How does multiple correlation (R) relate to R-squared in regression?

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AcadiFi · Accepted Answer

Multiple R is the correlation between the actual dependent variable Y and the predicted Ŷ from a multivariate regression. It measures how strongly the fitted model tracks the observed data. By construction, it's always between 0 and 1 (unlike simple correlation, which can be negative).

R-squared (R²) is simply (Multiple R)². It represents the fraction of variance in Y explained by the regression model. If R = 0.85, then R² = 0.72 — 72% of the variance in Y is captured by the model.

Example: Birchwood Asset Management runs a three-factor regression of fund returns on market, size, and value factors:

Fund Return = 0.2% + 1.05·MKT + 0.35·SMB - 0.18·HML

Results: Multiple R = 0.92, R² = 0.846. So 84.6% of the fund's monthly return variance is explained by the three factors. The remaining 15.4% is either alpha (skill) or residual noise.

Key cautions:

- R² always increases when you add predictors, even useless ones. Use Adjusted R² to penalize excess parameters.
- High R² does not imply causation, predictive power out-of-sample, or model correctness.
- R² can be inflated by overfitting small samples.

For CFA Level II, know the formula for Adjusted R² = 1 - [(1 - R²)(n - 1)/(n - k - 1)], where n is observations and k is predictors. This penalizes complexity and is the preferred metric when comparing models with different numbers of variables. Multiple R alone without context is rarely sufficient — always pair it with residual diagnostics and out-of-sample validation.

How does multiple correlation (R) relate to R-squared in regression?

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