27.7 Sharp Regression Discontinuity Design

A Sharp Regression Discontinuity Design occurs when treatment assignment follows a strict rule at a known cutoff. That is, units receive treatment if and only if their running variable $X_i$ crosses a threshold $c$:

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

Unlike Fuzzy RD, where treatment probability changes discontinuously but is not deterministic, Sharp RD ensures perfect compliance with the cutoff rule.

The key idea is that units just below and just above the cutoff are nearly identical in expectation, except for their treatment status. This mimics randomized experiments in a local neighborhood around $X = c$.

27.7.1 Assumptions for Identification

For a valid Sharp RD design, we assume:

1. Continuity of the Conditional Expectation of Potential Outcomes
   • The expected outcome given $X$ is smooth at $c$ in the absence of treatment: $$\lim_{x \uparrow c} E[Y(0) | X = x] = \lim_{x \downarrow c} E[Y(0) | X = x].$$
   • This ensures that any observed discontinuity in $E[Y | X]$ at $X = c$ is due to treatment, not pre-existing differences.
2. No Manipulation of the Running Variable
   • Agents cannot perfectly sort themselves around the cutoff (e.g., students manipulating test scores to qualify for a scholarship).
   • This is typically checked using the McCrary density test to detect discontinuities in the density of $X$ at $c$.
3. Local Randomization
   • Near the cutoff, individuals are as good as randomly assigned to treatment or control.

If these conditions hold, the Sharp RD estimator provides an unbiased estimate of the causal effect of treatment.

27.7.2 Estimating the Local Average Treatment Effect

The treatment effect at the cutoff is given by:

$$\tau = \lim_{x \downarrow c}E[Y | X = x] - \lim_{x \uparrow c} E[Y | X = x].$$

This represents the jump in the expected outcome at the cutoff.

27.7.3 Estimation Methods

1. Local Linear Regression

A common approach is to estimate separate linear regressions on each side of the cutoff:

For observations below the cutoff $(X < c)$:

$$Y_i = \alpha + \beta (X_i - c) + \epsilon_i.$$

For observations above the cutoff $(X \geq c)$:

$$Y_i = \gamma + \delta (X_i - c) + \tau D_i + \nu_i.$$

Here, the coefficient $\tau$ captures the treatment effect at $X = c$.

In practice, we estimate:

$$\hat{\tau} = \hat{E}[Y | X = c^+] - \hat{E}[Y | X = c^-].$$

This can be implemented using Weighted Least Squares with observations near the cutoff receiving higher weights.

2. Global Polynomial Regression

An alternative approach is to use a polynomial regression:

$$Y_i = \alpha + \sum_{k=1}^{K} \beta_k (X_i - c)^k + \tau D_i + \epsilon_i.$$

where:

• Higher-order terms $(X_i - c)^k$ capture nonlinear relationships.

• Typical choices for $K$ are 2 or 3, but higher orders may lead to overfitting.

3. Nonparametric Local Regression

Instead of assuming a linear or polynomial relationship, a local regression (kernel-based) method estimates:

$$E[Y | X = x] = \sum_{i=1}^{n} K_h (X_i - x) Y_i.$$

where $K_h$ is a kernel function (e.g., Epanechnikov), and $h$ is the bandwidth.

• A smaller $h$ captures local variation but increases variance.
• A larger $h$ smooths noise but risks bias.

27.7.4 Steps for Sharp RD

1. Visualization

• Graph the outcome variable:
  • Compute binned averages of $Y_i$ over intervals of $X$.
  • Choose bin sizes that balance smoothness and clarity.
  • Overlay a smoothed regression line on each side of $c$.
• Graph the running variable’s density:
  • Use histograms to check for manipulation.
  • Conduct a McCrary density test to detect discontinuities.

2. Estimation of the Treatment Effect

• Run local linear regression separately on both sides of the cutoff.
• Use nonparametric methods (e.g., kernel regression) for robustness.
• Estimate treatment effect using: $$\hat{\tau} = \hat{E}[Y | X = c^+] - \hat{E}[Y | X = c^-].$$

3. Robustness Checks

1. Check for Jumps in Other Covariates

• If any pre-determined covariate jumps at the cutoff, it suggests sorting or omitted variable bias.

• Run RD regressions for each covariate:

  $$W_i = \alpha + \beta (X_i - c) + \gamma D_i + \epsilon_i.$$

• A significant $\gamma$ suggests a violation of continuity assumptions.

2. McCrary Density Test (Checking for Manipulation)

• Run the McCrary test to examine whether the density of $X_i$ exhibits a discontinuity at $c$.
• A significant jump indicates sorting behavior, which can invalidate the RD design.

3. Placebo Tests

• Perform fake cutoff tests by estimating RD effects at arbitrary points $c^*$.
• If significant effects appear at non-cutoff points, the RD design may be picking up spurious trends.

4. Varying Bandwidth

• Re-run RD analysis using different bandwidths $h$.
• If results change dramatically, treatment effects may be highly sensitive to bandwidth choice.
• Use data-driven bandwidth selection methods (e.g., G. Imbens and Kalyanaraman (2012)).

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References

Imbens, Guido, and Karthik Kalyanaraman. 2012. “Optimal Bandwidth Choice for the Regression Discontinuity Estimator.” The Review of Economic Studies 79 (3): 933–59.