Linear Regression (OLS)

Ordinary Least Squares (OLS)

OLS regression fits a line $\hat{y} = \beta x + \alpha$ that minimizes the sum of squared residuals.

The closed-form solution for the slope is: $\beta = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} = \frac{SS_{xy}}{SS_{xx}}$

And the intercept: $\alpha = \bar{y} - \beta \bar{x}$

where $SS_{xy} = \sum x_i y_i - n \bar{x} \bar{y}$ and $SS_{xx} = \sum x_i^2 - n \bar{x}^2$.

OLS is used everywhere in quantitative finance: factor models, risk attribution, and return prediction all rely on OLS regression.

Your Task

Implement ols(xs, ys) which returns a tuple (slope, intercept).