R-squared & Residuals

R-squared & Residuals

After fitting an OLS line, we need to assess how well it fits the data.

Residuals are the differences between actual and predicted values: $e_i = y_i - \hat{y}_i = y_i - (\beta x_i + \alpha)$

R-squared (coefficient of determination) measures the proportion of variance explained: $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$

where:

• $SS_{res} = \sum (y_i - \hat{y}_i)^2$ — residual sum of squares
• $SS_{tot} = \sum (y_i - \bar{y})^2$ — total sum of squares

$R^2 = 1$ means perfect fit; $R^2 = 0$ means the regression explains nothing (as good as just predicting the mean). In factor models, $R^2$ represents how much of an asset's variance is explained by the factors.

Your Task

Implement:

• r_squared(xs, ys) — coefficient of determination using OLS fit
• residuals(xs, ys) — list of OLS residuals