C-RAFE-Tuned Eigenvalue Floor for Covariance

Cleans Sigma_hat by raising small eigenvalues to a floor set relative to the mean eigenvalue: with \(\hat\Sigma = Q\,\mathrm{diag}(\hat\lambda)\,Q^\top\), $$\tilde\Sigma(\tau) = Q\,\mathrm{diag}\bigl(\max(\hat\lambda_i,\ \tau\,\bar\lambda)\bigr)\,Q^\top,$$ where \(\bar\lambda\) is the mean of the sample eigenvalues. The relative parameterisation makes tau_rel scale-free.

Usage

sigma_crafe_floor(Sigma_hat, tau_rel = NULL)

Arguments

Sigma_hat

Forecast covariance.

tau_rel

Relative floor, typically in [0, 0.5]. A value of 0 returns Sigma_hat symmetrised and unchanged.

Value

A cleaned covariance matrix (same dimension as Sigma_hat), with the applied absolute floor attached as attribute tau_abs.

Details

tau_rel is tuned on inner-validation C-RAFE by sep_tune() and joint_trafe_tune(); this function applies a given floor.

References

Stöckl, S., Salcher, L., & Hanke, M. Post-Optimal Moment Correction for Mean-Variance Portfolios. Working paper.

Examples

set.seed(1)
R <- matrix(rnorm(30 * 5), 30, 5)
Sigma_hat <- stats::cov(R)

eigen(Sigma_hat, only.values = TRUE)$values
#> [1] 1.1935890 0.9484871 0.8390561 0.7601640 0.2978117
eigen(sigma_crafe_floor(Sigma_hat, 0.3), only.values = TRUE)$values
#> [1] 1.1935890 0.9484871 0.8390561 0.7601640 0.2978117