$\displaystyle=\begin{cases}\sigma_{zt}^{2}\sqrt{\frac{\mu_{\lambda}}{\lambda_{t}}}\frac{\exp(-h_{t}/2)\delta}{k(\delta)^{2}(1-\rho^{2})\sqrt{1-c^{2}\delta^{2}}}\left[y_{t}+\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)c\delta}{\sqrt{1-c^{2}\delta^{2}}}\right.\\\
\hskip
133.72786pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu]),&t=1,\ldots,T-1,\\\
\sigma_{zT}^{2}\sqrt{\frac{\mu_{\lambda}}{\lambda_{T}}}\frac{\exp(-h_{T}/2)\delta}{k(\delta)^{2}\sqrt{1-c^{2}\delta^{2}}}\left[y_{T}+\sqrt{\frac{\lambda_{T}}{\mu_{\lambda}}}\frac{\exp(h_{T}/2)c\delta}{\sqrt{1-c^{2}\delta^{2}}}\right],&t=T.\end{cases}$

####  A.1.8 Generation of $\bm{h}$

First, we rewrite the RSV-AZ-ST model (12), (13), (14), and (30) as

$\displaystyle x_{t}$ $\displaystyle=\mu_{x}+\alpha_{t}+u_{t},\quad
t=1,\ldots,n,$ (69) $\displaystyle y_{t}$
$\displaystyle=(\delta\bar{z}_{0t}+\sqrt{1-\delta^{2}}z_{t})\sqrt{\lambda_{t}}\exp(\alpha_{t}/2)\gamma,\quad
t=1,\ldots,n,$ (70) $\displaystyle\alpha_{t+1}$
$\displaystyle=\phi\alpha_{t}+\eta_{t},\quad t=0,\ldots,n-1,$ (71)

where

$\displaystyle\mu_{x}=\xi+\mu,\quad\alpha_{t}$
$\displaystyle=h_{t}-\mu,\quad\bar{z}_{0t}=z_{0t}-c,\quad\gamma=\frac{\exp(\mu/2)}{\sqrt{(1-c^{2}\delta^{2})\mu_{\lambda}}}.$
(72)

The log-likelihood of $(y_{t},x_{t})$ given $\alpha_{t}$, $\alpha_{t+1}$, and
other parameters excluding the constant term is

$\displaystyle l_{t}$
$\displaystyle=-\frac{\alpha_{t}}{2}-\frac{(y_{t}-\mu_{t})^{2}}{2\sigma_{t}^{2}}-\frac{(x_{t}-\mu_{x}-\alpha_{t})^{2}}{2\sigma_{u}^{2}},$
(73)

where

$\displaystyle\mu_{t}$
$\displaystyle=\begin{cases}\left[\delta\bar{z}_{0t}+\sqrt{1-\delta^{2}}\dfrac{\rho}{\sigma_{\eta}}(\alpha_{t+1}-\phi\alpha_{t})\right]\sqrt{\lambda_{t}}\exp(\alpha_{t}/2)\gamma,&t=1,\ldots,n-1,\\\\[5.0pt]
\delta\bar{z}_{0t}\sqrt{\lambda_{t}}\exp(\alpha_{t}/2)\gamma,&t=n,\end{cases}$
(74) $\displaystyle\sigma_{t}^{2}$
$\displaystyle=\begin{cases}(1-\rho^{2})(1-\delta^{2})\lambda_{t}\exp(\alpha_{t})\gamma^{2},&t=1,\ldots,n-1,\\\\[5.0pt]
(1-\delta^{2})\lambda_{t}\exp(\alpha_{t})\gamma^{2},&t=n.\end{cases}$ (75)

Using this log-likelihood, we can efficiently sample the latent variable
$(\alpha_{1},\ldots,\alpha_{n})$ via the multi-move sampler based on Omori et
al. (2007), Omori and Watanabe (2008, 2015). For further details, see
Takahashi et al. (2016, 2023).

####  A.1.9 Generation of $\bm{\lambda}$

The conditional probability density function of $\lambda_{t}$ is

$\displaystyle\pi(\lambda_{t}|\cdot)\propto$
$\displaystyle\lambda_{t}^{-(\frac{\nu+1}{2}+1)}\exp\left[-\frac{\nu}{2\lambda_{t}}-\frac{\mu_{\lambda}\exp(-h_{t})}{2\lambda_{t}k(\delta)^{2}(1-\rho^{2})}\left\\{y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right.\right.$
$\displaystyle\hskip
145.10922pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])I(t<T)\bigg{\\}}^{2}\bigg{]}$
$\displaystyle\propto$
$\displaystyle\lambda_{t}^{-(\frac{a}{2}+1)}\exp\left(-\frac{b_{t}}{2\lambda_{t}}\right)\times
g(\lambda_{t})$

where

$\displaystyle a$ $\displaystyle=\nu+1,$ $\displaystyle b_{t}$
$\displaystyle=\begin{cases}\nu+\frac{\mu_{\lambda}\exp(-h_{t})y_{t}^{2}}{k(\delta)^{2}(1-\rho^{2})},&t=1,\ldots,n-1,\\\
\nu+\frac{\mu_{\lambda}\exp(-h_{t})y_{t}^{2}}{k(\delta)^{2}},&t=n,\end{cases}$
$\displaystyle g(\lambda_{t})$
$\displaystyle=\begin{cases}\exp\left[\sqrt{\frac{\mu_{\lambda}}{\lambda_{t}}}\frac{\exp(-h_{t}/2)y_{t}}{k(\delta)^{2}(1-\rho^{2})}\left\\{\frac{\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}+\rho
k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])\right\\}\right],&t=1,\ldots,n-1,\\\
\exp\left[\sqrt{\frac{\mu_{\lambda}}{\lambda_{t}}}\frac{\exp(-h_{t}/2)y_{t}}{k(\delta)^{2}}\frac{\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right\\},&t=n,\end{cases}$

We propose a candidate from the inverse Gamma distribution,
$\lambda_{t}^{\dagger}\sim\mathcal{IG}(a/2,b_{t}/2)$, and accept it with
probability $\min\\{1,g(\lambda_{t}^{\dagger})/g(\lambda_{t})\\}$.

###  A.2 RSV-FS-ST Model

The joint probability density of $(y_{t},h_{t+1})$ is given by

$\displaystyle f(y_{t},h_{t+1}|\bm{\theta},\lambda_{t})=$ $\displaystyle
q_{T}\left(y_{t}\exp(-h_{t}/2)|\gamma,\nu\right)\exp\left(-\frac{1}{2}h_{t}\right)$
$\displaystyle\times\frac{1}{\sqrt{2\pi(1-\rho^{2})\sigma^{2}}}\exp\left[-\frac{1}{2(1-\rho^{2})\sigma^{2}}\left\\{h_{t+1}-\mu-\phi[h_{t}-\mu]-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}^{2}\right],$

where $q_{T}$ is defined in (46). Then, the joint posterior probability
density is given by

$\displaystyle\pi(\bm{h},\bm{\theta}|\bm{x},\bm{y})\propto$
$\displaystyle(1-\rho^{2})^{-\frac{T-1}{2}}\sigma^{-T}(1-\phi^{2})^{-\frac{1}{2}}\times\prod_{t=1}^{T}q_{T}\left(y_{t}\exp(-h_{t}/2)|\gamma,\nu\right)\times\exp\left(-\frac{1}{2}\sum_{t=1}^{T}h_{t}\right)$
$\displaystyle\times\exp\left[-\frac{1}{2(1-\rho^{2})\sigma^{2}}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\mu-\phi[h_{t}-\mu]-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}^{2}\right]$
$\displaystyle\times\exp\left[-\frac{(1-\phi^{2})}{2\sigma^{2}}(h_{1}-\mu)^{2}\right]\times\sigma_{u}^{-T}\times\exp\left\\{-\frac{1}{2\sigma_{u}^{2}}\sum_{t=1}^{T}(x_{t}-\xi-
h_{t})^{2}\right\\}\times\pi(\bm{\theta}).$

####  A.2.1 Generation of $\mu$

The conditional posterior distribution of $\mu$ is normal, $\mu|\cdot\sim
N(\mu_{1},\sigma_{\mu 1}^{2})$ where

$\displaystyle\sigma_{\mu 1}^{-2}$ $\displaystyle=\sigma_{\mu
0}^{-2}+\sigma^{-2}\left[1-\phi^{2}+(T-1)(1-\phi)^{2}(1-\rho^{2})^{-1}\right],$
$\displaystyle\mu_{1}$ $\displaystyle=\sigma_{\mu 1}^{2}\left[\sigma_{\mu
0}^{-2}\mu_{0}+\sigma^{-2}\left\\{(1-\phi^{2})h_{1}+\frac{1-\phi}{1-\rho^{2}}\sum_{t=1}^{T-1}\left(h_{t+1}-\phi
h_{t}-\rho\sigma\exp(-h_{t}/2)y_{t}\right)\right\\}\right].$

####  A.2.2 Generation of $\phi$

The conditional posterior density of $\phi$ is

$\displaystyle\pi(\phi|\cdot)\propto$ $\displaystyle(1-\phi)^{a_{\phi
0}-1}(1+\phi)^{b_{\phi 0}-1}(1-\phi^{2})^{\frac{1}{2}}$
$\displaystyle\times\exp\left[-\frac{1}{2(1-\rho^{2})\sigma^{2}}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\mu-\phi(h_{t}-\mu)-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}^{2}\right]$
$\displaystyle\times\exp\left[-\frac{(1-\phi^{2})}{2\sigma^{2}}(h_{1}-\mu)^{2}\right]$
$\displaystyle\propto$ $\displaystyle
k(\phi)\times\exp\left\\{-\frac{1}{2\sigma_{\phi}^{2}}(\phi-\mu_{\phi})^{2}\right\\},$

where

$\displaystyle k(\phi)$ $\displaystyle=(1-\phi)^{a_{\phi
0}-\frac{1}{2}}(1+\phi)^{b_{\phi 0}-\frac{1}{2}},$
$\displaystyle\sigma_{\phi}^{2}$
$\displaystyle=\frac{\sigma^{2}(1-\rho^{2})}{\rho^{2}(h_{1}-\mu)^{2}+\sum_{t=2}^{T-1}(h_{t}-\mu)^{2}},$
$\displaystyle\mu_{\phi}$
$\displaystyle=\frac{\sigma_{\phi}^{2}}{\sigma^{2}(1-\rho^{2})}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\mu-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}(h_{t}-\mu).$

Thus, we apply the MH algorithm. We propose a candidate $\phi^{\dagger}\sim
N(\mu_{\phi},\sigma_{\phi}^{2})$ and accept it with probability
$\min\\{1,k(\phi^{\dagger})/k(\phi)\\}$.

####  A.2.3 Generation of $\rho$

The conditional posterior density of $\rho$ is

$\displaystyle\pi(\rho|\cdot)\propto$ $\displaystyle(1+\rho)^{a_{\rho
0}-1}(1-\rho)^{b_{\rho 0}-1}(1-\rho^{2})^{-\frac{T-1}{2}}$
$\displaystyle\times\exp\left[-\frac{1}{2(1-\rho^{2})\sigma^{2}}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\mu-\phi(h_{t}-\mu)-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}^{2}\right].$

We approximate the conditional posterior distribution by
$N(\hat{\rho},\sigma_{\hat{\rho}}^{2})$ where $\hat{\rho}$ is the mode of the
conditional density and
$\sigma_{\hat{\rho}}^{-2}=-\frac{\partial^{2}\log\pi(\rho|\cdot)}{\partial\rho^{2}}\big{|}_{\rho=\hat{\rho}}$.
Propose a candidate
$\rho^{\dagger}\sim\mathcal{TN}_{(-1,1)}(\hat{\rho},\sigma_{\hat{\rho}}^{2})$
and accept it with probability
$\min\left[1,\frac{\pi(\rho^{\dagger}|\cdot)f_{N}(\rho|\hat{\rho},\sigma_{\hat{\rho}}^{2})}{\pi(\rho|\cdot)f_{N}(\rho^{\dagger}|\hat{\rho},\sigma_{\hat{\rho}}^{2})}\right]$.

####  A.2.4 Generation of $\sigma^{2}$

The conditional posterior density of $\sigma^{2}$ is

$\displaystyle\pi(\sigma^{2}|\cdot)\propto$
$\displaystyle(\sigma^{2})^{-\frac{n_{0}+T}{2}-1}\exp\left(-\frac{S_{0}}{2\sigma^{2}}\right)\times\exp\left[-\frac{(1-\phi^{2})}{2\sigma^{2}}(h_{1}-\mu)^{2}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2(1-\rho^{2})\sigma^{2}}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\mu-\phi(h_{t}-\mu)-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}^{2}\right]$
$\displaystyle\propto$ $\displaystyle
k(\sigma^{2})\times(\sigma^{2})^{-\frac{n_{1}}{2}-1}\exp\left(-\frac{S_{1}}{2\sigma^{2}}\right),$

where

$\displaystyle n_{1}$ $\displaystyle=n_{0}+T,$ $\displaystyle S_{1}$
$\displaystyle=S_{0}+(1-\phi^{2})(h_{1}-\mu)^{2}+\frac{1}{1-\rho^{2}}\sum_{t=1}^{T-1}\\{h_{t+1}-\mu-\phi(h_{t}-\mu)\\}^{2},$
$\displaystyle k(\sigma^{2})$
$\displaystyle=\exp\left[\sigma^{-1}\sum_{t=1}^{T-1}\frac{\rho
y_{t}\exp(-h_{t}/2)}{1-\rho^{2}}\left\\{h_{t+1}-\mu-\phi(h_{t}-\mu)\right\\}\right].$

Thus, we propose a candidate
$\sigma^{2\dagger}\sim\mathcal{IG}(n_{1}/2,S_{1}/2)$ and accept it with
probability $\min\\{1,k(\sigma^{2\dagger})/k(\sigma^{2})\\}$.

####  A.2.5 Generation of $\xi$

The conditional posterior distribution of $\xi$ is normal,
$N(\mu_{\xi},\sigma_{\xi}^{2})$, where

$\displaystyle\sigma_{\xi}^{-2}$ $\displaystyle=\sigma_{\xi
0}^{-2}+T\sigma_{u}^{-2},$ $\displaystyle\mu_{\xi}$
$\displaystyle=\sigma_{\xi}^{2}\left[\sigma_{\xi 0}^{-2}\mu_{\xi
0}+\sigma_{u}^{-2}\sum_{t=1}^{T}(x_{t}-h_{t})\right].$

####  A.2.6 Generation of $\sigma_{u}^{2}$

The conditional posterior distribution of $\sigma_{u}^{2}$ is
$\mathcal{IG}(n_{u1}/2,S_{u1}/2)$ where

$\displaystyle n_{u1}$ $\displaystyle=n_{u0}+T,$ $\displaystyle S_{u1}$
$\displaystyle=S_{u0}+\sum_{t=1}^{T}(x_{t}-\xi-h_{t})^{2}.$

####  A.2.7 Generation of $\gamma$

The conditional posterior density of $\gamma$ is

$\displaystyle\pi(\gamma|\cdot)\propto$ $\displaystyle\exp\left\\{(n_{\gamma
0}-1)\log\gamma-S_{\gamma 0}\gamma+\sum_{t=1}^{T}\log
q_{T}\left(y_{t}\exp(-h_{t}/2)|\gamma,\nu\right)\right\\}.$

We consider a transformation $\tilde{\gamma}=\log\gamma$, and conduct a random
walk MH algorithm for $\tilde{\gamma}$ (so that we add a Jacobian term
$\tilde{\gamma}$ to the log conditional posterior density).

####  A.2.8 Generation of $\nu$

The conditional probability density function of $\nu$ is

$\displaystyle\pi(\nu|\cdot)\propto\exp\left\\{(n_{\nu 0}-1)\log\nu-S_{\nu
0}\nu+\sum_{t=1}^{T}\log
q_{T}\left(y_{t}\exp(-h_{t}/2)|\gamma,\nu\right)\right\\}.$

We consider a transformation $\tilde{\nu}=\log(\nu-2)$, and conduct a random
walk MH algorithm for $\tilde{\nu}$ (so that we add a Jacobian term
$\tilde{\nu}$ to the log conditional posterior density).

####  A.2.9 Generation of $\bm{h}$

First we define

$\displaystyle g(h_{t}|\gamma,\nu)$
$\displaystyle=-\frac{\nu+1}{2}\log\left[1+\frac{\left(\sigma_{*}y_{t}\exp(-h_{t}/2)+\mu_{*}\right)^{2}}{\nu}\left\\{\gamma^{-2}I\left(y_{t}\exp(-h_{t}/2)\geq-\frac{\mu_{*}}{\sigma_{*}}\right)\right.\right.$
$\displaystyle\hskip
170.71652pt\left.\left.+\gamma^{2}I\left(y_{t}\exp(-h_{t}/2)<-\frac{\mu_{*}}{\sigma_{*}}\right)\right\\}\right].$

(1) For $t=1$, the conditional posterior density of $h_{1}$ is

$\displaystyle\pi(h_{1}|\cdot)$
$\displaystyle\propto\exp\left[g(h_{1}|\gamma,\nu)-\frac{1}{2(1-\rho^{2})\sigma^{2}}\left\\{h_{2}-\mu-\phi(h_{1}-\mu)-\rho\sigma\exp(-h_{1}/2)y_{1}\right\\}^{2}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}h_{1}-\frac{(1-\phi^{2})}{2\sigma^{2}}(h_{1}-\mu)^{2}-\frac{1}{2\sigma_{u}^{2}}(x_{1}-\xi-
h_{1})^{2}\right].$

Define

$\displaystyle\sigma_{h1}^{-2}$
$\displaystyle=\sigma^{-2}\left\\{1-\phi^{2}+\frac{\phi^{2}}{1-\rho^{2}}\right\\}+\sigma_{u}^{-2},$
$\displaystyle\mu_{h1}$
$\displaystyle=\sigma_{h1}^{2}\left[-\frac{1}{2}+\frac{\phi\\{h_{2}-(1-\phi)\mu\\}}{\sigma^{2}(1-\rho^{2})}+\frac{(1-\phi^{2})\mu}{\sigma^{2}}+\frac{x_{1}-\xi}{\sigma_{u}^{2}}\right],$
$\displaystyle k(h_{1})$
$\displaystyle=\exp\left[g(h_{1}|\gamma,\nu)+\frac{\rho
y_{1}\exp(-h_{1}/2)}{(1-\rho^{2})\sigma}\left\\{h_{2}-\mu-\phi(h_{1}-\mu)\right\\}-\frac{\rho^{2}y_{1}^{2}\exp(-h_{1})}{2(1-\rho^{2})}\right].$

Thus we propose a candidate $h_{1}^{\dagger}\sim N(\mu_{h1},\sigma_{h1}^{2})$
and accept it with probability $\min\\{1,k(h_{1}^{\dagger})/k(h_{1})\\}$.

(2) For $t=2,\ldots,T-1$, the conditional posterior density of $h_{t}$ is

$\displaystyle\pi(h_{t}|\cdot)$
$\displaystyle\propto\exp\left[g(h_{t}|\gamma,\nu)-\frac{1}{2(1-\rho^{2})\sigma^{2}}\left\\{h_{t+1}-\mu-\phi(h_{t}-\mu)-\rho\sigma\exp(-h_{t}/2)y_{t}\right\\}^{2}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2(1-\rho^{2})\sigma^{2}}\left\\{h_{t}-\mu-\phi(h_{t-1}-\mu)-\rho\sigma\exp(-h_{t-1}/2)y_{t-1}\right\\}^{2}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}h_{t}-\frac{1}{2\sigma_{u}^{2}}(x_{t}-\xi-
h_{t})^{2}\right].$

Define

$\displaystyle\sigma_{ht}^{-2}$
$\displaystyle=\sigma^{-2}\left(\frac{1+\phi^{2}}{1-\rho^{2}}\right)+\sigma_{u}^{-2},$
$\displaystyle\mu_{ht}$
$\displaystyle=\sigma_{ht}^{2}\left[-\frac{1}{2}+\frac{\phi\\{h_{t+1}-(1-\phi)\mu\\}}{\sigma^{2}(1-\rho^{2})}+\frac{\\{(1-\phi)\mu+\phi
h_{t-1}\\}}{\sigma^{2}(1-\rho^{2})}+\frac{\rho
y_{t-1}\exp(-h_{t-1}/2)}{(1-\rho^{2})\sigma}+\frac{x_{t}-\xi}{\sigma_{u}^{2}}\right],$
$\displaystyle k(h_{t})$
$\displaystyle=\exp\left[g(h_{t}|\gamma,\nu)-\frac{\rho^{2}y_{t}^{2}\exp(-h_{t})}{2(1-\rho^{2})}+\frac{\rho
y_{t}\exp(-h_{t}/2)}{(1-\rho^{2})\sigma}\left\\{h_{t+1}-\mu-\phi(h_{t}-\mu)\right\\}\right].$

Thus we propose a candidate $h_{t}^{\dagger}\sim N(\mu_{ht},\sigma_{ht}^{2})$
and accept it with probability $\min\\{1,k(h_{t}^{\dagger})/g(h_{t})\\}$.

(3) For $t=T$, the conditional posterior density of $h_{T}$ is

$\displaystyle\pi(h_{T}|\cdot)$
$\displaystyle\propto\exp\left[g(h_{T}|\gamma,\nu)-\frac{1}{2(1-\rho^{2})\sigma^{2}}\left\\{h_{T}-\mu-\phi(h_{T-1}-\mu)-\rho\sigma\exp(-h_{T-1}/2)y_{T-1}\right\\}^{2}\right]$
$\displaystyle\times\exp\left[-\frac{1}{2}h_{T}-\frac{1}{2\sigma_{u}^{2}}(x_{T}-\xi-
h_{T})^{2}\right].$

Define

$\displaystyle\sigma_{hT}^{-2}$
$\displaystyle=\sigma^{-2}(1-\rho^{2})^{-1}+\sigma_{u}^{-2},$
$\displaystyle\mu_{hT}$
$\displaystyle=\sigma_{hT}^{2}\left[-\frac{1}{2}+\frac{(1-\phi)\mu+\phi
h_{T-1}}{\sigma^{2}(1-\rho^{2})}+\frac{\rho
y_{T-1}\exp(-h_{T-1}/2)}{(1-\rho^{2})\sigma}+\frac{x_{T}-\xi}{\sigma_{u}^{2}}\right],$
$\displaystyle k(h_{T})$
$\displaystyle=\exp\left\\{g(h_{T}|\gamma,\nu)\right\\}.$

Thus we propose a candidate $h_{T}^{\dagger}\sim N(\mu_{hT},\sigma_{hT}^{2})$
and accept it with probability $\min\\{1,k(h_{T}^{\dagger})/k(h_{T})\\}$.

##  Appendix B Prediction Results for Nikkei 225 Index

###  B.1 Volatility Forecasts

Tables 9 and 10 present the volatility forecasts for N225. While the SV-AZ-SN
model boasts the lowest MSE, the RSV-GH-ST model exhibits the lowest QLIKE.
Notably, the MSE for the SV-GH-ST model is significantly elevated compared to
other models, indicating an over-prediction of volatility towards the end of
2018. Both RSV and REGARCH models outperform the SV and EGARCH models,
emphasizing that the incorporation of RV refines volatility forecasts. These
results mirror those obtained for DJIA. Furthermore, notable disparities
emerge among models within the same category but with distinct return
distributions. For instance, the RSV-FS-SN model surpasses the RSV-T, AZ-SN,
and AZ-ST models in terms of performance, whereas the REGARCH-T model stands
out against the REGARCH-N model. Such distinctions are visualized in Fig. 11.

Table 9: Average losses of volatility forecasts for N225 | MSE | $p_{MCS}$ | QLIKE | $p_{MCS}$
---|---|---|---|---
SV-N | $0.4983$ | $1.00^{**}$ | $0.1970$ | $0.00^{\hskip 5.60002pt}$
SV-T | $0.5026$ | $1.00^{**}$ | $0.2079$ | $0.00^{\hskip 5.60002pt}$
SV-AZ-SN | $0.4844$ | $1.00^{**}$ | $0.1951$ | $0.00^{\hskip 5.60002pt}$
SV-AZ-ST | $0.5057$ | $1.00^{**}$ | $0.2065$ | $0.00^{\hskip 5.60002pt}$
SV-FS-SN | $0.5402$ | $0.88^{**}$ | $0.2119$ | $0.00^{\hskip 5.60002pt}$
SV-FS-ST | $0.5428$ | $0.75^{**}$ | $0.2175$ | $0.00^{\hskip 5.60002pt}$
SV-GH-ST | $0.6408$ | $0.58^{**}$ | $0.2169$ | $0.00^{\hskip 5.60002pt}$
RSV-N | $0.4962$ | $1.00^{**}$ | $0.1434$ | $0.20^{*\hskip 2.79996pt}$
RSV-T | $0.5011$ | $1.00^{**}$ | $0.1456$ | $0.00^{\hskip 5.60002pt}$
RSV-AZ-SN | $0.4968$ | $1.00^{**}$ | $0.1442$ | $0.00^{\hskip 5.60002pt}$
RSV-AZ-ST | $0.5017$ | $1.00^{**}$ | $0.1459$ | $0.00^{\hskip 5.60002pt}$
RSV-FS-SN | $0.4932$ | $1.00^{**}$ | $0.1422$ | $1.00^{**}$
RSV-FS-ST | $0.4964$ | $1.00^{**}$ | $0.1436$ | $0.11^{*\hskip 2.79996pt}$
RSV-GH-ST | $0.5134$ | $1.00^{**}$ | $0.1416$ | $1.00^{**}$
EGARCH-N | $0.5715$ | $0.12^{*\hskip 2.79996pt}$ | $0.2228$ | $0.00^{\hskip 5.60002pt}$
EGARCH-T | $0.5690$ | $0.00^{\hskip 5.60002pt}$ | $0.2386$ | $0.00^{\hskip 5.60002pt}$
REGARCH-N | $0.5236$ | $1.00^{**}$ | $0.1551$ | $0.00^{\hskip 5.60002pt}$
REGARCH-T | $0.5205$ | $1.00^{**}$ | $0.1532$ | $0.00^{\hskip 5.60002pt}$
Note: Refer to Table 3 for additional details.
Table 10: Summary of GW tests on QLIKE of volatility forecasts for N225

| SV-N | T | AZ-SN | AZ-ST | FS-SN | FS-ST | GH-ST | RSV-N | T | AZ-SN | AZ-ST | FS-SN | FS-ST | GH-ST | EG-N | T | REG-N | T
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
SV-N |  | $\hskip 3.49998pt18^{**}$ | $\hskip 3.49998pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt15^{**}$ | $\hskip 3.49998pt11^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt8^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt97^{**}$ | $\hskip 3.49998pt11^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt8^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$
SV-T | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt88^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt93^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt19^{\hskip 2.8pt}$ | $\hskip 3.49998pt14^{\hskip 2.8pt}$ | $\hskip 3.49998pt16^{\hskip 2.8pt}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt19^{\hskip 2.8pt}$ | $\hskip 3.49998pt11^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$
SV-AZ-SN | $-1.0^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt5^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt15^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt12^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt12^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt9^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt92^{**}$
SV-AZ-ST | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt17^{\hskip 2.8pt}$ | $\hskip 3.49998pt14^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt17^{\hskip 2.8pt}$ | $\hskip 3.49998pt12^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$
SV-FS-SN | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt17^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt9^{\hskip 2.8pt}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt20^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$
SV-FS-ST | $\hskip 3.49998pt2.4^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.6^{**}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt78^{\hskip 2.8pt}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt41^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt9^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$
SV-GH-ST | $\hskip 3.49998pt3.5^{**}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt3.9^{**}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $-0.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt23^{\hskip 2.8pt}$ | $\hskip 3.49998pt10^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt99^{**}$ | $\hskip 3.49998pt99^{**}$
RSV-N | $-3.7^{**}$ | $-3.4^{**}$ | $-3.7^{**}$ | $-3.5^{**}$ | $-5.8^{**}$ | $-5.7^{**}$ | $-4.6^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt93^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$
RSV-T | $-3.5^{**}$ | $-3.3^{**}$ | $-3.5^{**}$ | $-3.3^{**}$ | $-5.7^{**}$ | $-5.6^{**}$ | $-4.4^{**}$ | $\hskip 3.49998pt4.1^{**}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt7^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{**}$ | $\hskip 3.49998pt100^{**}$ | $\hskip 3.49998pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$
RSV-AZ-SN | $-3.7^{**}$ | $-3.4^{**}$ | $-3.6^{**}$ | $-3.4^{**}$ | $-5.7^{**}$ | $-5.7^{**}$ | $-4.5^{**}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $-2.4^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{**}$ | $\hskip 3.49998pt99^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$
RSV-AZ-ST | $-3.5^{**}$ | $-3.3^{**}$ | $-3.5^{**}$ | $-3.3^{**}$ | $-5.6^{**}$ | $-5.6^{**}$ | $-4.4^{**}$ | $\hskip 3.49998pt4.1^{**}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.6^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt100^{**}$ | $\hskip 3.49998pt100^{**}$ | $\hskip 3.49998pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$
RSV-FS-SN | $-3.8^{**}$ | $-3.5^{**}$ | $-3.8^{**}$ | $-3.6^{**}$ | $-5.8^{**}$ | $-5.8^{**}$ | $-4.7^{**}$ | $-1.9^{\hskip 2.8pt}$ | $-3.0^{**}$ | $-2.6^{*\hskip 1.39998pt}$ | $-3.1^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt79^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$
RSV-FS-ST | $-3.7^{**}$ | $-3.4^{**}$ | $-3.7^{**}$ | $-3.5^{**}$ | $-5.8^{**}$ | $-5.7^{**}$ | $-4.6^{**}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $-3.2^{**}$ | $-1.4^{\hskip 2.8pt}$ | $-3.3^{**}$ | $\hskip 3.49998pt2.5^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt93^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$
RSV-GH-ST | $-3.8^{**}$ | $-3.5^{**}$ | $-3.7^{**}$ | $-3.5^{**}$ | $-5.9^{**}$ | $-5.8^{**}$ | $-4.5^{**}$ | $-0.8^{\hskip 2.8pt}$ | $-1.9^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $-0.3^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{**}$
EGARCH-N | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.3^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt6.9^{**}$ | $\hskip 3.49998pt6.7^{**}$ | $\hskip 3.49998pt6.9^{**}$ | $\hskip 3.49998pt6.7^{**}$ | $\hskip 3.49998pt6.9^{**}$ | $\hskip 3.49998pt6.9^{**}$ | $\hskip 3.49998pt6.7^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt97^{**}$ | $\hskip 3.49998pt97^{**}$
EGARCH-T | $\hskip 3.49998pt4.0^{**}$ | $\hskip 3.49998pt2.6^{**}$ | $\hskip 3.49998pt4.2^{**}$ | $\hskip 3.49998pt2.7^{**}$ | $\hskip 3.49998pt3.4^{**}$ | $\hskip 3.49998pt3.2^{**}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt6.6^{**}$ | $\hskip 3.49998pt6.4^{**}$ | $\hskip 3.49998pt6.6^{**}$ | $\hskip 3.49998pt6.4^{**}$ | $\hskip 3.49998pt6.6^{**}$ | $\hskip 3.49998pt6.6^{**}$ | $\hskip 3.49998pt6.5^{**}$ | $\hskip 3.49998pt2.8^{**}$ |  | $\hskip 3.49998pt98^{**}$ | $\hskip 3.49998pt98^{**}$
REGARCH-N | $-2.8^{**}$ | $-2.7^{**}$ | $-2.8^{**}$ | $-2.8^{**}$ | $-5.0^{**}$ | $-5.0^{**}$ | $-3.8^{**}$ | $\hskip 3.49998pt4.9^{**}$ | $\hskip 3.49998pt4.8^{**}$ | $\hskip 3.49998pt4.6^{**}$ | $\hskip 3.49998pt4.8^{**}$ | $\hskip 3.49998pt4.4^{**}$ | $\hskip 3.49998pt4.6^{**}$ | $\hskip 3.49998pt4.6^{**}$ | $-6.4^{**}$ | $-6.0^{**}$ |  | $\hskip 3.49998pt98^{**}$
REGARCH-T | $-3.0^{**}$ | $-2.8^{**}$ | $-2.9^{**}$ | $-2.9^{**}$ | $-5.2^{**}$ | $-5.1^{**}$ | $-3.9^{**}$ | $\hskip 3.49998pt4.6^{**}$ | $\hskip 3.49998pt4.4^{**}$ | $\hskip 3.49998pt4.3^{**}$ | $\hskip 3.49998pt4.4^{**}$ | $\hskip 3.49998pt4.1^{**}$ | $\hskip 3.49998pt4.3^{**}$ | $\hskip 3.49998pt4.2^{**}$ | $-6.5^{**}$ | $-6.1^{**}$ | $-5.8^{**}$ |

Note: Refer to Table 4 for additional details.

Figure 11: Cumulative loss differences (QLIKE) in comparison to the SV-N model
for N225.

###  B.2 VaR and ES Forecasts

The prediction results for the N225 index are presented in Tables 11–13 and
Figures 12–13. We observe that the VaR violations are closely aligned with the
target rates across all models. For both significance levels $\alpha=1\%$ and
$5\%$, the RSV-GH-ST model demonstrates the lowest FZ0 loss. Notably, the RSV
and REGARCH models are included in the 90% MCS for both $\alpha=1\%$ and
$5\%$. However, the SV and EGARCH models do not consistently meet this
criterion.

The unconditional GW tests, summarized in the lower triangular elements of the
tables, indicate that the RSV and REGARCH models frequently outperform the SV
and EGARCH models, but not vice versa. Conversely, the conditional GW tests,
presented in the upper triangular elements, suggest that the RSV models
occasionally surpass the REGARCH models in performance. The CLDs reveal that
for $\alpha=1\%$, the RSV and REGARCH models outperform the SV and EGARCH
models throughout the entire period. Similarly, for $\alpha=5\%$, the RSV
models maintain superiority almost throughout the entire period. Specifically,
the RSV-GH-ST model surpasses other models in performance. These results
underscore the efficacy of enhancing the RSV model by integrating a skewed-
heavy tail distribution into the daily return for VaR and ES forecasts.

Table 11: Violation rates and FZ0 losses of VaR and ES forecasts for N225 | $\alpha=1\%$ |  | $\alpha=5\%$
---|---|---|---
| $\hat{\alpha}$ | FZ0 | $p_{MCS}$ |  | $\hat{\alpha}$ | FZ0 | $p_{MCS}$
SV-N | $1.02$ | $1.2034$ | $0.90^{**}$ |  | $3.90$ | $0.8163$ | $0.42^{**}$
SV-T | $0.85$ | $1.2158$ | $0.53^{**}$ |  | $3.39$ | $0.7973$ | $1.00^{**}$
SV-AZ-SN | $0.85$ | $1.2047$ | $0.89^{**}$ |  | $3.56$ | $0.8178$ | $0.47^{**}$
SV-AZ-ST | $0.85$ | $1.2053$ | $0.94^{**}$ |  | $3.56$ | $0.8156$ | $0.59^{**}$
SV-FS-SN | $0.85$ | $1.1979$ | $1.00^{**}$ |  | $3.39$ | $0.8105$ | $0.83^{**}$
SV-FS-ST | $0.85$ | $1.1870$ | $1.00^{**}$ |  | $3.39$ | $0.8093$ | $0.92^{**}$
SV-GH-ST | $0.68$ | $1.2720$ | $0.30^{**}$ |  | $4.24$ | $0.8846$ | $0.00^{\hskip 5.60002pt}$
RSV-N | $1.53$ | $1.1447$ | $1.00^{**}$ |  | $5.08$ | $0.7844$ | $1.00^{**}$
RSV-T | $1.36$ | $1.1292$ | $1.00^{**}$ |  | $5.08$ | $0.7806$ | $1.00^{**}$
RSV-AZ-SN | $1.36$ | $1.1555$ | $1.00^{**}$ |  | $4.75$ | $0.7863$ | $1.00^{**}$
RSV-AZ-ST | $1.19$ | $1.1260$ | $1.00^{**}$ |  | $4.75$ | $0.7822$ | $1.00^{**}$
RSV-FS-SN | $1.19$ | $1.1533$ | $1.00^{**}$ |  | $5.25$ | $0.7887$ | $1.00^{**}$
RSV-FS-ST | $1.19$ | $1.1301$ | $1.00^{**}$ |  | $5.25$ | $0.7865$ | $1.00^{**}$
RSV-GH-ST | $1.19$ | $1.1126$ | $1.00^{**}$ |  | $4.58$ | $0.7685$ | $1.00^{**}$
EGARCH-N | $1.02$ | $1.2818$ | $0.60^{**}$ |  | $3.73$ | $0.8450$ | $0.00^{\hskip 5.60002pt}$
EGARCH-T | $1.02$ | $1.2344$ | $0.27^{**}$ |  | $3.73$ | $0.8292$ | $0.25^{*\hskip 2.79996pt}$
REGARCH-N | $1.69$ | $1.1844$ | $1.00^{**}$ |  | $4.41$ | $0.7997$ | $1.00^{**}$
REGARCH-T | $1.02$ | $1.1306$ | $1.00^{**}$ |  | $4.92$ | $0.7959$ | $1.00^{**}$
Note: Refer to Table 5 for additional details.

Table 12: Summary of GW tests on FZ0 loss of VaR and ES forecasts
($\alpha=1\%$) for N225

| SV-N | T | AZ-SN | AZ-ST | FS-SN | FS-ST | GH-ST | RSV-N | T | AZ-SN | AZ-ST | FS-SN | FS-ST | GH-ST | EG-N | T | REG-N | T
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
SV-N |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
SV-T | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
SV-AZ-SN | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
SV-AZ-ST | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.0^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
SV-FS-SN | $-0.1^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
SV-FS-ST | $-0.3^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt77^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
SV-GH-ST | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.3^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt94^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{*\hskip 1.39998pt}$
RSV-N | $-1.0^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$
RSV-T | $-1.6^{\hskip 2.8pt}$ | $-1.9^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $-0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt11^{\hskip 2.8pt}$
RSV-AZ-SN | $-0.8^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
RSV-AZ-ST | $-1.7^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-2.2^{*\hskip 1.39998pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-FS-SN | $-1.2^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.5^{*\hskip 1.39998pt}$ | $-0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
RSV-FS-ST | $-1.5^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-2.1^{*\hskip 1.39998pt}$ | $-0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt72^{\hskip 2.8pt}$
RSV-GH-ST | $-1.9^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $-2.1^{*\hskip 1.39998pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-2.2^{*\hskip 1.39998pt}$ | $-0.9^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-1.8^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
EGARCH-N | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
EGARCH-T | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $-0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
REGARCH-N | $-0.3^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-0.2^{\hskip 2.8pt}$ | $0.0^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{*\hskip 1.39998pt}$
REGARCH-T | $-1.7^{\hskip 2.8pt}$ | $-2.2^{*\hskip 1.39998pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $-2.1^{*\hskip 1.39998pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-2.7^{**}$ | $-0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $-1.0^{\hskip 2.8pt}$ |

Note: Refer to Table 4 for additional details.

Table 13: Summary of GW tests on FZ0 loss of VaR and ES forecasts
($\alpha=5\%$) for N225

| SV-N | T | AZ-SN | AZ-ST | FS-SN | FS-ST | GH-ST | RSV-N | T | AZ-SN | AZ-ST | FS-SN | FS-ST | GH-ST | EG-N | T | REG-N | T
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
SV-N |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt5^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt53^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
SV-T | $-2.0^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt3^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt15^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
SV-AZ-SN | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt3.2^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$
SV-AZ-ST | $-0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt3.2^{**}$ | $-0.3^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt86^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
SV-FS-SN | $-0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$
SV-FS-ST | $-0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt6^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt94^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$
SV-GH-ST | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt3.0^{**}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.4^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.4^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.5^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
RSV-N | $-1.2^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-2.6^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt3^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-T | $-1.4^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-2.9^{**}$ | $-0.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-AZ-SN | $-1.2^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-2.6^{**}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt31^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-AZ-ST | $-1.4^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-2.8^{**}$ | $-0.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-FS-SN | $-1.1^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-2.6^{**}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-FS-ST | $-1.2^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-2.7^{**}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-GH-ST | $-1.9^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.9^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-3.1^{**}$ | $-1.2^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$
EGARCH-N | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.3^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.4^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
EGARCH-T | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $-1.5^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
REGARCH-N | $-0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt3.2^{**}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt3.0^{**}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.4^{*\hskip 1.39998pt}$ | $-1.8^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$
REGARCH-T | $-0.9^{\hskip 2.8pt}$ | $-0.1^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-2.6^{**}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.6^{**}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.4^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $-1.8^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ |

Note: Refer to Table 4 for additional details.

|
---|---
|
Figure 12: VaR and ES forecasts for N225: RSV-GH-ST model (left) vs. REGARCH-T
model (right). $\alpha=1\%$
---
$\alpha=5\%$
Figure 13: Cumulative loss differences (FZ0) in comparison to the SV-N model
for N225