured intermediate health score $M_i$, and that each one-unit increase in $M_i$ predicts a $2$-unit increase in the final outcome $Y_i$. A local unadjusted RD analysis estimates a $5$-unit outcome jump. The variable $M_i$ is measured after treatment, so it is not predetermined.

If a regression subtracts the fitted contribution of a $1.5$-unit jump in $M_i$, it produces

5-2(1.5)=2\text{ outcome units}.

The result is not simply a more precise estimate of the same total cutoff effect. Because treatment can cause $M_i$, conditioning on it removes part of a pathway through which treatment affects $Y_i$, and can introduce additional bias depending on the causal structure. The $2$-unit coefficient cannot be interpreted as the original total RD effect just because it comes from a regression containing more variables. For estimating the total effect, use covariates that are determined before treatment assignment.

Implementation, Diagnostics, and Alternatives

A defensible analysis starts by defining the cutoff estimand and the local estimation choices, then considers whether a small set of substantively motivated predetermined covariates can improve precision. Keep the local polynomial structure and weights aligned between adjusted and unadjusted fits. If a covariate is included, its measurement and timing should be clear, and the model should allow the running-variable relationship to differ across the cutoff as required by the chosen RD specification.

Choose covariates before examining the treatment estimate.
Use variables measured before assignment that have a plausible relationship with the outcome. Avoid choosing them solely because they make the estimated jump larger, smaller, or statistically significant.

Fit comparable local models.
Keep the cutoff, bandwidth, kernel weighting, and running-variable polynomial structure comparable when assessing adjusted and unadjusted estimates. Otherwise, a difference between estimates may reflect several changes at once.

Compare estimates and uncertainty.
Report whether adjustment materially changes the jump and whether it changes the standard error. A smaller standard error is not proof of valid identification, and a changed estimate calls for explanation rather than automatic preference.

Check the role and behavior of each covariate.
Confirm that it is genuinely predetermined and examine whether its behavior near the cutoff is compatible with the design. A covariate discontinuity is a warning to investigate, not a problem that regression adjustment automatically solves.

Use inference matched to the estimator.
Covariate adjustment must be included in the variance and bias-correction procedure used for inference. Do not attach an uncertainty estimate from an incompatible regression to a different local estimator.

The main alternative is the unadjusted local polynomial estimator. It is transparent and avoids dependence on a covariate outcome model, but may be less precise when strong predetermined predictors are available. A covariate-adjusted local fit may gain precision, but introduces modeling choices and does not solve failures of the RD design. A global regression with many covariates is not a substitute for local estimation: it may impose a functional form far from the cutoff that drives the estimated jump.

Adjustment has limitations even when performed carefully. Precision gains depend on the covariates' predictive value and their relationship with the local estimation residuals; adding irrelevant variables can increase uncertainty. A large change in the cutoff estimate can reflect a covariate discontinuity, a misspecified covariate relationship, or differences between the compared models. If slopes relating covariates to outcomes vary across the cutoff, a common-coefficient adjustment may also be inadequate. No single adjusted coefficient resolves these issues without assumptions about the local model.

Common Mistakes

“More covariates always mean less bias.” Adding variables does not repair a failure of continuity or assignment at the cutoff. Covariates are principally a precision tool under a credible RD design.
“A smaller standard error proves the adjusted estimate is better.” Precision is only one consideration. The covariate must be predetermined, the specification must be suitable, and the identifying conditions must still be credible.
“Any variable related to the outcome is safe to control for.” A variable measured after treatment can be affected by treatment. Conditioning on it can change the effect being estimated or introduce bias.
“A covariate jump can be regressed away.” Adjustment can alter the estimate by subtracting a modeled covariate contribution, but does not explain or remove the design implications of a discontinuity in that covariate.
“The adjusted and unadjusted estimates should always be identical.” They coincide in the stated common-effect relationship when the covariate is continuous at the cutoff. In finite samples, and when covariate limits differ or the specification changes, they need not match.

Key takeaway: Predetermined covariates can improve precision in RD by explaining outcome variation near the cutoff. They do not identify the effect. Preserve the local cutoff comparison, avoid post-treatment variables, and investigate rather than conceal substantial changes caused by adjustment.

Check Your Understanding

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

A predetermined covariate has a fitted outcome coefficient of $1.5$ and an estimated cutoff difference of $0.20$. If the unadjusted jump is $4.0$, what adjusted jump follows from the common-effect relationship?
Why can a covariate reduce the standard error without changing the population cutoff effect?
A variable is measured after treatment and predicts the outcome. Why is it generally not a safe adjustment variable for estimating the total RD effect?
An adjusted estimate differs substantially from the unadjusted estimate. Give two reasons to investigate before choosing one estimate.

Solutions. (1) The fitted covariate contribution is $1.5(0.20)=0.30$, so the adjusted jump is $4.0-0.30=3.70$ outcome units. (2) If the covariate predicts outcome variation and its conditional mean is continuous at the cutoff, it can explain residual variation while contributing no population jump of its own. (3) Treatment may cause the variable; conditioning on it can remove part of the treatment pathway or introduce bias, so the resulting coefficient need not represent the total effect. (4) For example, the covariate may itself be discontinuous at the cutoff, or the covariate-outcome relationship or local model may be misspecified. Also verify that the two fits use comparable bandwidths, weights, and running-variable specifications.