Linear Regression (OLS) in Factor Models
Estimate factor loadings (betas) by regressing asset returns on one or more risk factors.
In a simple factor model r_t = α + β f_t + ε_t, β measures how sensitive the asset is to the factor.
Ordinary least squares (OLS) estimates α and β by minimising the sum of squared residuals between model and observed returns.
In practice, sampling noise, limited history, and correlated factors all affect how stable estimated betas and R² appear.
This scatter plot shows a single-factor model rₜ = α + β fₜ + εₜ. The blue line is the OLS fit; the amber line is the true factor relation. Noise and sample size determine how close the estimated beta is to the true beta.
The amber dashed line is the true factor relation rₜ = α + β fₜ. The blue solid line is the OLS regression fit α̂ + β̂ fₜ to the simulated points.
When idiosyncratic volatility is small and n is large, β̂ tracks β closely and R² is high. With more noise or fewer observations, the cloud thickens, β̂ wanders, and R² falls. The core idea: in a factor model, regression is the mechanism that turns a noisy scatter of (fₜ, rₜ) into an estimated beta and a measure of how much of the asset’s variation the factor actually explains.