Linear factor model
y = X beta + epsilonReturns are decomposed into factor-linked variation and a residual.
Foundations
Linear Regression (OLS) in Factor Models
Estimate factor loadings (betas) by regressing asset returns on one or more risk factors.
Mental model
Regression estimates how much observed return variation is associated with chosen 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.
Explore the mechanism
Which part of an asset's return variation is associated with each factor, and how uncertain is that estimate?
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.
β̂ ≈ 0.794 · R² ≈ 0.398n = 80 · corr(f, r) ≈ 0.631
Numbers
True β = 1.000 · estimated β̂ ≈ 0.794
α̂ ≈ 0.0009 (true α = 0)
corr(f, r) ≈ 0.631 · R² ≈ 0.398
Relative beta error ≈ -20.6%
Interpretation
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.
UNDERSTAND
What to notice
In practice, sampling noise, limited history, and correlated factors all affect how stable estimated betas and R² appear.
The quantitative structure
Notation
y- vector of asset returns
X- matrix of factor returns and optional intercept
beta- vector of factor loadings
epsilon- unexplained residual
OLS estimator
beta_hat = (X'X)^(-1) X'yOLS chooses coefficients that minimize the sum of squared residuals when the inverse exists.
This model assumes
- The relationship is linear in the included regressors.
- Regressors have sufficient independent variation and the error has conditional mean zero for unbiased estimation.
Where it appears
- estimating market and style-factor exposures
- attributing portfolio returns
- building hedge ratios
Where the intuition breaks
What this view leaves out
- OLS coefficients can be unstable or misleading when relationships are nonlinear, residuals are dependent or heteroskedastic, regressors are endogenous, or regimes change.
Common mistakes
“A significant regression coefficient proves that the factor causes the return.”
Not quite.OLS estimates conditional association; causal interpretation requires additional identification assumptions.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Linear Regression (OLS) in Factor Models
Linear Regression (OLS) in Factor ModelsYOU ARE HERE
Built onReturns and Log-ReturnsFactor regressions typically explain asset returns using market or factor-return series.
Used byTemperature–Load RelationshipA regression is a simple baseline for quantifying how load changes with temperature and other explanatory variables.
UsesCovariance Estimation and CorrelationFactor-model regression relates returns through estimated betas and residual covariance structure.
Built on · Returns and Log-Returns
Factor regressions typically explain asset returns using market or factor-return series.
Explore concept →