Covariance Forecast Error (C-RAFE) — compute

Computes the operator-norm precision distortion \(\mathrm{C\text{-}RAFE} = \|\Sigma^{1/2} \hat\Sigma^{-1} \Sigma^{1/2} - I\|_2\).

Usage

compute_crafe(Sigma, Sigma_hat, variant = c("none", "cc05", "cc0", "cv", "i"))

Arguments

Sigma: Realised covariance matrix.
Sigma_hat: Forecast covariance matrix.
variant: Covariance restriction applied to Sigma, from the nested sequence of Section 3.3; see restrict_cov().

Value

A length-1 numeric.

Details

The metric is zero if and only if \(\hat\Sigma = \Sigma\), and is invariant to the common scaling of both arguments.

Argument order

The realised covariance comes first, matching compute_crafe() in the replication code of Salcher, Stöckl & Hanke (2026) so that paper code runs unchanged against this package. The metric is not symmetric in its arguments, so the order matters.

References

Salcher, L., Stöckl, S., & Hanke, M. (2026). Lost in Translation? Risk-Adjusting RMSE for Economic Forecast Performance. Journal of Forecasting. doi:10.1002/for.70134

Examples

Sigma <- diag(c(1, 2, 3))
compute_crafe(Sigma, Sigma)          # exactly 0
#> [1] 0
compute_crafe(Sigma, 2 * Sigma)      # forecast covariance twice too large
#> [1] 0.5

# Variants from the nested sequence of Section 3.3:
compute_crafe(Sigma, 2 * Sigma, variant = "cc0")
#> [1] 0.5