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 estimate causal effects by assigning a cutoff point to a continuous variable. In the context of Brex, consider a financial technology company that offers a credit score threshold to determine eligibility for a premium business credit card. Let’s denote the credit score as $ X $ and the cutoff score as $ c $. Individuals with $ X \geq c $ qualify for the premium card, while those with $ X < c $ do not. The goal is to analyze the impact of qualifying for the premium card on business outcomes, such as monthly spending or revenue growth.

1. **Identify the cutoff**: Set $ c $ as the threshold credit score, e.g., 700.
2. **Collect Data**: Gather data on individuals’ credit scores and their corresponding business outcomes (e.g., monthly revenue).
3. **Model the relationship**: Use a regression model to estimate the relationship between credit score and business outcomes, focusing on individuals near the cutoff.
4. **Estimate the causal effect**: Compare the average outcomes for those just above and just below the cutoff using a local linear regression or polynomial regression.
5. **Check for continuity**: Ensure that other covariates are continuous around the cutoff, to validate the RD design.

Suppose we have the following data for 10 businesses:

| Credit Score | Revenue Before | Revenue After |
|--------------|----------------|---------------|
| 695          | $4,000         | $4,500        |
| 698          | $4,200         | $4,600        |
| 699          | $4,300         | $5,000        |
| 700          | $4,800         | $5,500        |
| 702          | $5,000         | $6,000        |
| 703          | $5,200         | $6,200        |
| 704          | $5,300         | $6,500        |
| 705          | $5,700         | $6,800        |
| 706          | $5,900         | $7,000        |
| 707          | $6,000         | $7,200        |

- Calculate the average revenue after for those just below the cutoff (695-699) and above (700-707).
- Below cutoff average revenue after: $ (4500 + 4600 + 5000) / 3 = 4700 $
- Above cutoff average revenue after: $ (5500 + 6000 + 6200 + 6500 + 6800 + 7000 + 7200) / 7 = 6350 $
- The estimated causal effect of qualifying for the premium card is $ 6350 - 4700 = 1650 $.

The result indicates that, on average, businesses whose credit scores are just above the threshold of 700 experience an increase of $1,650 in their monthly revenue after obtaining the premium credit card compared to those just below the threshold. This suggests that the premium card may have a significant positive impact on business performance for those who qualify.

Regression Discontinuity connects to other statistical concepts such as:
1. **Multiple Testing Correction**: When evaluating several thresholds or treatments, it’s vital to correct for multiple comparisons to avoid false positives. The Bonferroni correction can be applied if multiple cutoffs are tested.
2. **Central Limit Theorem**: As the sample size increases, the distribution of the sample mean approaches a normal distribution, which supports the validity of the RD estimates under certain conditions.
3. **Assumptions**: The key assumptions in RD include the continuity of potential outcomes at the cutoff and no manipulation of the assignment variable (credit score) around the cutoff. Violating these can lead to biased estimates.

Explain regression discontinuity with an example

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