27.8 Regression Kink Design

The Regression Kink Design (RKD) extends the logic of RD by exploiting changes in the slope of the treatment intensity function at a known threshold rather than a discontinuous jump in treatment assignment.

Instead of an RD jump in treatment probability at $X = c$, the treatment function $b(X)$ exhibits a kink at the cutoff:

• In Sharp RKD, the kink is deterministic, meaning the treatment function $b(X)$ changes its slope exactly at $X = c$.
• In Fuzzy RKD, treatment assignment remains probabilistic, requiring an instrumental variable approach similar to Fuzzy RD.

Example: Unemployment Benefits

Consider an unemployment insurance program where benefits increase at a diminishing rate as prior earnings increase. The function governing benefits, $b(X)$, exhibits a kink at a threshold $X = c$. The RKD framework allows us to estimate the marginal causal effect of additional benefits on employment duration.

27.8.1 Identification in Sharp Regression Kink Design

In a Sharp RKD, the treatment intensity function $b(X)$ exhibits a known change in slope at $X = c$, formally:

$$D_i = b(X_i), \quad \text{where } b(X) \text{ has a kink at } X = c.$$

The key identification assumption is that the potential outcome function $E[Y(d) | X]$ is smooth in $X$. Thus, any observed change in the slope of $E[Y | X]$ at $X = c$ can be attributed to the change in $b(X)$.

The causal effect of interest is:

$$\alpha_{KRD} = \frac{\lim\limits_{x \downarrow c} \frac{d}{dx}E[Y |X = x]- \lim\limits_{x \uparrow c} \frac{d}{dx}E[Y |X = x]}{\lim\limits_{x \downarrow c} \frac{d}{dx}b(x) - \lim\limits_{x \uparrow c} \frac{d}{dx}b(x)}.$$

where:

• $b(X)$ is a known function determining treatment intensity.

• The numerator captures the discontinuous change in the slope of the expected outcome at $X = c$.

• The denominator captures the change in the slope of the treatment function at $X = c$.

If $b(X)$ is known and deterministic, the denominator is non-random, allowing for precise estimation of $\alpha_{KRD}$.

Assumptions for Identification

1. Continuity of Potential Outcomes
   • The expected potential outcomes $E[Y(d)|X]$ are smooth in $X$ (no jumps).
2. No Manipulation of the Running Variable
   • The density of $X$ is continuous at $X = c$, implying that agents cannot sort themselves based on the kink.
3. First-Stage Validity
   • The slope of $b(X)$ must change at $X = c$ (i.e., the kink must exist).

If these assumptions hold, $\alpha_{KRD}$ represents the marginal causal effect of treatment intensity on the outcome.

27.8.2 Identification in Fuzzy Regression Kink Design

In Fuzzy RKD, the treatment function $D_i$ does not directly follow a deterministic function $b(X)$ but instead exhibits a kink in its probability distribution:

$$E[D | X] \text{ has a kink at } X = c.$$

The treatment intensity function is unknown, requiring an instrumental variable (IV) strategy, analogous to Fuzzy RD.

The causal effect is given by:

$$\alpha_{KRD} = \frac{\lim\limits_{x \downarrow c} \frac{d}{dx}E[Y |X = x]- \lim\limits_{x \uparrow c} \frac{d}{dx}E[Y |X = x]}{\lim\limits_{x \downarrow c} \frac{d}{dx}E[D |X = x]- \lim\limits_{x \uparrow c} \frac{d}{dx}E[D |X = x]}.$$

where:

• The numerator measures the kink in the expected outcome.

• The denominator measures the kink in the probability of treatment.

• The ratio provides a local instrumental variable estimate of the causal effect for compliers.

Identification Assumptions

In addition to the Sharp RKD assumptions, Fuzzy RKD requires:

1. Monotonicity
   • No individuals decrease their treatment intensity while others increase at the kink (analogous to Fuzzy RD monotonicity).
2. Relevance of the Kink
   • There must be a statistically significant slope change in $E[D | X]$ at $X = c$.

If these assumptions hold, the Fuzzy RKD estimator identifies a local treatment effect.

27.8.3 Estimation of RKD Effects

RKD estimation involves three main steps:

Step 1: Estimating the Kink in the Outcome Function

Estimate the left- and right-hand derivatives of $E[Y | X]$:

$$\frac{d}{dx}E[Y | X] = \lim_{h \to 0} \frac{E[Y | X = c + h] - E[Y | X = c - h]}{h}.$$

This can be done using:

• Local linear regression on either side of the kink.

• Higher-order polynomial regression for improved flexibility.

Step 2: Estimating the Kink in the Treatment Function

For Sharp RKD, the kink in $b(X)$ is known.

For Fuzzy RKD, estimate the kink in $E[D | X]$:

$$\frac{d}{dx}E[D | X] = \lim_{h \to 0} \frac{E[D | X = c + h] - E[D | X = c - h]}{h}.$$

Use local regression or piecewise polynomials to estimate this slope.

Step 3: Compute RKD Estimator

For Sharp RKD:

$$\hat{\alpha}_{KRD} = \frac{\hat{\tau}_Y}{\tau_b},$$

where:

• $\hat{\tau}_Y$ is the estimated kink in $E[Y | X]$.

• $\tau_b$ is the known slope change in $b(X)$.

For Fuzzy RKD:

$$\hat{\alpha}_{KRD} = \frac{\hat{\tau}_Y}{\hat{\tau}_D}.$$

where:

• $\hat{\tau}_D$ is the estimated kink in $E[D | X]$.

27.8.4 Robustness Checks

1. Assess Covariate Smoothness
   • Verify that pre-determined covariates (e.g., age, education) do not exhibit kinks at $X = c$.
2. Check for Manipulation of the Running Variable
   • Perform a McCrary test to ensure the density of $X$ is continuous at $X = c$.
3. Placebo Kinks
   • Test for spurious kinks at other arbitrary values of $X$.
4. Bandwidth Sensitivity
   • Estimate RKD effects with varying bandwidths to check for consistency.

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