Risk-Adjusted Forecast Error (RAFE) — compute

Computes \(\mathrm{RAFE} = \sqrt{(\hat\mu - \mu)^\top \Sigma^{-1} (\hat\mu - \mu)}\), the risk-adjusted distance between a forecast mean and the realised mean.

Usage

compute_rafe(
  mu_hat,
  mu,
  Sigma = NULL,
  Sigma_inv = NULL,
  variant = c("none", "cc05", "cc0", "cv", "i")
)

Arguments

mu_hat: Numeric vector of forecast means.
mu: Numeric vector of realised means (same length as mu_hat).
Sigma: Realised covariance matrix. The argument order matches the paper's replication code, so compute_rafe(mu_hat, mu, Sigma) does the same thing here as it does there.
Sigma_inv: Optional precision matrix, supplied instead of Sigma when it is already available (it avoids an inversion). Must be a precision, not a covariance. Ignored unless variant is "none".
variant: Covariance restriction from the nested sequence of Section 3.3; see restrict_cov(). "i" returns the paper's RMSE.

Value

A length-1 numeric.

Details

The metric is a Mahalanobis distance in the population-risk metric, with no \(1/N\) normalisation. Consequently, at \(\Sigma = I\) it reduces to the Euclidean norm of the forecast error, which is \(\sqrt{N}\) times the conventional RMSE — see the examples.

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

set.seed(1)
mu     <- rnorm(5)
mu_hat <- mu + rnorm(5, sd = 0.1)

compute_rafe(mu_hat, mu, Sigma = diag(5))
#> [1] 0.1371373

# At Sigma = I the metric is sqrt(N) times RMSE:
sqrt(5) * sqrt(mean((mu_hat - mu)^2))
#> [1] 0.1371373