Regression Discontinuity Design (RDD)

• Small change difference -> Large treatment difference

Introduction

• Regression Discontinuity Design (RDD) help us identify causal effects with reasonable assumptions in some specific circumstances:
  • Situations where abrupt cutoffs on a running variable split units into exposed/treated and unesposed/untreated groups
  • If there's just a little bit of randomness in the running variable, this can be thought of as a random-like assignment
    • Where we end up just below or just above the cutoff is "random"
    • So these groups near the threshold are very similar, don't differ systematically on potential confounders -> "apples-to-apples" comparison.
• Terms:
  • $X$ is a continuous variable that determines whether or not the student receives the scholarship (treatment)
    • Focusing Variable or Running Variable: the continuous variable determining treatment assignment
  • We can always rescale $X$ to set the treshold at any value we like
    • Set the treshold at 0, so $T=1$ for all $X>0$ and $T=0$ for all $X<0$
• Treatment Effects:
  • As always, one particular value of $Y_1$ or $Y_0$ exists for every observation in our sample.
  • If we could observe them, then we could calculate $E[Y_{1_i}-Y_{0_i}]$
  • Instead of trying to estimate $E[Y_{1_i}-Y_{0_i}]$, we will, with some weaker assumptions, estimate $E[Y_{1_i}-Y_{0_i}\mid X_i=0]$
    • This is called the Local Average Treatment Effect (LATE): more generally, a unit with a running variable value right at the threshold for treatment.
  • To obtain LATE, we cannot just look at those with exactly $X_i=0$. Instead, we run two regressions on treated and untreated "near" the threshold.
    • $\lim_{X\to0+}E[Y_1\mid X]$: the limit of the expected value of the outcome in the treated group as we appraoch the threshold from above.
    • $\lim_{X\to0-}E[Y_0\mid X]$: the limit of expected value of the outcome in the untreated group as we approach the threshold from below.
    • Then, $\text{LSAT}=\lim_{X\to0+}E[Y_1\mid X]-\lim_{X\to0-}E[Y_0\mid X]$
    • With some assumptions, we can interpret this as the effect on the outcome of the exposure/treatment

The Continuity Assumption

• For the LATE to equal the causal treatment effect (at the threshold), we must make the continuity assumption:
  • $\lim_{X\to0+}E[Y_{1_i}\mid X]=\lim_{X\to0-}E[Y_{1_i}\mid X]$
  • $\lim_{X\to0-}E[Y_{0_i}\mid X]=\lim_{X\to0+}E[Y_{0_i}\mid X]$
    • $\lim_{X\to0-}E[Y_{1_i}\mid X]$ and $\lim_{X\to0+}E[Y_{0_i}\mid X]$ are unobserable, by the fundamental problem of casual inference (we can't observe both $Y_{1_i}$ and $Y_{0_i}$ for the same unit).
  • In other words, we should fit a smooth, continuous function for the potential outcomes across the threshold in both the treated and untreated groups.
    • No other risk factors for the outcome change sharply at threshold - otherwise any differences could be due to those rather than exposure (confounding).
• Continuity Violation:
  • The running variable can be associated with the outcome (and potential outcomes)
  • Units can have some control over their running variable.
• Assess the Continuity Assumption and Avoid Violations
  • Subject matter expertise:
  • Look at measruable pre-treatment characteristics and see if they are continuous at the threshold.
    • If instead there is bunching or clumping near the threshold on either side, many have a continuity issue.
    • Statistical tests for sorting

Methods to Estimate LATE

• Naïve Binned Averages:
  • Calculating mean in small bins on either side of the threshold
  • Issue: choosing bin width
    • Has potentially large effect on LATE estimate
    • Assuming our outcome is correlated with our running variable (which it typically is), any nonzero bin width will bias our LATE
    • As we stretch further out, the average will be further biased from what it actually is right at the threshold.
• Better choice: directly estimate limits using regression
• Option 2: Local Linear Regression
  • Choose small bins on each side of the threshold and fit a straight line linear regression
    • Can be done in a single regression model with interaction terms
  • The coefficient of our exposure represents the jump (or discontinuity) in the graph to the right. This is the LATE.
  • Key assumption: does the outcome change linearly in these bins?
• Option 3: Polynomial Regression
  • Fit a polynomial linear regression with a discontinuity at the threshold
    • Can use entire data or just small bins on either side
    • Can use different polynomials for each side
  • The coefficient of our exposure represents the jump (or discontinuity) in the graph to the right. This is the LATE.
  • Far more flexible, but also more complex.