27.6 Fuzzy Regression Discontinuity Design

A Fuzzy Regression Discontinuity Design occurs when the assignment rule at the cutoff does not perfectly determine treatment status but instead causes a discontinuity in the probability of treatment. Unlike a Sharp RD Design, where crossing the threshold fully determines treatment, in a fuzzy RD, some individuals on both sides of the threshold may or may not receive the treatment.

If treatment is not strictly assigned at the cutoff, the usual RD estimator (which assumes deterministic assignment) is not valid. Instead, we use the cutoff as an instrumental variable to estimate the treatment effect for compliers, i.e., individuals whose treatment status depends on whether they cross the threshold.

Define an indicator variable $Z_i$ (i.e., instrument for treatment assignment) that captures whether an individual is above or below the cutoff:

$$Z_i = \begin{cases} 1 & \text{if } X_i \geq c \\ 0 & \text{if } X_i < c \end{cases}$$

This variable $Z_i$ serves as an instrument for the treatment variable $D_i$ because:

• It strongly correlates with treatment ($D_i$).

• It is exogenous, meaning it affects the outcome only through its effect on treatment.

27.6.1 Compliance Types

Since treatment assignment is no longer deterministic, individuals can be classified into four groups based on how they respond to the cutoff:

1. Compliers ($C_0$): Individuals who receive treatment if and only if $X_i \geq c$.
2. Always-Takers ($A_0$): Individuals who always receive treatment, regardless of whether $X_i \geq c$.
3. Never-Takers ($N_0$): Individuals who never receive treatment, even if $X_i \geq c$.
4. Defiers (violating monotonicity, assumed to be zero): Individuals who receive treatment if $X_i < c$ but not if $X_i \geq c$.

The Fuzzy RD estimator identifies the treatment effect only for compliers, because their treatment status depends on $Z_i$.

27.6.2 Estimating the Local Average Treatment Effect

We estimate the LATE using a ratio of discontinuities:

$$\text{LATE} = \frac{\lim\limits_{x \downarrow c}E[Y|X = x] - \lim\limits_{x \uparrow c}E[Y|X = x]}{\lim\limits_{x \downarrow c } E[D |X = x] - \lim\limits_{x \uparrow c}E[D |X=x]}$$

Intuitively, this formula represents:

$$\text{LATE} = \frac{\text{Discontinuity in } E[Y|X]}{\text{Discontinuity in } E[D|X]}$$

where:

• The numerator captures the jump in the expected outcome at the cutoff.

• The denominator captures the jump in the probability of treatment at the cutoff.

This ratio is valid under three key assumptions:

1. Continuity in potential outcomes: $E[Y(d)|X]$ is continuous at $X = c$ for both $d \in \{0,1\}$.
2. Monotonicity: There are no defiers ($P(D^+ \geq D^- | X = c) = 1$).
3. First-stage relevance: There is a discontinuity in $P(D = 1 | X)$ at $X = c$.

If these conditions hold, the fuzzy RD estimator gives a valid estimate of the causal effect of treatment for compliers.

27.6.3 Equivalent Representation Using Expectations

We can also define LATE in terms of conditional expectations of treatment and outcome:

$$\lim_{\epsilon \to 0} \frac{E[Y |Z = 1] - E[Y |Z=0]}{E[D|Z = 1] - E[D|Z = 0]}$$

where $Z$ is the instrument (indicator for being above the cutoff). This approach highlights the IV nature of fuzzy RD.

27.6.4 Estimation Strategies

There are two equivalent ways to estimate the LATE in practice:

Approach 1: Two-Step Estimation

1. Estimate Sharp RD for the Outcome $Y$:
   • Regress $Y$ on $X$ using a local linear regression on either side of $c$.
   • Estimate the discontinuity in $E[Y|X]$ at $c$.
2. Estimate Sharp RD for the Treatment $D$:
   • Regress $D$ on $X$ using a local linear regression on either side of $c$.
   • Estimate the discontinuity in $E[D|X]$ at $c$.
3. Compute the Ratio:
   • Divide the estimated discontinuity in $E[Y|X]$ by the estimated discontinuity in $E[D|X]$.

Mathematically:

$$\widehat{\text{LATE}} = \frac{\widehat{E[Y | X = c^+]} - \widehat{E[Y | X = c^-]}}{\widehat{E[D | X = c^+]} - \widehat{E[D | X = c^-]}}.$$

Approach 2: Instrumental Variables Regression

• Subset the data to observations close to $c$.

• Use $Z_i$ (above/below cutoff indicator) as an instrument for $D_i$ in a two-stage least squares regression:

  1. First-stage regression (predicting treatment using the cutoff indicator):

     $$D_i = \alpha + \beta Z_i + \gamma X_i + \epsilon_i$$

     • This captures the effect of the cutoff on treatment assignment.

  2. Second-stage regression (estimating treatment effect using predicted $D_i$):

     $$Y_i = \delta + \tau \widehat{D}_i + \lambda X_i + \nu_i$$

     • The coefficient $\tau$ gives the LATE estimate.

27.6.5 Practical Considerations

• Bandwidth Selection: Only observations near the cutoff should be used. Methods like cross-validation or Calonico, Cattaneo, and Farrell (2020) optimal bandwidth selection can help.
• Polynomial Order: A local linear model is typically preferred, but higher-order polynomials may be used cautiously.
• Robust Inference: Standard errors should be computed using heteroskedasticity-robust and clustered standard errors if necessary.

27.6.6 Steps for Fuzzy RD

1. Visualization

1. Graph the Outcome Variable:
   • Compute the average outcome within bins of the running variable $X_i$.
   • Choose bins large enough to display smooth trends but small enough to reveal discontinuities at the cutoff.
   • Overlay a smoothed regression line on either side of the cutoff to visualize any jumps.
2. Graph the Probability of Treatment:
   • Compute the average treatment probability within the same bins.
   • Plot $E[D|X]$ to check for a discontinuity at $X = c$, confirming the first-stage relevance of the instrument.

2. Estimation of Treatment Effect

Use Two-Stage Least Squares to estimate the Local Average Treatment Effect:

1. First Stage (Predict Treatment Using Cutoff Indicator $Z_i$): $$D_i = \alpha + \beta Z_i + \gamma X_i + \epsilon_i$$
   • This regression captures how treatment probability changes at the cutoff.
   • The coefficient $\beta$ measures the jump in treatment probability at $X = c$.
2. Second Stage (Estimate Outcome Using Predicted Treatment): $$Y_i = \delta + \tau \widehat{D}_i + \lambda X_i + \nu_i$$
   • The coefficient $\tau$ gives the LATE, which estimates the treatment effect for compliers.

3. Robustness Checks

1. Assess Possible Jumps in Other Covariates:
   • Check whether other pre-determined covariates (e.g., age, income) exhibit discontinuities at the cutoff.
   • If covariates jump, this may indicate endogenous sorting or omitted variable bias.
2. Hypothesis Testing for Bunching (McCrary Test):
   • Test for manipulation of the running variable by examining whether the density of $X_i$ changes discontinuously at $c$.
   • A significant density jump suggests sorting behavior, which could invalidate RD assumptions.
3. Placebo Tests:
   • Repeat the analysis at fake cutoffs (values of $X$ where no intervention occurs).
   • If a treatment effect appears at a placebo cutoff, this suggests a spurious RD effect.
4. Varying Bandwidth Sensitivity:
   • Re-run the analysis using different bandwidths around the cutoff.
   • Check whether estimates remain stable as the window narrows or expands.

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References

Calonico, Sebastian, Matias D Cattaneo, and Max H Farrell. 2020. “Optimal Bandwidth Choice for Robust Bias-Corrected Inference in Regression Discontinuity Designs.” The Econometrics Journal 23 (2): 192–210.