Regression discontinuity identifies a local causal effect when treatment assignment changes discontinuously at a known threshold of a continuously distributed running variable. Identification comes from continuity of potential-outcome regressions at the cutoff, while estimation is a boundary nonparametric-regression problem.
Sharp regression discontinuity
Definition. Let running variable \(R\) have cutoff \(c\), with treatment \[ A=\mathbf1(R\ge c). \] The sharp RD estimand is \[ \tau_{\mathrm{RD}}= \mathbb E\{Y(1)-Y(0)\mid R=c\}. \]
Continuity identification
Theorem. If \[ \mu_a(r)=\mathbb E\{Y(a)\mid R=r\} \] is continuous at \(c\) for \(a=0,1\), and consistency holds, then \[ \tau_{\mathrm{RD}} = \lim_{r\downarrow c}\mathbb E(Y\mid R=r) - \lim_{r\uparrow c}\mathbb E(Y\mid R=r). \]
Proof. Above the cutoff, \(Y=Y(1)\), so the right limit equals \(\lim_{r\downarrow c}\mu_1(r)=\mu_1(c)\). Below the cutoff, the left limit equals \(\mu_0(c)\). Subtract.
The continuity assumption permits all smooth relationships between \(R\) and potential outcomes. It excludes another cause of \(Y\) that jumps at exactly the same threshold.
Fuzzy regression discontinuity
Theorem. Suppose treatment probability, rather than treatment itself, jumps at \(c\). Under continuity of potential outcomes and potential treatment decisions, exclusion, and monotonicity, the local complier effect is \[ \tau_{\mathrm{FRD}}= \frac{ \lim_{r\downarrow c}\mathbb E(Y\mid R=r)- \lim_{r\uparrow c}\mathbb E(Y\mid R=r)} { \lim_{r\downarrow c}\mathbb E(A\mid R=r)- \lim_{r\uparrow c}\mathbb E(A\mid R=r)}. \]
Proof outline. Treat threshold eligibility \(Z=\mathbf1(R\ge c)\) as a local instrument. Continuity replaces exact randomization by equality of limiting potential-outcome distributions at the cutoff. The reduced-form outcome jump equals the first-stage treatment jump times the local average effect among units whose treatment responds to eligibility. Divide as in the Wald theorem.
Local polynomial estimation
Definition. On each side of \(c\), local linear regression solves \[ \min_{\alpha_\pm,\beta_\pm} \sum_{i:R_i\gtrless c} K\left(\frac{R_i-c}{h}\right) \{Y_i-\alpha_\pm-\beta_\pm(R_i-c)\}^2. \] The RD estimate is \(\widehat\alpha_+-\widehat\alpha_-\).
Local linear fitting is preferred to high-order global polynomials because it targets the boundary limit, adapts to local curvature, and avoids unstable oscillations far from the cutoff.
Bias and variance orders
Proposition. With twice differentiable conditional means, positive density at \(c\), and local linear fitting, \[ \operatorname{Bias}(\widehat\tau)=O(h^2), \qquad \operatorname{Var}(\widehat\tau)=O((nh)^{-1}). \] Mean-squared-error balancing gives \(h\asymp n^{-1/5}\).
Proof outline. A Taylor expansion shows local linear regression removes constant and first-order terms, leaving second-derivative bias of order \(h^2\). Roughly \(nh\) observations fall in a bandwidth-\(h\) neighborhood, giving variance order \(1/(nh)\). Balance \(h^4\) against \(1/(nh)\).
Robust bias-corrected inference
At an MSE-optimal bandwidth, \(h^2\) bias is of the same order as standard error, so a naive centered Wald interval undercover. Robust bias correction estimates the leading curvature bias using a higher-order fit and enlarges the variance to account for this correction.
Manipulation of the running variable
Remark. If units can precisely manipulate \(R\) around \(c\), populations just below and above may not be comparable. A density discontinuity test can detect some sorting, but continuity of density is neither necessary nor sufficient for all identifying assumptions. Institutional knowledge and predetermined-covariate continuity remain essential.
Placebo and falsification analyses
- Check predetermined covariates for discontinuities at \(c\).
- Estimate effects at placebo cutoffs where treatment does not change.
- Examine sensitivity to bandwidth and polynomial order.
- Plot binned means for communication but fit the estimator to raw data.
- Report the treatment first-stage jump in fuzzy designs.
Discrete running variables
When \(R\) has few support points, continuity limits are not empirically approached arbitrarily closely. Identification then relies on functional-form restrictions, local randomization over a chosen window, or partial identification. Conventional continuous-RD standard errors can be misleading.
Local randomization interpretation
An alternative framework assumes that within a prespecified narrow window around \(c\), assignment is as-if randomized. It supports randomization inference but targets the window population and requires a defensible window-selection procedure. It is conceptually distinct from continuity-based local-polynomial identification.
External validity
RD identifies an effect at the cutoff for compliers or threshold units. Extrapolating away from \(c\) requires assumptions about treatment-effect heterogeneity. A precise local effect can still be scientifically valuable, but it should not be relabeled as a population ATE.