Regression Kink Design

Why a Change in Slope Can Identify a Treatment Effect

Many policies change treatment intensity gradually rather than switching eligibility on or off at one threshold. For example, a subsidy might decline as income rises, but decline more steeply above a specified income. If treatment intensity has a kink at that threshold, and other determinants of the outcome do not acquire a matching kink, the outcome’s slope change can reveal a local causal response to treatment intensity.

This is a regression kink design (RKD). Its central comparison is not the outcome level just below versus just above the threshold. Instead, it compares the local slopes of the outcome relationship and the treatment relationship. The design is credible only when the threshold creates the treatment kink without also creating an unrelated kink in the outcome relationship.

Definition: Let $X$ be a running variable, $c$ a known threshold, $D$ treatment intensity, and $Y$ an outcome. Write $m_D(x)=E[D\mid X=x]$ and $m_Y(x)=E[Y\mid X=x]$. The treatment kink and outcome kink are $$ K_D=m_D'(c+)-m_D'(c-),\qquad K_Y=m_Y'(c+)-m_Y'(c-), $$ where $c+$ and $c-$ denote the right- and left-hand limits. If $K_D\ne 0$, the regression kink ratio is $K_Y/K_D$.

The ratio has units of outcome per unit of treatment. For example, if $Y$ is a test score in points and $D$ is tutoring hours, the ratio is measured in score points per additional tutoring hour. It is not generally the effect for every person or for every treatment level; it is a local response associated with the policy-induced change in treatment intensity near $X=c$.

Assumptions and Identification

The assumptions must rule out explanations for the outcome kink other than the treatment kink. The following formulation makes the core identifying argument explicit. Suppose that, near $c$, the conditional means can be written as $$ m_D(x)=d(x),\qquad m_Y(x)=g(x)+R(d(x)), $$ where $d(x)$ is the treatment-intensity schedule, $g(x)$ collects non-treatment contributions to the outcome, and $R(d)$ is the causal response component. Assume $g$ is differentiable at $c$, $d$ is continuous at $c$, and $R$ is differentiable at the common treatment level $d(c)$, with the same derivative approached from both sides. Assume also that the one-sided derivatives of $d$ exist and are finite.

This representation is exact, for example, when the policy deterministically assigns treatment intensity from $X$, and the outcome mean consists of a smooth baseline plus a response to that assigned intensity. With realized treatment that varies among units at the same $X$, the representation needs additional justification. In general, $E[R(D)\mid X=x]$ need not equal $R(E[D\mid X=x])$. A ratio based on conditional mean treatment therefore does not automatically identify a causal dose response when there is nonlinear response or treatment-effect heterogeneity. The assumptions about the conditional means, or an alternative structural argument, must fit the application.

Identifying assumptions: (1) The policy produces a nonzero kink in treatment intensity, $K_D\ne 0$, but no treatment-level jump at $c$. (2) The conditional mean treatment is continuous at $c$, and the response derivative has a common limit at the treatment level $d(c)$. (3) The non-treatment component $g(x)$ has no kink at $c$. (4) No other policy, behavioral change, or measurement process creates an outcome kink at that same threshold. (5) There is adequate support for $X$ on both sides to estimate local slopes, and units cannot precisely manipulate $X$ in a way that undermines comparability near $c$.

The continuity requirement on $m_D(x)$ matters. A treatment-level jump is a different design feature: it can change the treatment level being evaluated on either side, so a ratio of slope changes need not represent a marginal response at one common dose. Smoothness of the outcome’s non-treatment component matters for a parallel reason: a kink in $g(x)$ would enter the numerator even if treatment had no causal effect.

Why the Ratio Works

Under the conditional-mean representation, apply the chain rule separately on each side of the threshold:

m_Y'(c\pm)=g'(c)+R'(d(c))d'(c\pm).

The same $g'(c)$ appears on both sides because $g$ is differentiable at $c$. The same $R'(d(c))$ appears because treatment is continuous at $c$ and the response derivative has a common limit there. Subtract the left derivative from the right:

K_Y=R'(d(c))\{d'(c+)-d'(c-)\}=R'(d(c))K_D.

If $K_D\ne 0$, division gives $K_Y/K_D=R'(d(c))$. Thus the ratio identifies the local marginal response to treatment intensity at the threshold under these assumptions. In applications with treatment variation conditional on $X$, this interpretation additionally requires assumptions that justify the corresponding conditional-mean representation; the ratio alone does not supply them.

Worked Example: A Continuous Dose Schedule With a Kink

Suppose $X$ is income in thousands of dollars relative to a policy threshold, and $D$ is an annual benefit measured in hundreds of dollars. For a hypothetical policy, the conditional mean benefit near the threshold is $m_D(x)=2+0.1x$ for $x<0$ and $m_D(x)=2+0.3x$ for $x>0$. The two formulas both equal 2 at $x=0$, so treatment is continuous. Suppose the conditional mean outcome, measured in points, is $m_Y(x)=D(x)^2+0.5x$.

The left and right treatment slopes are $0.1$ and $0.3$ hundred-dollar units per thousand dollars of income. Therefore, $$ K_D=0.3-0.1=0.2. $$ At the threshold, $D(0)=2$. On the left, the outcome slope is $2D(0)(0.1)+0.5=2(2)(0.1)+0.5=0.9$ points per thousand dollars. On the right, it is $2D(0)(0.3)+0.5=2(2)(0.3)+0.5=1.7$ points per thousand dollars. Hence $K_Y=1.7-0.9=0.8$.

The kink ratio is $K_Y/K_D=0.8/0.2=4$ outcome points per additional hundred-dollar unit of benefit. This matches the marginal response $R'(D)=2D$, evaluated at the common threshold dose $D=2$: $R'(2)=4$. The calculation relies on the continuous dose schedule and on the smooth baseline term $0.5x$, whose slope cancels from the kink contrast.

Estimating the Kink Ratio

In data, the one-sided slopes are estimated rather than known. A common approach fits separate local linear regressions on the two sides of $c$, using observations within a chosen bandwidth. The estimated slope change is the right-side slope estimate minus the left-side slope estimate, for both the outcome and treatment. The same sample restrictions and compatible specifications should be used for each relationship so their slope changes refer to the same neighborhood.

Local linear fits allow the outcome and treatment relationships to have different slopes on either side. The estimated ratio is meaningful only if the estimated first-stage kink is distinguishable from zero and the local approximations are adequate. Bandwidth choice involves a trade-off: a narrow window focuses on observations close to the threshold but may leave noisy slope estimates; a wide window gives more data but may make a linear approximation less credible.

Worked Example: Ratio Estimate and Approximate Uncertainty

In a hypothetical analysis, let $X$ be income in thousands of dollars, let $D$ be an annual subsidy in hundreds of dollars, and let $Y$ be a health score in points. Local fits estimate $K_D=-0.20$ hundred-dollar units per thousand dollars of income and $K_Y=-1.20$ health-score points per thousand dollars. The estimated ratio is $$ \widehat{\tau}_{RKD}=\frac{\widehat K_Y}{\widehat K_D} =\frac{-1.20}{-0.20}=6.00. $$

The estimate is 6.00 health-score points per additional hundred-dollar unit of subsidy near the threshold, under the identifying assumptions. Both slope changes are negative because the subsidy and score slopes become more negative by the amounts estimated; their ratio is positive.

For an illustrative delta-method interval, suppose the standard errors are $SE(\widehat K_Y)=0.30$ and $SE(\widehat K_D)=0.05$, with estimated covariance zero. For a ratio $q=K_Y/K_D$, the first-order variance estimate is $$ \widehat{\operatorname{Var}}(q)= \frac{\widehat{\operatorname{Var}}(\widehat K_Y)}{\widehat K_D^2} +\frac{\widehat K_Y^2\widehat{\operatorname{Var}}(\widehat K_D)}{\widehat K_D^4} -\frac{2\widehat K_Y\widehat{\operatorname{Cov}}(\widehat K_Y,\widehat K_D)}{\widehat K_D^3}. $$

Substitution gives $0.30^2/0.20^2+(-1.20)^2(0.05^2)/0.20^4=2.25+2.25=4.50$. Thus the approximate standard error is $\sqrt{4.50}=2.12$ points per hundred-dollar unit. A normal-approximation 95% interval is $6.00\pm1.96(2.12)=[1.84,10.16]$. This interval is approximate and depends on the ratio approximation being adequate; it does not account for bias from a failed design assumption. When the first-stage kink is weak, ratio intervals based on a simple delta approximation can be unreliable, so inference methods designed for weak denominators may be needed.

Inspect the policy rule.
Confirm that the rule predicts a slope change in treatment intensity at the specified threshold, not merely a smooth curve or an unintended treatment-level jump.

Estimate both conditional-mean slopes.
Fit the outcome and treatment relationships locally on each side, then calculate right-minus-left slope changes in their original units.

Assess the first stage and diagnostics.
Check that the estimated treatment kink is informative, that there is support on both sides, and that placebo cutoffs and outcomes do not show comparable unexplained kinks.

Form and interpret the ratio.
Divide the outcome kink by the treatment kink, report units and uncertainty, and state the assumptions that give the ratio a causal interpretation.

Diagnostics, Alternatives, and Limitations

A fitted ratio does not validate the design by itself. Plot the raw outcome and treatment against $X$, with separate local fits on either side, to see whether the estimated slope change is supported by the data rather than a few observations. Check the running-variable support and observation counts near $c$; sparse or highly uneven support can make local slopes unstable. Examine whether predetermined covariates show unexplained kinks, whether there is evidence of precise sorting around the threshold, and whether placebo outcomes or placebo cutoffs produce similar patterns. These are diagnostic checks, not proofs that all assumptions hold.

Bandwidth sensitivity is also important. Report whether the sign and approximate size of the first-stage kink and outcome ratio are stable across reasonable bandwidths and local specifications. A pattern that changes sharply when a few observations or a modest bandwidth adjustment is used may indicate weak support, curvature, or an overly influential observation. The earlier material on bandwidth sensitivity and support diagnostics is relevant to these checks.

Worked Example: A Treatment Jump Changes the Interpretation

Consider a hypothetical threshold $c=0$. For $x<0$, suppose $m_D(x)=1+0.1x$; for $x>0$, suppose $m_D(x)=2+0.5x$. Let the outcome be $Y=D^2$, with no other outcome channel.

The left and right treatment slopes are $0.1$ and $0.5$, so $K_D=0.5-0.1=0.4$. The left outcome slope approaches $2(1)(0.1)=0.2$, while the right outcome slope approaches $2(2)(0.5)=2$. Thus $K_Y=2-0.2=1.8$, and the kink ratio is $1.8/0.4=4.5$.

But treatment jumps from 1 to 2 at the threshold. The marginal response is $2D$: it is 2 at dose 1 and 4 at dose 2. There is no single common threshold dose at which the two one-sided response derivatives are evaluated, so the ratio 4.5 is not the marginal response at the threshold. This example shows why treatment continuity is part of the identifying conditions, not a cosmetic feature of the graph.

A sharp regression discontinuity design targets a jump in treatment at a cutoff and uses the corresponding outcome jump. An RKD instead targets a change in slope when treatment is continuous. A fuzzy regression discontinuity uses a treatment jump that is imperfectly followed and a Wald ratio of level jumps; that level-jump argument is not interchangeable with a kink ratio. Ordinary regression can estimate associations between $D$ and $Y$, but it does not by itself isolate the policy-induced variation in treatment or remove confounding. The RKD uses that variation, but only under its own smoothness, continuity, and exclusion conditions.

The effect is local to the threshold and to the treatment levels whose variation is generated by the policy kink. Extrapolating it to distant values of $X$, other populations, or large treatment changes requires additional evidence. If treatment response varies across people, the simple ratio need not equal a universal causal effect; a precise heterogeneous-effect interpretation requires further assumptions about how the policy changes treatment and how responses are distributed. Likewise, a competing policy that changes slope at the same threshold, a kink in the untreated outcome trend, poor measurement of $X$, or a weak first stage can undermine identification even if the ratio is computed correctly.

Common mistake: “The policy changes the slope of treatment, so dividing the outcome slope change by the treatment slope change always gives the causal effect.” The correction is that the ratio has that interpretation only when the treatment level is continuous at the threshold, the response derivative has a common limit, the non-treatment outcome component has no kink, and the conditional-mean model or other structural assumptions justify the causal response interpretation.

Check Your Understanding

Answer each question, then compare with the brief solutions.

If $K_D=0.25$ units of treatment per running-variable unit and $K_Y=1.00$ outcome points per running-variable unit, what is the ratio and its units?
Why does a treatment-level jump threaten the usual marginal-response interpretation of a kink ratio?
Suppose a smooth baseline outcome trend has the same derivative on both sides of the threshold. What happens to that derivative in the outcome kink?
Name two empirical checks that can reveal problems with a regression kink design, and explain what each can diagnose.

Solutions. (1) $K_Y/K_D=1.00/0.25=4.00$ outcome points per treatment unit. (2) The left and right derivatives may be evaluated at different treatment levels, so the ratio need not represent a response derivative at one common dose. (3) The common derivative cancels when the right slope is minus the left slope. (4) For example, plotting local fits and checking support can reveal unstable slopes or sparse data; placebo cutoffs or outcomes can reveal unexplained kinks that weaken the claim that the policy threshold uniquely generated the observed change.

What You'll Learn