1 A Motivating Example: The Scholarship Cutoff

Imagine you work for a university that offers merit scholarships to students who score above 80 on an entrance exam. You want to know: does the scholarship improve students' perfor- mance once they arrive? Comparing scholarship recipients to non-recipients is misleading recipients are, by definition, higher scorers, so they would likely outperform non-recipients even without the scholarship. But notice something interesting. A student who scores 81 and a student who scores 79 are almost identical in ability, background, and future potential. Yet one receives a scholarship and the other does not. The 2-point difference is essentially noise a small perturbation in how well the student felt that morning, or whether a lucky guess was made. Near the 80-point cutoff, assignment to scholarship status looks almost random. This intuition underlies the regression discontinuity (RD) design: exploit a thresh- old rule to make causal comparisons between nearly identical units just above and just below the cutoff.

2 The Fundamental Problem and the RD Solution

We want to estimate the causal effect of a treatment (scholarship) on an outcome (university grades). The fundamental problem is that treatment assignment is correlated with ability. Near the threshold, however, this correlation disappears: units just above and just below the cutoff are comparable on all factors except treatment. Formally, the RD design identifies the average effect of treatment at the cutoff: the average difference in potential outcomes for a student with exactly score с:

$\tau_{RD}=\mathbb{E}[Y(1)-Y(0)|X=c]$,

(1)

where X is the exam score (the running variable or forcing variable) and $c=80$ is the cutoff. The key assumption is continuity: in the absence of the treatment, potential outcomes would vary smoothly through the cutoff. Any jump in the observed outcome at e must therefore be caused by the treatment.

3 Step by Step: How RD Works

Step 1: Plot the data. The most important step is plotting average outcomes against the running variable, in bins, to see whether a jump at the cutoff is visible. A good RD should show:

• A smooth relationship between the running variable and the outcome on each side of the cutoff.
• A visible jump at c.

Figure 1: Hypothetical RD plot. Students just above the cutoff (scholarship recipients) have higher GPAs than students just below. The jump estimates the causal effect of the scholarship.

Step 2: Estimate the jump using local regression. We estimate the outcome mean just above and just below the cutoff using local linear regression fitting a separate straight line on each side of the cutoff, using only observations in a bandwidth h around c. For the right side $(X_{i}\ge c)$:

The

$\hat{\beta}_{0}^{+}$

$min_{\beta_{0}^{+},\beta_{1}^{+}}\sum_{i:c\le X_{i}\le c+h}(Y_{i}-\beta_{0}^{+}-\beta_{1}^{+}(X_{i}-c))^{2}.$                (2)

the average outcome just above the cutoff. Similarly, $\hat{\beta}_{0}^{-}$ estimates the average just below. The RD estimate is

$\hat{\tau}_{RD}=\hat{\beta}_{0}^{+}-\hat{\beta}_{0}^{-}$                                    (3)

Step 3: Choose the bandwidth. The bandwidth h controls how wide a window around the cutoff we use. A narrow bandwidth uses only the closest observations (more credible but noisier); a wide bandwidth uses more data (less noise but potentially more bias if the outcome relationship is curved). Modern practice uses the data-driven bandwidth selector of Calonico et al. [2014], implemented in the rdrobust package.

4 Sharp vs. Fuzzy RD

So far, we assumed treatment switches sharply from 0 to 1 at the cutoff a sharp RD. But what if the threshold only increases the probability of treatment? This is a fuzzy RD. Example: a scholarship cutoff at 80, but some students above 80 decline the scholarship (always-takers who do not need it) and some students below 80 receive it through appeals (never-takers who lobby for it). Treatment probability jumps at 80 but does not go from 0 to 1. In the fuzzy RD, we divide the jump in the outcome by the jump in treatment probability:

$\hat{\tau}_{FRD}=\frac{\hat{\beta}_{0,Y}^{+}-\hat{\beta}_{0,Y}^{-}}{\hat{\beta}_{0,D}^{+}-\hat{\beta}_{0,D}^{-}}.$                        (4)

where the numerator is the jump in the outcome and the denominator is the jump in treat- ment take-up. This is exactly an IV estimator near the cutoff the threshold indicator serves as the instrument.

5 Key Checks: Is This a Real RD?

1. No manipulation of the running variable. Can individuals sort around the cutoff? A student who knows the cutoff is 80 might dispute their score to push it above 80. If so, students just above are systematically different from students just below, and the RD fails. To check, plot the density of the running variable: it should be smooth at c, not spiked just above it. The McCrary [2008] test formalises this.
2. Predetermined covariates should not jump. Run the RD on baseline characteris- tics (family income, prior test scores) that should be unaffected by treatment. If they jump at the cutoff, the continuity assumption is violated.
3. No effect at placebo cutoffs. Estimate "jumps" at other values of X where there is no real cutoff. These should be small and insignificant.

6 Common Mistakes

Using global polynomial regression. An early practice was to regress Y on high-degree polynomials of $X(e.g.,X^{5})$ over the full sample and estimate the jump. Gelman and Imbens [2019] show this often produces spurious results due to overfitting at the boundary. Use local linear regression instead.

Ignoring the bandwidth. Reporting results for only one bandwidth is insufficient. Show robustness across a range of bandwidths.

Over-interpreting the estimate. The RD estimate applies at the cutoff for units near the cutoff. It does not tell us about the effect for units far from the threshold.

7 Famous Examples

• Class size and test scores [Angrist and Lavy, 1999]: Israeli schools must split classes when enrolment exceeds 40. Just above the threshold, class sizes drop, allowing identification of class-size effects on achievement.
• Electoral incumbency [Lee, 2008]: In close elections, barely-winning candidates are compared to barely-losing ones to estimate the incumbency advantage.
• Colonial institutions and poverty [Dell, 2010]: Geographic boundaries of a 16th- century Peruvian mining labour system serve as a sharp discontinuity. Communities that fell just inside the boundary are significantly poorer today.

8 Conclusion

Regression discontinuity designs are among the most transparent and convincing quasi- experimental methods available. They require minimal assumptions essentially, that potential outcomes vary smoothly through the threshold an assumption that is testable and often plausible. When a cutoff exists and cannot be manipulated, RD should be a researcher's first instinct for causal identification.

References

1. Angrist, J. D. and Lavy, V. (1999). Using Maimonides' rule to estimate the effect of class size on scholastic achievement. Quarterly Journal of Economics, 114(2):533-575.
2. Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica, 82(6):2295-2326.
3. Dell, M. (2010). The persistent effects of Peru's mining mita. Econometrica, 78(6):1863-1903. Gelman, A. and Imbens, G. W. (2019). Why high-order polynomials should not be used in regression discontinuity designs. Journal of Business & Economic Statistics, 37(3):447- 456.
4. Imbens, G. W. and Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. Journal of Econometrics, 142(2):615-635.
5. Lee, D. S. (2008). Randomized experiments from non-random selection in U.S. House elec- tions. Journal of Econometrics, 142(2):675-697.
6. Lee, D. S. and Lemieux, T. (2010). Regression discontinuity designs in economics. Journal of Economic Literature, 48(2):281-355.
7. McCrary, J. (2008). Manipulation of the running variable in the regression discontinuity design: A density test. Journal of Econometrics, 142(2):698-714.