Explain regression discontinuity with an example

Question

Explain regression discontinuity in simple terms and provide a concrete example.

Codemia · Accepted Answer

Regression Discontinuity (RD) is a quasi-experimental design used to identify causal effects by exploiting a cutoff point or threshold in a continuous variable. Formally, let $ Y $ be the outcome variable, $ X $ be the running variable (which determines whether an observation falls on one side of the cutoff or the other), and $ c $ be the cutoff. The potential outcomes are defined as:

- $ Y(1) $ if $ X \geq c $ (treatment group) 
- $ Y(0) $ if $ X < c $ (control group)

The key assumption is that individuals just above and just below the cutoff are similar in all aspects except for the treatment, allowing us to estimate the causal effect of the treatment on the outcome.

To analyze the causal effect using RD, we follow these steps:
1. **Data Collection**: Gather data for the running variable $ X $ and the outcome $ Y $.
2. **Define the Cutoff**: Identify the threshold $ c $ where treatment starts.
3. **Model Specification**: Use local linear regression or polynomial regression to model the relationship between $ Y $ and $ X $ on both sides of the cutoff.
4. **Estimate Treatment Effect**: Calculate the difference in the predicted values of $ Y $ just above and just below the cutoff to estimate the treatment effect.
5. **Statistical Testing**: Conduct hypothesis tests to assess the significance of the estimated treatment effect.

Consider a school that offers a scholarship based on test scores. Students with scores $ X \geq 75 $ receive the scholarship (treatment group), while those with scores $ X < 75 $ do not (control group). Assume we have the following average outcomes:
- Average test score for those receiving the scholarship: $ Y(1) = 85 $
- Average test score for those not receiving the scholarship: $ Y(0) = 80 $

Using local linear regression, we fit two models:
- For $ X < 75 $: $ Y = eta_0 + eta_1 X $
- For $ X \geq 75 $: $ Y = \alpha_0 + \alpha_1 X $

Estimating both equations, we find:
- Predicted $ Y(1) $ at the cutoff $ (X = 75) $ is approximately 84
- Predicted $ Y(0) $ at the cutoff $ (X = 75) $ is approximately 80

The estimated treatment effect is $ 84 - 80 = 4 $.

The estimated treatment effect of 4 suggests that receiving the scholarship increases the average test score by 4 points for students who are just above the cutoff compared to those just below it. This finding is critical for policymakers: it indicates that the scholarship program has a positive impact on student performance. However, we must consider potential confounders such as socioeconomic status or prior academic performance that could influence both test scores and the likelihood of receiving the scholarship.

Regression Discontinuity connects to various statistical concepts:
- **Causal Inference Basics**: RD allows us to make causal claims from observational data, which is crucial for effective policy evaluation.
- **Hypothesis Testing**: We can test the null hypothesis $ H_0: 	ext{Treatment Effect} = 0 $ against the alternative $ H_1: 	ext{Treatment Effect} > 0 $ to assess significance.
- **Power Analysis**: Understanding how many observations we need around the cutoff to detect a meaningful treatment effect.
- **Central Limit Theorem**: As sample sizes grow, the distribution of the estimated treatment effect will approach normality, which is important for constructing confidence intervals and conducting hypothesis tests.

Explain regression discontinuity with an example

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