Factor Model & Shrinkage
Estimate factor loadings and construct the shrinkage covariance matrix.
Factor Model and Shrinkage
A factor model is estimated using PCA to reduce the dimensionality of the covariance matrix. The shrinkage estimator combines the factor model (systematic risk) with diagonal residual variances (idiosyncratic risk) to produce a well-conditioned covariance matrix.
Key Steps:- Select number of components (95% variance threshold)
- Estimate factor loadings (betas)
- Calculate fitted covariance from factor model
- Extract residual variances
- Combine into shrinkage estimator
Beta Estimation:
$$
\beta = Q_{\text{port,comp}} \cdot Q_{\text{comp,comp}}^{-1}
$$
Fitted Covariance:
$$
Q_{\text{fit}} = \beta^\top Q_{\text{comp}} \beta
$$
Residual Covariance:
$$
Q_{\text{res}} = \text{diag}\left(\text{diag}(Q_{\text{emp}} - Q_{\text{fit}})\right)
$$
Shrinkage Covariance:
$$
Q_{\text{shrink}} = Q_{\text{fit}} + Q_{\text{res}} = \beta^\top Q_{\text{comp}} \beta + \text{diag}(\sigma_{\epsilon}^2)
$$