Negated QLIKE score for variance forecasts

Computes the positively oriented (negated) QLIKE quasi-likelihood loss for variance forecasts.

Usage

qlike_score(sigma2_hat, sigma2)

Arguments

sigma2_hat

Numeric vector. Forecast variance (strictly positive).

sigma2

Numeric vector. Realised variance (strictly positive).

Value

Numeric vector of negated QLIKE scores. Higher is better. Maximum value is 0, achieved at a perfect forecast sigma2_hat = sigma2. Unbounded below.

Details

Standard QLIKE loss is $$L_{QL}(\hat\sigma^2, \sigma^2) = \frac{\sigma^2}{\hat\sigma^2} - \log\frac{\sigma^2}{\hat\sigma^2} - 1.$$ This is loss-oriented (lower = better, minimum 0 at a perfect forecast), so the function negates it: \(S_{QL} = -L_{QL}\).

Literature note: some sources define QLIKE as log(sigma2_hat) + sigma2 / sigma2_hat, which differs by constants from the form above. Here the loss is normalised to have minimum 0 and is then negated for positive orientation.

Unbounded below

QLIKE is unbounded below. It should not be used directly with the finite-sample bounded-difference confidence sequences or e-processes. Use cs_asymptotic() for QLIKE-based confidence sequences, or use eprocess_predictable() only when genuine ex ante predictable bounds are available. QLIKE is not compatible with the Winkler construction because Winkler scores are restricted to binary outcomes and probability forecasts.

Examples

sigma2_hat <- c(1.0, 1.5, 2.0)
sigma2 <- c(1.1, 1.4, 2.2)
qlike_score(sigma2_hat, sigma2)
#> [1] -0.004689820 -0.002326205 -0.004689820