Regression and econometrics

The regression slope

The single most reused formula in quantitative finance: a beta is a correlation scaled by a volatility ratio.

β^=Cov(x,y)Var(x)=ρ σyσx,β^=(X⊤X)−1X⊤y\hat{\beta} = \frac{\mathrm{Cov}(x,y)}{\mathrm{Var}(x)} = \rho\,\frac{\sigma_y}{\sigma_x}, \qquad \hat{\beta} = (X^{\top}X)^{-1}X^{\top}y

Where

R2=ρ2R^2 = \rho^2
In simple regression only.
βy∣xβx∣y=ρ2\beta_{y|x}\beta_{x|y} = \rho^2
The two slopes are reciprocals only when ∣ρ∣=1|\rho| = 1.

Assumptions

  • E[ε∣X]=0\mathbb{E}[\varepsilon \mid X] = 0 for unbiasedness — this is what omitted variables break.

Sanity check. OLS is an orthogonal projection, so the residual is orthogonal to every regressor by construction.

Where this is taught