The scatter plots below depict the relation between excess returns of two stocks and the market excess return. In each graph, the market excess return is on the x-axis and the stock excess return is on the y-axis. The dashed line is estimated using a single-variable linear regression and represents the best linear fit:

Which stock has higher R²?

Question

The scatter plots below depict the relation between excess returns of two stocks and the market excess return. In each graph,  the market excess return is on the x-axis and the stock excess return is on the y-axis. The dashed line is estimated using a single-variable linear regression and represents the best linear fit:

Which stock has higher R²?

BlackTom AI · Accepted Answer

R² measures the fraction of variance in the dependent variable (stock excess return) that is explained by the independent variable (market excess return) in the linear regression r_i,t − r_f,t = α_i + β_i (r_M,t − r_f,t) + ε_i,t.

1) Relevant knowledge points and concepts
- R² = 1 − (SSR / SST), where SSR is the sum of squared residuals (unexplained variance) and SST is the total sum of squared deviations of the dependent variable from its mean. 
- A higher R² means the fitted line explains a larger share of the variability of y (stock excess returns); points tightly clustered around the regression line imply small SSR and high R². Points widely scattered around the line imply large SSR and low R².

2) Step-by-step reasoning
- Inspect the two scatterplots with their dashed fitted lines. For Stock A, the points lie very close to the dashed line across the range of market excess returns. For Stock B, points are more widely dispersed around its dashed line.
- Because the residuals (vertical distances from points to the line) are much smaller for Stock A than for Stock B, SSR_A << SSR_B (for comparable SST).
- Therefore R²_A = 1 − SSR_A/SST_A is much larger than R²_B = 1 − SSR_B/SST_B.

3) Illustrative calculation (intermediate steps)
- Conceptual example: if SST ≈ 1 for both series, and SSR_A ≈ 0.05 then R²_A ≈ 0.95; if SSR_B ≈ 0.5 then R²_B ≈ 0.5. This demonstrates how smaller residual variance yields higher R².

4) Why the correct answer is correct
- Stock A’s observations are tightly aligned with its fitted line, indicating the market excess return explains most of its variation → high R². Stock B shows substantial scatter, so the market explains less of its variation → lower R².

5) Why other options are incorrect
- “Stock B” is incorrect because its larger residual spread implies a smaller R². “Both have the same R²” is incorrect because the visual spreads around the lines differ noticeably.

6) Conclusion
- The correct choice is Stock A, which matches the provided answer.

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