Covariate Adjustment for Regression Discontinuity

Why Adjust for Covariates at a Regression Discontinuity Cutoff?

Local polynomial regression discontinuity (RD) estimates compare outcomes just below and just above an assignment threshold. Predetermined covariates—characteristics measured before treatment assignment—can explain some outcome variation in that neighborhood. Accounting for that variation may make the estimated cutoff jump more precise. The purpose of adjustment is not to create the RD design: identification still comes from the assignment rule and continuity at the cutoff.

Adjustment must also respect the timing of the design. A baseline characteristic can be used as a precision aid under suitable conditions. A variable changed by treatment is different: conditioning on it can block part of the treatment effect or introduce bias. Nor does adding variables automatically make a regression coefficient causal. The interpretation depends on the RD assumptions and on the conditional-mean model used.

Definition: Let $R_i$ be unit $i$'s running variable, $c$ the cutoff, $D_i=1\{R_i\ge c\}$ the sharp-RD treatment assignment, and $Y_i$ the outcome. The sharp RD estimand is the discontinuity in the outcome mean at the cutoff: $$ \tau_{\mathrm{RD}} = \lim_{r\downarrow c} E[Y_i\mid R_i=r] - \lim_{r\uparrow c} E[Y_i\mid R_i=r]. $$ Here the first limit is from the right and the second from the left. Under the sharp assignment rule and the relevant continuity assumptions, this jump identifies the treatment effect at the cutoff. A predetermined covariate $X_i$ is measured before assignment and cannot itself be caused by $D_i$.

The notation $X_i$ may represent several baseline variables. Examples include age measured before eligibility, a prior-period outcome, or a baseline health measure. In an RD design, a useful adjustment variable is typically predictive of $Y_i$, observed comparably on both sides of $c$, and not affected by crossing the cutoff.

A Local Polynomial Regression With Covariate Adjustment

Write $Z_i=R_i-c$, and restrict estimation to observations near the cutoff using a bandwidth $h$. A local polynomial regression fits separate polynomial trends on the two sides, weighting observations by their distance from the cutoff. One simple adjusted local-linear specification is:

Y_i = \alpha_-+\beta_- Z_i + D_i\{\tau+\left(\beta_+-\beta_-\right)Z_i\} + \gamma^\top(X_i-\bar X_h) + \varepsilon_i, \qquad |Z_i|\le h.

The term $\bar X_h$ is a local covariate reference value, such as the weighted mean in the estimation window. Centering makes the intercepts interpretable at that reference; it does not change the treatment coefficient. At $Z_i=0$, the fitted outcome difference between the right and left sides, holding the covariates at the same reference value, is $\tau$. Higher-order local polynomials replace the linear terms with powers of $Z_i$; the intercept difference remains the fitted jump at the cutoff.

This specification uses a common covariate slope $\gamma$ on both sides. It is a useful working model when that restriction is defensible locally. Other specifications can allow more flexible covariate relationships, but then the target should be defined carefully—for example, by comparing fitted conditional means on both sides and averaging their difference over the same covariate distribution at the cutoff. A treatment coefficient in a more flexible regression is not automatically the marginal RD effect just because covariates were included.

Conditions for a valid precision adjustment: The RD assignment and identification assumptions must hold; the adjusted variables must be predetermined; their relevant distributions or moments must vary continuously through the cutoff; and the local outcome model must be adequate for the adjustment being made. In a regression interpretation, a condition such as $E[\varepsilon_i\mid R_i,X_i]=0$ is needed for the specified conditional-mean model to recover its coefficients. Mean-zero errors without this conditional restriction are not enough. Adjustment does not repair a discontinuity in potential outcomes, sorting around the cutoff, or a misspecified conditional model.

The continuity requirement matters because the adjusted comparison should not manufacture a jump by comparing different covariate compositions on the two sides. If the distribution of baseline covariates has the same limit at $c$, a common covariate adjustment changes outcome predictions on both sides in a way that does not itself create an outcome discontinuity. If a covariate does jump at the cutoff, that fact requires scrutiny; subtracting its fitted contribution is not, by itself, proof that the original RD assumptions hold.

Worked Examples

Worked Example: Adjusting a Local Linear RD Estimate

Suppose a hypothetical program is assigned to people with a score at or above $c=70$. The outcome is a later test score, measured in points. Age, measured before assignment, is used as a covariate. In a local-linear fit with a fixed bandwidth, the unadjusted fitted jump is 8.4 points. The fitted local outcome relationship has a common slope of 0.60 test-score points per year of age, and the estimated right-minus-left age difference at the cutoff is 0.50 years. Assume these are hypothetical summaries from fits using the same local sample and weighting scheme.

Under the stated common-slope adjustment, the part of the unadjusted jump associated with the age difference is $0.60\times0.50=0.30$ test-score points. The adjusted jump is therefore:

\widehat{\tau}_{\mathrm{adj}} = \widehat{\tau}_{\mathrm{unadj}}-\widehat{\gamma}\,\widehat{\Delta X} = 8.4-(0.60)(0.50) = 8.10\text{ points}.

This arithmetic is appropriate for the specified common-slope model: it removes the fitted outcome difference associated with the estimated covariate difference. The adjusted estimate is still an RD estimate, and its causal interpretation still relies on the RD assumptions. The calculation would not justify adjustment if age were measured after program assignment or if the local conditional-mean model were unsuitable.

Worked Example: Interpreting Adjusted Robust Inference

In a separate hypothetical analysis of the same kind of sharp RD design, suppose the bias-corrected adjusted estimate is 8.10 test-score points and its robust standard error is 1.45 points. Treat these as outputs of an inference procedure appropriate for the chosen local polynomial and bandwidth; they are not obtained by simply substituting the standard error from the unadjusted regression.

The standardized statistic for testing $H_0:\tau_{\mathrm{RD}}=0$ is:

z=\frac{8.10-0}{1.45}=5.586.

A two-sided normal-reference test has $p<0.001$. A corresponding approximate 95% interval is:

8.10\pm1.96(1.45) = 8.10\pm2.842 = [5.258,\;10.942]\text{ points}.

Under the maintained RD assumptions and the stated inference procedure, the interval estimates the cutoff effect as between about 5.26 and 10.94 test-score points. It does not assign a 95% probability to the fixed effect being in this particular interval. The example also illustrates why uncertainty must be recomputed for the adjusted fit, including the relevant covariance and bias correction, rather than inferred from the point-estimate adjustment alone.

Worked Example: Why a Post-Treatment Variable Is Not a Routine Control

Consider a hypothetical sharp RD near its cutoff, and let $D$ denote assignment. Let $M$ be a mediator measured after assignment, with $M=3D+u$, where $u$ has mean zero. Suppose the outcome equation is $Y=5D+2M+\varepsilon$, with $\varepsilon=u$. The structural direct effect of $D$ holding $M$ fixed is 5 outcome units in this equation. But $M$ is post-treatment, and the regression coefficient on $D$ after controlling for $M$ need not equal that direct effect.

Substituting $u=M-3D$ into the outcome equation gives:

$$ Y=5D+2M+u =5D+2M+(M-3D) =2D+3M. $$

Consequently, a regression of $Y$ on $D$ and $M$ fits a coefficient of 2 on $D$, not 5. Although $E[\varepsilon]=E[u]=0$, the error is not conditionally mean-zero given $D$ and $M$: $E[\varepsilon\mid D,M]=M-3D$. Thus mean-zero errors alone do not establish that a controlled regression coefficient recovers a structural effect. In an RD application, conditioning on a mediator can also remove part of the total effect the cutoff design was intended to estimate. This is why baseline covariate adjustment and post-treatment conditioning must not be treated as interchangeable.

Why Adjustment Can Improve Precision

The basic precision argument is easiest to see under a local conditional-mean model. Suppose, in the estimation window, the outcome can be written as a local RD mean plus a predictable covariate component and a residual:

Y_i=m(R_i,D_i)+\gamma^\top(X_i-\mu_X)+\varepsilon_i,

where $\mu_X$ is a local reference value and the covariate relationship is correctly represented by $\gamma$. Removing the fitted covariate component leaves residual variation $\varepsilon_i$. If $X_i$ predicts $Y_i$, this residual variation can be smaller than the variation in $Y_i$ before adjustment. For a fixed set of local weights and otherwise suitable conditions, smaller outcome residual variance can reduce the sampling variance of the estimated jump.

This is a potential, not a guarantee. If the covariates are weakly predictive, noisy, or numerous relative to the local sample, adjustment may do little or can increase finite-sample uncertainty. The treatment coefficient and its standard error also depend on the covariate variation and the weighted design near the cutoff. Report the adjusted estimate with uncertainty from that adjusted specification. Do not claim a precision gain merely because covariates were added.

In practice, the local polynomial order, kernel, and bandwidth define the comparison; covariate adjustment is layered onto that comparison. If bias-corrected RD inference is used, the point estimate, bias correction, and variance calculation must all correspond to the adjusted procedure. A precision adjustment does not justify choosing a bandwidth or inference method that is otherwise inappropriate.

A Short Justification for the Adjustment

For a common-slope adjustment, the logic can be expressed with the fitted conditional mean. At the cutoff, write the local mean on each side as $\alpha_-+\gamma^\top(X-\bar X_h)$ and $\alpha_++\gamma^\top(X-\bar X_h)$, respectively. Comparing the two fitted means at the same covariate value cancels the common covariate component:

\{\alpha_++\gamma^\top(X-\bar X_h)\} - \{\alpha_-+\gamma^\top(X-\bar X_h)\} = \alpha_+-\alpha_-.

Thus, under this model, $\tau=\alpha_+-\alpha_-$ is the adjusted jump. If the covariate distribution has the same limit from both sides, the common-reference comparison corresponds to the same limiting covariate composition. This algebra explains why a valid baseline adjustment can preserve the cutoff contrast while accounting for predictable outcome variation. The conclusion depends on the model and continuity conditions; the cancellation is not a general proof that any regression controlling for any variable identifies a causal effect.

Common Mistakes

Controlling for a variable measured after treatment. A mediator or other post-treatment outcome can block part of the effect or create conditional associations. Use genuinely predetermined variables for routine precision adjustment.
Assuming mean-zero errors are sufficient. A structural disturbance can have unconditional mean zero while being related to included regressors. A regression coefficient has the intended conditional interpretation only under suitable exogeneity and specification conditions.
Assuming adjustment creates identification. Covariates do not replace the RD assignment rule, continuity assumptions, or a credible local comparison.
Assuming a smaller standard error is automatic. Predictive covariates can help, but the adjusted estimate and its standard error must be calculated using the adjusted design and inference procedure.
Interpreting a coefficient without defining its target. With side-specific covariate effects or treatment-effect heterogeneity, the regression coefficient may represent a conditional contrast rather than the marginal cutoff effect. Define the target and the covariate distribution over which it is averaged.

Key takeaway: Predetermined covariates can improve the precision of local polynomial RD estimation when the adjustment model is appropriate and the covariates behave continuously at the cutoff. They do not change the source of causal identification, and post-treatment variables are not routine precision controls.

Check Your Understanding

Answer each question using the assumptions and estimands defined above.

What does the sharp RD estimand compare at the cutoff, and which side corresponds to the right-hand limit?
A baseline covariate predicts the outcome but has a discontinuity at the cutoff. Why does including it not automatically validate the adjusted RD estimate?
In the worked adjustment example, calculate the fitted covariate contribution to the raw jump and the resulting adjusted jump.
Why can a disturbance with unconditional mean zero still fail to support the desired regression coefficient after conditioning on a post-treatment variable?
What must be recalculated when robust bias-corrected inference is applied to an adjusted local polynomial fit?