What does R-squared really tell you, and what are its limitations?

R² (coefficient of determination) is the most commonly reported regression statistic, but it's widely misunderstood. Let's get precise about what it does and doesn't tell you.

What R² Measures:

R² = 1 - (SSE / SST)

Where:

• SST (Total Sum of Squares) = Total variation in Y
• SSE (Sum of Squared Errors) = Unexplained variation
• SSR (Sum of Squares Regression) = Explained variation
• SST = SSR + SSE

R² = SSR / SST = Proportion of Y's variation explained by X

Example:

R² = 0.72 means 72% of the variation in the dependent variable is explained by the independent variable(s). The remaining 28% is unexplained.

When high R² is meaningful:

• Cross-sectional models with genuine economic relationships (e.g., company size explaining analyst coverage)
• The slope coefficient is statistically significant
• The model passes residual diagnostics

When high R² is misleading:

1. Spurious correlation:

Regressing US GDP on world population gives R² near 0.99 — both trend upward over time, but there's no causal link. Time-trending variables will always produce high R².

2. Overfitting:

Adding more variables to a regression always increases R² (even random noise variables). That's why we also check Adjusted R², which penalizes for additional variables:

Adj R² = 1 - [(1 - R²)(n - 1) / (n - k - 1)]

3. Non-linear relationships:

If Y and X have a U-shaped relationship, a linear regression may have low R² even though X strongly predicts Y.

For simple regression (one X variable):

R² = r² (the square of the correlation coefficient)

If r = 0.85, then R² = 0.7225 = 72.25%

If r = -0.90, then R² = 0.81 = 81% (R² is always positive)

Practical interpretation guide:

Exam tip: Don't evaluate a model on R² alone. The CFA exam may present a model with high R² but insignificant coefficients, residual patterns, or obvious spurious correlation — you need to recognize these red flags.

Practice regression interpretation in our CFA Level I question bank.