Regression Discontinuity With A Discrete Running Variable

Why Discrete Support Changes the Problem

Regression discontinuity (RD) designs use a threshold in a running variable to assign treatment. When the running variable has many distinct values close to the threshold, observations on either side can provide information about the outcome near the cutoff. If it takes only a few values, however, there may be a substantial gap between the nearest observed values on opposite sides. The analysis must then distinguish what the data show at those values from what a model extrapolates to the cutoff.

Let \(X\) be the running variable and \(c\) the cutoff. For clarity, write \(r=X-c\), so the cutoff is at \(r=0\). In a sharp RD design, treatment is \(D=1(X\geq c)\). Write \(Y(1)\) and \(Y(0)\) for the potential outcomes under treatment and control. The causal effect at the cutoff is \(\tau=E[Y(1)-Y(0)\mid X=c]\), when this conditional target is defined. Under the usual RD identification conditions, including continuity of the conditional mean potential outcomes at \(c\), the effect is the difference between the right and left limits of the observed conditional outcome mean at the cutoff.

Continuity of potential outcome means remains a substantive identification condition, but it does not fill in unobserved values between support points. A finite collection of observations can be consistent with many outcome curves that agree at every observed running-variable value and differ at the cutoff. Thus, a regression coefficient from a discrete-support RD analysis can rely more heavily on the modeling assumption than a local estimate with dense support.

Two-Sided Linear Extrapolation

One approach is to assume that the conditional mean outcome is linear on each side of the cutoff over the chosen range. This is a working model, not a consequence of discreteness or of the RD design itself.

The slopes may differ on the two sides. Estimating separate lines avoids imposing a common slope, but still imposes linearity within each side. In a discrete design, it is especially important not to describe the intercept contrast as “read directly from the data”: it is an extrapolated contrast unless observations actually occur at the cutoff on both sides, which a threshold rule generally precludes.

Deriving the Extrapolated Contrast and Its Uncertainty

With exactly two distinct support values on a side, the fitted line through the two sample cell means has a simple intercept. If the side’s support values are \(r_1\) and \(r_2\), and their sample cell means are \(\bar{Y}_1\) and \(\bar{Y}_2\), the slope is \((\bar{Y}_2-\bar{Y}_1)/(r_2-r_1)\). Substituting that slope into the line equation gives the intercept. For the left-side support \(-2,-1\), this reduces to \(2\bar{Y}_{-1}-\bar{Y}_{-2}\); for the right-side support \(1,2\), it reduces to \(2\bar{Y}_{1}-\bar{Y}_{2}\).

The same calculation shows why cell sizes and within-cell variation matter for uncertainty. If observations are independent across cells, with a common outcome variance \(\sigma^2\), each cell mean has variance \(\sigma^2/n_j\), where \(n_j\) is its cell size. For the two-point left intercept, the weights are \(-1\) on \(\bar{Y}_{-2}\) and \(2\) on \(\bar{Y}_{-1}\). Its variance is therefore the sum of the squared weights times the cell-mean variances. The right intercept has the corresponding expression, and the variance of their difference is the sum of the two intercept variances when the two sides are independent.

In practice, regression software can estimate the two lines and their covariance using individual-level observations. The model-based standard error reflects sampling variability conditional on the specified form; it does not account for bias from a wrongly specified form. A narrow confidence interval cannot make an unsupported extrapolation credible.

Support Diagnostics and Sensitivity to Functional Form

Begin by tabulating the distinct running-variable values on each side of the cutoff, the number of observations at each value, and the distance of the nearest support point from zero. A plot of cell means should show the actual support points rather than suggesting that the running variable varies smoothly where no observations exist. Report whether the cutoff itself is in the support and how the threshold convention assigns observations at \(X=c\).

The number of distinct values is not the same as the number of observations. A large sample concentrated at two values on a side can yield precise cell means, but it still provides only two locations for assessing the shape of the conditional mean. With exactly two values, a line fits those two cell means exactly; there is no remaining cell-level information to check whether the relationship is linear. Having more support points permits some assessment of curvature, but a few points over a short range may still provide weak evidence about the form between the points and the cutoff.

A useful sensitivity analysis compares a transparent primary model with reasonable alternatives over the same stated support range. For example, compare separate linear trends with a model that allows curvature when there are enough distinct values to estimate it. Large changes in the intercept contrast indicate dependence on functional form. High-order polynomials can be unstable, especially with sparse support, so adding terms does not automatically improve credibility. Also compare the extrapolated contrast with the adjacent-cell difference, clearly labeling the latter as a different estimand rather than treating agreement as proof of validity.

Alternatives, Limitations, and Common Pitfalls

The main alternatives answer different questions or make different assumptions. An adjacent-cell comparison is easy to explain and avoids fitting a slope, but it estimates a contrast across two running-variable values, not necessarily the cutoff effect. A separate linear model targets the cutoff through extrapolation and makes a clear, relatively simple shape assumption. More flexible models can represent curvature, but require enough distinct support values and can be unstable near the threshold. A saturated model that assigns an unrestricted mean to every support value describes observed cell means without imposing a smooth shape; when no observation provides both potential outcomes at the cutoff, it cannot by itself recover the unobserved cutoff limit.

The central limitations are therefore substantive as well as statistical. First, the cutoff effect may not be nonparametrically identified from a finite discrete support. Second, functional-form assumptions can be hard to test with few support points. Third, standard errors quantify sampling variation under the fitted model, not the uncertainty caused by choosing a model. Fourth, a running variable with very few values can make “local” estimation depend on observations that are not very close to the threshold.

• Pitfall: calling the nearest-cell difference the RD effect. The wrong reasoning is that the nearest observations are on opposite sides, so their outcome difference must be the cutoff jump. The correction is to recognize that this contrast is measured at two nonzero distances from the cutoff; interpreting it as the cutoff effect requires additional assumptions.
• Pitfall: treating a perfect fit as evidence for linearity. Two cell means determine a line exactly, so zero residuals at those points do not test the line between them or its extrapolation. Look for additional support points and report sensitivity to plausible forms.
• Pitfall: interpreting a small standard error as proof of identification. A large sample at a few support values can reduce sampling error while leaving extrapolation assumptions unchanged. Discuss precision and identification separately.