QLIKE loss function to evaluate forecasting model of log(realized volatility)

Quoted from Forex.com.bd-Editorial External answer — Quantitative Finance Stack Exchange Author: carry_and_pray Source score (net votes, not local likes): 0 Original post: https://quant.stackexchange.com/a/85610 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. In QLIKE, $h$ and $\hat{\sigma}_t^2$ should be on the variance/realized-variance scale and not on the log-scale. The standard QLIKE loss is, $$ L_t^\text{QLIKE} (h_t, \hat{\sigma}_t^2) = \frac{\hat{\sigma}_t^2}{h_t} - \log \left( \frac{\hat{\sigma}_t^2}{h_t} \right) - 1, $$ which is equivalent up to an additive term that does not affect forecast ranking to, $$ L_t^\text{QLIKE} (h_t, \hat{\sigma_t}^2) = \log h_t + \frac{\hat{\sigma}_t^2}{h_t}. $$ Patton shows that QLIKE is the loss that depends only on the standardized forecast error $\hat{\sigma}_t^2 / h_t$ and Liu-Patton-Sheppard use it exactly with $\hat{\sigma}_t^2$ as quadratic variation or a proxy for it, and $h_t$ as the volatility forecast. So if your model is for $y_t = \log(RV_t)$ then the output of the model is a forecast of $\log(RV_t)$ , say $\hat{y}_t$ . That is not what goes into QLIKE directly. Instead, QLIKE needs a positive forecast of $RV_t$ so you must back transform via $h_t = \exp(\hat{y}_t)$ . In particular, you should not plug $\hat{y}_t$ itself into QLIKE because QLIKE needs $h_t > 0$ and log forecasts can be negative even when the underlying variance forecast is perfectly sensible. There is one subtlety, that is, if $\hat{y}_t$ is a forecast of the conditional mean of log variance i.e., $\hat{y}_t \approx \mathbb{E}[ \log RV_t \mid \mathcal{F}_{t - 1} ]$ then $\exp(\hat{y}_t)$ is generally not the conditional mean of $RV_t$ because of Jensen's inequality. It is closer to a conditional median unless you make further assumptions. If you want a forecast of the conditional mean $RV_t$ and you assume that $\log RV_t \mid \mathcal{F}_{t - 1} \sim N(m_t, s_t^2)$ then the appropriate back-transform is $h_t = \exp \left( m_t + \frac{1}{2} s_t^2 \right)$ . More generally, some bias correction or smearing adjustment is needed whenever you forecast logs but evaluate in levels (correction about back-transformation and not about QLIKE itself).

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