When should I use Regression Kink Design (RKD)?

When the treatment variable changes slope (but not level) at a threshold -- e.g., tax brackets, unemployment insurance schedules, progressive benefit formulas.

What is the key assumption of Regression Kink Design (RKD)?

The conditional expectation of the outcome and the density of the running variable are smooth through the kink point. No bunching at the kink.

What is the most common mistake with Regression Kink Design (RKD)?

Applying standard RDD to a kink -- the level discontinuity is zero by construction, so RDD finds no effect. RKD requires estimating the ratio of derivatives.

Design-BasedModern

Regression Kink Design (RKD)

Identifies causal effects from a kink (slope change) in the treatment assignment function, estimating a ratio of derivatives rather than a level discontinuity.

Quick Reference

When to Use: When the treatment variable changes slope (but not level) at a threshold -- e.g., tax brackets, unemployment insurance schedules, progressive benefit formulas.
Key Assumption: The conditional expectation of the outcome and the density of the running variable are smooth through the kink point. No bunching at the kink.
Common Mistake: Applying standard RDD to a kink -- the level discontinuity is zero by construction, so RDD finds no effect. RKD requires estimating the ratio of derivatives.
Estimated Time: 2.5 hours

One-Line Implementation

Stata: rdrobust y running_var, c(kink_point) deriv(1) kernel(triangular)

R: rdrobust(Y, X, c = kink_point, deriv = 1, kernel = 'triangular')

Python: # rdrobust not available in Python; manual: local linear derivative estimation at kink

Download Full Analysis Code

Complete scripts with diagnostics, robustness checks, and result export.

Motivating Example

How much does a more generous unemployment insurance (UI) benefit extend the time workers spend unemployed? This question is central to labor economics and social insurance policy (Nielsen et al., 2010), but answering it is hard. Workers who receive higher UI benefits may differ systematically from those who receive lower benefits -- they may have had higher prior wages, different industries, or different job search strategies.

Card, Lee, Pei, and Weber found an elegant solution (Card et al., 2015). In Austria, the UI benefit formula replaces a fixed fraction of prior earnings up to an earnings cap, and then the replacement rate drops. At the earnings cap, the benefit function does not jump -- workers just below and just above the cap receive almost identical benefit levels. But the slope of the benefit function changes: below the cap, each additional euro of prior earnings increases benefits by a fixed amount; above the cap, additional earnings produce zero additional benefits.

This slope change -- a kink -- creates quasi-experimental variation. Workers whose prior earnings place them just below the cap are nearly identical to workers just above it. The key difference is that workers below the cap face a steeper relationship between earnings and benefits than workers above it. If higher benefits cause longer unemployment durations, the relationship between earnings and duration should also change slope at the kink.

Card et al. found that a 10% increase in UI benefits extends unemployment duration by about 1.5 weeks. This estimate comes entirely from the kink in the benefit formula -- no discontinuity in benefit levels, no arbitrary control variables, just the change in slope of a known policy rule.

A. Overview

What RKD Does

A exploits a known kink (slope change) in the function that maps a running variable to a treatment variable. Unlike a standard regression discontinuity design, which exploits a jump (level change) in treatment at a cutoff, RKD exploits a change in slope of the treatment function. The treatment is continuous at the kink -- there is no discontinuity in levels. But the derivative of the treatment function changes.

The Key Intuition

Consider a progressive tax system where the marginal tax rate increases from 20% to 30% at an income threshold of $50,000. A worker earning $49,999 pays almost exactly the same total tax as a worker earning $50,001. There is no jump in the tax burden. But the slope of the tax function changes: each additional dollar below $50,000 is taxed at 20 cents, while each additional dollar above $50,000 is taxed at 30 cents.

If we plot total tax paid against income, we see a line that bends at $50,000 -- it gets steeper. Now if we plot some outcome (say, labor supply) against income, and we see a corresponding bend at $50,000, we can attribute this bend in the outcome to the bend in the tax function. The ratio of the two bends -- the kink in the outcome divided by the kink in the treatment -- gives us the causal effect.

RKD vs. RDD

Feature	RDD	RKD
Treatment at the cutoff	Jumps (level discontinuity)	Continuous (no jump)
Treatment function slope	May or may not change	Changes (kink)
Estimand	Ratio of level jumps	Ratio of slope changes
`rdrobust` option	`deriv(0)` (default)	`deriv(1)`
Visual signature	Jump in the scatterplot	Bend in the scatterplot
Key diagnostic	Density discontinuity test	Density smoothness test

When to Use RKD

The treatment variable changes slope (but not level) at a known threshold of the running variable
The kink is created by an institutional rule, policy formula, or regulatory schedule
There are sufficient observations near the kink point
You can credibly argue that the density of the running variable is smooth at the kink

When NOT to Use RKD

There is a level discontinuity in treatment at the threshold -- use RDD instead
The kink point is not sharply defined or changes over time unpredictably
There is strong evidence of bunching -- agents sorting to one side of the kink
You want to estimate effects far from the kink -- the estimate is local to the kink point
The treatment function slope changes are tiny, giving a weak "first stage"

The Taxonomy Position

RKD is a method, closely related to sharp and fuzzy RDD. Its credibility comes from the institutional rule that creates the kink, not from a model of confounders. RKD is more demanding than RDD in several respects: it requires smoothness of the density (not just continuity), the identifying variation is subtler (a slope change rather than a level jump), and the estimates tend to be noisier because they rely on estimating derivatives rather than levels. Card et al. (2015) describe RKD as the "derivative-based analog" of RDD (Card et al., 2015).

Common Confusions

Honest questions, honest answers

Q: Why can't I just run a standard RDD at the kink point? Because there is no level discontinuity in treatment at the kink. If you run rdrobust with deriv(0) (the default), it will look for a jump in the outcome at the kink. But if the treatment function is continuous at the kink, the outcome function should also be continuous -- so the RDD estimate will be zero (or noise). The identifying variation is in the slopes, not the levels. You need deriv(1).
Q: What if there is both a kink and a jump at the threshold? This situation is a combined RDD+RKD setting. You should first estimate the standard RDD (level jump). If there is a significant level discontinuity, the RDD estimate identifies a causal effect. The RKD component can provide additional information, but separating the two requires care. See Card et al. (2017) for guidance (Card et al., 2015).
Q: Is the RKD estimate "local" like an RDD estimate? Yes. The RKD estimate applies only to units at the kink point -- workers whose earnings are exactly at the benefit cap, taxpayers whose income is exactly at the bracket threshold. The effect may be different for units far from the kink.
Q: Why does RKD require density smoothness rather than just density continuity? In RDD, continuity of the density at the cutoff suffices because the identification comes from level jumps. In RKD, identification comes from changes in derivatives, which requires the density to be not just continuous but smooth (differentiable) at the kink. A density that is continuous but has a kink itself (a "bunching" pattern) would invalidate the RKD.
Q: How does bunching relate to RKD? Bunching occurs when agents strategically cluster at the kink point (e.g., workers adjusting their earnings to stay just below a tax bracket threshold). Bunching creates a discontinuity in the density of the running variable at the kink, violating the smoothness assumption. If bunching is present, RKD is not valid without correction. The bunching estimator literature (Saez, 2010; Kleven & Waseem, 2013) offers alternative approaches for settings with bunching.

B. Identification

Assumption 1: Smoothness of Conditional Expectations

Plain language: The relationship between the running variable and the outcome, absent any treatment effect, changes smoothly (no sudden bends) as you pass through the kink point. The only reason the outcome-running variable relationship bends at the kink is because the treatment function bends there.

Formally: $E[Y(0) | X = x]$ is twice continuously differentiable in a neighborhood of the kink point $c$ . This condition means the first derivative of the conditional expectation of the untreated potential outcome is continuous at $c$ .

Derivative continuity is a stronger assumption than what RDD requires (Card et al., 2015). RDD needs only continuity of $E[Y(0) | X = x]$ at the cutoff; RKD needs continuity of its derivative.

Assumption 2: Density Smoothness (No Bunching)

Plain language: The distribution of the running variable is smooth at the kink point. There is no piling up (bunching) of observations at or near the kink. If workers can strategically adjust their earnings to position themselves on one side of a benefit threshold, the density will have a kink or spike, and the RKD is invalid.

Formally: The density $f_X(x)$ is continuously differentiable at $c$ . This rules out discontinuities in the density and also rules out kinks in the density.

Assumption 3: First-Stage Kink Exists

Plain language: The treatment function actually changes slope at the kink point. If the treatment function is linear through the kink (no slope change), there is no identifying variation and the denominator of the RKD estimand is zero.

Formally: $\lim_{x \downarrow c} T'(x) - \lim_{x \uparrow c} T'(x) \neq 0$ , where $T(x)$ is the treatment assignment function.

This requirement is analogous to the relevance condition in instrumental variables. A "weak first stage" -- a small kink in the treatment function -- leads to imprecise and potentially biased RKD estimates.

What the Estimand Is

Under Assumptions 1--3, the RKD identifies:

\tau_{RKD} = \frac{\lim_{x \downarrow c} E'[Y|X=x] - \lim_{x \uparrow c} E'[Y|X=x]}{\lim_{x \downarrow c} T'(x) - \lim_{x \uparrow c} T'(x)}

In a sharp RKD where $T(x)$ is a known formula (e.g., a tax schedule), the denominator is known. The researcher estimates only the numerator -- the kink in the conditional expectation of the outcome. In a fuzzy RKD, both numerator and denominator must be estimated, and inference follows the same logic as the Wald/IV estimator.

Sharp vs. Fuzzy RKD

In a sharp RKD, the treatment function $T(X)$ is a known, deterministic function of the running variable (e.g., the UI benefit formula). The denominator of the RKD estimand is known exactly, and only the numerator (kink in the outcome) needs to be estimated. The formal conditions for sharp RKD identification are laid out in Card, Lee, Pei, and Weber (Card et al., 2015).

In a fuzzy RKD, the actual treatment received may differ from the assigned treatment, or the treatment function is not perfectly known. Both the numerator and denominator must be estimated from data, and inference follows IV-like logic. Fuzzy RKD is to sharp RKD as fuzzy RDD is to sharp RDD.

C. Visual Intuition

Compare three approaches at a kink point. Standard RDD finds no level jump (because there is none), a global polynomial misses the local slope change, and the RKD derivative estimator correctly identifies the causal effect from the ratio of slope changes.

Interactive Simulation

Why the Kink Matters: RKD vs. Alternatives

DGP: Y = 1.0·T(X) + 0.5·X² + ε, where T(X) = X for X < 0 and T(X) = 2.0·X for X ≥ 0. The kink in the treatment slope at X = 0 is 1.00. N = 500.

Standard RDD (level jump)Global polynomialLocal derivative (RKD)

Estimation Results

Estimator	β̂	SE	95% CI	Bias
Standard RDD (level jump)	2.228	0.216	[1.80, 2.65]	+1.228
Global polynomialclosest	1.560	0.067	[1.43, 1.69]	+0.560
Local derivative (RKD)	2.362	0.436	[1.51, 3.22]	+1.362
True β	1.000	—	—	—

Sample size (N)500

Number of observations

Kink slope change2.00

Slope of T(X) for X >= 0 (slope is 1 for X < 0)

True causal effect (β)1.00

The causal effect of T on Y

Noise (σ)1.50

Standard deviation of the error term

Why the difference?

The standard RDD estimator looks for a level jump at the kink, but the treatment function is continuous there -- so it finds approximately a spurious value (estimate = 2.228). There is no discontinuity to exploit. The global polynomial fits a smooth curve through all the data, but it cannot capture the abrupt slope change at x = 0. It misses the kink entirely. The local derivative estimator (RKD) targets the slope change. With the current noise level, the estimate (2.362) deviates from the truth (1.000), but it is the only approach that can identify the effect in this setting. More data or less noise would improve precision.

D. Mathematical Derivation

Don't worry about the notation yet — here's what this means in words: The RKD estimand is identified as the ratio of the derivative of the conditional outcome expectation to the derivative of the treatment function at the kink point.

Setup. At the kink point $c$ , the treatment function $T(X)$ has a slope change: the derivative $T'(X)$ jumps from $\tau_0$ (left slope) to $\tau_1$ (right slope) at $X = c$ .

Step 1: Outcome kink. Under the causal model $Y = g(T(X), X, \varepsilon)$ , the outcome $E[Y | X]$ also exhibits a kink at $c$ . The change in the slope of $E[Y | X]$ at $c$ is:

\Delta_Y = \lim_{x \downarrow c} \frac{dE[Y|X=x]}{dX} - \lim_{x \uparrow c} \frac{dE[Y|X=x]}{dX}

Step 2: Treatment kink. The change in the slope of $T(X)$ at $c$ is:

\Delta_T = \tau_1 - \tau_0

Step 3: Derivative ratio. The causal effect is identified as:

\hat{\beta}_{RKD} = \frac{\Delta_Y}{\Delta_T}

This ratio is analogous to the Wald estimator in IV, but using derivatives (slope changes) instead of levels (jumps). The key identification assumption is that $E[Y | X]$ is smooth through $c$ in the absence of the kink in $T(X)$ — any kink in the outcome must be caused by the kink in treatment.

Step 4: Estimation. In practice, $\Delta_Y$ and $\Delta_T$ are estimated using local polynomial regression with deriv = 1 (derivative estimation) via rdrobust. The bandwidth is chosen by MSE-optimal procedures adapted for derivative estimation.

The Estimand: A Ratio of Derivatives

The RKD estimand is the :

\tau_{RKD} = \frac{\lim_{x \downarrow c} \frac{dE[Y|X=x]}{dx} - \lim_{x \uparrow c} \frac{dE[Y|X=x]}{dx}}{\lim_{x \downarrow c} \frac{dT(x)}{dx} - \lim_{x \uparrow c} \frac{dT(x)}{dx}}

The numerator is the change in the slope of the conditional expectation of the outcome at the kink point. The denominator is the change in the slope of the treatment function at the kink. This ratio is analogous to the Wald estimator in instrumental variables, but applied to derivatives rather than levels.

In a sharp RKD where the treatment function is known (e.g., a tax formula), the denominator is known and the researcher only needs to estimate the numerator. In a fuzzy RKD, both must be estimated from the data.

E. Implementation

Estimation Procedure

Step 1: Plot the Treatment Function

Before anything else, plot the treatment variable against the running variable. You should see a visible bend -- the slope changes at the kink point. If you cannot see the kink, the first stage is likely too weak to support an RKD.

Step 2: Plot the Outcome

Plot the outcome against the running variable. If the treatment effect exists, you should see a corresponding bend in the outcome at the kink point. Unlike RDD, where the effect shows up as a visible jump, in RKD the effect shows up as a visible bend. Bends are harder to see than jumps, which is one reason RKD estimates tend to be noisier.

Step 3: Estimate the Reduced Form (Outcome Kink)

Use rdrobust with deriv=1 to estimate the change in slope of the outcome at the kink. The deriv(1) option tells rdrobust to estimate the first derivative of the local polynomial on each side of the cutoff, and then take the difference. This gives you the numerator of the RKD estimand.

Step 4: Estimate the First Stage (Treatment Kink)

Estimate the change in slope of the treatment function at the kink. If the treatment function is known by formula (e.g., a published tax schedule), you can compute the denominator analytically rather than estimating it.

Step 5: Compute the RKD Estimate

The causal effect is the ratio:

\hat{\tau}_{RKD} = \frac{\text{kink in outcome}}{\text{kink in treatment}}

With a known treatment function, the standard error can be computed via the delta method:

SE(\hat{\tau}_{RKD}) \approx \frac{SE(\text{outcome kink})}{|\text{treatment kink}|}

Bandwidth Selection

Bandwidth selection for RKD follows the same principles as for RDD: rdrobust uses MSE-optimal bandwidth selection by default. However, because RKD estimates derivatives (which require estimating curvature), optimal bandwidths for RKD tend to be larger than for RDD with the same data.

1library(rdrobust)
2
3# RKD: derivative estimation at the kink
4# deriv=1 estimates the change in slope, not level
5rkd_fit <- rdrobust(Y, X, c = kink_point, deriv = 1,
6                  kernel = "triangular")
7summary(rkd_fit)
8
9# First stage: slope change in treatment
10fs_fit <- rdrobust(treatment, X, c = kink_point, deriv = 1)
11summary(fs_fit)
12
13# RKD ratio = outcome slope change / treatment slope change
14rkd_estimate <- rkd_fit$coef[1] / fs_fit$coef[1]

Requiresrdrobust

F. Diagnostics

Diagnostic 1: Density Smoothness at the Kink

The density of the running variable must be smooth at the kink (Dong, 2015). Use the Cattaneo-Jansson-Ma density test (rddensity), but keep in mind that the standard test checks for a jump in the density. For RKD, you also need smoothness of the derivative. In R: rddensity(X, c = kink_point). In Stata: rddensity X, c(kink_point). If the density test rejects (significant p-value), bunching may be present.

Diagnostic 2: No Level Discontinuity

In a pure RKD, the outcome should be continuous at the kink. Run a standard RDD (deriv=0) to check. If you find a significant level discontinuity, you may have a combined RDD+RKD setting. In R: rdrobust(Y, X, c = kink_point, deriv = 0).

Diagnostic 3: Covariate Balance

Predetermined covariates should not exhibit a kink at the threshold. For each covariate, run rdrobust with deriv=1. A significant kink in a predetermined covariate suggests that the smoothness assumption is violated.

Diagnostic 4: First-Stage Strength

The kink in the treatment function must be large enough to provide identifying variation. A small first-stage kink leads to noisy estimates and potential bias (analogous to the weak instrument problem in IV). Report the first-stage kink estimate and its significance.

Interpreting Your Results

Reading the Output

The rdrobust output with deriv=1 reports the change in the slope of the conditional expectation at the kink point. This quantity is the reduced-form kink -- the numerator of the RKD estimand. To get the causal effect, divide by the first-stage kink (the slope change in the treatment function).

Component	What it estimates	How to get it
Reduced-form kink	$\Delta E'[Y \mid X=c]$	`rdrobust(Y, X, c=kink, deriv=1)`
First-stage kink	$\Delta T'(c)$	Known from policy formula, or `rdrobust(T, X, c=kink, deriv=1)`
RKD estimate ( $\hat\tau$ )	Causal effect	Reduced-form / First-stage

What to Report

A well-reported RKD analysis should include:

The treatment function: a clear description (and ideally a plot) of the policy rule creating the kink
The first-stage kink: the change in slope of the treatment function, with standard error and significance
The reduced-form kink: the change in slope of the outcome, with standard error and significance
The RKD estimate: the ratio of reduced form to first stage, with standard error
Density smoothness test: evidence that bunching is not present
Level continuity check: evidence that there is no level discontinuity in the outcome
Bandwidth sensitivity: how the estimate changes across bandwidth choices
Covariate balance: predetermined covariates should not exhibit kinks at the threshold

Example write-up

"We estimate the effect of unemployment insurance benefit levels on unemployment duration using a regression kink design (Card et al., 2015). The Austrian UI benefit formula replaces 55% of prior daily wages up to a cap, creating a kink in the benefit function at the earnings cap. Below the cap, each additional euro of daily wages increases daily benefits by 0.55 euros; above the cap, the marginal increase is zero -- a first-stage slope change of 0.55.

Using local linear derivative estimation (rdrobust with deriv=1, triangular kernel, MSE-optimal bandwidth of 8.3 euros), we estimate a reduced-form kink in unemployment duration of 0.083 weeks per euro (robust SE = 0.024, p < 0.001). The RKD estimate of the causal effect is 0.083 / 0.55 = 0.15 weeks per euro of daily benefits, or approximately 1.5 weeks for a 10% benefit increase (SE = 0.44).

The density smoothness test yields p = 0.42, consistent with no bunching. The level continuity check shows no significant jump in duration at the kink (p = 0.78). Results are robust to alternative bandwidths (0.5h to 2h), polynomial orders (p = 1 to 3), and kernel functions."

Bandwidth Considerations

RKD estimation is inherently noisier than RDD because estimating derivatives requires fitting curvature, which demands more data near the kink. In practice:

MSE-optimal bandwidths for RKD are typically larger than for RDD with the same data
The RKD estimate is more sensitive to bandwidth choice than a typical RDD estimate
Bandwidth sensitivity analysis is even more important for RKD than for RDD

G. What Can Go Wrong

Assumption Failure Demo

Applying Standard RDD to a Kink

RKD estimation using deriv=1 in rdrobust at the UI benefit cap kink point

The change in slope of unemployment duration at the kink is 0.15 weeks per euro (SE = 0.04, p < 0.001). With a known first-stage kink of 0.10, the RKD estimate is 1.5 weeks per 10% benefit increase.

Assumption Failure Demo

Bunching at the Kink Violates Identification

RKD applied to a kink where the running variable density is smooth -- workers cannot precisely control their prior earnings relative to the benefit cap

Density smoothness test p = 0.45. The density is smooth at the kink. RKD estimate: 1.5 weeks per 10% benefit increase (SE = 0.4).

Assumption Failure Demo

Weak First Stage -- Tiny Kink in Treatment

RKD at a benefit cap where the replacement rate drops from 55% to 0% -- a large and clearly identifiable slope change in the treatment function

First-stage kink: 0.55 (SE = 0.02, p < 0.001). The treatment function has a large, precisely estimated kink. The RKD estimate is 1.5 (SE = 0.4) -- informative and precise.

Assumption Failure Demo

Nonlinear Outcome Confounds the Kink

The conditional expectation of the outcome is approximately linear on each side of the kink, with a clear change in slope at the kink point

Local linear derivative estimation (p=1) produces a kink estimate of 0.12 (SE = 0.03). Local quadratic (p=2) gives 0.11 (SE = 0.04). Results are stable across polynomial orders, indicating the kink is genuine.

H. Practice

H.1 Concept Checks

Concept Check

A researcher studies the effect of UI benefits on unemployment duration. The UI formula replaces 55% of earnings below a cap and 0% above it. She runs rdrobust(duration, earnings, c = cap) with the default settings (deriv=0) and finds no significant effect (p = 0.82). She concludes that UI benefits do not affect unemployment duration. What went wrong?

The sample is too small. She needs more observations near the cutoff.She used deriv=0 (RDD) when she should have used deriv=1 (RKD). There is no level discontinuity at the cap because benefits are continuous -- the identifying variation comes from the change in slope of the benefit function, not a jump.She should use a different kernel function. The triangular kernel is inappropriate for kink designs.She should use a broader bandwidth to capture more variation around the cutoff.

Concept Check

You estimate an RKD at a tax bracket threshold. The density smoothness test yields p = 0.02, and a histogram of income shows a clear spike just below the bracket threshold. What should you conclude?

The RKD is valid because p = 0.02 only indicates a small effect. Proceed with the analysis.The density smoothness assumption is violated. Bunching at the kink means that units on either side of the threshold are not comparable on unobservables. The standard RKD is not valid without correction.The bunching is a good sign -- it shows that taxpayers respond to the tax kink, confirming a strong first stage.Add covariates to control for the selection created by bunching.

Concept Check

An RKD estimate using polynomial order p=1 (local linear) yields a kink of 0.18 (SE = 0.04, p < 0.001). When you increase to p=2 (local quadratic), the kink estimate drops to 0.03 (SE = 0.07, p = 0.68). What does this pattern suggest?

The p=1 estimate is correct and the p=2 estimate is too noisy. Stick with local linear.The apparent kink at p=1 may be driven by global curvature in the outcome-running variable relationship, not by a genuine treatment effect. The local quadratic specification absorbs this curvature, and the kink disappears.You should average the two estimates: (0.18 + 0.03) / 2 = 0.105.This proves the treatment has no effect.

H.2 Guided Exercise

Guided Exercise

Interpreting RKD Output from a UI Benefits Study

You estimate the effect of unemployment insurance benefit levels on unemployment duration using an RKD at the earnings cap in Austria. The benefit formula replaces 55% of prior daily wages below the cap and 0% above the cap. Your output: **First stage (treatment function kink):** rdrobust(benefits, earnings, c = cap, deriv = 1) Kink estimate (bias-corrected): 0.548 (Robust SE = 0.031, p < 0.001) Bandwidth: 12.4 euros. Effective N: 4,230. **Reduced form (outcome kink):** rdrobust(duration, earnings, c = cap, deriv = 1) Kink estimate (bias-corrected): 0.082 (Robust SE = 0.025, p = 0.001) Bandwidth: 9.8 euros. Effective N: 3,150. **Diagnostics:** - Density smoothness test: p = 0.52 - Level continuity (outcome): level jump = 0.15 weeks, p = 0.71 - Level continuity (treatment): level jump = 0.03 euros, p = 0.89 **Bandwidth sensitivity (reduced form):** | Mult | Estimate | SE | p-value | | 0.50 | 0.095 | 0.041 | 0.021 | | 0.75 | 0.088 | 0.030 | 0.003 | | 1.00 | 0.082 | 0.025 | 0.001 | | 1.25 | 0.079 | 0.022 | < 0.001 | | 1.50 | 0.075 | 0.020 | < 0.001 | | 2.00 | 0.071 | 0.018 | < 0.001 |

H.3 Error Detective

Error Detective

Read the analysis below carefully and identify the errors.

A labor economist studies the effect of a minimum wage kink on employment. At a revenue threshold of $500,000, firms face a higher minimum wage ($15/hour instead of $12/hour). She runs: rdrobust employment revenue, c(500000) kernel(triangular) She reports: "The RKD estimate shows that the higher minimum wage reduces employment by 3.2 workers per firm (SE = 1.1, p = 0.004). The density test is not significant (p = 0.34). Bandwidth sensitivity shows stable results across 0.5h to 2h." She plots the employment-revenue relationship and it shows a clear downward jump at the $500,000 threshold.

Select all errors you can find:

Using deriv=0 (default) instead of deriv=1 for a kink design(Estimation command / method choice)

The policy creates a level discontinuity, not a kink(Research design classification)

No first-stage kink estimation reported(Reporting / causal estimate)

Error Detective

Read the analysis below carefully and identify the errors.

A public finance researcher estimates the effect of progressive income taxation on labor supply using an RKD at a tax bracket boundary. At $80,000 of taxable income, the marginal tax rate increases from 22% to 24%. She runs: rdrobust hours_worked income, c(80000) deriv(1) kernel(triangular) She reports: "The RKD reduced-form estimate is -0.45 hours per week per dollar of income (SE = 0.12, p < 0.001). The density test shows no bunching (p = 0.08). Covariate balance tests using standard RDD (deriv=0) show no significant jumps." She does not report the first-stage kink, polynomial sensitivity, or a level continuity check on the outcome.

Select all errors you can find:

Density test p = 0.08 is borderline and deserves discussion(Density smoothness test interpretation)

Covariate balance tests use deriv=0 instead of deriv=1(Covariate balance diagnostic)

No first-stage kink reported(First stage / causal estimate)

No polynomial order sensitivity reported(Robustness / polynomial order)

H.4 You Are the Referee

Referee Exercise

Read the paper summary below and write a brief referee critique (2-3 sentences) of the identification strategy.

Paper Summary

The authors study how housing subsidy generosity affects residential mobility using a regression discontinuity design at an income threshold where the subsidy amount changes. At a gross income threshold of EUR 25,000, the housing subsidy drops from 30% to 20% of rent. The authors use rdrobust with the default deriv=0 setting and report a significant level discontinuity in mobility rates at the threshold. They claim this is an RKD because the subsidy amount changes as a function of income.

Key Table

Variable	Coefficient	SE	p-value
RD estimate (deriv=0)	-0.042	0.015	0.005
Bandwidth	1,850
Effective N	6,420
Density test p-value	0.41
Covariate balance	Pass
N	42,000

Authors' Identification Claim

The authors argue that the income threshold creates a kink in the subsidy-income relationship, and they use regression discontinuity methods to estimate the causal effect of subsidy generosity on residential mobility at the threshold.

I. Swap-In: When to Use Something Else

Sharp RDD: when the treatment function has a level discontinuity at the threshold rather than (or in addition to) a slope change. If treatment jumps, use RDD. If treatment only bends, use RKD.
Fuzzy RDD: when crossing a threshold changes the probability of treatment in a discontinuous way. If the jump in treatment probability is the source of variation, use fuzzy RDD, not RKD.
Instrumental Variables: when you have an instrument that is not based on a kink or discontinuity. RKD can be viewed as a special case of IV where the instrument is the change in slope of the treatment function at the kink. If your instrument operates through a different mechanism, standard IV may be more appropriate.
Bunching Estimator (Saez 2010; Kleven & Waseem 2013): when the running variable shows bunching at the kink point. If agents can sort around the kink, the density smoothness assumption is violated and RKD is invalid. The bunching estimator directly uses the excess mass at the kink to estimate behavioral responses.

Combining RDD and RKD

Some policy settings create both a level discontinuity and a kink at the same threshold. For example, a program might provide a lump-sum payment plus an income-dependent benefit, with the benefit formula kinking at a threshold. In such cases:

Estimate the RDD component (level jump) using deriv=0
Estimate the RKD component (slope change) using deriv=1
Be transparent that identification comes from two sources

J. Reviewer Checklist

Critical Reading Checklist

Is the treatment function kink clearly documented? Is the policy rule creating the kink described precisely?
Is the first-stage kink estimated and reported? Is it statistically significant and economically meaningful?
Is deriv=1 used in rdrobust (not the default deriv=0)?
Is the density smoothness test reported and passed? Is bunching at the kink addressed?
Is the level continuity of the outcome verified? Is there evidence that the outcome does NOT jump at the kink?
Are covariate balance tests conducted using deriv=1 (not deriv=0)?
Is bandwidth sensitivity analysis reported? Does the estimate remain stable across bandwidths?
Is the RKD estimate reported as the ratio of reduced-form to first-stage, with appropriate standard errors?
Is polynomial order sensitivity reported? Does the kink survive higher-order polynomials?
Are plots provided showing the treatment function kink and the outcome kink?

Paper Library

Has replication code

Foundational (2)

Card, D., Lee, D. S., Pei, Z., & Weber, A. (2015). Inference on Causal Effects in a Generalized Regression Kink Design.

EconometricaDOI: 10.3982/ECTA11224

Card, Lee, Pei, and Weber formalize the regression kink design, establishing conditions under which a kink in the treatment assignment function identifies causal effects. They show that the estimand is the ratio of the change in the slope of the conditional expectation of the outcome to the change in the slope of the treatment function at the kink point. The paper develops inference procedures and applies the method to estimate the effect of unemployment insurance benefits on unemployment duration.

Dong, Y. (2015). Regression Discontinuity Applications with Rounding Errors in the Running Variable.

Journal of Applied EconometricsDOI: 10.1002/jae.2369

Dong examines regression kink designs when the running variable is subject to rounding or heaping, a common practical concern. She develops diagnostic tests and estimation corrections for settings where the running variable is discrete or rounded, extending the applicability of RKD to settings with imperfect measurement of the running variable.

Application (2)

Nielsen, H. S., Sorensen, T., & Taber, C. (2010). Estimating the Effect of Student Aid on College Enrollment: Evidence from a Government Grant Policy Reform.

American Economic Journal: Economic PolicyDOI: 10.1257/pol.2.2.185

Nielsen, Sorensen, and Taber apply a regression kink design to estimate the effect of student financial aid on college enrollment in Denmark. The Danish student aid formula creates a kink in the relationship between parental income and aid received. They exploit this kink to identify causal effects, providing one of the earliest applications of the RKD methodology.

Landais, C. (2015). Assessing the Welfare Effects of Unemployment Benefits Using the Regression Kink Design.

American Economic Journal: Economic PolicyDOI: 10.1257/pol.20130248

Landais uses the regression kink design to decompose the moral hazard and liquidity effects of unemployment insurance benefits using US data. The progressive UI benefit formula creates kinks that provide quasi-experimental variation in benefit levels. This paper demonstrates the power of RKD for evaluating social insurance programs where benefits change slope at known thresholds.

Quick Reference

One-Line Implementation

Download Full Analysis Code

Motivating Example#

A. Overview#

What RKD Does#

The Key Intuition#

RKD vs. RDD#

When to Use RKD#

When NOT to Use RKD#

The Taxonomy Position#

Common Confusions#

B. Identification#

Assumption 1: Smoothness of Conditional Expectations#

Assumption 2: Density Smoothness (No Bunching)#

Assumption 3: First-Stage Kink Exists#

What the Estimand Is#

Sharp vs. Fuzzy RKD#

C. Visual Intuition#

Why the Kink Matters: RKD vs. Alternatives

D. Mathematical Derivation#

The Estimand: A Ratio of Derivatives#

E. Implementation#

Estimation Procedure#

Step 1: Plot the Treatment Function

Step 2: Plot the Outcome

Step 3: Estimate the Reduced Form (Outcome Kink)

Step 4: Estimate the First Stage (Treatment Kink)

Step 5: Compute the RKD Estimate

Bandwidth Selection

F. Diagnostics#

Diagnostic 1: Density Smoothness at the Kink#

Diagnostic 2: No Level Discontinuity#

Diagnostic 3: Covariate Balance#

Diagnostic 4: First-Stage Strength#

Interpreting Your Results#

Reading the Output

What to Report

Bandwidth Considerations#

G. What Can Go Wrong#

Applying Standard RDD to a Kink

Bunching at the Kink Violates Identification

Weak First Stage -- Tiny Kink in Treatment

Nonlinear Outcome Confounds the Kink

H. Practice#

H.1 Concept Checks#

H.2 Guided Exercise#

H.3 Error Detective#

H.4 You Are the Referee#

Paper Summary

Key Table

Authors' Identification Claim

I. Swap-In: When to Use Something Else#

Combining RDD and RKD#

J. Reviewer Checklist#

Critical Reading Checklist

Paper Library

Foundational (2)

Application (2)

Tags

Motivating Example

A. Overview

What RKD Does

The Key Intuition

RKD vs. RDD

When to Use RKD

When NOT to Use RKD

The Taxonomy Position

Common Confusions

B. Identification

Assumption 1: Smoothness of Conditional Expectations

Assumption 2: Density Smoothness (No Bunching)

Assumption 3: First-Stage Kink Exists

What the Estimand Is

Sharp vs. Fuzzy RKD

C. Visual Intuition

D. Mathematical Derivation

The Estimand: A Ratio of Derivatives

E. Implementation

Estimation Procedure

F. Diagnostics

Diagnostic 1: Density Smoothness at the Kink

Diagnostic 2: No Level Discontinuity

Diagnostic 3: Covariate Balance

Diagnostic 4: First-Stage Strength

Interpreting Your Results

Bandwidth Considerations

G. What Can Go Wrong

H. Practice

H.1 Concept Checks

H.2 Guided Exercise

H.3 Error Detective

H.4 You Are the Referee

I. Swap-In: When to Use Something Else

Combining RDD and RKD

J. Reviewer Checklist