Regression Kink Design

Definition

A quasi-experimental identification strategy that recovers a local average treatment effect at a point where a policy assignment rule has a kink — a discontinuous change in slope — rather than a jump in level. Where regression discontinuity design (RDD) exploits a jump in the level of treatment at a threshold, the regression kink design (RKD) exploits a change in the rate at which treatment varies with a running variable. The foundational reference is Card, Lee, Pei, and Weber (2015).

Key Ideas

The running variable must continuously determine both the treatment level and (at the kink) the rate of treatment assignment. At the kink point, the treatment function changes slope but remains continuous — there is no jump.
The key identification assumption is that the density of the running variable is smooth at the kink and potential outcomes are continuously differentiable. Intuitively: people cannot precisely sort themselves to just above or below the kink (unlike RDD, where a level jump creates stronger sorting incentives).
A fuzzy RKD generalizes the design to settings where assignment is not deterministic — the kink in the conditional mean of treatment given the running variable is used as an instrument for the actual treatment received. Analogous to the distinction between sharp and fuzzy RDD.
The estimand is a weighted local average treatment effect at the kink point — causal for the population near the kink, not for the population as a whole.

How It Works — Mechanics

The Basic Setup

Let $b$ be a running variable (e.g., prior earnings) and $T$ be the treatment (e.g., benefit amount). The assignment rule $T = B(b)$ has a kink at $b_0$ : the function is continuous but its derivative changes at $b_0$ . The RKD estimate is:

$\hat{\tau}_{RKD} = \frac{\lim_{b \downarrow b_0} \frac{dE[Y \mid b]}{db} - \lim_{b \uparrow b_0} \frac{dE[Y \mid b]}{db}}{\lim_{b \downarrow b_0} \frac{dB(b)}{db} - \lim_{b \uparrow b_0} \frac{dB(b)}{db}}$

Numerator: the change in slope of the outcome–running-variable relationship at the kink.
Denominator: the change in slope of the treatment–running-variable relationship at the kink (known from the policy formula).

The DI Benefit Formula Application

Social Security disability insurance (DI) benefits are determined by the Primary Insurance Amount (PIA), which is a piecewise-linear function of Average Indexed Monthly Earnings (AIME). The PIA-AIME schedule has known bend points where the marginal replacement rate changes — these are kinks in the benefit formula:

Lower bend point ( $\approx 4$ th percentile of AIME): replacement rate drops from $90\%$ to $32\%$
Upper bend point ( $\approx 84$ th percentile of AIME): replacement rate drops from $32\%$ to $15\%$
Family maximum bend points: further kinks capping total household benefits

At each bend point, the level of the benefit is continuous, but the slope of benefit with respect to AIME changes discontinuously. This creates exogenous variation in benefit amount conditional on the AIME level — identified separately from any variation in receipt of DI. Workers on either side of a bend point have similar earnings histories and similar health; the only discontinuous difference is how many additional dollars of DI they receive per additional dollar of AIME.

Why RKD Complements the Examiner/Judge Instrumental Variable (IV)

The examiner/judge IV (Maestas, Mullen, and Strand 2013; French and Song 2014) estimates the effect of DI receipt — being on the program vs. not. The RKD estimates the effect of DI benefit amount — conditional on already being a beneficiary, how does an additional dollar of DI affect outcomes? This distinction maps directly onto the income vs. substitution effect decomposition:

Substitution effect (examiner IV): DI receipt as an outside option changes the attractiveness of work relative to DI. Identified at the margin of program entry.
Income effect (RKD): Additional DI income changes consumption capacity without changing program status or the Substantial Gainful Activity (SGA) incentive. Identified within the beneficiary population, at the bend points.

The finding that the RKD detects no labor supply effect at the bend points (Gelber, Moore, and Strand 2016) while the examiner IV detects a large work-disincentive effect implies the entire work disincentive is a substitution effect, not an income effect — receipt changes the DI-vs.-work margin, but additional benefit dollars within the program do not.

Key Results in the DI Literature

Gelber, Moore, and Strand (2016) — Income Effect on Earnings

Upper bend point; $N = 610{,}271$ new DI beneficiaries $2001$ – $2007$
$\$1$ in annual DI income $\to$ $-\$0.20$ in annual earnings (income effect only; stable across four post-award years)
No significant effect at lower bend point → validates the upper bend as the active margin
This income effect nearly equals the full work-disincentive estimate from Maestas, Mullen, and Strand and French/Song, implying substitution effects from the SGA threshold account for little of the total crowdout

Gelber, Moore, and Strand (2017) — Income Effect on Mortality

Lower and family maximum bend points; $N = 3.65$ million new DI beneficiaries $1997$ – $2009$
$\$1{,}000$ /year higher DI income $\to$ $-0.26$ percentage points (pp) annual mortality at the lower bend point; elasticity $-0.56$
Cost per statistical life-year: $\approx \$59{,}000$ — within the commonly used $\$50{,}000$ – $\$100{,}000$ threshold
No labor supply effect at bend points → mortality reduction is a pure income channel
Largest effects for Black beneficiaries, women, and Disability Determination Services (DDS)-allowed (most severely impaired) subgroups
Mechanism (suggestive): higher income $\to$ higher consumption of basic life-sustaining goods (food, housing, healthcare, transportation)

Why It Matters

The RKD in the DI context establishes that DI income causally saves lives, adding a welfare benefit to the program's accounting that prior work had not quantified. Combined with the Deshpande and Lockwood (2022) finding that $63\%$ of DI's insurance value covers nonhealth risk, the mortality result implies the program's benefit side is substantially larger than a pure health-targeting framework would suggest.

The income vs. substitution effect decomposition also has policy implications: efforts to reduce DI rolls by cutting benefit levels (income effect) would eliminate the mortality-reduction benefit without addressing the program-entry margin (substitution effect). The efficient instrument for reducing rolls is eligibility standards, not benefit generosity.

Open Questions

The RKD estimates are local to the bend-point population — workers near the lower bend point are very low earners ( $\approx 4$ th AIME percentile). Extrapolation to higher-earning beneficiaries requires functional form assumptions.
The mortality effect is identified in the $1997$ – $2009$ period; it may have changed as the DI population's diagnostic mix shifted toward musculoskeletal/mental conditions.
The absence of a labor supply effect at the bend points holds for current beneficiaries. Whether additional income at the margin of program entry (for potential applicants) has a different substitution effect is not identified by the RKD.