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R-squared for an intercept return fit

For this equal-weight OLS fit, R²=1−SSE/Syy when asset-return variation is positive.

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Define the two squared sums

SSE sums squared residuals around the fitted line. Syy sums squared asset-return deviations around the asset mean. Both must use the same response-return unit.

Use a defined denominator

R2=1−∑i(yi−y^i)2∑i(yi−yˉ)2R^2=1-\frac{\sum_i(y_i-\widehat{y}_i)^2}{\sum_i(y_i-\bar{y})^2}

The denominator must be positive; this identity is scoped to the intercept fit.

Reconcile the original score

SSE is 1.2 and Syy 45.3 in percentage points squared. The resulting R² is 147/151≈97.350993%. For simple OLS with an intercept and nonconstant series, it also equals squared sample correlation.

Avoid an outcome-probability interpretation

This is an in-sample variation ratio. It is not a probability that the next observation will lie on the line, evidence of causation or a significance result. Squaring correlation also removes its sign.

Further references