Local Polynomial Estimation for RD

Why Fit Separate Regressions at the Cutoff?

In a regression discontinuity design (RD), treatment eligibility changes at a known threshold in a running variable. The central empirical quantity is the difference between the outcome just to the right of the cutoff and the outcome just to its left. Observations near the threshold provide information about those two limits, but the observations generally do not lie exactly at the cutoff. Separate local polynomial regressions estimate each side’s trend near the threshold and evaluate the fitted trends at the cutoff.

The word separate matters: the outcome trend may differ on the two sides, while the RD estimand is the difference between their limiting values, not the difference between arbitrary observations farther from the threshold. The procedure applies to sharp RD, where treatment changes from zero to one at the cutoff, and fuzzy RD, where treatment probability changes but does not necessarily jump from zero to one.

Fitting and Evaluating the Two Sides

Choose a polynomial order \(p\) and a neighborhood around \(c\), often described by a bandwidth \(h>0\). On side \(s\), where \(s\) is left or right, fit a polynomial in the centered running variable:

For example, a degree-zero fit is a constant, a degree-one fit is a line, and a degree-two fit is a quadratic. The fit may be obtained by least squares, possibly giving observations nearer the cutoff greater weight. In general, for outcome observations \(Y_i\) and fitting weights \(w_i\), the side-specific coefficients minimize

The fitted value at the cutoff is \(\widehat m_s(0)=\widehat\beta_{s,0}\), since every term involving a positive power of \(x\) is zero at \(x=0\). Thus the estimated outcome jump is

Centering the running variable makes this evaluation direct: the intercept on each side is the fitted limit at the threshold. If the uncentered variable \(R\) is used instead, the fitted value must be evaluated at \(R=c\), not at zero. The observations used for the left fit must be below the cutoff, and those for the right fit must be at or above it according to the assignment convention. Each side needs enough distinct running-variable values to estimate its chosen polynomial.

Connecting the Fitted Jump to RD Estimands

Write \(D\) for treatment receipt and \(Y(1)\) and \(Y(0)\) for the potential outcomes under treatment and no treatment. In a sharp design, \(D=1\) exactly when \(R\geq c\). The observed outcome is \(Y=D Y(1)+(1-D)Y(0)\). Immediately to the right of the cutoff, the observed conditional mean follows the treated potential-outcome mean; immediately to the left, it follows the untreated potential-outcome mean.

For the fitted jump to have a causal interpretation, the relevant potential-outcome conditional means must each be continuous at the cutoff. In particular, continuity of the untreated potential-outcome mean alone is not enough: the right-side limit of the treated potential-outcome mean must also equal its value at the threshold. Under sharp assignment and these continuity conditions, with no other discontinuous change at the threshold that affects the outcome, the population jump is

Here the first limit approaches \(c\) from above and the second from below. The right side is the average treatment effect at the cutoff. The fitted intercept difference estimates the left side using observed data and a local polynomial approximation.

In a fuzzy design, crossing the cutoff changes the probability of treatment but does not deterministically assign treatment. Fit one pair of side-specific regressions for the outcome and another pair for treatment receipt. The outcome jump is the reduced-form discontinuity, and the treatment jump is the first-stage discontinuity. Provided the treatment jump is nonzero, the fuzzy RD estimate is their ratio:

This ratio has a local complier-effect interpretation only under additional assumptions. In particular, the threshold must affect the outcome only through treatment (exclusion), crossing the threshold must not make anyone less likely to receive treatment (monotonicity), the relevant potential-outcome and potential-treatment conditional means must be continuous at the cutoff, and the first-stage jump must be nonzero. Under these assumptions, the population ratio identifies the average treatment effect for compliers at the cutoff, not necessarily the effect for everyone near the threshold.

Polynomial Order and the Meaning of the Intercept

Polynomial order determines how flexibly each side’s mean relationship with the running variable can bend within the selected neighborhood. The cutoff estimate always comes from evaluating each fitted polynomial at zero. For degree two, for example, the side-specific fit is \(\widehat m_s(x)=\widehat\beta_{s,0}+\widehat\beta_{s,1}x+\widehat\beta_{s,2}x^2\); at the cutoff, the slope and curvature terms vanish, leaving \(\widehat\beta_{s,0}\). This does not mean that slopes or curvature are irrelevant to fitting the intercept: they affect how the regression allocates the observed variation across its coefficients.

The following hypothetical data show the calculation for quadratic fits. They are constructed to lie exactly on separate quadratics; the example isolates how the two fitted limits are obtained, rather than recommending a particular polynomial order or neighborhood.

Justification: Why the Intercept Difference Is the Estimated Jump

A polynomial fit on either side defines an estimated function for values of \(x\) near zero. Evaluating that function at zero gives its fitted one-sided limit. For any polynomial of order \(p\),

Subtracting the left limit from the right limit therefore yields \(\widehat\beta_{\mathrm{right},0}-\widehat\beta_{\mathrm{left},0}\), the intercept difference. In a fuzzy design, the same reasoning applies separately to the outcome and treatment regressions, after which their estimated jumps are divided. This establishes the algebra of the estimator; it does not establish that either fitted limit equals the true conditional-mean limit. That depends on the data, the local approximation, and the identifying assumptions.

What the Procedure Does—and Does Not—Establish

Local polynomial estimation turns observations near the cutoff into estimates of two one-sided limits. For a sharp RD, the outcome jump is the estimated discontinuity. For a fuzzy RD, the ratio of the outcome jump to the treatment jump is the estimated Wald-type RD effect. The assumptions that justify causal interpretations concern the assignment process and behavior of potential outcomes around the threshold; fitting separate regressions does not verify those assumptions.

A common mistake is to say, “The right-side fitted outcome is 5 points higher, so treatment caused a 5-point increase.” The 5-point value is an estimated outcome jump in the sharp example. It estimates a causal effect at the cutoff only if treatment follows the sharp rule and the conditional means of both potential outcomes are continuous there, along with the other relevant RD conditions. In a fuzzy design, the outcome jump alone is not the Wald estimate; it must be divided by the treatment jump, and the assumptions for a local complier interpretation must also hold.

Another important distinction is between evaluating a regression at the cutoff and reporting the fitted value at a nearby observed score. The RD estimand concerns one-sided limits at \(R=c\); a fitted value at \(R=c+1\), for example, is not that limit. Centering \(R\) at \(c\) and evaluating each fitted polynomial at \(x=0\) keeps the target explicit.