R-Squared: Definition, Calculation, and Interpretation

What is R-squared?

R-squared (R²), or the coefficient of determination, measures the proportion of variance in a dependent variable that is explained by one or more independent variables in a regression model. It is commonly reported as a value between 0 and 1 (or 0%–100%), where higher values indicate a greater share of explained variation.

Formula

R² = 1 − (SS_res / SS_tot)

• SS_res (sum of squared residuals): the unexplained variation (sum of squared differences between actual and predicted values).
• SS_tot (total sum of squares): the total variation (sum of squared differences between actual values and their mean).

How to calculate R-squared (brief)

1. Fit a regression model and obtain predicted values.
2. Compute residuals (actual − predicted) and square them; sum these to get SS_res.
3. Compute deviations of actual values from their mean, square them, and sum to get SS_tot.
4. Apply the formula above.

Interpretation

• R² is the fraction of total variation explained by the model. An R² of 0.50 means roughly half the observed variation is explained by the predictors.
• R² does not indicate causation, nor does it alone show whether a model is appropriate or unbiased.
• Context matters: what counts as a “good” R² depends on the field and the problem (e.g., social sciences vs. physics).

Practical uses

• In investing, R² is used to describe how much of a fund’s or security’s price movements are explained by a benchmark index. Expressed as a percentage, an R² of 90% means 90% of movements align with the index.
• R² is often paired with other metrics (like beta) to evaluate performance and risk characteristics.

R-squared vs. Adjusted R-squared

• R² always increases (or stays the same) when you add predictors, even if they add no real explanatory power.
• Adjusted R² penalizes unnecessary predictors and only increases when a new variable improves the model more than would be expected by chance. It is more appropriate for comparing models with different numbers of predictors.

R-squared vs. Beta

• R² measures the strength of the relationship between an asset and a benchmark (how well movements align).
• Beta measures relative volatility (how large those movements are compared with the benchmark).
• Used together, R² and beta give a fuller picture: high R² with a beta near 1 means the asset tracks the benchmark closely; high R² with beta > 1 means it generally follows the benchmark but with greater swings.

Limitations

• A high R² does not guarantee a good or unbiased model; it may reflect overfitting or omitted variable bias.
• A low R² does not necessarily mean a model is useless—some phenomena are inherently noisy.
• R² is sensitive to outliers, sample range, and model specification.
• Note: while R² is normally between 0 and 1 for models with an intercept, certain definitions or models (e.g., no-intercept regressions) can produce negative R² values.

Improving R-squared (safely)

• Select relevant features through exploratory analysis, domain knowledge, or techniques like stepwise selection.
• Engineer informative variables and consider transformations or interaction terms to capture nonlinear relationships.
• Address multicollinearity (e.g., VIF analysis, principal component analysis) to stabilize coefficient estimates.
• Use regularization (ridge, lasso) to balance fit and generalization—be cautious: optimizing R² alone can encourage overfitting.

Common questions

Can R-squared be negative?
– In typical OLS regressions with an intercept, R² lies between 0 and 1. However, with certain model formulations (no intercept) or alternative R² definitions, negative values can occur, indicating the model performs worse than using the mean as a predictor.

Why is my R-squared so low?
– Possible reasons: missing important predictors, dominant random variation, inappropriate functional form (nonlinearity), measurement error, or small sample size.

What is a “good” R-squared?
– Depends on context. In finance, R² > 0.7 often indicates strong correlation with a benchmark; in other fields, lower values may still be informative. Evaluate R² alongside domain expectations and other diagnostics.

Is a higher R-squared always better?
– Not necessarily. For forecasting or explanatory modeling, higher R² is desirable, but extremely high R² can signal overfitting. In active investment management, a low R² may indicate the manager is taking returns that are not simply benchmark-driven.

Bottom line

R-squared is a useful summary of how much variation a model explains, but it should not be used in isolation. Combine R² with adjusted R², residual analysis, validation on new data, and domain knowledge to assess model quality and reliability.