# Realized Stochastic Volatility Model with Skew-t Distributions for Improved
Volatility and Quantile Forecasting††thanks: The authors gratefully
acknowledge the Ministry of Education, Culture, Sports, Science and Technology
of the Japanese Government through Grant-in-Aid for Scientific Research (Nos.
19H00588, 22K13376, and 23H00048), the Hitotsubashi Institute for Advanced
Study, and Hosei University’s Innovation Management Research Center.

Makoto Takahashi Corresponding author. Email<EMAIL_ADDRESS>Faculty
of Business Administration, Hosei University  Yuta Yamauchi  Graduate School
of Economics, Nagoya University  Toshiaki Watanabe  Graduate School of Social
Data Science, Hitotsubashi University  Yasuhiro Omori  Faculty of Economics,
University of Tokyo

###### Abstract

Forecasting volatility and quantiles of financial returns is essential for
accurately measuring financial tail risks, such as value-at-risk and expected
shortfall. The critical elements in these forecasts involve understanding the
distribution of financial returns and accurately estimating volatility. This
paper introduces an advancement to the traditional stochastic volatility
model, termed the realized stochastic volatility model, which integrates
realized volatility as a precise estimator of volatility. To capture the well-
known characteristics of return distribution, namely skewness and heavy tails,
we incorporate three types of skew-$t$ distributions. Among these, two
distributions include the skew-normal feature, offering enhanced flexibility
in modeling the return distribution. We employ a Bayesian estimation approach
using the Markov chain Monte Carlo method and apply it to major stock indices.
Our empirical analysis, utilizing data from US and Japanese stock indices,
indicates that the inclusion of both skewness and heavy tails in daily returns
significantly improves the accuracy of volatility and quantile forecasts.

Keywords: Bayesian estimation, Markov chain Monte Carlo, Realized volatility,
Skew-t distribution, Stochastic volatility

##  1 Introduction

Financial volatility, defined as the standard deviation or variance of asset
returns, exhibits stochastic variation over time, making its accurate
forecasting crucial for financial risk management. Historically, time-varying
volatility estimation and forecasting relied on two classes of time-series
models using financial returns: the generalized autoregressive conditional
heteroskedasticity (GARCH) methodologies, as per Engle (1982) and Bollerslev
(1986), and the stochastic volatility (SV) model, introduced by Taylor (1994).
These models, known for capturing volatility clustering and high volatility
persistence, have evolved to accommodate volatility asymmetry or the leverage
effect – the observed negative correlation between today’s return and
tomorrow’s volatility in stock markets. One notable development is the
exponential GARCH (EGARCH) model by Nelson (1991).

In more recent years, realized volatility (RV) has emerged as a predominant
method for volatility estimation, replacing traditional models. RV, computed
daily as the sum of squared intraday returns, is thoroughly reviewed by
Andersen and Benzoni (2009) and McAleer and Medeiros (2008). To model RV
dynamics, various studies, such as Andersen et al. (2003), have shown that
daily RV may follow a long-memory process, leading to the use of
autoregressive fractionally integrated moving average (ARFIMA) models (refer
to Beran (1994) for a detailed discussion on long-memory and ARFIMA models).
Additionally, the heterogeneous autoregressive (HAR) model, proposed by Corsi
(2009), is widely used for modeling the dynamics of RV. Although not a long-
memory model per se, the HAR model effectively approximates long-memory
processes.

Despite the improved volatility forecasts offered by ARFIMA and HAR models
over GARCH and SV models, RV is susceptible to biases from market
microstructure noise and non-trading hours. Techniques such as the multiscale
estimators by Zhang et al. (2005) and Zhang (2006a), the pre-averaging
approach by Jacod et al. (2009), and the realized kernel (RK) method by
Barndorff-Nielsen et al. (2008, 2009) have been developed to mitigate these
biases. For further details, see Aït-sahalia and Mykland (2009), Ubukata and
Watanabe (2014), and Liu et al. (2015).

Addressing the bias in RV, two hybrid model classes have been introduced: the
realized stochastic volatility (RSV) model, as proposed by Takahashi et al.
(2009), Dobrev and Szerszen (2010), and Koopman and Scharth (2013), and the
realized GARCH (RGARCH) and realized EGARCH (REGARCH) models, introduced by
Hansen et al. (2012) and Hansen and Huang (2016), respectively. Although
previous studies have compared the volatility forecasting abilities of these
models within their respective classes, a comprehensive cross-class comparison
has been limited. Notable exceptions are Takahashi et al. (2021) and Takahashi
et al. (2023), which evaluate the forecasting abilities of various models,
including SV, EGARCH, RSV, and REGARCH models.

In addition to volatility forecasting, accurately predicting financial tail
risks, such as value-at-risk (VaR) and expected shortfall (ES), is essential.
Financial return distributions often exhibit heavier tails and more peakedness
than the normal distribution, a phenomenon known as leptokurtosis. While time-
varying volatility partly explains this heavy-tailed nature, the conditional
return distribution on volatility may still exhibit leptokurtic
characteristics and skewness. Consequently, various skew Student’s $t$
distributions have been incorporated into volatility models (Abanto-Valle et
al., 2015; Kobayashi, 2016). Notably, the generalized hyperbolic (GH) skew
Student’s $t$ distribution, as discussed in Aas and Haff (2006), has been
applied to both SV (Nakajima and Omori, 2012; Leão et al., 2017) and RSV
models (Trojan, 2013; Nugroho and Morimoto, 2014, 2016; Takahashi et al.,
2016).

Building on this framework, the RSV model is extended to include three types
of skew-$t$ distributions, and a Bayesian estimation scheme using a Markov
chain Monte Carlo (MCMC) method is developed. These skew-$t$ distributions are
those proposed by Azzalini (1985) and Fernández and Steel (1995), along with
the GH skew-$t$ distribution. In our empirical analysis using the Dow Jones
Industrial Average (DJIA) and the Nikkei 225 (N225) data, we evaluate the
volatility, VaR, and ES forecasts of RSV models as well as SV, EGARCH, and
REGARCH models. Forecasts are assessed using mean squared error and quasi
likelihood loss functions for volatility, and the joint loss function proposed
by Fissler and Ziegel (2016), with specifications suggested by Patton et al.
(2019), for VaR and ES. Implementing the predictive ability test by Giacomini
and White (2006) and the model confidence set by Hansen et al. (2011), our
results indicate that RV significantly improves forecasting performance. The
RSV models, especially those incorporating skew-$t$ distributions, demonstrate
superior performance, corroborating and extending findings from previous
studies.

The remainder of the paper is structured as follows. Section 2 introduces the
basic SV model and the RSV models, which incorporate skew-$t$ distributions,
along with a brief overview of RV. Section 3 delves into the Bayesian
estimation methodology, utilizing MCMC simulations, and discusses various
measures for evaluating forecast performance. Section 4 presents an empirical
study employing financial stock indices data. Finally, Section 5 provides a
conclusion, summarizing key findings and implications of the study.

##  2 Model

###  2.1 Stochastic Volatility Model

First, let’s define the daily return of an asset as

$y_{t}=\log p_{t}-\log p_{t-1},$

where $p_{t}$ represents the closing price on day $t$. We further define the
volatility as a variance of $y_{t}$ as

$\text{E}[y_{t}^{2}]=\sigma_{t}^{2}=\exp(h_{t}).$

The term $h_{t}$ denotes the logarithm of volatility. This logarithmic
volatility, $h_{t}$, is considered a latent variable. It is presumed to follow
a stationary first-order autoregressive (AR(1)) process characterized by the
autoregressive parameter $\phi$ and a mean of $\mu$. Additionally, the
absolute value of $\phi$ should be less than one to ensure stationarity. With
$h_{t}$ given at time $t$, the SV model can be detailed as:

$\displaystyle y_{t}$
$\displaystyle=\epsilon_{t}\exp\left(\frac{h_{t}}{2}\right),\quad
t=1,\ldots,n,$ (1) $\displaystyle h_{t+1}$
$\displaystyle=\mu+\phi(h_{t}-\mu)+\eta_{t},\quad
t=1,\ldots,n-1,\quad|\phi|<1,$ (2) $\displaystyle h_{1}$ $\displaystyle\sim
N\left(\mu,\frac{\sigma_{\eta}^{2}}{1-\phi^{2}}\right),$ (3)
$\displaystyle\epsilon_{t}$ $\displaystyle\sim N(0,1),$ (4)
$\displaystyle\eta_{t}|\epsilon_{t}$ $\displaystyle\sim
N(\rho\sigma_{\eta}\epsilon_{t},(1-\rho^{2})\sigma_{\eta}^{2}).$ (5)

Here, we assume a stationary distribution for the initial log latent
volatility, $h_{1}$. The AR(1) process for $h_{t}$ is intended to describe
volatility clustering. This is the phenomenon where significant price
fluctuations tend to occur in close succession. The disturbance vector
$(\epsilon_{t},\eta_{t})^{\prime}$ is believed to obey a bivariate normal
distribution with a correlation $\rho$ between $\epsilon_{t}$ and $\eta_{t}$
(or equivalently, between $y_{t}$ and $h_{t+1}$). A negative correlation,
where $\rho<0$, indicates an asymmetry or a leverage effect. This suggests
that a decline in $y_{t}$ is often followed by an increase in $h_{t+1}$.

###  2.2 Realized Volatility

In the SV model, the volatility $\exp(h_{t})$ is considered a latent variable
since $h_{t}$ is unobserved. Recently, due to the availability of high-
frequency data, RV has been proposed as an estimator for true volatility using
intraday data. We first define the true volatility at time $t$. Let $p(s)$
denote the logarithm of the price of the asset at time $s$, and assume it
follows the continuous time diffusion process given by

$dp(s)=\mu(s)ds+\sigma(s)dW(s),$ (6)

where $\mu(s)$, $\sigma^{2}(s)$, and $W(s)$ represent the drift, the
volatility, and the Wiener process, respectively. The true volatility for day
$t$ is defined as

$IV_{t}=\int_{t}^{t+1}\sigma^{2}(s)ds,$ (7)

which is also referred to as the integrated volatility. Suppose we have $n$
intraday observations for day $t$, given by
$\left\\{r_{t},r_{t+1/n},\ldots,r_{t+(n-1)/n}\right\\}$. Then the RV, as an
estimator of $IV_{t}$, is defined by

$RV_{t}=\sum_{i=0}^{n-1}r_{t+i/n}^{2}.$ (8)

Under certain conditions, it can be shown that $RV_{t}$ converges to $IV_{t}$
as $n\rightarrow\infty$. However, these conditions are often not satisfied in
practice with high-frequency data, resulting in a bias in the estimator.

One source of this bias arises from non-trading hours during which we lack
intraday observations. If we ignore the non-trading hours to compute the RV,
we underestimate the true volatility for a whole day (24 hours). To address
the non-trading hours, Hansen and Lunde (2005) proposed adjusting $RV_{t}$ as
follows:

$\displaystyle RV_{t}^{HL}=c_{HL}RV_{t},\quad
c_{HL}=\frac{\sum_{t=1}^{n}(y_{t}-\overline{y})^{2}}{\sum_{t=1}^{n}RV_{t}}\quad\text{and}\quad\overline{y}=\frac{1}{n}\sum_{t=1}^{n}y_{t},$
(9)

where $RV_{t}$ represents the RV when non-trading hours are ignored. The
average of $RV_{t}^{HL}$ is set equal to the sample variance of the daily
returns.

Another source of bias is the microstructure noise. The actual asset prices
are observed with microstructure noise, which results in autocorrelations in
the asset returns and violates these assumptions. The effect of microstructure
noise is known to become larger as the time interval between observed prices
becomes smaller. The optimal time intervals to compute the RV have been
investigated in the literature (Aït-Sahalia et al., 2005; Bandi and Russell,
2006, 2008; Liu et al., 2015). Alternative estimators, such as the two or
multi-scale estimator (Zhang et al., 2005; Zhang, 2006b) and the preaveraging
estimator (Jacod et al., 2009), have also been proposed. In the empirical
study in Section 4, we use the RK (Barndorff-Nielsen et al., 2008) as an
unbiased volatility estimator. The RK for day $t$ is defined as

$\displaystyle RK_{t}=\sum_{h=-H}^{H}k\left(\frac{h}{H+1}\right)\gamma_{h},$
(10)

where $H$ is the bandwidth, $k(\cdot)\in[0,1]$ is a non-stochastic weight
function, and

$\displaystyle\gamma_{h}=\sum_{i=|h|+1}^{m}r_{t-1+i/m}r_{t-1+(i-|h|)/m}.$ (11)

For the choice of $k(\cdot)$, Barndorff-Nielsen et al. (2009) have suggested
the Parzen kernel function and provided a detailed procedure for the choice of
$H$.

Alternatively, these biases can be incorporated into two classes of hybrid
models: the RSV and RGARCH models. In particular, the RSV model proposed by
Takahashi et al. (2009) adjusts the bias of the RV by estimating the model
parameter as discussed in Section 2.3.

###  2.3 Realized Stochastic Volatility Model

Takahashi et al. (2009) introduced an additional source of information
regarding the true volatility. Besides the measurement equation for the daily
return $y_{t}$, they also considered that of the log RV, $x_{t}=\log RV_{t}$.
Let $h_{t}$ denote the logarithm of the true volatility, $h_{t}=\log IV_{t}$.
The RSV model is defined as follows:

$\displaystyle x_{t}$ $\displaystyle=\xi+h_{t}+u_{t},\quad t=1,\ldots,n,$ (12)
$\displaystyle y_{t}$
$\displaystyle=\epsilon_{t}\exp\left(\frac{h_{t}}{2}\right),\quad
t=1,\ldots,n,$ (13) $\displaystyle h_{t+1}$
$\displaystyle=\mu+\phi(h_{t}-\mu)+\eta_{t},\quad
t=1,\ldots,n-1,\quad|\phi|<1,$ (14) $\displaystyle h_{1}$ $\displaystyle\sim
N\left(\mu,\frac{\sigma_{\eta}^{2}}{1-\phi^{2}}\right),$ (15) $\displaystyle
u_{t}$ $\displaystyle\sim N(0,\sigma_{u}^{2}),$ (16)
$\displaystyle\epsilon_{t}$ $\displaystyle\sim N(0,1),$ (17)
$\displaystyle\eta_{t}|\epsilon_{t}$ $\displaystyle\sim
N(\rho\sigma_{\eta}\epsilon_{t},(1-\rho^{2})\sigma_{\eta}^{2}),$ (18)

where $u_{t}$ is assumed to be independent of $(\epsilon_{t},\eta_{t})$ for
simplicity. In general, equation (12) could be replaced by

$x_{t}=\xi+\psi h_{t}+u_{t}.$

However, empirical studies have found that this extension does not improve the
forecasting performance of the volatilities, leading to the assumption
$\psi=1$ as in Takahashi et al. (2009).

The parameter $\xi$ serves as a bias correction factor in $\log RV_{t}$, where
$\xi=0$ implies that there is no bias. Ignoring the non-trading hours while
computing the RV leads to an underestimation of $IV_{t}$, expecting a negative
bias. Hansen and Lunde (2006) demonstrated that microstructure noise may
induce either a positive or negative bias. Therefore, $\xi>0$ if the positive
bias caused by microstructure noise dominates the negative bias from ignoring
non-trading hours, and $\xi<0$ otherwise.

The distribution of $\epsilon_{t}$ corresponds to the return distribution
conditional on the effect of the time-varying volatility. Although the
inclusion of stochastic volatility $h_{t}$ captures the heavy tails of
financial returns, the distribution of $\epsilon_{t}$ can be further assumed
to have heavy tails, such as a Student’s $t$ distribution. Moreover, financial
returns may also exhibit skewness. The excess kurtosis and negative skewness
can be accounted for by adopting an asymmetric heavy-tailed distribution for
daily returns.

For the SV model, Nakajima and Omori (2012) and Leão et al. (2017) adapted the
GH skew Student’s $t$ distribution proposed by Aas and Haff (2006).
Additionally, Abanto-Valle et al. (2015) employed the skew Student’s $t$
distribution proposed by Azzalini and Capitanio (2003), while Kobayashi (2016)
utilized the skew exponential power distribution. Furthermore, Steel (1998)
extended the SV model using the method proposed by Fernández and Steel (1995).
For the RSV model, Trojan (2013), Nugroho and Morimoto (2014, 2016), and
Takahashi et al. (2016) adapted the GH skew Student’s $t$ distribution. In the
following, we extend the RSV model in equations (12)–(18), henceforth referred
to as the RSV-N model, by employing such distributions for $\epsilon_{t}$.

###  2.4 RSV Model with Student’s $t$ Distribution

The RSV model, when augmented with the Student’s $t$ distribution, will
henceforth be referred to as the RSV-T model. This model is derived by
modifying the distributions of $\epsilon_{t}$ and $\eta_{t}$ as described in
equations (17)–(18), to:

$\displaystyle\epsilon_{t}$
$\displaystyle=z_{t}\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}},$ (19)
$\displaystyle z_{t}$ $\displaystyle\sim N(0,1),$ (20)
$\displaystyle\eta_{t}|z_{t}$ $\displaystyle\sim
N(\rho\sigma_{\eta}z_{t},(1-\rho^{2})\sigma_{\eta}^{2}),$ (21)

where

$\displaystyle\lambda_{t}$ $\displaystyle\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}IG\left(\frac{\nu}{2},\frac{\nu}{2}\right),\quad\mu_{\lambda}=E[\lambda_{t}]=\frac{\nu}{\nu-2},\quad\nu>2.$
(22)

The parameter $\nu$ is the degrees of freedom, which governs the excess
kurtosis not captured by the stochastic volatility $h_{t}$. As $\nu$
approaches infinity, the model reverts to the RSV-N model described in
equations (12)–(18). It’s worth noting that the correlation between
$\epsilon_{t}$ and $\eta_{t}$ now becomes a function of both $\rho$ and $\nu$,
expressed as:

$\displaystyle\text{Corr}[\epsilon_{t},\eta_{t}]=\frac{E\left[\sqrt{\lambda_{t}}\right]}{\sqrt{\mu_{\lambda}}}\rho,$
(23)

where

$\displaystyle
E\left[\sqrt{\lambda_{t}}\right]=\sqrt{\frac{\nu}{2}}\frac{\Gamma\left(\frac{\nu-m}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)}.$
(24)

###  2.5 RSV Model with GH Skew-$t$ Distribution

Takahashi et al. (2016) extend the RSV-N model delineated in (12)–(18) by
incorporating the GH skew-$t$ distribution, hereafter referred to as the RSV-
GH-ST model. This model is attained by substituting $\epsilon_{t}$ and
$\eta_{t}$ in (17)–(18) with the ensuing mixture representation:

$\displaystyle\epsilon_{t}$
$\displaystyle=\frac{\beta(\lambda_{t}-\mu_{\lambda})+\sqrt{\lambda_{t}}z_{t}}{\sqrt{\beta^{2}\sigma_{\lambda}^{2}+\mu_{\lambda}}},$
(25) $\displaystyle z_{t}$ $\displaystyle\sim N(0,1),$ (26)
$\displaystyle\eta_{t}|z_{t}$ $\displaystyle\sim
N(\rho\sigma_{\eta}z_{t},(1-\rho^{2})\sigma_{\eta}^{2}),$ (27)

where $\lambda_{t}$ and $\mu_{\lambda}$ are defined as in (22), and

$\displaystyle\sigma_{\lambda}^{2}=V[\lambda_{t}]=E\left[\lambda_{t}^{2}\right]-\\{E[\lambda_{t}]\\}^{2}=\frac{2\nu^{2}}{(\nu-2)^{2}(\nu-4)},\quad\nu>4.$
(28)

The term $\sqrt{\beta^{2}\sigma_{\lambda}^{2}+\mu_{\lambda}}$ standardizes the
return ensuring that the variance of the return remains $\exp(h_{t})$. It is
noteworthy that the correlation between $\epsilon_{t}$ and $\eta_{t}$ is now a
function of $\rho$, $\nu$, and $\beta$, expressed by:

$\displaystyle\text{Corr}[\epsilon_{t},\eta_{t}]=\frac{E\left[\sqrt{\lambda_{t}}\right]}{\sqrt{\beta^{2}\sigma_{\lambda}^{2}+\mu_{\lambda}}}\rho,$
(29)

where $E\left[\sqrt{\lambda_{t}}\right]$ is specified in (24). When $\beta=0$,
this model reverts to the RSV-T model.

The skewness and tail thickness of $\epsilon_{t}$ are concurrently determined
by the parameter values of $\beta$ and $\nu$. To exhibit how the distribution
shape varies with the values, we render the simulated density function of
$\epsilon_{t}$ in Figure 1. We discern from (i) of the figure that a lower
value of $\beta$ yields a more negative or left-skewed distribution with
heavier tails. As stipulated, $\beta=0$ corresponds to a symmetric Student’s
$t$ density. Conversely, from (ii), the density appears less skewed and has
lighter tails as $\nu$ increases. Additionally, as $\nu\to\infty$ or
$\lambda_{t}=1$ for all $t$, the distribution morphs into a normal
distribution regardless of the value of $\beta$. Therefore, $\nu$ and $\beta$
can be interpreted as parameters regarding the tail thickness and skewness,
respectively, albeit the overall shape of the distribution is determined by
their joint values.

---
Figure 1: Simulated density function of the standardized GH skew-$t$
distribution in (25). (i) $\beta=0,-1,-2$ with $\nu=10$ fixed. (ii)
$\nu=24,16,8$ with $\beta=-1$ fixed.

###  2.6 RSV Model with Azzalini Skew-$t$ Distribution

We extend the RSV-N model as presented in Equations (12)–(18) by employing the
mixture representation of the skew-$t$ distribution discussed in Azzalini
(1985).111Also see Azzalini and Capitanio (2003) and Sahu et al. (2003).
Hereafter, the extended RSV model will be referred to as the RSV-AZ-ST model,
which is realized by modifying $\epsilon_{t}$ and $\eta_{t}$ in Equations
(17)–(18) as follows:

$\displaystyle\epsilon_{t}$
$\displaystyle=\frac{\delta(z_{0t}-c)+\sqrt{1-\delta^{2}}\hskip
1.99997ptz_{t}}{\sqrt{1-c^{2}\delta^{2}}}\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}},\quad|\delta|<1,$
(30) $\displaystyle z_{t}$ $\displaystyle\sim N(0,1),$ (31) $\displaystyle
z_{0t}$ $\displaystyle\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}TN_{(0,\infty)}(0,1),$ (32) $\displaystyle\eta_{t}|z_{t}$
$\displaystyle\sim N(\rho\sigma_{\eta}z_{t},(1-\rho^{2})\sigma_{\eta}^{2}),$
(33)

where $\lambda_{t}$ and $\mu_{\lambda}$ are defined in Equation (22),
$TN_{(0,\infty)}$ denotes a truncated normal distribution for regions greater
than 0, and

$\displaystyle c=E[z_{0t}]=\sqrt{\frac{2}{\pi}},\quad
V[z_{0t}]=E[z_{0t}^{2}]-E[z_{0t}]^{2}=1-c^{2}.$ (34)

The term $\sqrt{1-c^{2}\delta^{2}}$ standardizes the return so that the
variance of the return remains $\exp(h_{t})$. It is notable that the
correlation between $\epsilon_{t}$ and $\eta_{t}$ is now expressed as a
function of $\rho$, $\nu$, and $\delta$, given by

$\displaystyle\text{Corr}[\epsilon_{t},\eta_{t}]=\sqrt{\frac{1-\delta^{2}}{(1-c^{2}\delta^{2})\mu_{\lambda}}}E\left[\sqrt{\lambda_{t}}\right]\rho,$
(35)

where $E\left[\sqrt{\lambda_{t}}\right]$ is provided in Equation (24). When
$\delta=0$, this model reduces to the RSV-T model. Alternatively, when
$\nu\to\infty$ or $\lambda_{t}=1$ for all $t$, the model reduces to the RSV
model with a skew normal distribution, which will henceforth be referred to as
the RSV-AZ-SN model.

The skewness and kurtosis of $\epsilon_{t}$ in (30) are determined by both the
values of $\delta$ and $\nu$. Upon examining the moments of $\lambda_{t}$,

$\displaystyle
E[\lambda_{t}^{\frac{m}{2}}]=\left(\frac{\nu}{2}\right)^{\frac{m}{2}}\frac{\Gamma\left(\frac{\nu-m}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)},\quad\nu>m,$
(36)

we obtain

$\displaystyle E[\epsilon_{t}^{3}]$
$\displaystyle=\frac{4-\pi}{2}\frac{c^{3}\delta^{3}}{(1-c^{2}\delta^{2})^{\frac{3}{2}}}\left(\frac{\nu-2}{2}\right)^{\frac{3}{2}}\frac{\Gamma\left(\frac{\nu-3}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)},$
(37) $\displaystyle E[\epsilon_{t}^{4}]$
$\displaystyle=\left(3+2(\pi-3)\frac{c^{4}\delta^{4}}{(1-c^{2}\delta^{2})^{2}}\right)\frac{\nu-2}{\nu-4}.$
(38)

Similarly to the GH skew-$t$ distribution, we may interpret $\nu$ and $\delta$
as parameters concerning the tail thickness and skewness, respectively.
However, it’s notable that the overall shape of the distribution is determined
by their joint values.

Figure 2 demonstrates how the skewness and tail thickness of $\epsilon_{t}$
depend on the parameter values of $\delta$ and $\nu$. Sub-figure (i) shows
that a lower value of $\delta$ results in a more negative skewness or left-
skewness, with $\delta=0$ corresponding to a symmetric Student’s $t$ density.
In sub-figure (ii), it is observed that the density has lighter tails as $\nu$
increases. Yet, even when $\nu\to\infty$ or $\lambda_{t}=1$ for all $t$, the
distribution remains negatively skewed as long as $\delta$ is negative,
transitioning to a skew normal density as mentioned.

---
Figure 2: Simulated density function of the standardized AZ skew-$t$
distribution in (30). (i) $\delta=0,-0.6,-0.9$ with $\nu=10$ fixed. (ii)
$\nu=15,10,5$ with $\delta=-0.9$ fixed.

###  2.7 RSV Model with Fernández-Steel Skew-$t$ Distribution

Employing the method outlined in Fernández and Steel (1995), we define the
Fernández-Steel (FS) skew-$t$ density222The scale parameter is set to one as
it is arbitrary. as follows:

$\displaystyle
p_{T}(w|\gamma,\nu)=\frac{2}{\gamma+\gamma^{-1}}\left(f_{T}\left(\frac{w}{\gamma}\bigg{|}\nu\right)\mathbbm{1}\left\\{w\geq
0\right\\}+f_{T}\left(\gamma
w|\nu\right)\mathbbm{1}\left\\{w<0\right\\}\right),\quad\gamma>0,$ (39)

where $f_{T}(\cdot|\nu)$ denotes the probability density function of the
Student’s $t$ distribution with $\nu$ degrees of freedom, expressed as

$\displaystyle
f_{T}(\tilde{w}|\nu)=c_{\nu}\left(1+\frac{\tilde{w}^{2}}{\nu}\right)^{-\frac{\nu+1}{2}},\quad
c_{\nu}=\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)\Gamma\left(\frac{1}{2}\right)\sqrt{\nu}},\quad\nu>2,$
(40)

and $\mathbbm{1}\\{A\\}$ represents an indicator function such that
$\mathbbm{1}\\{A\\}=1$ when $A$ is true and 0 otherwise. The mean and variance
of $w$ following the FS skew-$t$ distribution are given by

$\displaystyle\mu_{*}$ $\displaystyle=E[w]=M_{1}(\gamma-\gamma^{-1}),$ (41)
$\displaystyle\sigma_{*}^{2}$
$\displaystyle=V[w]=E[w^{2}]-E[w]^{2}=M_{2}\frac{\gamma^{3}+\gamma^{-3}}{\gamma+\gamma^{-1}}-\\{M_{1}(\gamma-\gamma^{-1})\\}^{2},$
(42)

where

$\displaystyle M_{1}$
$\displaystyle=2\int_{0}^{\infty}\tilde{w}f_{T}(\tilde{w}|\nu)d\tilde{w}=\frac{2c_{\nu}\nu}{\nu-1}$
(43) $\displaystyle M_{2}$
$\displaystyle=2\int_{0}^{\infty}\tilde{w}^{2}f_{T}(\tilde{w}|\nu)d\tilde{w}=\frac{\nu}{\nu-2}.$
(44)

It is noteworthy that $\mu_{*}$ and $\sigma_{*}^{2}$ are functions of $\gamma$
and $\nu$. To standardize the density, we let $w_{*}=(w-\mu_{*})/\sigma_{*}$.
Subsequently, we have $E[z]=0$, $V[z]=E[z^{2}]=1$, and its probability density
function is

$\displaystyle q_{T}(w_{*}|\gamma,\nu)$
$\displaystyle=\frac{2\sigma_{*}}{\gamma+\gamma^{-1}}\left(f_{T}\left(\frac{\sigma_{*}w_{*}+\mu_{*}}{\gamma}\bigg{|}\nu\right)\mathbbm{1}\left\\{w_{*}\geq-\frac{\mu_{*}}{\sigma_{*}}\right\\}\right.$
(45) $\displaystyle\hskip
55.00008pt\left.+f_{T}\left(\gamma(\sigma_{*}w_{*}+\mu_{*})\bigg{|}\nu\right)\mathbbm{1}\left\\{w_{*}<-\frac{\mu_{*}}{\sigma_{*}}\right\\}\right)$
(46)

Thus, $q_{T}(\cdot|\gamma,\nu)$ is obtained as the standardized FS skew-$t$
density. By replacing $f_{T}(\cdot|\gamma,\nu)$ with the standard normal
density $f_{N}(\cdot)$, we arrive at the standardized FS skew normal density,
denoted by $q_{N}(\cdot|\gamma)$.

Now, by employing the FS skew-$t$ distribution, we extend the RSV model, which
will be referred to as the RSV-FS-ST henceforth. This extension is achieved by
modifying $\epsilon_{t}$ and $\eta_{t}$ in equations (17)–(18) as follows

$\displaystyle\epsilon_{t}\sim q_{T}(\epsilon_{t}|\gamma,\nu),$ (47)
$\displaystyle\eta_{t}|\epsilon_{t}\sim
N(\rho\sigma_{\eta}\epsilon_{t},(1-\rho^{2})\sigma_{\eta}^{2}),$ (48)

where $q_{T}(\cdot|\gamma,\nu)$ is the standardized FS skew-$t$ density
defined in (46). By replacing $q_{T}(\cdot|\gamma,\nu)$ with
$q_{N}(\cdot|\gamma)$, we derive the RSV model with skew normal distribution,
referred to as the RSV-FS-SN henceforth. Note that $\rho$ symbolizes the
correlation between $\epsilon_{t}$ and $\eta_{t}$, akin to the RSV-N model.

The parameter $\gamma$ controls the skewness of the distribution. When
$\gamma=1$, the distribution is symmetric and resembles the standardized
Student’s $t$ distribution. For $\gamma<1$, the distribution becomes left-
skewed, whereas for $\gamma>1$, it turns right-skewed. Figure 3 elucidates the
dependence of the skewness and tail thickness of $\epsilon_{t}$ on the
parameters $\gamma$ and $\nu$. In sub-figure (i), a lower value of $\gamma$
yields more pronounced negative skewness or left-skewness, with $\gamma=1$
corresponding to a symmetric Student’s $t$ density. Sub-figure (ii) reveals
that the density exhibits lighter tails as $\nu$ escalates. Nevertheless, even
as $\nu\to\infty$, the distribution retains its negative skewness provided
$\gamma$ remains below unity, transitioning towards a skew normal density as
previously mentioned.

---
Figure 3: Density function of the standardized FS skew-$t$ distribution
defined in (46). (i) $\gamma=1,0.8,0.6$ with $\nu=10$ fixed. (ii)
$\nu=20,15,10$ with $\gamma=0.5$ fixed.

##  3 Bayesian Estimation Methodology

Each of the RSV models defined in Section 2 is characterized as a nonlinear
Gaussian state space model with numerous latent variables. This complexity
makes the evaluation of the likelihood function, given the parameters, a
challenging task, as is the implementation of maximum likelihood estimation.
To circumvent these challenges, we adopt a Bayesian approach for parameter
estimation using MCMC simulation, as has been done in previous literature.

For the prior distribution of
$(\mu,\phi,\sigma_{\eta}^{2},\rho,\xi,\sigma_{u}^{2})$, the parameters common
across the RSV models, we assume

$\displaystyle\mu$ $\displaystyle\sim
N(m_{\mu},s_{\mu}^{2}),\quad\frac{\phi+1}{2}\sim B(a_{\phi 0},b_{\phi 0}),$
(49) $\displaystyle\quad\frac{\rho+1}{2}$ $\displaystyle\sim B(a_{\rho
0},b_{\rho 0}),\quad\sigma_{\eta}^{2}\sim
IG\left(\frac{n_{\eta}}{2},\frac{S_{\eta}}{2}\right)$ (50) $\displaystyle\xi$
$\displaystyle\sim N(m_{\xi},s_{\xi}^{2}),\quad\sigma_{u}^{2}\sim
IG\left(\frac{n_{u}}{2},\frac{S_{u}}{2}\right)$ (51)

where $B(\cdot,\cdot)$ and $IG(\cdot,\cdot)$ denote beta and inverse gamma
distributions, respectively. The parameters of the prior distributions
$(m_{\mu},s_{\mu},a_{\phi 0},b_{\phi 0},a_{\rho 0},b_{\rho
0},n_{\eta},S_{\eta},m_{\xi},s_{\xi},n_{u},S_{u})$ are treated as
hyperparameters and are chosen to reflect any prior information regarding the
parameters.

For notational simplicity, let $\bm{y}=(y_{1},\ldots,y_{n})^{\prime}$,
$\bm{x}=(x_{1},\ldots,x_{n})^{\prime}$, and
$\bm{h}=(h_{1},\ldots,h_{n})^{\prime}$. Let $\pi(\bm{\theta})$ denote the
joint prior probability density of $\bm{\theta}$. Furthermore, let
$f(\bm{y},\bm{x},\bm{h}|\bm{\theta})$ denote the likelihood of
$(\bm{y},\bm{x})$ given $\bm{\theta}$.

In what follows, we illustrate the MCMC algorithm for the RSV-AZ-ST and RSV-
FS-ST models.333See Takahashi et al. (2009) and Takahashi et al. (2016) for
algorithms of the RSV-N and RSV-GH-ST models, respectively. Additionally, we
describe the methodology for obtaining one-day-ahead forecasts of volatility
and daily return, from which we derive the one-day-ahead VaR and ES.

###  3.1 RSV-AZ-ST Model

Let
$\bm{\theta}=(\mu,\phi,\rho,\sigma_{\eta}^{2},\delta,\xi,\sigma_{u}^{2},\nu)^{\prime}$,
$\bm{\lambda}=(\lambda_{1},\ldots,\lambda_{n})^{\prime}$, and
$\bm{z}_{0}=(z_{01},\ldots,z_{0n})^{\prime}$. For the prior distribution of
$\bm{\theta}$, excluding
$(\mu,\phi,\sigma_{\eta}^{2},\rho,\xi,\sigma_{u}^{2})$, we assume

$\displaystyle\frac{\delta+1}{2}\sim B(a_{\delta 0},b_{\delta 0}),\quad\nu\sim
G(n_{\nu 0},S_{\nu 0}),$ (52)

where $G(\cdot,\cdot)$ denotes the gamma distribution. We then implement the
MCMC simulations as follows.

  1. 1.

Initialize $\bm{h}$, $\bm{\theta}$, $\bm{z}_{0}$, and $\bm{\lambda}$

  2. 2.

Generate $\mu|\bm{\theta}_{-\mu},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  3. 3.

Generate
$\phi|\bm{\theta}_{-\phi},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  4. 4.

Generate
$(\rho,\sigma_{\eta}^{2})|\bm{\theta}_{-(\rho,\sigma_{\eta}^{2})},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  5. 5.

Generate
$\delta|\bm{\theta}_{-\delta},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  6. 6.

Generate $\xi|\bm{\theta}_{-\xi},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  7. 7.

Generate
$\sigma_{u}^{2}|\bm{\theta}_{-\sigma_{u}^{2}},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  8. 8.

Generate $\nu|\bm{\theta}_{-\nu},\bm{h},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  9. 9.

Generate $\bm{z}_{0}|\bm{\theta},\bm{h},\bm{\lambda},\bm{x},\bm{y}$

  10. 10.

Generate $\bm{h}|\bm{\theta},\bm{z}_{0},\bm{\lambda},\bm{x},\bm{y}$

  11. 11.

Generate $\bm{\lambda}|\bm{\theta},\bm{h},\bm{z}_{0},\bm{x},\bm{y}$

  12. 12.

Return to step 2

The specific details for each sampling scheme are provided in Appendix A.1.

###  3.2 RSV-FS-ST Model

Let
$\bm{\theta}=(\mu,\phi,\rho,\sigma_{\eta}^{2},\gamma,\xi,\sigma_{u}^{2},\nu)^{\prime}$.
For the prior distribution of $\bm{\theta}$, excluding
$(\mu,\phi,\sigma_{\eta}^{2},\rho,\xi,\sigma_{u}^{2})$, we assume

$\displaystyle\gamma\sim G(n_{\gamma 0},S_{\gamma 0}),\quad\nu\sim G(n_{\nu
0},S_{\nu 0}).$ (53)

We then implement the MCMC simulations as follows.

  1. 1.

Initialize $\bm{h}$ and $\bm{\theta}$.

  2. 2.

Generate $\mu|\bm{\theta}_{-\mu},\bm{h},\bm{x},\bm{y}$.

  3. 3.

Generate $\phi|\bm{\theta}_{-\phi},\bm{h},\bm{x},\bm{y}$.

  4. 4.

Generate $\rho|\bm{\theta}_{-\rho},\bm{h},\bm{x},\bm{y}$.

  5. 5.

Generate
$\sigma_{\eta}^{2}|\bm{\theta}_{-\sigma_{\eta}^{2}},\bm{h},\bm{x},\bm{y}$.

  6. 6.

Generate $\xi|\bm{\theta}_{-\xi},\bm{h},\bm{x},\bm{y}$.

  7. 7.

Generate $\sigma_{u}^{2}|\bm{\theta}_{-\sigma_{u}^{2}},\bm{h},\bm{x},\bm{y}$.

  8. 8.

Generate $\gamma|\bm{\theta}_{-\gamma},\bm{h},\bm{x},\bm{y}$.

  9. 9.

Generate $\nu|\bm{\theta}_{-\nu},\bm{h},\bm{x},\bm{y}$.

  10. 10.

Generate $\bm{h}|\bm{\theta},\bm{x},\bm{y}$.

  11. 11.

Return to step 2

The specific details for each sampling scheme are provided in Appendix A.2.

###  3.3 One-day-ahead Forecast

To obtain one-day-ahead forecasts of financial returns and volatilities, we
utilize the predictive distribution within each state-space model. Let
$\bm{\theta}^{(i)}$ and $\bm{h}^{(i)}$ represent the $i$th sample of
parameters and latent log-volatilities in the MCMC simulation, respectively.
We can then generate the one-step-ahead predictive samples as follows.

  1. 1.

Generate $h_{n+1}^{(i)}|\bm{x},\bm{y},\bm{h}^{(i)},\bm{\theta}^{(i)}\sim
N(\mu_{n+1}^{(i)},(\sigma_{n+1}^{(i)})^{2})$ where

$\displaystyle\mu_{n+1}^{(i)}$
$\displaystyle=\mu^{(i)}+\phi^{(i)}h_{n}^{(i)}-\mu^{(i)}+\rho^{(i)}\sigma_{\eta}^{(i)}y_{n}\exp\left(-\frac{h_{n}^{(i)}}{2}\right),$
(54) $\displaystyle(\sigma_{n+1}^{(i)})^{2}$
$\displaystyle=(1-(\rho^{(i)})^{2})(\sigma_{\eta}^{(i)})^{2}.$ (55)

  2. 2.

Generate $\epsilon_{n+1}^{(i)}$ from the distribution corresponding to the
chosen RSV model given $\bm{\theta}^{(i)}$ and $\bm{h}^{(i)}$.

  3. 3.

Compute $y_{n+1}^{(i)}$ as

$\displaystyle
y_{n+1}^{(i)}=\epsilon_{n+1}^{(i)}\exp\left(-\frac{h_{n+1}^{(i)}}{2}\right).$
(56)

By repeating the above procedure for $M$ times, we obtain the generated
samples $\\{h_{n+1}^{(i)}\\}_{i=1}^{M}$ and $\\{y_{n+1}^{(i)}\\}_{i=1}^{M}$.

One-day-ahead quantile forecasts, such as VaR and ES, can be derived from the
distribution of $\\{y_{n+1}^{(i)}\\}_{i=1}^{M}$ for each model. The one-day-
ahead VaR forecasts with probability $\alpha$ at time $n+1$, denoted by
$\mbox{VaR}_{n+1}(\alpha)$, are defined as

$\displaystyle\mbox{Pr}(y_{n+1}<\mbox{VaR}_{n+1}(\alpha)|\mathcal{I}_{n})=\alpha,$
(57)

where $\mathcal{I}_{n}$ denotes the information set available at time $n$. The
corresponding ES forecasts, denoted by $\mbox{ES}_{n+1}(\alpha)$, are then
given as

$\displaystyle\mbox{ES}_{n+1}(\alpha)=E[y_{n+1}|y_{n+1}<\mbox{VaR}_{n+1}(\alpha),\mathcal{I}_{n}]$
(58)

The one-day-ahead VaR and ES can be obtained as the $(1-\alpha)$th quantile
and conditional average of $\\{y_{n+1}^{(i)}\\}_{i=1}^{M}$, respectively.

We evaluate the obtained volatility, VaR, and ES forecasts by employing
suitable loss or scoring functions. Specifically, following Takahashi et al.
(2023), we compute the Mean Squared Error (MSE) and Gaussian Quasi-Likelihood
(QLIKE) loss functions for volatility forecasts, and the Fissler and Ziegel
(2016) (FZ) joint loss function for VaR and ES forecasts. To utilize the FZ
loss function for forecast evaluation, we adhere to Patton et al. (2019) and
employ the following form, referred to as the FZ0 loss function hereafter,

$\displaystyle L_{FZ0}(y,v,e;\alpha)=-\frac{1}{\alpha e}\mathbbm{1}\\{y\leq
v\\}(v-y)+\frac{v}{e}+\log(-e)-1,$ (59)

where $y$, $v$, and $e$ denote a return, VaR, and ES, respectively. Patton et
al. (2019) demonstrate that the FZ0 loss function is the only FZ loss function
generating loss differences that are homogeneous of degree zero.

While average losses derived from the aforementioned loss functions offer
initial insights into the forecast performance of the models in contention,
they do not indicate if the differences in losses are statistically
significant. To ascertain this, we utilize the conditional and unconditional
predictive ability tests by Giacomini and White (2006), hereinafter referred
to as GW tests, on the loss differences. The GW tests are particularly
pertinent to this paper’s objectives because they emphasize the rolling window
methods we adopt in Section 4. This allows for a cohesive assessment of both
nested and non-nested models, including the RSV models described in Section 2.

For the unconditional predictive ability, the GW test statistic conforms to
that proposed by Diebold and Mariano (2002) and is, under the null hypothesis,
asymptotically standard normally distributed. For the conditional predictive
ability, the GW test defines the null hypothesis as

$\displaystyle H_{0,h}:E[\Delta L_{t+1}|\mathcal{I}_{t}]=0,\quad
t=n,n+1,\ldots,n+n_{f}-1,$ (60)

where $\Delta L_{t}$ is the loss difference between two models at forecast
date $t$, and $\mathcal{I}_{t}$ is the information set at time $t$.

Given a test function $\mathbbm{h}_{t}$, a $q\times 1$ vector measurable with
respect to $\mathcal{I}_{t}$ (Stinchcombe and White, 1998), the null
hypothesis translates to

$\displaystyle H_{0,h}:E[\mathbbm{h}_{t}\Delta L_{t+1}]=0,\quad
t=n,n+1,\ldots,n+n_{f}-1.$ (61)

Defining $Z_{t+1}=\mathbbm{h}_{t}\Delta L_{t+1}$ and
$\bar{Z}_{n_{f}}=n_{f}^{-1}\sum_{t=n}^{n+n_{f}-1}Z_{t+1}$, the alternative
hypothesis becomes

$\displaystyle H_{A,h}:E[\bar{Z}_{n_{f}}^{\prime}]E[\bar{Z}_{n_{f}}]\geq
d>0,\quad\mbox{for sufficiently large $n_{f}$}.$ (62)

We employ the Wald-type test statistic

$\displaystyle
T_{n_{f}}=n\bar{Z}_{n_{f}}^{\prime}\hat{\Omega}_{n_{f}}^{-1}\bar{Z}_{n_{f}},$
(63)

where
$\hat{\Omega}_{n_{f}}=n_{f}^{-1}\sum_{t=n}^{n+n_{f}-1}Z_{t+1}Z_{t+1}^{\prime}$
consistently estimates the variance of $Z_{t+1}$. Under the null, $T_{n_{f}}$
follows a chi-square distribution with $q$ degrees of freedom, $\chi_{q}^{2}$.
For significance level $p$, the null hypothesis is rejected when
$T_{n_{f}}>\chi_{q,1-p}^{2}$, where $\chi_{q,1-p}^{2}$ is the $(1-p)$ quantile
of $\chi_{q}^{2}$.

In practice, the selection of $\mathbbm{h}_{t}$ should discern between the
forecast performances of the models. Focusing on one-day-ahead forecasts, we
follow the specifications of Takahashi et al. (2023) and define
$\mathbbm{h}_{t}=(1,\Delta L_{t})^{\prime}$ for $t=n,n+1,\ldots,n+n_{f}-1$. If
the null is rejected, it implies that the lagged loss differences, $\Delta
L_{t}$, can forecast the subsequent loss differences, $\Delta L_{t+1}$. Using
$\hat{b}$ to represent the regression coefficient of $\Delta L_{t+1}$ on
$\mathbbm{h}_{t}$, the predicted loss differences
$\\{\hat{b}^{\prime}\mathbbm{h}_{t}\\}_{t=n}^{n+n_{f}-1}$ serve as indicators
for assessing model performances across different time points. In Section 4,
we measure relative performance by the frequency with which one model
forecasts larger losses than the other, as expressed by

$\displaystyle
I_{n_{f},0}=\frac{1}{n_{f}}\sum_{t=n}^{n+n_{f}-1}I(\hat{b}^{\prime}\mathbbm{h}_{t}>0).$
(64)

##  4 Empirical Study

###  4.1 Data and Descriptive Statistics

We implement the SV and RSV models, as described in Section 2, on the daily
(close-to-close) returns and Realized Volatilities (RVs) of the U.S. and
Japanese stock indices, Dow Jones Industrial Average (DJIA) and Nikkei 225
(N225). Following Liu et al. (2015), we employ a 5-minute RV, computed
excluding the non-trading hours, among various RV estimators. The DJIA data
are sourced from the Oxford-Man Institute’s Realized Library
(https://realized.oxford-man.ox.ac.uk/), while the N225 data are derived from
the Nikkei NEEDS-TICK dataset.444Refer to Ubukata and Watanabe (2014) for the
construction of the N225 data. The analysis spans from June 2009 to September
2019 for both indices. Specifically, the DJIA sample encompasses 2,596 trading
days from June 1, 2009, to September 27, 2019, and the N225 sample covers
2,532 trading days from June 1, 2009, to September 30, 2019.

Figure 4 displays the time series plots and histograms of the daily returns
for DJIA and N225. We notice that the daily returns fluctuate considerably
around zero for both series. Additionally, the returns tend to exhibit larger
values in the negative direction, indicating a negative skewness.

DJIA | N225
---|---
|
|
Figure 4: Time series plots (top) and histograms (bottom) of the daily returns
in percentage points for DJIA and N225.

Table 1 displays the descriptive statistics of the daily returns. For DJIA,
the means of daily returns significantly deviate from zero, while for N225,
they do not. Given the minuscule magnitude of the mean, we opt not to adjust
the mean of daily returns for both series in subsequent analyses. The
$p$-value of the Ljung and Box (1978) statistic, modified for
heteroskedasticity following Diebold (1988) to examine the null hypothesis of
no autocorrelation up to 10 lags, suggests that the null hypothesis stands
unrejected. This result enables us to estimate the models utilizing the daily
returns without making adjustments for the autocorrelation. The kurtosis of
daily returns highlights a leptokurtic distribution, a characteristic commonly
observed in financial returns, and the skewness of daily returns is
significantly negative for both series. Consequently, the Jarque-Bera (JB)
statistic repudiates its normality. The excess kurtosis and negative skewness
are anticipated to be accounted for by adopting the skew-$t$ distributions
detailed in Section 2 for daily returns.

Table 1: Descriptive statistics of the daily returns for DJIA and N225

| Mean | SD | Skew | Kurt | Min | Max | JB | LB
---|---|---|---|---|---|---|---|---
DJIA | $0.044$ | $0.895$ | $-0.448$ | $6.675$ | $-5.562$ | $4.857$ | $0.00$ | $0.52$
| $(0.018)$ |  | $(0.048)$ | $(0.096)$ |  |  |  |
N225 | $0.033$ | $1.321$ | $-0.544$ | $8.101$ | $-11.153$ | $7.426$ | $0.00$ | $0.55$
| $(0.026)$ |  | $(0.049)$ | $(0.097)$ |  |  |  |

Note: Parenthetical figures indicate standard errors. JB denotes the $p$-value
of the Jarque-Bera statistic. LB refers to the $p$-value of the Ljung and Box
(1978) statistic, adapted as per Diebold (1988).

Figure 5 illustrates the time series plots of the 5-minute RVs alongside their
logarithms, as well as histograms for DJIA and N225. The RVs exhibit
substantial variance, occasional clusters, and some pronounced spikes for both
series.555For DJIA, the single largest spike in the RV signifies the
”upheaval” on August 24, 2015 (refer to, for instance, The New York Times
archive at https://www.nytimes.com/2015/08/25/business/dealbook/daily-stock-
market-activity.html). For N225, the paramount spike on March 15, 2011,
denotes the impact of Japan’s earthquake on March 11, 2011. Such sporadic
large values of RVs lead to a positive skew in their logarithms.

DJIA | N225
---|---
|
|
|
Figure 5: Time series plots of 5-minute RVs (top) and their logarithms
(middle), along with histograms of the logarithms (bottom) for DJIA and N225.

Table 2 delineates the descriptive statistics of the 5-minute RVs and their
logarithms. For both DJIA and N225, the LB statistic repudiates the null
hypothesis of no autocorrelation, aligning with the widely acknowledged
persistence of volatility termed as volatility clustering. The skewness is
markedly positive, and the kurtosis indicates a leptokurtic distribution.
Consequently, the JB statistic refutes the normality of the logarithms of
5-minute RVs. This refutation contradicts the assumed normality of the error
term $u_{t}$ in the RSV model denoted by equation (16), albeit we adhere to
this assumption herein. Addressing the non-normality within the model emerges
as a compelling avenue for prospective research.

Table 2: Descriptive statistics of the logarithms of 5-minute RVs for DJIA and
N225

| Mean | SD | Skew | Kurt | Min | Max | JB | LB
---|---|---|---|---|---|---|---|---
DJIA | $-0.987$ | $1.048$ | $0.317$ | $3.177$ | $-3.953$ | $4.087$ | $0.00$ | $0.00$
| $(0.021)$ |  | $(0.048)$ | $(0.096)$ |  |  |  |
N225 | $-0.858$ | $0.844$ | $0.589$ | $4.219$ | $-3.321$ | $3.258$ | $0.00$ | $0.00$
| $(0.017)$ |  | $(0.049)$ | $(0.097)$ |  |  |  |

Note: Refer to Table 1 for additional details.

###  4.2 Competing Models

We evaluate the forecast performance of the RSV models as outlined below:

  1. 1.

RSV-N: RSV model in which $\epsilon_{t}$ follows a normal distribution.

  2. 2.

RSV-T: RSV model in which $\epsilon_{t}$ follows Student’s $t$ distribution.

  3. 3.

RSV-GH-ST: RSV model in which $\epsilon_{t}$ follows the GH skew-$t$
distribution.

  4. 4.

RSV-AZ-SN: RSV model in which $\epsilon_{t}$ follows the AZ skew normal
distribution.

  5. 5.

RSV-AZ-ST: RSV model in which $\epsilon_{t}$ follows the AZ skew-$t$
distribution.

  6. 6.

RSV-FS-SN: RSV model in which $\epsilon_{t}$ follows the FS skew normal
distribution.

  7. 7.

RSV-FS-ST: RSV model in which $\epsilon_{t}$ follows the FS skew-$t$
distribution.

We set the priors for the standard parameters as follows

$\displaystyle\mu\sim N(0,100),\quad\frac{\phi+1}{2}\sim
B(1,1),\quad\frac{\rho+1}{2}\sim B(1,1),\quad\sigma_{\eta}^{2}\sim
IG(0.05,0.05),$ (65) $\displaystyle\xi\sim N(0,10),\quad\sigma_{u}^{2}\sim
IG(2.5,0.1).$ (66)

And for the remaining parameters, we have

$\displaystyle\nu\sim G(5,0.5),\quad\beta\sim
N(0,1),\quad\frac{\delta+1}{2}\sim B(1,1),\quad\gamma\sim G(1,1).$ (67)

Considering that GARCH models—including ARCH (Engle, 1982), GARCH (Bollerslev,
1986), and their extensions—are frequently employed to capture time-varying
volatility, we assess the efficacy of RSV models by contrasting them with
GARCH and SV models. From the myriad of GARCH models available,666Refer to
Andersen et al. (2013) and the references therein for more details. we select
the EGARCH model as introduced by Nelson (1988). Incorporating RV, both Hansen
et al. (2012) and Hansen and Huang (2016) extended the GARCH and EGARCH
models, respectively. Given that the REGARCH model is a more comprehensive
version than the RGARCH, we opt for the REGARCH model. For model descriptions,
see Takahashi et al. (2023), for instance.

In our out-of-sample forecasting, we adopt the standard normal and Student’s
$t$ distributions for returns. We then estimate these models’ parameters using
the QML method and predict the one-day-ahead volatility, VaR, and ES forecasts
by integrating the QML estimates into the parameters. For other realized GARCH
models employing Bayesian estimation techniques and their applications to
volatility, VaR, and ES forecasts, refer to Chen et al. (2021).

###  4.3 Out-of-sample Forecasting

Adhering to the approach of Takahashi et al. (2016), we employ a rolling
window estimation strategy to forecast volatility, VaR, and ES. The window
size remains constant throughout the analysis. For the DJIA, this size is set
at 1,993 observations, with the last observation dates spanning from April 28,
2017, to September 26, 2019. For the N225, the window encompasses 1,942
observations, and the final observation dates range from April 28, 2017, to
September 27, 2019. We produce one-day ahead forecasts after each estimation.
Specifically, we generate 15,000 prediction samples, then calculate the means
and medians of volatility forecasts for the SV and RSV models.777We opt for
the median when forecasting volatility rather than the mean, as the mean
values for the SV models tend to be excessively large on days that have a few
outliers in the posterior samples. Additionally, we compute the quantiles of
return forecasts. Ultimately, from the start of May 2017 to the close of
September 2019, we amass 603 and 590 prediction samples for the DJIA and N225,
respectively.

####  4.3.1 Volatility forecasts

To appraise the accuracy of the volatility forecasts, we compute their average
loss using the MSE and QLIKE loss metrics, as elaborated in Section 3.3. Both
the MSE and QLIKE losses are robust functions and offer rankings congruent
with those derived using the true volatility, provided the volatility proxy
serves as a conditionally unbiased estimator of the actual volatility. In
order to counteract the bias introduced by market microstructure noise, we
employ the RK utilizing the Parzen kernel function, accompanied by a bandwidth
as recommended by Barndorff-Nielsen et al. (2009), to approximate the
underlying volatility. Additionally, to correct for biases arising from non-
trading hours, we adopt the method proposed by Hansen and Lunde (2005),
adjusting the influence of these non-trading periods on the RK according to
equation (9). Specifically, the adjustment term $c_{HL}$ for the RK at time
$t$ is computed as

$\displaystyle
c_{HL}=\frac{\sum_{s=t-n+1}^{t}(y_{s}-\bar{y})^{2}}{\sum_{s=t-n+1}^{t}RK_{s}},\quad\bar{y}=\frac{1}{n}\sum_{s=t-n+1}^{t}y_{s},$
(68)

where $n$ signifies the fixed window size.

---
|
|
Figure 6: Volatility forecasts and adjusted RKs for DJIA: Overall period (top)
and sub-periods (bottom).
---
|
|
Figure 7: Volatility forecasts and adjusted RKs for N225: Overall period (top)
and sub-periods (bottom).

Figure 6 displays the volatility forecasts alongside the adjusted RKs for
DJIA. We note that the forecasts from the SV models present several spikes.
Notably, the SV-GH-ST model significantly over-predicts the volatility towards
the end of 2018 relative to other models. The general observation suggests
that the volatility forecasts from the RSV models tend to be more accurate
(closer to the RK) compared to the SV models, highlighting the enhancing role
of RV in prediction accuracy. Figure 7 depicts analogous observations for the
N225.

Table 3 presents the average losses of volatility forecasts for DJIA along
with the $p$-values of the model confidence set (MCS) proposed by Hansen et
al. (2011).888Our rolling window estimation scheme adheres to the stationarity
assumption of the MCS procedure with bootstrap methods. We compute the MCS
using 1,000 bootstrap replications with a block size of 10. For details, refer
to Section 4.3 in Hansen et al. (2011). The RSV-FS-SN model yields the lowest
MSE and QLIKE scores. With ∗∗ and ∗ denoting models within the 75% and 90%
MCS, respectively, we generally find that the volatility forecasts of the RSV
and REGARCH models surpass those of the SV and EGARCH models. Moreover, the
MSE for the SV-GH-ST model is substantially higher than for other models,
echoing the over-prediction of volatility towards the end of 2018. It is
essential to note that the MSE and QLIKE are loss functions with homogeneous
degrees of 1 and 0, respectively. As a result, the QLIKE for the SV-GH-ST
model is higher than for other models but to a lesser degree.

Table 3: Average losses of volatility forecasts for DJIA

| MSE | $p_{MCS}$ | QLIKE | $p_{MCS}$
---|---|---|---|---
SV-N | $0.2132$ | $1.00^{**}$ | $0.3019$ | $0.00^{\hskip 5.60002pt}$
SV-T | $0.2280$ | $1.00^{**}$ | $0.2990$ | $0.00^{\hskip 5.60002pt}$
SV-AZ-SN | $0.2085$ | $1.00^{**}$ | $0.2951$ | $0.00^{\hskip 5.60002pt}$
SV-AZ-ST | $0.2251$ | $1.00^{**}$ | $0.2932$ | $0.00^{\hskip 5.60002pt}$
SV-FS-SN | $0.2263$ | $1.00^{**}$ | $0.3243$ | $0.00^{\hskip 5.60002pt}$
SV-FS-ST | $0.2445$ | $1.00^{**}$ | $0.3327$ | $0.00^{\hskip 5.60002pt}$
SV-GH-ST | $0.8092$ | $0.50^{**}$ | $0.3213$ | $0.00^{\hskip 5.60002pt}$
RSV-N | $0.2075$ | $1.00^{**}$ | $0.2526$ | $1.00^{**}$
RSV-T | $0.2156$ | $1.00^{**}$ | $0.2587$ | $1.00^{**}$
RSV-AZ-SN | $0.2168$ | $1.00^{**}$ | $0.2538$ | $1.00^{**}$
RSV-AZ-ST | $0.2214$ | $1.00^{**}$ | $0.2563$ | $1.00^{**}$
RSV-FS-SN | $0.2069$ | $1.00^{**}$ | $0.2516$ | $1.00^{**}$
RSV-FS-ST | $0.2109$ | $1.00^{**}$ | $0.2541$ | $1.00^{**}$
RSV-GH-ST | $0.2137$ | $1.00^{**}$ | $0.2567$ | $1.00^{**}$
EGARCH-N | $0.2207$ | $1.00^{**}$ | $0.3580$ | $0.00^{\hskip 5.60002pt}$
EGARCH-T | $0.2359$ | $1.00^{**}$ | $0.3615$ | $0.00^{\hskip 5.60002pt}$
REGARCH-N | $0.2100$ | $1.00^{**}$ | $0.2767$ | $0.28^{**}$
REGARCH-T | $0.2133$ | $1.00^{**}$ | $0.2794$ | $0.29^{**}$

Notes: $p_{MCS}$ denotes the MCS $p$-value. ∗∗ and ∗ represent models within
the 75% and 90% MCS, respectively.

The outcomes of the GW tests on the loss differences can be found in Table
4.999In line with the observations made by Patton and Sheppard (2009), we
report only the results for QLIKE given its superior power over MSE in the
unconditional GW tests. The lower triangular sections of the table provide the
test statistics for the unconditional GW test. A negative value in this
context signifies that the row model outperforms the column model. Conversely,
the upper triangular sections display the percentage of predicted losses
stemming from the conditional GW test, as delineated in (64). In this case, a
higher percentage indicates superior performance by the column model compared
to the row model. Significance of the GW tests’ rejection is marked by symbols
∗ and ∗∗ for the 1% and 5% levels, respectively. Our evaluation reveals the
RSV models notably outperforming the SV and EGARCH models. The REGARCH models,
meanwhile, show improved performance over the EGARCH models, underscoring the
benefits of incorporating RV into volatility forecasts. Furthermore, the
superiority of the SV models over the EGEARCH models is consistent with the
insights of Takahashi et al. (2021) and Takahashi et al. (2023). While both
the statistics and proportions suggest that RSV models have a slight edge over
REGARCH models in terms of reduced forecasting losses, this difference is not
statistically significant.

Table 4: Summary of GW tests on QLIKE of volatility forecasts for DJIA

| 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 |  | $\hskip 3.49998pt72^{\hskip 2.8pt}$ | $\hskip 3.49998pt93^{\hskip 2.8pt}$ | $\hskip 3.49998pt84^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt90^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt87^{\hskip 2.8pt}$ | $\hskip 3.49998pt87^{\hskip 2.8pt}$
SV-T | $-0.5^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt3^{**}$ | $\hskip 3.49998pt10^{\hskip 2.8pt}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt7^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt89^{\hskip 2.8pt}$ | $\hskip 3.49998pt88^{\hskip 2.8pt}$
SV-AZ-SN | $-1.6^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt90^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt86^{\hskip 2.8pt}$ | $\hskip 3.49998pt85^{\hskip 2.8pt}$
SV-AZ-ST | $-1.6^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $-0.5^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt7^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt0^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt96^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt85^{\hskip 2.8pt}$ | $\hskip 3.49998pt84^{\hskip 2.8pt}$
SV-FS-SN | $\hskip 3.49998pt3.7^{**}$ | $\hskip 3.49998pt3.4^{**}$ | $\hskip 3.49998pt3.8^{**}$ | $\hskip 3.49998pt4.0^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt75^{\hskip 2.8pt}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt92^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt10^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt95^{\hskip 2.8pt}$ | $\hskip 3.49998pt95^{\hskip 2.8pt}$
SV-FS-ST | $\hskip 3.49998pt4.1^{**}$ | $\hskip 3.49998pt4.5^{**}$ | $\hskip 3.49998pt4.3^{**}$ | $\hskip 3.49998pt4.9^{**}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt93^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt94^{**}$ | $\hskip 3.49998pt24^{**}$ | $\hskip 3.49998pt11^{**}$ | $\hskip 3.49998pt95^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt95^{\hskip 2.8pt}$
SV-GH-ST | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.6^{**}$ | $\hskip 3.49998pt2.6^{**}$ | $\hskip 3.49998pt3.2^{**}$ | $-0.3^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt98^{**}$ | $\hskip 3.49998pt98^{**}$ | $\hskip 3.49998pt97^{**}$ | $\hskip 3.49998pt97^{**}$ | $\hskip 3.49998pt98^{**}$ | $\hskip 3.49998pt97^{**}$ | $\hskip 3.49998pt97^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt95^{\hskip 2.8pt}$ | $\hskip 3.49998pt95^{\hskip 2.8pt}$
RSV-N | $-3.5^{**}$ | $-3.8^{**}$ | $-3.5^{**}$ | $-3.5^{**}$ | $-4.4^{**}$ | $-4.8^{**}$ | $-4.3^{**}$ |  | $\hskip 3.49998pt20^{\hskip 2.8pt}$ | $\hskip 3.49998pt31^{\hskip 2.8pt}$ | $\hskip 3.49998pt16^{\hskip 2.8pt}$ | $\hskip 3.49998pt83^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt9^{\hskip 2.8pt}$ | $\hskip 3.49998pt21^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{\hskip 2.8pt}$
RSV-T | $-2.8^{**}$ | $-3.2^{**}$ | $-2.7^{**}$ | $-2.7^{**}$ | $-3.8^{**}$ | $-4.3^{**}$ | $-4.0^{**}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt81^{\hskip 2.8pt}$ | $\hskip 3.49998pt61^{\hskip 2.8pt}$ | $\hskip 3.49998pt77^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt76^{\hskip 2.8pt}$ | $\hskip 3.49998pt64^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$
RSV-AZ-SN | $-3.1^{**}$ | $-3.4^{**}$ | $-3.0^{**}$ | $-3.1^{**}$ | $-3.9^{**}$ | $-4.4^{**}$ | $-3.9^{**}$ | $\hskip 3.49998pt0.4^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt77^{\hskip 2.8pt}$ | $\hskip 3.49998pt42^{\hskip 2.8pt}$ | $\hskip 3.49998pt28^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{\hskip 2.8pt}$
RSV-AZ-ST | $-2.9^{**}$ | $-3.2^{**}$ | $-2.8^{**}$ | $-2.9^{**}$ | $-3.8^{**}$ | $-4.2^{**}$ | $-3.8^{**}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.9^{**}$ |  | $\hskip 3.49998pt87^{\hskip 2.8pt}$ | $\hskip 3.49998pt83^{\hskip 2.8pt}$ | $\hskip 3.49998pt50^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$
RSV-FS-SN | $-3.6^{**}$ | $-3.9^{**}$ | $-3.5^{**}$ | $-3.6^{**}$ | $-4.4^{**}$ | $-4.8^{**}$ | $-4.4^{**}$ | $-1.4^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt10^{\hskip 2.8pt}$ | $\hskip 3.49998pt14^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$
RSV-FS-ST | $-3.3^{**}$ | $-3.6^{**}$ | $-3.2^{**}$ | $-3.3^{**}$ | $-4.2^{**}$ | $-4.6^{**}$ | $-4.1^{**}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt26^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt1^{\hskip 2.8pt}$
RSV-GH-ST | $-2.9^{**}$ | $-3.1^{**}$ | $-2.8^{**}$ | $-2.8^{**}$ | $-3.8^{**}$ | $-4.2^{**}$ | $-4.0^{**}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt5^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{**}$ | $\hskip 3.49998pt\hskip 1.99997pt6^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt4^{\hskip 2.8pt}$
EGARCH-N | $\hskip 3.49998pt4.1^{**}$ | $\hskip 3.49998pt4.4^{**}$ | $\hskip 3.49998pt4.3^{**}$ | $\hskip 3.49998pt4.5^{**}$ | $\hskip 3.49998pt2.9^{**}$ | $\hskip 3.49998pt2.3^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.3^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt5.2^{**}$ | $\hskip 3.49998pt4.9^{**}$ | $\hskip 3.49998pt5.0^{**}$ | $\hskip 3.49998pt4.8^{**}$ | $\hskip 3.49998pt5.2^{**}$ | $\hskip 3.49998pt5.1^{**}$ | $\hskip 3.49998pt4.7^{**}$ |  | $\hskip 3.49998pt36^{\hskip 2.8pt}$ | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt96^{**}$
EGARCH-T | $\hskip 3.49998pt4.4^{**}$ | $\hskip 3.49998pt4.7^{**}$ | $\hskip 3.49998pt4.6^{**}$ | $\hskip 3.49998pt4.8^{**}$ | $\hskip 3.49998pt3.0^{**}$ | $\hskip 3.49998pt2.5^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.3^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt5.5^{**}$ | $\hskip 3.49998pt5.1^{**}$ | $\hskip 3.49998pt5.2^{**}$ | $\hskip 3.49998pt5.1^{**}$ | $\hskip 3.49998pt5.4^{**}$ | $\hskip 3.49998pt5.4^{**}$ | $\hskip 3.49998pt4.9^{**}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt95^{**}$ | $\hskip 3.49998pt96^{**}$
REGARCH-N | $-1.0^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-1.8^{\hskip 2.8pt}$ | $-2.2^{*\hskip 1.39998pt}$ | $-1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $-3.1^{**}$ | $-3.4^{**}$ |  | $\hskip 3.49998pt\hskip 1.99997pt0^{\hskip 2.8pt}$
REGARCH-T | $-0.9^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.9^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $-2.9^{**}$ | $-3.1^{**}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ |

Notes: EG and REG refer to the EGARCH and REGARCH models, respectively. ∗∗ and
∗ signify rejections of the GW test at the 1% and 5% significance levels,
respectively. In the lower triangular sections, test statistics from the
unconditional GW test are provided; a negative value implies the row model’s
superiority over the column model. Conversely, the upper triangular sections
show the percentage of predicted losses, as detailed in (64); a higher
percentage suggests the column model performs better than the row model.

Examining the temporal variations in volatility forecasts across models is
insightful. Figure 8 presents the cumulative loss differences (CLDs) of the
models relative to the SV-N model. For each time instance, the cumulative
QLIKE of a given model is deducted from that of the SV-N model.101010MSE
results are excluded due to their susceptibility to outliers, making model
comparisons challenging. A greater CLD indicates enhanced accuracy. Notably,
both RSV and REGARCH models consistently outperform the SV and EGARCH models
throughout the forecast horizon, underscoring the benefit of incorporating RV
in predicting volatility. The distinction between the RSV and REGARCH models
is primarily evident in late 2017 forecasts.111111Specifically, the RK value
on December 1, 2017, stood at 1.29. The SV-N model projected a value close to
0.24, while the RSV and REGARCH models estimated around 0.19 and 0.07,
respectively.

Figure 8: Cumulative loss differences (QLIKE) in comparison to the SV-N model
for DJIA.

The prediction results for N225 mirror those obtained for DJIA, emphasizing
that the incorporation of RV refines volatility forecasts. The details are
provided in Appendix B.1.

####  4.3.2 VaR and ES forecasts

Figure 9 depicts the VaR and ES forecasts of the RSV-FS-ST and REGARCH-T
models for DJIA. Notably, VaR violations (instances where actual returns drop
below the VaR forecasts) manifest consistently across models. These violations
are evident not only during turbulent phases like early 2018 and late 2019 but
also during calmer periods. While the VaR and ES forecasts of other models
exhibit similar trajectories with minor level variations, they are excluded
here for conciseness.

|
---|---
|
Figure 9: VaR and ES forecasts for DJIA: RSV-FS-ST model (left) vs. REGARCH-T
model (right).

Table 5 reports the violation rates of the VaR forecasts, denoted as
$\hat{\alpha}$, along with the average losses determined by the FZ0 loss
function as described by Patton et al. (2019) in equation (59). The SV-GH-ST
and EGARCH-T models yield violation rates nearest to the target rates of
$\alpha=1\%$ and 5%, respectively. For the target rate $\alpha=1\%$, every
model demonstrates violation rates above 1%, suggesting that actual returns
surpass the VaR forecasts more frequently than anticipated. When $\alpha=5\%$,
the violation rates gravitate closely around 5%. Conversely, the SV-AZ-SN and
RSV-AZ-ST models display the minimal FZ0 loss for $\alpha=1\%$ and 5%,
respectively, insinuating their superior performance in forecasting the return
distribution’s tail. It is worth noting that, with the exception of the SV-GH-
ST model for $\alpha=5\%$, all models fall within the 75% MCS for both
$\alpha=1\%$ and 5%.

Table 5: Violation rates and FZ0 losses of VaR and ES forecasts for DJIA

| $\alpha=1\%$ |  | $\alpha=5\%$
---|---|---|---
| $\hat{\alpha}$ | FZ0 | $p_{MCS}$ |  | $\hat{\alpha}$ | FZ0 | $p_{MCS}$
SV-N | $2.32$ | $1.0714$ | $1.00^{**}$ |  | $4.64$ | $0.5965$ | $0.76^{**}$
SV-T | $1.99$ | $1.0483$ | $1.00^{**}$ |  | $4.31$ | $0.5809$ | $1.00^{**}$
SV-AZ-SN | $1.82$ | $0.9698$ | $1.00^{**}$ |  | $4.31$ | $0.5858$ | $0.99^{**}$
SV-AZ-ST | $1.82$ | $1.0180$ | $1.00^{**}$ |  | $4.31$ | $0.5830$ | $1.00^{**}$
SV-FS-SN | $1.49$ | $0.9786$ | $1.00^{**}$ |  | $4.15$ | $0.5947$ | $0.90^{**}$
SV-FS-ST | $1.49$ | $0.9899$ | $1.00^{**}$ |  | $4.15$ | $0.5954$ | $0.87^{**}$
SV-GH-ST | $1.33$ | $1.1224$ | $1.00^{**}$ |  | $4.31$ | $0.7012$ | $0.12^{*\hskip 2.79996pt}$
RSV-N | $2.99$ | $1.2445$ | $0.45^{**}$ |  | $5.64$ | $0.5584$ | $1.00^{**}$
RSV-T | $2.82$ | $1.1419$ | $0.98^{**}$ |  | $5.80$ | $0.5465$ | $1.00^{**}$
RSV-AZ-SN | $2.32$ | $1.1282$ | $1.00^{**}$ |  | $5.47$ | $0.5414$ | $1.00^{**}$
RSV-AZ-ST | $2.16$ | $1.0780$ | $1.00^{**}$ |  | $5.14$ | $0.5299$ | $1.00^{**}$
RSV-FS-SN | $2.32$ | $1.0490$ | $1.00^{**}$ |  | $5.64$ | $0.5411$ | $1.00^{**}$
RSV-FS-ST | $2.49$ | $1.0958$ | $1.00^{**}$ |  | $5.31$ | $0.5365$ | $1.00^{**}$
RSV-GH-ST | $2.16$ | $1.0689$ | $1.00^{**}$ |  | $5.64$ | $0.5387$ | $1.00^{**}$
EGARCH-N | $2.65$ | $1.2928$ | $0.39^{**}$ |  | $4.81$ | $0.6226$ | $0.35^{**}$
EGARCH-T | $1.82$ | $1.0797$ | $1.00^{**}$ |  | $4.98$ | $0.5855$ | $1.00^{**}$
REGARCH-N | $3.15$ | $1.3273$ | $0.30^{**}$ |  | $5.47$ | $0.5652$ | $1.00^{**}$
REGARCH-T | $2.99$ | $1.3202$ | $0.32^{**}$ |  | $5.47$ | $0.5621$ | $1.00^{**}$

Notes: $\hat{\alpha}$ represents violation rate (%). FZ0 denotes the average
FZ0 loss. $p_{MCS}$ indicates the MCS $p$-value. ${}^{**},{}^{*}$ correspond
to models within the 75% and 90% MCS, respectively.

Tables 6 and 7 present the GW test results for the FZ0 loss. The lower and
upper triangular elements are interpreted as in Table 4. The test statistics
of the unconditional GW tests from the lower triangular elements indicate that
the SV-AZ-SN model significantly outperforms the SV-N model, while the RSV-N
model is significantly inferior to the other RSV models at $\alpha=1\%$.
Furthermore, the EGARCH-T model exhibits superior performance compared to the
EGARCH-N model. The proportions of predicted losses from the conditional GW
tests in the upper triangular elements echo these findings. These outcomes
underscore the significance of accounting for heavy tails when forecasting
return distributions. Additionally, it is observed that all RSV models, with
the exception of RSV-N and RSV-GH-ST, perform significantly better than the
REGARCH models at $\alpha=1\%$.

Table 6: Summary of GW tests on FZ0 loss of VaR and ES forecasts
($\alpha=1\%$) for DJIA

| 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 |  | $\hskip 3.49998pt\hskip 1.99997pt94^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt6^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt8^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt54^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt12^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt67^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt23^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$
SV-T | $-0.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt35^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt7^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt14^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
SV-AZ-SN | $-2.2^{*\hskip 1.39998pt}$ | $-1.6^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt6^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
SV-AZ-ST | $-0.9^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt47^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt59^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
SV-FS-SN | $-1.5^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt11^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt5^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt16^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
SV-FS-ST | $-1.4^{\hskip 2.8pt}$ | $-0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
SV-GH-ST | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt89^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt57^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt92^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt71^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
RSV-N | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$
RSV-T | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{**}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt93^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$
RSV-AZ-SN | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.0^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $-0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt81^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$
RSV-AZ-ST | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-2.3^{*\hskip 1.39998pt}$ | $-1.8^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt90^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt67^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt62^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$
RSV-FS-SN | $-0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-2.1^{*\hskip 1.39998pt}$ | $-1.5^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt6^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$
RSV-FS-ST | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $-0.2^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt83^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{*\hskip 1.39998pt}$
RSV-GH-ST | $0.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt36^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
EGARCH-N | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$
EGARCH-T | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $-0.3^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $-0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-2.1^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$
REGARCH-N | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$
REGARCH-T | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $-1.8^{\hskip 2.8pt}$ |

Note: Refer to Table 4 for additional details.

Table 7: Summary of GW tests on FZ0 loss of VaR and ES forecasts
($\alpha=5\%$) for DJIA

| 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 |  | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt95^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt92^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt80^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt95^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$
SV-T | $-0.6^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt7^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt5^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt7^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt5^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt25^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$
SV-AZ-SN | $-0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt86^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt38^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$
SV-AZ-ST | $-0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $-0.2^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt8^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt3^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt32^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$
SV-FS-SN | $-0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt34^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt63^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt95^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$
SV-FS-ST | $-0.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt3^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt53^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt96^{\hskip 2.8pt}$
SV-GH-ST | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt98^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
RSV-N | $-1.0^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ | $-2.1^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt97^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt8^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt11^{\hskip 2.8pt}$
RSV-T | $-1.3^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-1.0^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-2.3^{*\hskip 1.39998pt}$ | $-1.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt58^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt86^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt81^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt5^{\hskip 2.8pt}$
RSV-AZ-SN | $-1.5^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $-1.5^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt74^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt81^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt74^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt3^{\hskip 2.8pt}$
RSV-AZ-ST | $-1.9^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $-2.6^{**}$ | $-1.9^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-2.3^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt12^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt0^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$
RSV-FS-SN | $-1.7^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $-1.5^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $0.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt100^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt89^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt1^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt8^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt9^{\hskip 2.8pt}$
RSV-FS-ST | $-1.8^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-1.6^{\hskip 2.8pt}$ | $-2.5^{*\hskip 1.39998pt}$ | $-1.8^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $-0.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt12^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt5^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt7^{\hskip 2.8pt}$
RSV-GH-ST | $-1.7^{\hskip 2.8pt}$ | $-1.2^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-1.3^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-2.4^{*\hskip 1.39998pt}$ | $-1.0^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $-0.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt2^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt4^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt\hskip 1.99997pt9^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt10^{\hskip 2.8pt}$
EGARCH-N | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $-1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.9^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.8^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$
EGARCH-T | $-0.4^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.2^{\hskip 2.8pt}$ | $0.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.1^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ | $-1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.8^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $-2.3^{*\hskip 1.39998pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt99^{\hskip 2.8pt}$ | $\hskip 3.49998pt100^{\hskip 2.8pt}$
REGARCH-N | $-0.8^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt0.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.7^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.2^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.2^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.6^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.0^{\hskip 2.8pt}$ | $-1.4^{\hskip 2.8pt}$ | $-0.4^{\hskip 2.8pt}$ |  | $\hskip 3.49998pt\hskip 1.99997pt97^{*\hskip 1.39998pt}$
REGARCH-T | $-0.8^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.6^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-0.7^{\hskip 2.8pt}$ | $-2.0^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt0.3^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt2.1^{*\hskip 1.39998pt}$ | $\hskip 3.49998pt1.1^{\hskip 2.8pt}$ | $\hskip 3.49998pt1.5^{\hskip 2.8pt}$ | $\hskip 3.49998pt0.9^{\hskip 2.8pt}$ | $-1.5^{\hskip 2.8pt}$ | $-0.5^{\hskip 2.8pt}$ | $-2.5^{*\hskip 1.39998pt}$ |

Note: Refer to Table 4 for additional details.

Figure 10 displays the CLDs of the models in comparison to the SV-N models. A
higher CLD value indicates greater accuracy. Notable spikes can be observed,
especially in early 2018 and late 2019, which align with the pronounced
volatility spikes shown in Fig. 6. For $\alpha=1\%$, many models, notably the
REGARCH models, underperformed in early 2018 and were substantially inferior
to the SV-N model. Following this spike, the SV-AZ-SN model consistently
surpassed its counterparts. Conversely, for $\alpha=5\%$, the RSV and REGARCH
models showed superior performance compared to the SV and EGARCH models from
July 2018 onwards.

Given that the FZ0 loss evaluates both the VaR and ES, accounting for tail
information which is not reflected by the violation rate, our findings suggest
that both daily return attributes, skewness and heavy tails, are beneficial
for forecasting VaR and ES.

$\alpha=1\%$
---
$\alpha=5\%$
Figure 10: Cumulative loss differences (FZ0) in comparison to the SV-N model
for DJIA.

The prediction results for N225 are qualitatively the same as for DJIA.
Namely, they underscore the efficacy of enhancing the RSV model by integrating
a skewed-heavy tail distribution into the daily return for VaR and ES
forecasts. The details are provided in Appendix B.2.

####  4.3.3 Summary

Table 8 presents the rankings of average losses for volatility, VaR, and ES
forecasts. The average ranking in the last column indicates superior
performance of the RSV models compared to other models, with the RSV-FS-SN
model ranking as the best. In summary, our analysis reveals that RV is a
valuable predictor for volatility, VaR, and ES forecasts. Additionally, the
RSV model shows greater effectiveness than the REGARCH model. Furthermore,
incorporating skewed $t$-distributions into the RSV model appears to enhance
forecast accuracy.

Table 8: Ranking of average losses for volatility, VaR, and ES forecasts

| DJIA |  | N225 |  |
---|---|---|---|---|---
| MSE | QLIKE |  FZ01% |  FZ05% |  | MSE | QLIKE |  FZ01% |  FZ05% |  | Avg
SV-N |  6 | 13 |  8 | 16 |  |  6 | 11 | 12 | 14 |  | $10.75$
SV-T | 15 | 12 |  5 | 10 |  |  9 | 13 | 15 |  9 |  | $11.00$
SV-AZ-SN |  3 | 11 |  1 | 13 |  |  1 | 10 | 13 | 15 |  | $8.38$
SV-AZ-ST | 13 | 10 |  4 | 11 |  | 10 | 12 | 14 | 13 |  | $10.88$
SV-FS-SN | 14 | 15 |  2 | 14 |  | 14 | 14 | 11 | 12 |  | $12.00$
SV-FS-ST | 17 | 16 |  3 | 15 |  | 15 | 16 | 10 | 11 |  | $12.88$
SV-GH-ST | 18 | 14 | 12 | 18 |  | 18 | 15 | 17 | 18 |  | $16.25$
RSV-N |  2 |  2 | 15 |  7 |  |  3 |  3 |  6 |  4 |  | $5.25$
RSV-T |  9 |  7 | 14 |  6 |  |  7 |  6 |  3 |  2 |  | $6.75$
RSV-AZ-SN | 10 |  3 | 13 |  5 |  |  5 |  5 |  8 |  5 |  | $6.75$
RSV-AZ-ST | 12 |  5 |  9 |  1 |  |  8 |  7 |  2 |  3 |  | $5.88$
RSV-FS-SN |  1 |  1 |  6 |  4 |  |  2 |  2 |  7 |  7 |  | $3.75$
RSV-FS-ST |  5 |  4 | 11 |  2 |  |  4 |  4 |  4 |  6 |  | $5.00$
RSV-GH-ST |  8 |  6 |  7 |  3 |  | 11 |  1 |  1 |  1 |  | $4.75$
EGARCH-N | 11 | 17 | 16 | 17 |  | 17 | 17 | 18 | 17 |  | $16.25$
EGARCH-T | 16 | 18 | 10 | 12 |  | 16 | 18 | 16 | 16 |  | $15.25$
REGARCH-N |  4 |  8 | 18 |  9 |  | 13 |  9 |  9 | 10 |  | $10.00$
REGARCH-T |  7 |  9 | 17 |  8 |  | 12 |  8 |  5 |  8 |  | $9.25$

Notes: MSE denotes Mean Squared Error for volatility forecasts; QLIKE
represents Gaussian Quasi Likelihood for volatility forecasts; FZ01% and FZ05%
are FZ0 loss functions for VaR and ES forecasts at $\alpha=1\%$ and $5\%$,
respectively; Avg indicates the average rank across all categories.

##  5 Conclusion

This study has systematically analyzed the performance of various models in
forecasting volatility, VaR, and ES for financial indices, specifically
focusing on the DJIA and N225 indices. Our comprehensive evaluation, as
presented in Table 8, demonstrates that the RSV models consistently outperform
other competing models in terms of average ranking losses for volatility, VaR,
and ES forecasts.

Particularly noteworthy is the superior performance of the RSV-FS-SN model,
which ranks as the best among all evaluated models. This finding underscores
the effectiveness of incorporating RV into forecasting models. Moreover, the
comparative analysis between RSV and REGARCH models reveals that the RSV
models exhibit enhanced predictive capabilities.

An important contribution of our study is the validation of the benefit of
integrating skewed $t$-distributions into these forecasting models. This
integration has shown a marked improvement in forecast accuracy, thereby
providing a significant insight into the methodology of financial forecasting.

In conclusion, our results suggest that for accurate and reliable forecasts of
volatility, VaR, and ES, the RSV models, particularly when augmented with
skewed $t$-distributions, offer a robust and effective approach. These
findings have important implications for risk management and financial
decision-making, highlighting the critical role of model selection and the
incorporation of appropriate statistical distributions in enhancing forecast
precision.

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## Appendix

##  Appendix A MCMC Sampling Scheme

###  A.1 RSV-AZ-ST Model

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

$\displaystyle f(y_{t},h_{t+1}|z_{0t},h_{t},\bm{\theta})=$
$\displaystyle\frac{1}{2\pi
k(\delta)\sqrt{{(1-\rho^{2})\sigma^{2}}}}\sqrt{\frac{\mu_{\lambda}}{\lambda_{t}}}\exp\left[-\frac{1}{2}h_{t}\right]$
$\displaystyle\times\exp\left[-\frac{\mu_{\lambda}\exp(-h_{t})}{2\lambda_{t}k(\delta)^{2}(1-\rho^{2})}\left\\{y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right.\right.$
$\displaystyle\hskip
128.0374pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])\bigg{\\}}^{2}\Bigg{]}$
$\displaystyle\times\exp\left[-\frac{1}{2\sigma^{2}}\\{h_{t+1}-\mu-\phi[h_{t}-\mu]\\}^{2}\right],$

where

$k(\delta)=\sqrt{\frac{1-\delta^{2}}{1-c^{2}\delta^{2}}}.$

Then, the augmented joint posterior probability density is given by

$\displaystyle\pi(\bm{h},\bm{\theta},\bm{z}_{0},\bm{\lambda}|\bm{x},\bm{y})\propto$
$\displaystyle
k(\delta)^{-T}(1-\rho^{2})^{-\frac{T-1}{2}}\sigma^{-T}(1-\phi^{2})^{\frac{1}{2}}$
$\displaystyle\times\exp\left\\{-\frac{1}{2}\sum_{t=1}^{T}h_{t}-\frac{\mu_{\lambda}\exp(-h_{T})}{2\lambda_{T}k(\delta)^{2}}\left[y_{T}-\sqrt{\frac{\lambda_{T}}{\mu_{\lambda}}}\frac{\exp(h_{T}/2)\delta[z_{0T}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right]^{2}\right\\}$
$\displaystyle\times\exp\left[-\sum_{t=1}^{T-1}\frac{\mu_{\lambda}\exp(-h_{t})}{2\lambda_{t}k(\delta)^{2}(1-\rho^{2})}\left\\{y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right.\right.$
$\displaystyle\hskip
145.10922pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])\bigg{\\}}^{2}\bigg{]}$
$\displaystyle\times\exp\left[-\frac{1}{2\sigma^{2}}\sum_{t=1}^{T-1}\\{h_{t+1}-\mu-\phi[h_{t}-\mu]\\}^{2}-\frac{(1-\phi^{2})}{2\sigma^{2}}(h_{1}-\mu)^{2}\right]$
$\displaystyle\times\exp\left\\{-\frac{1}{2}\sum_{t=1}^{T}z_{0t}^{2}I(z_{0t}>0)\right\\}\times\prod_{t=1}^{T}\frac{\mu_{\lambda}^{\frac{1}{2}}\left(\frac{\nu}{2}\right)^{\frac{\nu}{2}}}{\Gamma\left(\frac{\nu}{2}\right)}\lambda_{t}^{-\frac{\nu+1}{2}-1}\exp\left(-\frac{\nu}{2\lambda_{t}}\right)$
$\displaystyle\times\sigma_{u}^{-T}\times\exp\left\\{-\frac{1}{2\sigma_{u}^{2}}\sum_{t=1}^{T}(x_{t}-\xi-
h_{t})^{2}\right\\}\times\pi(\bm{\theta}),$

where $\pi(\bm{\theta})$ is a prior probability density function of
$\bm{\theta}$.

####  A.1.1 Generation of $\mu$

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

$\displaystyle\sigma_{\mu 1}^{-2}$ $\displaystyle=\sigma_{\mu
0}^{-2}+\sigma^{-2}[1-\phi^{2}+(T-1)(1-\phi)^{2}(1-\rho^{2})^{-1}],$
$\displaystyle\mu_{1}$ $\displaystyle=\sigma_{\mu 1}^{2}\left\\{\sigma_{\mu
0}^{-2}\mu_{0}+\sigma^{-2}\left[(1-\phi^{2})h_{1}+\frac{1-\phi}{1-\rho^{2}}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\phi
h_{t}\right.\right.\right.$ $\displaystyle\hskip
133.72786pt\left.\left.\left.-\sqrt{\frac{\mu_{\lambda}}{\lambda_{t}}}\frac{\rho\sigma\exp(-h_{t}/2)}{k(\delta)}\left(y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right)\right\\}\right]\right\\}.$

####  A.1.2 Generation of $\phi$

The conditional posterior density of $\phi$ is

$\displaystyle\pi(\phi|\cdot)\propto$ $\displaystyle(1+\phi)^{a_{\phi
0}-1}(1-\phi)^{b_{\phi 0}-1}(1-\phi^{2})^{\frac{1}{2}}$
$\displaystyle\times\exp\left[-\sum_{t=1}^{T-1}\frac{\mu_{\lambda}\exp(-h_{t})}{2\lambda_{t}k(\delta)^{2}(1-\rho^{2})}\left\\{y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right.\right.$
$\displaystyle\hskip
142.26378pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])\bigg{\\}}^{2}\Bigg{]}$
$\displaystyle\times\exp\left[-\frac{1}{2\sigma^{2}}\sum_{t=1}^{T-1}\\{h_{t+1}-\mu-\phi[h_{t}-\mu]\\}^{2}-\frac{(1-\phi^{2})}{2\sigma^{2}}(h_{1}-\mu)^{2}\right],$
$\displaystyle\propto$ $\displaystyle
g(\phi)\times\exp\left\\{-\frac{1}{2\sigma_{\phi}^{2}}(\phi-\mu_{\phi})^{2}\right\\},$

where

$\displaystyle g(\phi)$ $\displaystyle=(1+\phi)^{a_{\phi
0}-\frac{1}{2}}(1-\phi)^{b_{\phi 0}-\frac{1}{2}},$
$\displaystyle\sigma_{\phi}^{2}$
$\displaystyle=\frac{\sigma^{2}(1-\rho^{2})}{\rho^{2}(h_{1}-\mu)^{2}+\sum_{t=2}^{T-1}(h_{t}-\mu)^{2}},$
$\displaystyle\mu_{\phi}$
$\displaystyle=\frac{\sigma_{\phi}^{2}}{\sigma^{2}(1-\rho^{2})}\sum_{t=1}^{T-1}\left\\{h_{t+1}-\mu-\sqrt{\frac{\mu_{\lambda}}{\lambda_{t}}}\frac{\rho\sigma\exp(-h_{t}/2)}{k(\delta)}\left(y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right)\right\\}(h_{t}-\mu).$

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

#### Generation of $\sigma_{\eta}^{2}$ and $\rho$

Instead of sampling $\sigma_{\eta}^{2}$ and $\rho$ separately, we sample them
jointly. Define $\vartheta=(\sigma_{\eta},\rho)^{\prime}$. Then, the
conditional posterior density of $\vartheta$ is

$\displaystyle\pi(\vartheta|\cdot)$
$\displaystyle\propto(\sigma_{\eta}^{2})^{-\frac{n_{0}+T}{2}-1}\exp\left(-\frac{S_{0}}{2\sigma_{\eta}^{2}}\right)(1+\rho)^{a_{\rho
0}-1}(1-\rho)^{b_{\rho 0}-1}(1-\rho^{2})^{-\frac{T-1}{2}}$
$\displaystyle\quad\times\exp\left[-\sum_{t=1}^{T-1}\frac{\mu_{\lambda}\exp(-h_{t})}{2\lambda_{t}k(\delta)^{2}(1-\rho^{2})}\left\\{y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right.\right.$
$\displaystyle\quad\hskip
145.10922pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma_{\eta}^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])\Bigg{\\}}^{2}\Bigg{]}$
$\displaystyle\quad\times\exp\left[-\frac{1}{2\sigma_{\eta}^{2}}\sum_{t=1}^{T-1}\\{h_{t+1}-\mu-\phi[h_{t}-\mu]\\}^{2}-\frac{(1-\phi^{2})}{2\sigma_{\eta}^{2}}(h_{1}-\mu)^{2}\right].$

To handle the restrictions on $\sigma_{\eta}$ and $\rho$, namely,
$\sigma_{\eta}>0$ and $|\rho|<1$, we change the variables as
$\omega_{1}=\log\sigma_{\eta}$ and $\omega_{2}=\log(1+\rho)-\log(1-\rho)$.
Then, we can sample $\omega_{1}$ and $\omega_{2}$ without any restriction. Let
$\hat{\vartheta}$ be the mode of the conditional density and $\hat{\omega}$ be
the corresponding value. Then, we approximate the conditional posterior
distribution by $N(\omega_{*},\sigma_{\eta*})$ where

$\displaystyle\omega_{*}=\hat{\omega}+\sigma_{\eta*}\frac{\partial\log\pi(\omega|\cdot)}{\partial\omega}\Bigg{|}_{\omega=\hat{\omega}},\quad\sigma_{\eta*}^{-1}=-\frac{\partial^{2}\log\pi(\omega|\cdot)}{\partial\omega\partial\omega^{\prime}}\Bigg{|}_{\omega=\hat{\omega}}.$

Thus, we propose a candidate $\omega^{\dagger}\sim
N(\omega_{*},\sigma_{\eta*})$ and accept it with probability

$\displaystyle\min\left[1,\frac{\pi(\vartheta^{\dagger}|\cdot)f_{N}(\omega|\omega_{*},\sigma_{\eta*})|J(\vartheta)|}{\pi(\vartheta|\cdot)f_{N}(\omega^{\dagger}|\omega_{*},\sigma_{\eta*})|J(\vartheta^{\dagger})|}\right],$

where $f_{N}(\cdot|\mu,\sigma_{\eta}^{2})$ denotes the probability density of
a normal distribution with mean $\mu$ and variance $\sigma_{\eta}^{2}$, and
$J(\cdot)$ is the Jacobian, and $\vartheta^{\dagger}$ is the inverse
transformation of $\omega^{\dagger}$.

####  A.1.3 Generation of $\delta$

The conditional posterior density of $\delta$ is

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

We approximate the conditional posterior distribution by
$N(\hat{\delta},\sigma_{\hat{\delta}}^{2})$ where $\hat{\delta}$ is the mode
of the conditional density, and
$\sigma_{\hat{\delta}}^{-2}=-\frac{\partial^{2}\log\pi(\delta|\cdot)}{\partial\delta^{2}}\Big{|}_{\delta=\hat{\delta}}$.
A candidate
$\delta^{\dagger}\sim\mathcal{TN}_{(-1,1)}(\hat{\delta},\sigma_{\hat{\delta}}^{2})$
is proposed and accepted with probability
$\min\left[1,\left\\{\frac{\pi(\delta^{\dagger}|\cdot)f_{N}(\delta|\hat{\delta},\sigma_{\hat{\delta}}^{2})}{\pi(\delta|\cdot)f_{N}(\delta^{\dagger}|\hat{\delta},\sigma_{\hat{\delta}}^{2})}\right\\}\right]$,
where $\mathcal{TN}_{(a,b)}(\mu,\sigma^{2})$ denotes a normal distribution
$N(\mu,\sigma^{2})$ truncated over $(a,b)$, and $f_{N}(\cdot|\mu,\sigma^{2})$
denotes the probability density of a normal distribution with mean $\mu$ and
variance $\sigma^{2}$.

####  A.1.4 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.1.5 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.1.6 Generation of $\nu$

The conditional probability density function of $\nu$ is

$\displaystyle\pi(\nu|\cdot)\propto$
$\displaystyle\exp\left\\{\frac{T}{2}\log\mu_{\lambda}+\frac{\nu
T}{2}\log\left(\frac{\nu}{2}\right)-T\log\Gamma\left(\frac{\nu}{2}\right)-\frac{\nu}{2}\sum_{t=1}^{T}(\lambda_{t}^{-1}+\log\lambda_{t})+(n_{\nu
0}-1)\log\nu-S_{\nu 0}\nu\right\\}$
$\displaystyle\times\exp\left[-\sum_{t=1}^{T-1}\frac{\mu_{\lambda}\exp(-h_{t})}{2\lambda_{t}k(\delta)^{2}(1-\rho^{2})}\left\\{y_{t}-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\frac{\exp(h_{t}/2)\delta[z_{0t}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right.\right.$
$\displaystyle\hskip
145.10922pt-\sqrt{\frac{\lambda_{t}}{\mu_{\lambda}}}\rho\exp(h_{t}/2)k(\delta)\sigma^{-1}(h_{t+1}-\mu-\phi[h_{t}-\mu])\Bigg{\\}}^{2}\Bigg{]}$
$\displaystyle\times\exp\left[-\frac{\mu_{\lambda}\exp(-h_{T})}{2\lambda_{T}k(\delta)^{2}}\left\\{y_{T}-\sqrt{\frac{\lambda_{T}}{\mu_{\lambda}}}\frac{\exp(h_{T}/2)\delta[z_{0T}-c]}{\sqrt{1-c^{2}\delta^{2}}}\right\\}^{2}\right].$

We approximate the conditional density by a normal density around the mode and
conduct a Metropolis-Hastings (MH) algorithm as in the generation of $\delta$.

####  A.1.7 Generation of $\bm{z}_{0}$

The conditional posterior distribution of $z_{0t}$ is
$\mathcal{TN}_{(0,\infty)}(\mu_{zt},\sigma_{zt}^{2})$ where

$\displaystyle\sigma_{zt}^{-2}$
$\displaystyle=\begin{cases}1+\frac{\delta^{2}}{(1-\delta^{2})(1-\rho^{2})},&t=1,\ldots,T-1,\\\
\frac{1}{1-\delta^{2}},&t=T,\end{cases}$ $\displaystyle\mu_{zt}$