1 Motivation

‍The regression discontinuity design (RDD) exploits sharp jumps in the level of treatment at a threshold to identify causal effects. But many policies are not designed with sharp jumps in treatment level: they instead feature sharp changes in the slope of treatment assignment at a threshold. Unemployment insurance (UI) benefit schedules, for instance, typically tie benefit amounts to prior earnings with a replacement rate that changes at a kink point rather than with a discrete jump.

The regression kink design (RKD), formalised by Card et al. [2015], exploits such kinks in the treatment assignment function to identify causal effects. Like the RDD, the RKD is a local identification strategy: it recovers a treatment effect for individuals near the kink point. Unlike the RDD, the key variation is in the slope of treatment, not its level.

2 The Setup: Sharp and Fuzzy RKD

‍Let X be a running variable (e.g., prior earnings), B(X) the assigned benefit amount (the treatment), and Y the outcome. In a sharp RKD, the assignment function B(X) has a known kink at X = x₀:

The change in slope is Δb ≡ b+ - b_. If Y is a smooth function of X in the absence of treatment, then any kink in the regression of Y on X at x_0 must be caused by the kink in B. The sharp RKD estimator is:

The numerator is the change in slope of the conditional expectation of the outcome at the kink; the denominator is the change in slope of the treatment at the kink. The ratio delivers the causal effect of a marginal unit of treatment, dY/dB evaluated at X = x₀.

      In a fuzzy RKD, agents have some discretion over actual treatment receipt. The received treatment W(X) does not kink by a known amount even though the assigned treatment B(X) does. The fuzzy estimator is the ratio of the kink in the outcome regression to the kink in the actual treatment regression:

analogous to the fuzzy RDD ratio of reduced-form to first-stage estimates.

3 Identifying Assumptions

Card et al. [2015] provide nonparametric identification results under the following conditions.

1. Smoothness of the conditional distribution. The conditional density of the running variable f(x|ε) and the conditional mean of potential outcomes E[Y(w)|X=x] are continuously differentiable in x at x_0 for all values of the potential treatment w. This rules out a kink in the outcome regression caused by sorting of agents with particular potential outcomes to the kink point.
2. ‍No manipulation of the running variable. This is the density continuity assumption analogous to that in the RDD [McCrary, 2008]. If agents can precisely sort to one side of x_0, the running variable is endogenous. The standard McCrary test checks for a discontinuity in the density of X at x_0; in the RKD context, Card et al. [2015] note that a kink (not a jump) in the density at x_0 would also indicate manipulation.‍
3. Differentiability of the assignment rule. The assignment function B(X) must be differentiable everywhere except at x_0, and the change in slope Δb must be non-zero (relevance condition).

Under these conditions, the sharp RKD estimator identifies a weighted average derivative:

the average marginal effect of treatment at the kink point, for individuals at X = x₀.

4 Estimation in Practice

Estimation follows the local polynomial approach analogous to the RDD. Fix a bandwidth h around x₀ and fit local linear (or higher-order polynomial) regressions on each side of the kink:

where D_i = 1[X_i ≥ x₀] is an indicator for being above the kink. The coefficient β₁ captures the change in slope. Dividing by the known kink in B(X) gives the RKD estimate. Inference uses the bias-corrected robust standard errors developed for the RDD by Calonico et al. [2014], adapted to the derivative context.

Bandwidth selection follows MSE-optimal criteria. Card et al. [2015] derive the optimal bandwidth for the RKD, which is proportional to n^(-1/5) (as in the RDD), but with a different optimal constant reflecting the slower rate of convergence of slope estimators relative to level estimators.

5 Application: Unemployment Insurance and Employment Duration

The motivating application in Card et al. [2015] uses Austrian administrative data on unemployment spells. Austrian UI benefits are determined by a formula linking benefit amounts to prior earnings, with a known kink in the benefit schedule. Specifically, workers earning above a threshold receive a lower replacement rate (benefit as a fraction of prior earnings) than those below it; the replacement rate kinks at the threshold.

The outcome of interest is the duration of unemployment spells. The RKD estimate implies that a 10 percentage point increase in the UI replacement rate extends unemployment duration by approximately 1.5 weeks. This effect is local it applies to workers near the kink in the benefit schedule and represents the causal effect of UI generosity on job search intensity for that population.

(Figure 1 illustrates the key patterns: the kink in the benefit schedule and the corresponding kink in the outcome regression.)

6 Relation to RDD and Extensions

‍The RKD is related to the RDD in the following sense: the RDD identifies a treatment effect from a level discontinuity (jump in E[Y|X] at x₀) [Imbens and Lemieux, 2008], while the RKD identifies a marginal treatment effect from a derivative discontinuity. The RKD produces estimates with a slower rate of convergence than the RDD because estimating derivatives is harder than estimating levels: the optimal bandwidth is larger and the standard error is larger for a given sample size.

Extensions include: (i) the generalised RKD of Card et al. [2015] that allows for a kink in the density of X; (ii) applications to nonlinear treatment effects using local quadratic approximations; (iii) combining RDD and RKD when both a level jump and a slope change occur at the same threshold.

7 Available Software

‍The rdrobust package in R and Stata (Calonico et al. 2014) supports RKD estimation via the deriv = 1 option, which estimates the first derivative of the regression function rather than the level. The rdbwselect function computes MSE-optimal bandwidths for derivative estimation. Researchers should also apply rddensity to test for manipulation of the running variable.

8 Conclusion

‍The regression kink design provides a credible identification strategy for settings where policy assigns treatment in proportion to a running variable, with a change in that proportion at a known threshold. It requires the same no-manipulation assumption as the RDD, plus smoothness of the conditional distribution of potential outcomes. The estimator identifies a local marginal effect of treatment, making it well-suited to welfare analysis of continuous policy instruments such as tax rates, benefit replacement rates, and subsidy intensities.

Applied researchers working with administrative data from social insurance programmes will find the RKD a valuable complement to the standard RDD toolkit.

References

1. Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica, 82(6):2295-2326.
2. Card, D., Lee, D. S., Pei, Z., and Weber, A. (2015). Inference on causal effects in a generalized regression kink design. Econometrica, 83(6):2453-2483.
3. Imbens, G. W. and Lemieux, T. (2008). Regression discontinuity designs: a guide to practice. Journal of Econometrics, 142(2):615-635.
4. McCrary, J. (2008). Manipulation of the running variable in the regression discontinuity design: a density test. Journal of Econometrics, 142(2):698-714.[cite: 10]

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