#  A semi-parametric financial risk forecasting framework incorporating
multiple realized measures

Author 1  The University of Sydney, Australia  Author 2  The University of
Sydney, Australia  Author 3  The University of Sydney, Australia  Author 4
The University of Sydney, Australia

###### Abstract

A semi-parametric joint Value-at-Risk (VaR) and Expected Shortfall (ES)
forecasting framework employing multiple realized measures is developed. The
proposed model incorporates a measurement equation that models the
contemporaneous dependence between the ES and multiple realized measures.
Further, the information from multiple realized measures is used to model the
time varying dynamic relationship between VaR and ES. A quasi-likelihood,
built on the asymmetric Laplace distribution, enables the Bayesian inference
for the proposed model. An adaptive Markov Chain Monte Carlo method is used
for model estimation, whose finite sample properties are compared to maximum
likelihood estimation via a simulation study. The empirical section evaluates
the model’s performance with six stock markets from 2000 to 2022, covering the
period of COVID-19. Three realized measures, including 5-minute realized
variance, bi-power variation, and realized kernel, are incorporated and
evaluated in the proposed framework. One-step-ahead VaR and ES forecasting
results of the proposed model are compared to a range of parametric and semi-
parametric models. The forecasting results lend support to the effectiveness
of the proposed framework.

_Keywords_ : Semi-parametric; realized measures; Markov chain Monte Carlo;
Value-at-Risk; Expected Shortfall.

##  1 Introduction

Quantitative financial risk forecasting measurement is a tool for calculating
capital requirements for market risk, investment decisions and external
regulations. For several decades, Value-at-Risk (VaR) has been employed as a
risk measurement tool to measure and control financial risk
(duffie1997overview). In 1993, J. P. Morgan introduced VaR to represent market
risk as one number, and since then, it has become a standard measurement for
risk management and capital allocation.

If $\mathcal{L}$ is the information available at time $t$ and

$F_{t}(r)=Pr(r_{t}\leq r\mid\mathcal{L}_{t-1})$

be the Cumulative Distribution Function (CDF) of the return $r_{t}$
conditional on $\mathcal{L}_{t-1}$. Here, we assume that $F_{t}(\cdot)$ is
strictly increasing and continuous on the real line $\mathbb{R}$ and that all
conditional and unconditional distributions are to be understood in an “almost
sure” sense (wang2023semi). Under these assumptions, the on-step-ahead
$\alpha$ level Value-at-Risk at time $t$ can be defined as:

$Q_{t}=F_{t}^{-1}(\alpha),\quad 0<\alpha<1$

Using VaR as a tool to measure risk, banks could control the risk more
efficiently, and later, it was recognised by the Basel Committee. In 2008,
observed violations to evaluate and compare VaR models were used to implement
Basel II. Though the VaR approach is frequently investigated as it is an
important and popular method, it is also criticised as it cannot measure the
magnitude of the loss for violations. It is not a mathematically coherent risk
measure, meaning that it is not a sub-additive measure and can favour non-
diversification (yamai2005value). artzner1999coherent propose an alternative
coherent measure called expected shortfall (ES), also called conditional VaR
or tail VaR. The new measure, ES, calculates the expected loss conditional on
exceeding a VaR threshold and is a coherent measure. The one-step-ahead
$\alpha$ level Expected Shortfall can be shown to be equal to the tail
conditional expectation of $r_{t}$ using the same framework above:

$\textup{ES}_{t}=E(r_{t}\mid r_{t}\leq Q_{t,}\mathcal{L}_{t-1})$

The global financial crisis (GFC) and its repercussions affected many changes
in banking supervision and regulations in financial markets. To enhance
financial stability worldwide, BCBS mandate strengthening banks’ regulation,
supervision, and practices. One activity the BCBS seeks to achieve its mandate
is establishing and promoting global standards for the regulation and
supervision of banks as well as guidelines and sound practices. Basel III
Accords is the recent change introduced by BCBS where the ES is introduced as
the primary market risk measure for international banking regulations. The
implementation date of the Basel III standards finalized in December 2017 has
been deferred by one year to 1 January 2023 (de2020Implementation). Within
this new market discipline, banks will regularly disclose ES and VaR, and in
calculating ES, banks must use a 97.5th percentile, one-tailed confidence
level (de2019minimum). The suggested 97.5th percentile cannot be considered as
it is in this paper as we use a different sign convention, and the left tail
forecasting is focused. Therefore, a 2.5% probability level is employed in
this paper.

There are three approaches in volatility estimating literature: non-
parametric, parametric and semi-parametric. In the parametric approach, an
assumption for the return error distribution is made. Under a non-parametric
approach, a historical simulation method is used for VaR and ES forecasts.
Here, sample percentiles are employed as VaR estimations and sample averages
beyond these percentiles are used as ES forecasts. Though it is easy to
calculate, it puts more weight on historical data when calculating VaR;
therefore, the accuracy is very low. Hence, this paper will only consider
parametric and semi-parametric approaches in evaluating the proposed semi-
parametric model.

The availability of high-frequency intra-day data has been used to propose a
wide range of accurate and efficient realized measures of volatility, which
have been very useful in evaluating volatility models. Realized variance (RV)
(andersen1998answering, andersen2003modeling), Realized Range (RR)
(christensen2007realized, martens2007measuring), Realized Kernel (RK)
(barndorff2009realized, Bi-power variation (BV) (barndorff2004power, sub-
sampled realized variance and sub-sampled realized range (wang2019bayesian)
and realized variance in different sampling frequencies (hansen2006realized)
can be found as frequently used realized measures in volatility modelling
literature. However, in this paper, we consider only RV, RK and BV as realized
measures and the derivations are given in Appendix (D).

The parametric approach uses GARCH-type models for volatility and risk
forecasting. Conventional GARCH models, such as the Autoregressive Conditional
Heteroskedastic (ARCH) by engle1982autoregressive and Generalized ARCH (GARCH)
by bollerslev1986generalized rely on daily squared returns when modelling the
volatility. Since then, numerous models such as EGARCH by
nelson1991conditional and GJR-GARCH by glosten1993relation were introduced to
capture the well-known leverage effect (wang2019bayesian). However, these
conventional GARCH models depend only on the daily returns, which are weak
signals of the volatility level. Therefore, when the volatility bounds to a
new level within a short period, these conventional GARCH-type models take
several periods to catch up with the new conditional variance level
(hansen2016exponential).

Therefore, to alleviate this problem, hansen2012realized include realized
measures in GARCH models. In 2012, hansen2012realized proposed a framework
called Realized GARCH, which enables joint modelling of returns and realized
volatility measures by using a measurement equation. Later
hansen2016exponential extend the realized GARCH framework to include multiple
realized measures, and the Realized Exponential GARCH (REGARCH) model is
introduced and dominates the models that rely on a single realized measure.
Since the tail risk forecast performance of these parametric models depends
heavily on the choice of error distribution, semi-parametric models, which
avoid this choice of error distribution, are considered in the literature.

engle2004caviar introduce the conditional auto-regressive VaR (CAViaR) model,
which directly estimates VaR, the quantile of the return distribution via
minimising the quantile loss function. However, in CAViaR models, no loss
function is uniquely minimised by ES in expectation, so they do not estimate
ES as it is not elicitable. The issue that ES is not elicitable is the
challenge in a semi-parametric context as ES has gained increasing attention
from investors, banking supervisors, regulators, counterparts of banks, and
risk managers (patton2019dynamic). After the implementation of the Basel III
accords in 2023, there is a need to find an effective and efficient model for
ES. For a risk measure to be elicitable, there should be a loss function such
that the risk measure is the solution to minimizing the expected loss.

fissler2016higher find a solution to the elicitable problem of ES and show
that the ES and VaR are jointly elicitable though it is not individually. This
finding has significant implications in risk forecasting literature. It opens
up new paths, especially for the researchers in the semi-parametric approach
to focus more on ES modelling. In the recent literature, researchers focus
more on modelling the dynamics of VaR and ES using the classes joint of loss
functions (fissler2016higher), which are strictly consistent for VaR and ES.
patton2019dynamic propose semi-parametric dynamic ES-VaR models based on the
Generalized autoregressive score models (GAS) framework and asymptotic
distribution theory for estimating these models.

In the same year, taylor2019forecasting proposes a new model to jointly
estimate dynamic models for VaR and ES called the ES-CAViaR model. He
constructs a quasi-likelihood that allows the joint estimation of conditional
VaR and ES by employing an asymmetric Laplace (AL) distribution with a time-
varying scale. In this paper, taylor2019forecasting shows that the quasi-
likelihood function corresponds to one of the loss functions suggested by
fissler2016higher. The MLE AL density in ES, which can estimate the
conditional scale and conditional quantile, is proposed there. Further, the
CAViaR model by engle2004caviar is used to estimate the VaR component directly
and proposed two new versions to estimate the ES component as ES-CAViaR-Add
and ES-CAViaR-Mult that use additive and multiplicative VaR to ES
relationships respectively. In 2020, gerlach2020semi incorporate realized
measure as an exogenous variable to ES-CAViaR models and propose the ES-X-
CAViaR-X model, which shows better results in forecasting VaR and ES.

wang2023semi introduce the Realized-ES-CAViaR framework considering
measurement equation and leverage effect in the model where the realized
measure separately influences VaR and ES, and it includes Realized-GARCH as a
special case. They extend the models in taylor2019forecasting and
gerlach2020semi in two methods when proposing the Realized-ES-CAViaR
framework. One links realized measures and tail risk using a measurement
equation, and the second uses separate VaR and ES dynamics. The quasi-
likelihood approach based on AL is employed, and Bayesian MCMC is used for
parameter estimation there. Further, this framework jointly forecasts VaR and
ES in one step, making practical implementation more convenient.

The main contribution of this paper is as follows. Firstly, we propose a new
semi-parametric joint VaR and ES forecasting framework incorporating multiple
realized measures under Bayesian estimation. The new framework is named
“Realized-ES-CAViaR-M”. Here, we consider a log specification by modifying the
ES-X-CAViaR-X model (gerlach2020semi) and Realized-ES-CAViaR framework
(wang2023semi). Secondly, multiple realized measures are used to jointly model
the relationship between VaR and ES as motivated by hansen2016exponential and
finally, followed by Realized EGARCH hansen2016exponential model, a
measurement equation is included in the framework to model the joint
dependence between the quantile and realized measures. In addition, an
adaptive Bayesian MCMC algorithm (chen2022dynamic) is used to estimate the
proposed model, including the parameters in the variance-covariance matrix.
Then, the VaR and ES forecasts from the proposed models are compared with
other models in parametric and semi-parametric approaches. Empirical data from
the 01st of January 2000 to the 30th of June 2022, including the forecasting
period of nearly a decade (2012 - 2022), supports the results of the proposed
model.

This paper is organized as follows. Section 2 reviews the relevant existing
parametric and semi-parametric models. Section 3 proposes the model. The
likelihood function and the adaptive Bayesian MCMC algorithm are presented in
section 4. Section 5 presents the empirical results. Section 6 concludes the
paper with future work.

##  2 Background models

This section describes the existing parametric and semi-parametric models used
to model VaR and ES in the literature. The properties of each model are
considered in developing the new framework. Some fundamental concepts used in
the model-developing approach are also discussed here.

Initially, we consider the GARCH(1,1) model the most popular among
practitioners. Consider a series of returns ${r_{t}}$ as follows.

$r_{t}=\mu_{t}+a_{t}=\mu_{t}+\sigma_{t}z_{t},$

$\sigma^{2}_{t}=\omega_{0}+\alpha_{0}a^{2}_{t-1}+\beta_{0}\sigma^{2}_{t-1},$

where $\sigma_{t}^{2}\equiv Var(a_{t}|\mathcal{L}_{t-1})$ and
$\mathcal{L}_{t-1}$ is the available information at time $t-1$, $a_{t}$ be the
mean ($\mu_{t}$) corrected return process. Here, it is assumed that $z_{t}$ is
a sequence of iid random variables with 0 mean and variance 1. Following a
GARCH-type volatility model in tail risk estimation, a new concept of risk
parameter in conditional volatility models is introduced by francq2015risk and
gao2008estimation where VaR and ES are considered constant multiples of the
latent volatility. In this form, it is assumed a zero mean, constant
conditional return distribution.

$\displaystyle Q_{t}$ $\displaystyle=a_{\alpha}\sigma_{t},$ $\displaystyle
ES_{t}$ $\displaystyle=b_{\alpha}\sigma_{t},$
$\displaystyle\frac{ES_{t}}{\sigma_{t}}$
$\displaystyle=\frac{b_{\alpha}}{a_{\alpha}}=c_{\alpha},$ (1)

where $a_{\alpha},b_{\alpha}$ and $c_{\alpha}\geq 1$ are constant and depend
on the return distribution and $Q_{t}$ is the quantile and ES at time t is the
expected shortfall. To estimate $a_{\alpha}$ and $b_{\alpha}$, historical
simulation (HS) is applied to the standardized residuals. The $\sigma_{t}$ is
estimated using quasi-maximum likelihood (QML) with no assumption for the
distribution for returns (gao2008estimation).

###  2.1 Realized EGARCH model

hansen2016exponential propose the realized EGARCH model (REGARCH), which
considers multiple realized measures of volatility, where K denotes the number
of realized measures included in the model. In this model, the GARCH equation
is $k$-dimensional vectors, which enable multiple realised volatility
measures. A standard deviation REGARCH specification can be defined as;

$\displaystyle r_{t}$ $\displaystyle=\mu_{t}+\sigma_{t}z_{t},$
$\displaystyle\mathrm{log}(\sigma_{t})$
$\displaystyle=\omega+\beta\mathrm{log}(\sigma_{t-1})+\tau_{1}z_{t-1}+\tau_{2}\left(z_{t-1}^{2}-1\right)+\boldsymbol{\gamma}^{T}\mathbf{u}_{t-1},$
$\displaystyle\mathrm{log}(x_{j,t})$
$\displaystyle=\xi_{j}+\varphi_{j}\mathrm{log}(\sigma_{t})+\delta_{11}z_{t}+\delta_{12}(z_{t}^{2}-1)+\mathbf{u}_{j,t},\;$
(2)

where j = 1, 2, …, k. $\mathbf{x}_{t}=(x_{1,t},...,x_{K,t})^{T}$ is the square
root of the realized measure on the same scale as volatility $\sigma_{t}$.
$\mu_{t}$ is the conditional mean and $\sigma_{t}$ is the conditional
volatility.
$z_{t}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,1\right)$ and
${u}_{t}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,\Sigma\right)$
where $\mathbf{u}_{t}=(u_{1,t},u_{2,t},...,u_{K,t})^{T}$. In this model,
$z_{t}$ and ${u}_{t}$ are mutually and serially independent. Further, the
coefficient $\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{K})^{T}$ of
${u}_{t-1}$ represents how informative the realized measures are about future
volatility. In this model, they have used two sets of leverage functions
$\tau(\cdot)$ and $\delta_{k}(\cdot)$ ($k=1,2,...,K$) to facilitate modelling
of the dependence between $z_{t}$ and $u_{t}$ realistic in practice. The
quadratic version for the leverage functions are adopted in the empirical
analysis. Furthermore, the Quasi-maximum likelihood estimation (QMLE) is used
in the estimation after some modifications and extensions to the previous work
of hansen2012realized. In the above equation (2), the first equation is the
return equation, the next one is the GARCH equation, and the last equation is
the measurement equation. The most important feature of the REGARCH model is
the measurement equation, which enables the incorporation of multiple realized
measures into the model.

###  2.2 ES-X-CAViaR-X model

taylor2019forecasting proposes a joint model for the conditional quantile and
conditional ES where parameters are estimated by maximum likelihood based on
an AL density, which we will explain later. For the VaR component, they
propose a CAViaR model introduced by engle2004caviar; for the ES component,
they use two formulations. The first ES formulation (ES-CAViaR-mult) expresses
ES as the product of the quantile and a constant multiplicative factor with an
exponential function of an unconstrained parameter to avoid VaR and ES
crossing. In the second ES formulation (ES-CAViaR-Add) the difference between
ES and the quantile is modelled using an autoregressive (AR) term to smooth
the magnitude of exceedances beyond the quantile.

gerlach2020semi extend the ES-CAViaR-Add model by allowing the influence of
realized measure to both VaR and ES separately. Incorporating the realized
measure into the ES dynamics has improved the ES estimation and forecasting
performances. Though they have different specifications for $Q_{t}$ and
$ES_{t}$, the AL-based likelihood formation for the ES-X-CAViaR-X model is
identical to the ES-CAViaR models. ES-X-CAViaR-X model can be defined as
follows.

$\displaystyle Q_{t}$
$\displaystyle=\beta_{0}+\beta_{1}X_{t-1}+\beta_{2}Q_{t-1},$
$\displaystyle\omega_{t}$
$\displaystyle=\gamma_{0}+\gamma_{1}X_{t-1}+\gamma_{2}\omega_{t-1},$
$\displaystyle ES_{t}$ $\displaystyle=Q_{t}-\omega_{t},$ (3)

where $\gamma_{0}\geq 0,\gamma_{1}\geq 0,\gamma_{2}\geq 0,$ to ensure that VaR
and ES do not cross. They claim that the ES-X-CAViaR-X model with sub-sampled
Realized variance (RV) and sub-sampled Realized range (RR) produces the most
accurate VaR and ES forecasting results in their empirical study.

###  2.3 Realized-ES-CAViaR models

wang2023semi propose a Realized-ES-CAViaR framework, including various model
versions that can jointly forecast VaR and ES in one step, which is very
convenient in practical implementation. However, out of all the proposed
models, we only consider the Realized-ES-CAViaR-Add-X model as it relates to
our focused area in this paper. In this model, the realized measure can
influence VaR and ES separately through the dynamics of $\omega_{t}$, which
includes the realized measure. Also, the ratio of ES to VaR is allowed to
change at each observation and allows conditional return distribution to
change over time. Here, they have substituted
$\sigma_{t}=\frac{Q_{t}}{a_{\alpha}}$ and $z_{t}=\epsilon_{t}a_{\alpha}$ into
the GARCH and measurement equation in standard deviation form of the Realized
GARCH model (hansen2012realized). The realized-ES-CAViaR-Add-X model can be
defined as follows.

$\displaystyle Q_{t}$
$\displaystyle=\beta_{0}+\beta_{1}X_{t-1}+\beta_{2}Q_{t-1},$ $\displaystyle
X_{t}$
$\displaystyle=\xi+\phi|ES_{t}|+\tau_{1}\epsilon_{t}+\tau_{2}\left(\epsilon^{2}_{t}-E(\epsilon^{2})\right)+u_{t},$
$\displaystyle ES_{t}$ $\displaystyle=Q_{t}-\omega_{t},$
$\displaystyle\omega_{t}$
$\displaystyle=\gamma_{0}+\gamma_{1}X_{t-1}+\gamma_{2}\omega_{t-1},$ (4)

where $\gamma_{0}\geq 0,\gamma_{1}\geq 0,\gamma_{2}\geq 0$ to ensure VaR and
ES do not cross. In this model, VaR and ES are influenced by the realized
measure separately through the quantile, the $\omega_{t}$, and the ES
equation. The specification of $\omega_{t}$ is directly generalized from a
GARCH-type model, allowing the conditional return distribution to change over
time by employing different time-varying ES to VaR relationships. Compared to
the ES-X-CAViaR-X model (gerlach2020semi), this model includes the measurement
equation and the leverage effect is also considered.

##  3 Proposed Model

This paper proposes a new Realized-ES-CAViaR-M model, including multiple
realized measures to jointly and semiparametrically estimate VaR and ES. When
deriving the model, it extracts the features of the ES-X-CAViaR-X model of
gerlach2020semi where the realized measure influence to both VaR and ES
separately, and the Realized-ES-CAViaR-Add model of wang2023semi.

In this research, we consider $\alpha=2.5\%$; therefore, the quantile $Q_{t}$
is negative for the $\alpha$ value we consider. Therefore, in the Realized-ES-
CAViaR-M model, we include the log specification with the negative quantile to
be consistent with the log definition. The proposed Realized-ES-CAViaR-M model
contains four equations as quantile equation (5), measurment equation (6),
$\omega_{t}$ equation (7), and ES equation (8) and the model framework is as
follows.

$\displaystyle\mathrm{log}(|Q_{t}|)$
$\displaystyle=\beta_{0}^{*}+\beta_{1}^{*}\mathrm{log}{(|Q_{t-1}|)}+\tau_{1}^{*}\epsilon_{t-1}+\tau_{2}^{*}(\epsilon_{t-1}^{2}-E(\epsilon^{2}))+\gamma_{j}^{*}\mathbf{u}_{j,t-1},$
(5) $\displaystyle\mathrm{log}(X_{j,t})$
$\displaystyle=\xi_{j}+\phi^{*}_{j}\mathrm{log}(|Q_{t}|)+\delta_{j,1}^{*}\epsilon_{t}+\delta_{j,2}^{*}(\epsilon_{t}^{2}-E(\epsilon^{2}))+\mathbf{u}_{j,t},$
(6) $\displaystyle\omega_{t}$
$\displaystyle=\nu_{0}+\nu_{1}\omega_{t-1}+\psi_{j}|\mathbf{u}_{j,t-1}|,$ (7)
$\displaystyle ES_{t}$ $\displaystyle=Q_{t}-\omega_{t},$ (8)

where $j=1,2,...,k$. $X_{t}$ is the square root of the realized measure on the
same scale as volatility $\sigma_{t}$. $\mu_{t}$ is the conditional mean and
$\sigma_{t}$ is the conditional volatility. Further, the coefficient
$\boldsymbol{\gamma}=(\gamma_{1},...,\gamma_{K})^{T}$ of $\mathbf{u}_{t-1}$
represents how informative the realized measures are about future volatility.

In the quantile equation (5), we consider the log specification of the
absolute quantile value. In this research, we consider $\alpha=2.5\%$.
Therefore, deriving this equation from the standard deviation REGARCH equation
(2), we substitute $\sigma_{t}=\frac{-Q_{t}}{-a_{\alpha}}$ to be consistent
with the log definition and $z_{t}=\epsilon_{t}a_{\alpha}$ in the leverage
effect. Following the same specifications of Realized-ES-CAViaR model of
wang2023semi,
$z_{t}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,1\right)$ and
${u}_{t}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,\Sigma\right)$.
$\Sigma$ is the variance-covariance matrix where the dimensions depend on the
number of realized measures considered. In this model, $z_{t}$ and ${u}_{t}$
are mutually and serially independent. Though the quantile equation of the
Realized-ES-CAViaR model doesn’t include the leverage effect, we keep the
quadratic version of the leverage effect here as it is empirically important
to model the dependence between return shocks and volatility shocks. K
dimensional $\mathbf{u}_{j,t-1}$ enables the use of multiple realized measures
of volatility and $\gamma_{j}^{*}$ shows how informative the realized measures
about future volatility as explained by REGARCH model.

The measurement equation (6) completes the model by providing a way to model
the joint dependence between the return and realized measures
(hansen2016exponential). The leverage function in the measurement equation
generates an asymmetric response in volatility to return shocks and uses the
same specifications as the quantile equation. However, as the Realized-ES-
CAViaR model, we don’t include the time-varying $ES_{t}$ in the measurement
equation. Instead, we use the $Q_{t}$ here, which is obtained in the process
of deriving the equation from the model (2) under the log specifications. For
the measurement equation errors $\epsilon_{t}$, we use the specifications as
in Realized-ES-CAViaR models wang2023semi.

The novelty of this research is in $\omega_{t}$ equation (7), which is not
included in the REGARCH model though we derived it by substituting
$\sigma_{t}=\frac{\omega_{t}}{g_{\alpha}}$, to the GARCH equation in (2) when
$Q_{t}=a_{\alpha}\sigma_{t}$, $ES_{t}=b_{\alpha}\sigma_{t}$ and
$\omega_{t}=Q_{t}-ES_{t}=(a_{\alpha}-b_{\alpha})\sigma_{t}=g_{\alpha}\sigma_{t}$.
However, to simplify the framework, we didn’t consider the log form for
$\omega_{t}$ and also ignored the leverage effect. More importantly, after
considering several versions of this model, we introduce the absolute value of
the error $\mathbf{u}_{j,t-1}$ here, which differs from the Realized-ES-
CAViaR-Add-X model specifications. The proposed model’s parameters in the
$\omega_{t}$ are restricted by convention to ensure that the VaR and ES do not
cross. Therefore, $\nu_{0}\geq 0,\nu_{1}\geq 0,\psi_{j}\geq 0$. With the new
specification for $\omega_{t}$, the multiple realized measures impact both
quantile $Q_{t}$ and ES as $ES_{t}$ is a function of $Q_{t}$.

ES equation (8) is the additive difference $\omega_{t}$, so the multiple
realized measures can affect and influence the VaR and ES separately. But the
other versions, such as a multiplicative version of this equation, could also
be developed, like the models proposed in the Realized-ES-CAViaR framework.
Since the main objective of this research is to use multiple realized measures
in the model, we limit our focus to the additive version of the model. By
allowing multiple realized measures, we expect to improve the ES estimation
and forecasting performance further.

A stationary condition is enforced during the estimation to guarantee that the
series $Q_{t}$ does not diverge. When k is the number of realized measures in
the model, the condition, $\beta_{1}-\gamma_{k}^{1}\phi_{k}<1$, is derived by
substituting the measurement equation into the quantile equation and the
detailed proof is given in Appendix A.

##  4 Likelihood and model estimation

engle2004caviar mention that the solutions to optimising the CAViaR type
models depend on the chosen initial values. For the ES-CAViaR models,
taylor2019forecasting uses the same approach as engle2004caviar to generate
good initial values for the AL-based QML estimation. However, via simulation
study, gerlach2020semi claim that the Bayesian estimator is more accurate than
the AL-based QML estimator. Therefore, following the work of gerlach2020semi,
this research mainly focuses on using Bayesian estimation as the estimation
methodology. However, maximum likelihood estimation (MLE) is also used to
compare the model performances. Appendix C presents the simulation study
details to compare proposed model performances in MLE and Bayesian estimation
based on the parameter estimation and one-step-ahead VaR and ES forecast
accuracy.

###  4.1 Likelihood function for the proposed model

taylor2019forecasting uses the following log-likelihood function to jointly
estimate the VaR and ES semi-parametrically in his ES-CAViaR models. There, he
notes that the negative logarithm of the resulting likelihood function is
strictly consistent for $Q_{t}$ and $ES_{t}$ jointly, meaning that it fits
into the class of strictly consistent loss functions developed by
fissler2016higher. Though the likelihood function in equation (9) estimates
VaR and ES jointly, the observations $r_{t}$ are not assumed to follow an AL
distribution. Later, the same AL-based likelihood function is employed by
wang2023semi in their Realized-ES-CAViaR model and gerlach2020semi in their
ES-X-CAViaR-X model.

$\displaystyle L(r:\theta)$
$\displaystyle=\sum_{t=1}^{n}\left(\log\frac{(\alpha-1)}{ES_{t}}+\frac{(r_{t}-Q_{t})(\alpha-I(r_{t}\leq
Q_{t}))}{\alpha ES_{t}}\right),$ (9)

where $r=\left\\{r_{1},r_{2},...r_{n}\right\\}$ and $n$ is the in-sample size
and parameter vector is represented by $\theta$.

However, the proposed model framework consists of a measurement equation to
add the multiple realized measures into the framework. Therefore, the AL-based
likelihood estimation of ES-CAViaR models (taylor2019forecasting) is combined
with the likelihood of the measurement equation in the joint likelihood
function. Also, the variance-covariance matrix is considered in the likelihood
function.

$\displaystyle L(r,X:\theta)=L(r:\theta)+L(x|r:\theta),$

Following hansen2016exponential the likelihood function of the measurement
equation, $L(x|r;\theta)$ is shown as follows:

$\displaystyle L(x|r:\theta)$
$\displaystyle=-\frac{1}{2}\left(\sum_{t=1}^{n}k\log(2\pi)+\log(\left|\Sigma\right|)+U_{T}^{t}\Sigma^{-1}U_{t}\right),$
(10)

By including a measurement equation in the proposed model with multiple
realized measures, we expect to derive more accurate VaR and ES forecasts.
Finally, the log-likelihood of the proposed models can be written as follows:

$\begin{split}L(r,x:\theta)=&\sum_{t=1}^{n}\left(\log\frac{(\alpha-1)}{ES_{t}}+\frac{(r_{t}-Q_{t})(\alpha-I(r_{t}\leq
Q_{t}))}{\alpha ES_{t}}\right)\\\
&-\frac{1}{2}\left(\sum_{t=1}^{n}k\log(2\pi)+\log(\left|\Sigma\right|)+U_{T}^{t}\Sigma^{-1}U_{t}\right),\end{split}$
(11)

In this model, we estimate the model parameters by maximizing the above log-
likelihood function for $\theta$ and $\Sigma$. hansen2016exponential show that
the above function has a convenient structure where the partial maximization
concerning $\Sigma$ for a given value of $\theta$ has a simple solution as
follows:

$\hat{\Sigma}(\theta)=\frac{1}{n}\sum_{t=1}^{n}U_{t}(\theta){U_{t}(\theta)}^{\prime}$

hansen2016exponential point out that $U_{t}$ in the above equation depends on
$\theta$ but not depends on the covariance matrix $\Sigma$. Following the same
proof by them, the below statement is used to calculate the log-likelihood
$L(x|r:\theta)$.

$\sum_{t-1}^{n}U_{t}^{T}(\theta)\Sigma(\theta)^{-1}U_{t}(\theta)=tr\left\\{\sum_{t=1}^{n}\hat{\Sigma}(\theta)^{-1}U_{t}(\theta)U_{t}^{T}(\theta)\right\\}$

This research employs Maximum likelihood and Bayesian estimation and compares
the performance via a simulation study. Results favour the Bayesian
estimation; hence, it is utilised subsequently for estimation and forecasting
for the proposed model with the above likelihood function $L(r,x:\theta)$ and
prior distribution $\pi(\theta)$ specification, which is explained in the
below section.

###  4.2 Maximum likelihood estimation

The proposed model uses multiple realized measures, and the number of
parameters increases with the number of realized measures. For example, with
three realized measures, the total number of parameters in the model is 30.
Appendix C compares the parameter estimates for MLE and MCMC methods for the
proposed Realized-ES-CAViaR-M model with one and two realized measures. The
results favour the MCMC method in terms of precision and bias. Therefore, we
consider Bayesian estimation in this research.

###  4.3 Bayesian estimation

taylor2019forecasting uses AL-based QML estimation for the ES-CAViaR models.
Via a simulation study, gerlach2020semi prove that the Bayesian estimator is
more accurate than the AL-based QML estimator. Therefore, we use the adaptive
Bayesian MCMC approach as chen2022dynamic to estimate the parameters in the
proposed model.

Priors are chosen as mostly uninformative over the parameters’ possible
stationarity and positivity region A, with a standard Jeffreys prior for the
scale parameters, $\Sigma_{jj}$. The indicator functions $I(A)=1$ over the
region A and zero otherwise. This is in line with the previous work of
gerlach2016forecasting, contino2017bayesian, gerlach2017semi, and
chen2022dynamic, which have used Bayesian MCMC for model estimation.

$\displaystyle\pi(\theta)\propto\frac{1}{\Sigma_{jj}}I(A),$ (12)

where $\Sigma_{jj}$ is the variance and covariance matrix. The stationary
condition discussed in Appendix A is applied for Bayesian and MLE estimations.

Following chen2022dynamic, to tune the proposal distribution with a
satisfactory level of mixing, we update the parameter vector in blocks. Our
$\theta$ has 30 dimensions when considering the proposed model with three
realized measures, and using the blocking structure helps to avoid the
difficulties in tuning. Blocks are chosen so that parameters between blocks
are less correlated in the posterior, and parameters within a block tend to be
more correlated in the posterior. Appendix B describes the details of the
blocking structure, including the parameters in a block and the number of
blocks used based on the number of realised measures in the model. The block-
wise proposals are generated and accepted with the usual Metropolis acceptance
probability as described in chen2022dynamic.

Here, we chose the proposal density to be a mixture of multivariate normals
with a different scale for each component. After examining several
combinations, we finally choose the vector of mixing weights $(w)$ as (0.7,
0.15, 0.15) and the corresponding vector of component scales $(s)$ to be (1,
100, 0.01), which is applied irrespective of the number of realized measures
in the proposed model. This prevents the chain from getting stuck altogether
or in specific dimensions of the parameter space. With $w$, $s$ and a fixed
covariance matrix, the scale is automatically tuned for each block during a
tuning epoch using the function explained in chen2022dynamic. Here, we use the
target acceptance rate according to the size of the block by following the
standards of the roberts1997weak where the acceptance rate of 0.44% for
$d_{i}=1$, 0.35 when $2\leq d_{i}\leq 4$ and 0.234 for $d_{i}>4$. The initial
scale of each block is set to $\frac{2.4}{\sqrt{d_{i}}}I_{d_{i}}$ where
$d_{i}$ is the dimension of the $i^{th}$ block and $I_{d_{i}}$ is the identity
matrix of dimension $d_{i}$. After tuning the scale of the proposal
distribution effectively, the MCMC sampler is run for N = 150,000 to enhance
the convergence and considers the average of the last 5,000 results to
calculate the optimum parameter set and tail risk forecasts.

The simulation study in Appendix C compares the properties and performances of
the Bayesian and MLE methods for the proposed Realized-ES-CAViaR-M Model for
the one-step-ahead VaR and ES forecasting accuracy and parameter estimation.
Results suggest that using the MCMC estimator is more beneficial, and the
proposed model is more effective in forecasting VaR and ES.

##  5 Data and Empirical study

###  5.1 Realized measures

This research aims to check the changes in model performance when including
multiple realized measures. Therefore, the correlations between the realized
measures should be considered in selecting realized measures for the empirical
study. Table 1 shows the correlations between all the realized measures
considered for S&P500. There are strong positive correlations between the
realized measures. However, we avoided using the combinations of measures with
the correlations coefficient = 1 as we could not get the converged results in
Bayesian estimation with those combinations.

Table 1: Pearson correlation coefficients within the realized measures | RV5 | RV5$\\_$SS | Rk$\\_$Parzen | BV | BV$\\_$SS | RV10 | RV10$\\_$SS
---|---|---|---|---|---|---|---
RV5 | 1 |  |  |  |  |  |
RV5$\\_$SS | 1 | 1 |  |  |  |  |
Rk$\\_$Parzen | 0.93 | 0.93 | 1 |  |  |  |
BV | 0.97 | 0.97 | 0.92 | 1 |  |  |
BV$\\_$SS | 0.97 | 0.97 | 0.92 | 1 | 1 |  |
RV10 | 0.98 | 0.98 | 0.94 | 0.96 | 0.96 |  |
RV10$\\_$SS | 0.98 | 0.98 | 0.94 | 0.96 | 0.96 | 1 | 1

Finally, considering all the issues and our limited time, we used only three
realized measures for the empirical study. Therefore, Realized variance (5
minutes) (RV5), Realized kernel (RK), and Bi-power variation (BV) are selected
for this research. Details of the selected realized measures are presented in
Appendix D.

###  5.2 Data description

Daily closing prices and realized measure data are downloaded from Oxford-man
Institute’s realized library (heber2009omi) from January 2000 to June 2022.
Daily return values are calculated using daily price data. Six market indices
are used in this research, including S&P500 and NASDAQ in the US, FTSE 100
(UK), DAX (Germany), SMI (Swiss), and HSI (Hong Kong). Each data series is
split into an initial in-sample period, from January 2000 to December 2011,
covering 12 years, and a forecasting period from 2012 January to June 2022 for
nearly ten years. Our in-sample period includes the global financial crisis
(GFC), and our forecast period includes COVID-19. Sample sizes and forecasting
periods are different for each series due to different non-trading days in
each market over the period we have considered. Table 2 represents The total
number of sample observations for each market considered and the number of
observations in in-sample and out-of-sample.

Table 2: Summary of the selected sample Index | Total no of observations | In-sample observations (n) | Out-of-sample observations (m)
---|---|---|---
S&P500 | 5634 | 3008 | 2626
FTSE | 5667 | 3020 | 2647
NASDAQ | 5636 | 3006 | 2630
HSI | 5504 | 2937 | 2567
DAX | 5697 | 3050 | 2647
SMI | 5634 | 3013 | 2621

Figure 1 displays the time series plot of the absolute value of daily return,
RV, BV, and RK of S&P500 as an example.

Figure 1: S&P500 Absolute return series and realized measures

Table 3 shows the descriptive statistics for each of the six markets
considered in this study. Considering the standard deviation values, the
emerging market NASDAQ has the highest standard deviation compared to other
matured markets. NASDAQ has the highest mean return; for all markets, mean
returns are positive. Further, negative skewness is observed for all the
markets where SMI has the lowest skewness.

Table 3: Summary statistics of the return series. Market | N | Mean | Std | Skewness | Kurtosis | Min | Max
---|---|---|---|---|---|---|---
S&P500 | 4744 | 0.0119 | 1.1972 | -0.1988 | 8.1992 | -9.6883 | 10.642
FTSE | 5667 | 0.0018 | 1.1617 | -0.3006 | 7.1731 | -10.1365 | 9.48487
NASDAQ | 5636 | 0.0176 | 1.5926 | -0.1867 | 6.3717 | -13.1409 | 13.2782
HSI | 5504 | 0.0040 | 1.4673 | -0.0812 | 7.4245 | -13.5820 | 13.4068
DAX | 5697 | 0.0118 | 1.4564 | -0.1576 | 6.2969 | -11.8631 | 12.0268
SMI | 5634 | 0.0070 | 1.1539 | -0.3258 | 8.2299 | -10.1339 | 10.7876

###  5.3 Tail risk forecasting

Daily one-step-ahead VaR and ES forecasts are considered for all six series
using the quantile level $\alpha=2.5\%$ recommended by the Basel III Capital
Accord de2019minimum. A rolling window, with fixed in-sample size (n), one-
step-ahead forecasts of VaR and ES at 2.5% quantile level is employed for each
series. Table 2 shows the details for each market.

On average, around m = 2626, One-step-ahead, VaR and ES forecasts are
generated for each return series from 29 models, with the average in-sample
size around n = 3008. Table 4 shows the summary of the competing models. In
our proposed Realized-ES-CAViaR-M framework, We use RV5 (5 minutes), RK and BV
as the three main realized measures to test the performance of multiple
realized measures. In total, there are seven different versions of the
proposed model itself, including the three models with one realized measure
(RV, RK, BV) each, three models with two realized measures each with the
combinations of RV5-RK, RV5-BV, RK-BV and one model with three realized
measures (RV5-RK-BV) in the model. For each market, these seven models are
compared with the existing models in the tail risk forecasting literature.

We use conventional GARCH (bollerslev1986generalized), EGARCH
(nelson1991conditional), and GJR-GARCH models (glosten1993relation) all with
student-t errors and Gaussian errors and used MLE for estimation using the
Econometrics toolbox in Matlab. However, these models only depend on the
return series and do not include realized measures. Further, we use the
parametric REGARCH model (hansen2016exponential) with the same realized
measures RV5, RK, and BV. Gaussian specifications and Bayesian estimation were
employed for parameter estimation in the REGARCH model.

From the semi-parametric field, we use the recent developments in the
literature and consider the ES-CAViaR-add-X model, ES-X-CAVIaR-Add-X models
(gerlach2020semi) and Realized-ES-CAViaR-Add-X model (wang2023semi) with
adaptive MCMC for the comparison. Thus, there are a total of 29 models to be
compared, including the seven models in the proposed Realized-ES-CAViaR-M.

Table 4: A summary of the competing models in the empirical section is based on the model type, realized measures used, and the number of realized measures (k). ’NA’ represents that the model does not use any realized measures. Model | Type | Realized measures | K
---|---|---|---
GARCH Models |  |  |
GARCH-t | Parametric | NA | 0
EGARCH-t | Parametric | NA | 0
GJR-GARCH-t | Parametric | NA | 0
GARCH-QML | Parametric | NA | 0
EGARCH-QML | Parametric | NA | 0
GJR-GARCH-QML | Parametric | NA | 0
REGARCH Models |  |  |
REGARCH-GG-RV5 | Parametric | RV5 | 1
REGARCH-GG-RK | Parametric | RK | 1
REGARCH-GG-BV | Parametric | BV | 1
REGARCH-GG-RV5-RK | Parametric | RV5, RK | 2
REGARCH-GG-RV5-BV | Parametric | RV5, BV | 2
REGARCH-GG-RK-BV | Parametric | RKBV | 2
REGARCH-GG-RK-RV5-BV | Parametric | RV5, RK, BV | 3
ES-CAViaR-Add-X Models |  |  |
ES-CAViaR-Add-RV5 | Semi-parametric | RV5 | 1
ES-CAViaR-Add-RK | Semi-parametric | RK | 1
ES-CAViaR-Add-BV | Semi-parametric | BV | 1
ES-X-CAViaR-X-Add Models |  |  |
ES-RV5-CAViaR-Add-RV5 | Semi-parametric | RV5 | 1
ES-RK-CAViaR-Add-RK | Semi-parametric | RK | 1
ES-BV-CAViaR-Add-BV | Semi-parametric | BV | 1
Realized-ES-CAViaR-Add-X Models |  |  |
RESCAViaR-ADD-RV5 | Semi-parametric | RV5 | 1
RESCAViaR-ADD-RK | Semi-parametric | RK | 1
RESCAViaR-ADD-BV | Semi-parametric | BV | 1
Realized-ES-CAViaR-M Models |  |  |
Realized-ES-CAViaR-M-RV5 | Semi-parametric | RV5 | 1
Realized-ES-CAViaR-M-RK | Semi-parametric | RK | 1
Realized-ES-CAViaR-M-BV | Semi-parametric | BV | 1
Realized-ES-CAViaR-M-RV5-RK | Semi-parametric | RV5, RK | 2
Realized-ES-CAViaR-M-RV5-BV | Semi-parametric | RV5, BV | 2
Realized-ES-CAViaR-M-RK-BV | Semi-parametric | RK, BV | 2
Realized-ES-CAViaR-M-RV5-BV-RK | Semi-parametric | RV5, RK, BV | 3

####  5.3.1 Assessing Value-at-Risk

This section shows the results of assessing VaR. All semi-parametric and
REGARCH models are estimated using the MCMC and include realized measures. In
contrast, the GARCH-type models are estimated using the MLE approach.

First, the VaR violation rate (VRate) is initially employed in the literature
to assess VaR forecasting accuracy. Though it is an informal test, we use it
as a model evaluation criterion to compare our results with the previous
models. VRate is the proportion of returns in the forecast period that exceed
the forecast VaR level.

$\displaystyle\textup{VRate
}=\frac{1}{m}\sum_{t=n+1}^{n+m}I(r_{t}<\hat{Q}_{t})$ (13)

where n is the in-sample size and m is the forecasting sample size. We prefer
models with VRate values close to our nominal quantile level, $\alpha=2.5\%$
or equivalently $\frac{\textup{VRate}}{\alpha}$ closest to 1. Table 5
summarizes the VRates divided by the target violation rates, $\alpha=2.5\%$
for each model over the six return series. The mean absolute deviation in the
“MAD” column employs $2.5\%$ as the target VRate across the six indices. A box
indicates the model with the ratio of VRate to $\alpha$ is closest to one in
each market. “Average Rank” is the average of the ranks across the six
markets, calculated by considering the absolute deviation of each VRate from
$2.5\%$. The mean absolute values of these deviations (MAD) are presented in
the “MAD” columns. Boxes indicate the best models for these measures, and the
blue text represents the second-ranked models.

Table 5: $2.5\%$ VaR Forecasting $\frac{VRate}{\alpha=2.5\%}$ with different
models on seven indices

Model | S&P500 | FTSE | NASDAQ | HSI | DAX | SMI | MAD | Avg Rank
---|---|---|---|---|---|---|---|---
GARCH Models |  |  |  |  |  |  |  |
GARCH-t | 1.4318 | 1.5867 | 1.5361 | 1.4647 | 1.6018 | 1.3583 | 1.2414 | 21.75
EGARCH-t | 1.3709 | 1.4054 | 1.5057 | 1.4336 | 1.3298 | 1.4040 | 1.0206 | 19.08
GJR-GARCH-t | 1.2947 | 1.4809 | 1.5057 | 1.3401 | 1.3298 | 1.3430 | 0.9559 | 17.25
GARCH-QML | 1.4928 | 1.6320 | 1.6274 | 1.5115 | 1.6925 | 1.4193 | 1.4065 | 25.17
EGARCH-QML | 1.3557 | 1.4507 | 1.5361 | 1.4492 | 1.3449 | 1.3735 | 1.0459 | 19.75
GJR-GARCH-QML | 1.3861 | 1.5111 | 1.5209 | 1.3245 | 1.3903 | 1.3735 | 1.0443 | 18.92
REGARCH Models |  |  |  |  |  |  |  |
GG-RV5 | 1.8126 | 1.6169 | 1.6274 | 1.4024 | 1.8587 | 1.5261 | 1.6017 | 26.25
GG-RK | 1.7060 | 1.6471 | 1.5665 | 1.3401 | 1.7832 | 1.3888 | 1.4299 | 23.75
GG-BV | 1.7517 | 1.6169 | 1.5665 | 1.3401 | 0.7556 | 1.5872 | 1.2945 | 20.67
GG-RV5-RK | 1.7365 | 1.6169 | 1.5665 | 1.3557 | 1.8285 | 1.5261 | 1.5126 | 24.83
GG-RV5-BV | 1.7060 | 1.6320 | 1.5665 | 1.3401 | 1.8738 | 1.5872 | 1.5440 | 25.33
GG-RK-BV | 1.7060 | 1.6774 | 1.5665 | 1.3401 | 1.7832 | 1.6177 | 1.5379 | 25.25
GG-RK-RV5-BV | 1.6603 | 1.6169 | 1.4905 | 1.4024 | 1.8285 | 1.5261 | 1.4686 | 23.08
ES-CAViaR-Add Models |  |  |  |  |  |  |  |
RV5 | 1.2490 | 1.2845 | 1.2471 | 1.0908 | 1.4507 | 1.1751 | 0.6238 | 15.75
RK | 1.1577 | 1.3147 | 1.2471 | 0.9661 | 1.4356 | 1.0683 | 0.5239 | 12.17
BV | 1.1272 | 1.2694 | 1.3080 | 0.9817 | 1.4205 | 1.2209 | 0.5685 | 13.08
ES-X-CAViaR-X-Add Models |  |  |  |  |  |  |  |
RV5 | 1.0206 | 1.2694 | 1.1559 | 1.1063 | 1.3449 | 1.0530 | 0.3959 | 12.00
RK | 1.0510 | 1.2391 | 1.0798 | 1.0596 | 1.3147 | 0.9920 | 0.3134 | 9.08
BV | 1.0358 | 1.2694 | 1.0342 | 1.0284 | 1.3147 | 1.0683 | 0.3128 | 9.17
Realized-ES-CAViaR-Add-X Models |  |  |  |  |  |  |  |
RV5 | 0.9596 | 1.2240 | 1.0951 | 1.0596 | 1.3600 | 1.0836 | 0.3595 | 10.17
RK | 1.0206 | 1.1485 | 1.0342 | 1.0908 | 1.3147 | 1.0072 | 0.2567 | 8.00
BV | 1.0053 | 1.2694 | 1.0038 | 0.9973 | 1.2694 | 1.1141 | 0.2770 | 8.67
Realized-ES-CAViaR-M Models |  |  |  |  |  |  |  |
RV5 | 0.9596 | 1.1787 | 0.9734 | 1.0908 | 1.3298 | 1.0836 | 0.3125 | 8.67
RK | 0.9901 | 1.1334 | 0.9886 | 1.0596 | 1.2543 | 0.9462 | 0.2177 | 4.83
BV | 0.9901 | 1.1938 | 0.9430 | 1.0440 | 1.2694 | 1.0988 | 0.2804 | 6.75
RV5-RK | 0.9596 | 1.1636 | 0.9582 | 1.0596 | 1.2694 | 1.0530 | 0.2616 | 5.42
RV5-BV | 0.9292 | 1.1636 | 0.9582 | 1.0752 | 1.2694 | 1.0988 | 0.2998 | 6.58
RK-BV | 0.9596 | 1.2089 | 0.9582 | 1.0284 | 1.2391 | 1.0378 | 0.2485 | 4.42
RV5-BV-RK | 0.9292 | 1.1787 | 0.9430 | 1.0440 | 1.2391 | 1.0836 | 0.2805 | 4.50

Note: Boxes indicate the favoured models, and the blue text indicates the
second-ranked model in each column.

Table 5 shows that among all the models considered, irrespective of the
market, the overall best-ranked models are the proposed Realized-ES-CAViaR-M
models based on the MAD and average rank. All seven models in the proposed
Realized-ES-CAViaR-M, including 1, 2, and 3 realized measures, are among the
top 10 models when considering the MAD and average ranks.

Overall, in the proposed Realized-ES-CAViaR-M model, when considering RK and
BV individually and combined, they perform better than the models, including
RV5. Only for the DAX market, the proposed model with all three realized
measures and the proposed model with RK and BV have been identified as the
best models. Therefore, RK and BV have a more significant impact on the model
performance than the RV5.

In addition to the VRate, the standard quantile loss function in equation (14)
is also employed to compare the VaR forecast accuracy. The model that gives
the minimum quantile loss function value is considered the most accurate VaR
forecast.

$\displaystyle\frac{1}{m}\sum_{t=n+1}^{n+m}\left(\alpha-I(r_{t}\leq\widehat{Q}_{t})\right)\left(r_{t}-\widehat{Q}_{t}\right),$
(14)

where $\widehat{Q}_{n+1},...,\widehat{Q}_{n+m}$ is a series of quantile
forecasts at level $\alpha$ for observations $r_{n+1},...,r_{n+m}$. Table 6
represents the 2.5% VaR quantile loss function results. The average ranking
based on rankings of the quantile loss across seven markets is also included
in the table.

Table 6: $2.5\%$ VaR Forecasting quantile loss on seven indices

Model | S&P500 | FTSE | NASDAQ | HSI | DAX | SMI | Avg Rank
---|---|---|---|---|---|---|---
GARCH Models |  |  |  |  |  |  |
GARCH-t | 181.16 | 183.01 | 218.51 | 210.94 | 225.96 | 174.99 | 27
EGARCH-t | 175.34 | 176.91 | 213.38 | 203.70 | 217.22 | 168.55 | 15
GJR-GARCH-t | 174.62 | 176.78 | 211.11 | 204.74 | 219.88 | 168.00 | 16
GARCH-QML | 182.04 | 183.95 | 219.82 | 211.94 | 227.33 | 175.39 | 28
EGARCH-QML | 176.66 | 177.88 | 214.78 | 204.81 | 217.64 | 169.04 | 20
GJR-GARCH-QML | 175.54 | 177.61 | 212.25 | 205.42 | 220.31 | 168.44 | 19
REGARCH Models |  |  |  |  |  |  |
GG-RV5 | 183.58 | 183.32 | 212.55 | 202.97 | 229.64 | 175.96 | 25
GG-RK | 182.07 | 182.33 | 212.96 | 203.85 | 227.81 | 175.47 | 23
GG-BV | 180.50 | 181.82 | 212.06 | 202.38 | 264.92 | 176.93 | 29
GG-RV5-RK | 184.07 | 183.79 | 213.32 | 203.10 | 229.54 | 174.78 | 26
GG-RV5-BV | 181.05 | 181.41 | 211.76 | 203.44 | 228.82 | 175.17 | 21
GG-RK-BV | 181.74 | 182.18 | 211.96 | 202.94 | 228.98 | 177.25 | 24
GG-RV5-RK-BV | 181.15 | 182.09 | 211.75 | 203.22 | 229.52 | 175.01 | 22
ES-CAViaR-Add-X Models |  |  |  |  |  |  |
RV5 | 174.63 | 178.92 | 205.63 | 202.98 | 221.04 | 172.39 | 17
RK | 174.13 | 178.41 | 206.84 | 203.33 | 222.62 | 173.16 | 18
BV | 173.50 | 175.28 | 205.03 | 202.74 | 219.76 | 174.92 | 14
ES-X-CAViaR-X-Add Models |  |  |  |  |  |  |
RV5 | 169.68 | 177.18 | 197.95 | 200.96 | 218.75 | 164.82 | 10
RK | 168.21 | 176.31 | 197.28 | 202.40 | 218.79 | 166.44 | 11
BV | 168.12 | 174.96 | 196.17 | 200.55 | 219.34 | 166.18 | 8
Realized-ES-CAViaR-Add-X Models |  |  |  |  |  |  |
RV5 | 170.7 | 178.27 | 198.02 | 200.58 | 218.94 | 166.60 | 12
RK | 171.32 | 179.02 | 197.67 | 201.84 | 218.60 | 168.02 | 13
BV | 165.76 | 176.83 | 196.07 | 200.91 | 218.24 | 167.89 | 9
Realized-ES-CAViaR-M Models |  |  |  |  |  |  |
RV5 | 165.16 | 173.09 | 195.23 | 198.02 | 214.70 | 161.96 | 5
RK | 165.32 | 172.59 | 195.62 | 197.93 | 214.23 | 163.52 | 7
BV | 162.95 | 171.87 | 193.95 | 198.44 | 214.10 | 163.10 | 1
RV5-RK | 165.58 | 173.09 | 194.93 | 198.23 | 214.57 | 162.05 | 6
RV5-BV | 163.16 | 171.86 | 195.07 | 198.53 | 214.30 | 162.15 | 2
RK-BV | 164.08 | 172.37 | 194.12 | 198.19 | 214.39 | 163.08 | 4
RV5-BV-RK | 163.61 | 172.35 | 194.78 | 198.31 | 214.46 | 162.47 | 3

Note: Boxes indicate the favoured models, and the blue text indicates the
second-ranked model in each column.

Table 6 presents the quantile loss values. In the last column, the ranks of
the average loss values for each model are presented. Based on the average
ranking, the proposed Realized-ES-CAViaR-M models have a ranking of 1-7. In
contrast, the Realized-ES-CAViaR-M-BV is the best model out of all models
considered, and Realized-ES-CAViaR-M-RV5-BV with two realized measures is the
second and Realized-ES-CAViaR-M-RV5-BV-RK, with all three, realized measures,
is the third-ranking model.

For S$\&$P market, Realized-ES-CAViaR-M-BV with one realized measure has the
smallest loss value hence ranks as the best model, and Realized-ES-CAViaR-M-
RV5-BV model with two realized measures is the second best-ranked model, and
the Realized-ES-CAViaR-M-RV5-BV-RK with all three realized measures considered
is the third-ranked model. For the FTSE market, Realized-ES-CAViaR-M-RV5-BV
with two realized measures is the best model, while Realized-ES-CAViaR-M-BV
with one realized measure is the second-ranked model, and the Realized-ES-
CAViaR-M-RV5-BV-RK model with all three realized measures is the third-ranked
model. Similar to S$\&$P, for the NASDAQ market, Realized-ES-CAViaR-M-BV is
the best model, and Realized-ES-CAViaR-M-RK-BV and Realized-ES-CAViaR-M-
RV5-BV-RK are the second and third-ranking models respectively. For HSI,
Realized-ES-CAViaR-M-RK is the best model, and Realized-ES-CAViaR-M-RV5 and
Realized-ES-CAViaR-M-RK-BV are the second and third-ranked models. Realized-
ES-CAViaR-M-BV is the best model for DAX, as the Realized-ES-CAViaR-M-RK is
the second-ranked model, and Realized-ES-CAViaR-M-RV5-BV is the third-ranked
model. For SMI, Realized-ES-CAViaR-M-RV5 is the best model, and Realized-ES-
CAViaR-M-RV5-RK and Realized-ES-CAViaR-M-RV5-BV are the second and third-
ranked models. Except for FTSE, proposed models with one realized measure have
been identified as the best-performed model, and in FTSE, the proposed model
with two realized measures is the best. In addition, out of all six markets
considered, a proposed model with only BV has been identified as the best
model for three markets, highlighting that BV has a significant contribution
to the model performances in terms of the quantile loss function.

####  5.3.2 Expected Shortfall

This research incorporates several backtests for evaluating the ES forecasts,
including VaR & ES joint loss function (fissler2016higher) and model
confidence set (MCS) (taylor2019forecasting). The VaR and ES joint loss
function (fissler2016higher) is used to compare the VaR and ES forecasts
jointly. Also, it helps to quantify and clarify any extra efficiency gains
from the proposed model compared to its competitors.

$\displaystyle
S_{t}\left(r_{t},VaR_{t},ES_{t}\right)=-\log\frac{(\alpha-1)}{ES_{t}}-\frac{(r_{t}-Q_{t})(\alpha-I(r_{t}\leq
Q_{t}))}{\alpha ES_{t}},$ (15)

taylor2019forecasting refers the above equation (15) as AL log score where the
loss function $S=\sum_{t-1}^{T}S_{t}$ which is exactly the negative of the AL
log-likelihood in equation (11) and a strictly consistent scoring rule that is
jointly minimized by the true VaR and ES series. We use this function to
compare and jointly assess all models’ VaR and ES forecasts. Table 7 presents
the VaR and ES joint loss function values $S$ calculated using equation (15)
for the forecast period of each market.

Table 7: $2.5\%$ VaR and $2.5\%$ VaR & ES Joint loss function values across the markets Model | S&P500 | FTSE | NASDAQ | HSI |  DAX | SMI | AVg Rank
---|---|---|---|---|---|---|---
GARCH Models |  |  |  |  |  |  |
GARCH-t | 5274.1 | 5325.9 | 5810.0 | 5641.2 | 5871.3 | 5138.3 | 18
EGARCH-t | 5187.0 | 5265.3 | 5810.0 | 5553.3 | 5789.0 | 5070.5 | 15
GJR-GARCH-t | 5207.1 | 5234.5 | 5762.1 | 5565.3 | 5819.2 | 5053.7 | 14
GARCH-QML | 5373.5 | 5422.0 | 5900.4 | 5701.6 | 5962.6 | 5206.3 | 26
EGARCH-QML | 5246.3 | 5314.7 | 5841.6 | 5590.7 | 5824.0 | 5110.3 | 16
GJR-GARCH-QML | 5273.4 | 5287.6 | 5811.0 | 5600.9 | 5867.4 | 5097.3 | 17
REGARCH Models |  |  |  |  |  |  |
GG-RV5 | 5420.3 | 5436.5 | 5823.0 | 5566.9 | 6133.5 | 5250.4 | 28
GG-RK | 5341.7 | 5397.8 | 5837.0 | 5559.4 | 6089.0 | 5239.3 | 24
GG-BV | 5367.8 | 5363.8 | 5831.9 | 5552.3 | 6497.5 | 5262.3 | 29
GG-RV5-RK | 5391.8 | 5437.6 | 5855.0 | 5565.0 | 6134.5 | 5233.9 | 27
GG-RV5-BV | 5348.5 | 5361.6 | 5818.6 | 5571.2 | 6090.4 | 5245.8 | 23
GG-RK-BV | 5331.8 | 5371.2 | 5832.9 | 5560.9 | 6111.8 | 5273.7 | 25
GG-RV5-RK-BV | 5309.0 | 5386.8 | 5797.6 | 5574.1 | 6106.4 | 5244.4 | 22
ES-CAViaR-Add Models |  |  |  |  |  |  |
RV5 | 5254.7 | 5390.9 | 5668.3 | 5651.1 | 6021.1 | 5270.3 | 21
RK | 5219.6 | 5362.9 | 5697.3 | 5613.8 | 6011.6 | 5241.6 | 20
BV | 5195.1 | 5302.3 | 5666.4 | 5645.3 | 5961.8 | 5322.2 | 19
ES-X-CAViaR-X-Add Models |  |  |  |  |  |  |
RV5 | 4905.8 | 5146.3 | 5382.0 | 5494.9 | 5748.8 | 4905.0 | 10
RK | 4900.7 | 5141.0 | 5380.2 | 5496.0 | 5742.2 | 4951.3 | 12
BV | 4900.5 | 5117.4 | 5366.6 | 5491.2 | 5745.2 | 4923.0 | 9
Realized-ES-CAViaR-Add-X Models |  |  |  |  |  |  |
RV5 | 4908.1 | 5154.8 | 5381.6 | 5482.7 | 5731.0 | 4945.1 | 11
RK | 4949.7 | 5159.4 | 5389.6 | 5488.7 | 5724.3 | 4981.7 | 13
BV | 4840.6 | 5130.4 | 5366.8 | 5489.8 | 5710.1 | 4959.1 | 8
Realized-ES-CAViaR-M Models |  |  |  |  |  |  |
RV5 | 4842.7 | 5085.4 | 5355.5 | 5452.8 | 5681.4 | 4876.0 | 6
RK | 4861.8 | 5080.6 | 5359.8 | 5444.7 | 5683.9 | 4899.7 | 7
BV | 4810.3 | 5056.1 | 5351.6 | 5451.6 | 5661.2 | 4892.0 | 2
RV5-RK | 4844.0 | 5086.0 | 5351.0 | 5454.8 | 5679.0 | 4871.2 | 5
RV5-BV | 4812.7 | 5054.8 | 5358.0 | 5460.4 | 5664.0 | 4872.7 | 1
RK-BV | 4815.9 | 5063.0 | 5345.5 | 5453.6 | 5667.9 | 4884.7 | 3
RV5-BV-RK | 4815.9 | 5063.0 | 5355.0 | 5458.2 | 5670.7 | 4873.4 | 4

Table 7 shows the 2.5% VaR and ES joint loss function values for each model
and each market. On this measure, proposed Realized-ES-CAViaR-M models with
one, two, and three realized measures do best overall, having lower joint
forecast loss values than all the other models considered. Out of all the
models, the proposed Realized-ES-CAViaR-M-RV5-BV with two realized measures is
the best model following Realized-ES-CAViaR-M-BV and Realized-ES-CAViaR-M-RK-
BV is the second and third-ranked model. The proposed model with three
realized measures is the fourth-ranked model.

For S&P and DAX, the proposed Realized-ES-CAViaR-M-BV is the best model, and
Realized-ES-CAViaR-M-RV5-BV is the second-best model whereas, for FTSE,
Realized-ES-CAViaR-M-RV5-BV is the best model and the Realized-ES-CAViaR-M-BV
is the second-best model. Realized-ES-CAViaR-M-RK-BV is the best model for
NASDAQ, and Realized-ES-CAViaR-M-RV5-RK is the second-ranked model. For HSI,
Realized-ES-CAViaR-M-RK is the best-ranked model, following Realized-ES-
CAViaR-M-BV. For SMI, Realized-ES-CAViaR-M-RV5-RK is the best model following
Realized-ES-CAViaR-M-RV5-BV. Out of the six markets considered, the proposed
framework with one realized measure is the best model for three markets, and
the proposed model with two realized measures is the best model for the other
three markets.

The proposed model, which has BV individually and combined with other realized
measures, shows better performances in all the markets considered. However,
the proposed models with RK individually are the best for only one market. The
individual performance of RV5 is not significant in the proposed model, but
sometimes, when it is combined with other realized measures, it performs
better.

hansen2011model introduced a Model confidence set(MCS) to statistically
compare the group of forecast models via a loss function. This test is
developed to contain the best model with a given confidence level, which is
90%.

We use the Matlab code for MCS testing downloaded from Kevin Sheppard’s web
page (sheppard2009mfe). The downloaded codes are adapted in two ways. One way
is incorporating the VaR and ES joint loss function in equation (15) in the
MCS calculation, which allows us to compare the ES forecasts’ performance.
Then, we incorporate the quantile loss function values into the equation (14).
Table 8 represents the 2.5% VaR and ES joint loss MCS results and quantile
loss MCS results.

Table 8: 90% model confidence set with R and SQ methods  Model | Joint Loss Function | Quantile Loss Function
---|---|---
| 2.5% R method | 2.5% SQ method | 2.5% R method | 2.5% SQ method
GARCH Models |  |  |  |
GARCH-t | 1 | 0 | 1 | 0
EGARCH-t | 2 | 2 | 4 | 2
GJR-GARCH-t | 2 | 1 | 4 | 2
GARCH-QML | 0 | 0 | 1 | 0
EGARCH-QML | 2 | 1 | 4 | 1
GJR-GARCH-QML | 1 | 1 | 4 | 1
REGARCH Models |  |  |  |
GG-RV5 | 1 | 1 | 2 | 1
GG-RK | 1 | 0 | 2 | 1
GG-BV | 1 | 1 | 2 | 1
GG-RV5-RK | 1 | 1 | 2 | 0
GG-RV5-BV | 1 | 1 | 2 | 1
GG-RK-BV | 1 | 1 | 2 | 0
GG-RV5-RK-BV | 0 | 0 | 3 | 0
ES-CAViaR-Add Models |  |  |  |
RV5 | 0 | 0 | 4 | 0
RK | 0 | 0 | 4 | 0
BV | 0 | 0 | 4 | 2
ES-X-CAViaR-X-Add Models |  |  |  |
RV5 | 6 | 4 | 6 | 5
RK | 5 | 5 | 6 | 4
BV | 6 | 5 | 6 | 5
Realized-ES-CAViaR-Add-X Models |  |  |  |
RV5 | 6 | 5 | 6 | 4
RK | 5 | 5 | 6 | 4
Bv | 6 | 5 | 6 | 5
Realized-ES-CAViaR-M Models |  |  |  |
RV5 | 6 | 6 | 6 | 6
RK | 6 | 6 | 6 | 6
BV | 6 | 6 | 6 | 6
RV5-RK | 6 | 6 | 6 | 6
RV5-BV | 6 | 6 | 6 | 6
RK-BV | 6 | 6 | 6 | 6
RV5-BV-RK | 6 | 6 | 6 | 6

Table 8 summarizes the results of 90% MCS for joint and quantile loss
functions. The R and SQ methods columns show the total number of times each
model is included in the $90\%$ MCS across the six return series. All models
of the proposed Realized-ES-CAViaR-M framework produce very competitive
results concerning the joint loss and quantile loss results. MCS includes all
proposed models with 1, 2, and 3 realized measures for both R and SQ methods.

Overall, across several VaR and ES measures and tests for forecast accuracy
and model comparisons over six markets, the performances of proposed Realized-
ES-CAViaR-M models, even with one realized measure, were generally highly
favourable relative to a range of competing models. When considering VRates,
quantile loss, VaR & ES joint loss, and MCS, the proposed framework with two
realized measures is the most favourable model overall.

##  6 Conclusion

This paper proposes a new semi-parametric realized conditional autoregression
framework to jointly estimate VaR and ES with multiple realized measures
called Realized-ES-CAViaR-M. This proposed framework includes Realized-EGARCH
as a special case. Through incorporating only three realized measures,
realized variance (5 min), realized kernel, and bi-power variation, the
proposed Realized-ES-CAViaR-M framework produces improved out-of-sample tail
risk forecasts over a range of competing models with comprehensive empirical
study.

The proposed Realized-ES-CAViaR-M-BV and Realized-ES-CAViaR-M-RV5-BV generate
highly competitive VaR and ES forecasts regarding VaR and ES joint and
quantile loss function. Further, the proposed framework with one and two
realized measures performs better than the three realized measures. More
importantly, in most cases, the models, including BV and RK, perform better
individually and together than the models, including RV5. Therefore, selecting
the best-realized measures and properly combining the realized measures is
important to improve the model’s performance. Further, considering all
performance criteria, the proposed framework with two realized measures
outperforms its counterparts.

To conclude, the proposed Realized-ES-CAViaR-M models with multiple realized
measures work effectively and perform better than its parametric counterpart,
the REGARCH model. Further, the model performances are higher than most semi-
parametric models, including single-realized measurements in jointly
estimating VaR and ES. The proposed Realized-ES-CAViaR-M models with two
realized measures, especially employing BV individually and combined with RK
or RV5, should be considered in developing financial applications in tail risk
forecasting. So, the financial institutions will be able to produce more
accurate capital allocations under the Basel Capital Accords to be safe from
the extreme market movements expected in the future.

This work can be improved by considering more realized measures, including
their sub-sampled measures and different frequencies. Different combinations
of realized measures can be utilized when selecting the best model. Realized
range and sub-sampled realized range can be used in future studies. Moreover,
the proposed framework includes single lags only, which can be extended to
multiple lags. Finally, a different version of the model, such as a
multiplicative form of the proposed model, can be considered in future
studies.

##  7 Appendix

###  7.1 Appendix A: Stationary condition for the proposed model

Here, we prove the stationary condition for the proposed Realized-ES-CAViaR-M
model.

$\displaystyle\textup{log}\left(-Q_{t}\right)$
$\displaystyle=\beta_{0}+\beta_{1}\textup{log}\left(-Q_{t-1}\right)+\tau_{1}\varepsilon_{t-1}+\tau_{2}\left(\varepsilon_{t-1}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)+\gamma_{j}U_{j,t-1},$
$\displaystyle\textup{log}\left(X_{j,t}\right)$
$\displaystyle=\xi_{j}+\phi_{j}\textup{log}\left(-Q_{t}\right)+\delta_{j,1}\varepsilon_{t}+\delta_{j,2}\left(\varepsilon_{t}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)+U_{j,t},$
$\displaystyle ES_{t}$ $\displaystyle=Q_{t}-\omega_{t},$
$\displaystyle\omega_{t}$
$\displaystyle=\nu_{0}+\nu_{1}\omega_{t-1}+\psi_{j}\left|U_{j,t-1}\right|,$

where $j=1,2,...,k$. Following equation can be derived When subject the
$U_{j,t-1}$ values from measurement equation $X_{j,t}$ and substitute it to
quantile equation $Q_{t}$:

$\textup{log}\left(-Q_{t}\right)=\beta_{0}+\beta_{1}\textup{log}\left(-Q_{t-1}\right)+\tau_{1}\varepsilon_{t-1}+\tau_{2}\left(\varepsilon_{t-1}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)+\gamma_{j}\left(\textup{log}\left(X_{j,t}\right)-\xi_{j}-\phi_{j}\textup{log}\left(-Q_{t}\right)-\delta_{j,1}\varepsilon_{t}-\delta_{j,2}\left(\varepsilon_{t}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)\right)$

When expanding the brackets in the above equation, we get the below equation.

$\textup{log}\left(-Q_{t}\right)=\beta_{0}+\beta_{1}\textup{log}\left(-Q_{t-1}\right)+\tau_{1}\varepsilon_{t-1}+\tau_{2}\left(\varepsilon_{t-1}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)+\gamma_{j}\textup{log}\left(X_{j,t}\right)-\gamma_{j}\xi_{j}-\gamma_{j}\phi_{j}\textup{log}\left(-Q_{t}\right)-\gamma_{j}\delta_{j,1}\varepsilon_{t}-\gamma_{j}\delta_{j,2}\left(\varepsilon_{t}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)$

When $Q_{t-1}<0$, then $Q_{t-1}$ is always $<0$ for $\alpha$ =(0.025, 0.01),
we have considered $\textup{log}\left(-Q_{t}\right)$ hence there is no change
for the signs.
Following the AR(1) stationary condition,

$\textup{log}\left(-Q_{t}\right)=\beta_{0}+\beta_{1}\textup{log}\left(-Q_{t-1}\right)+\tau_{1}\varepsilon_{t-1}+\tau_{2}\left(\varepsilon_{t-1}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)+\gamma_{j}\textup{log}\left(X_{j,t}\right)-\gamma_{j}\xi_{j}-\gamma_{j}\phi_{j}\textup{log}\left(-Q_{t}\right)-\gamma_{j}\delta_{j,1}\varepsilon_{t}-\gamma_{j}\delta_{j,2}\left(\varepsilon_{t}^{2}-\textup{E}\left(\varepsilon^{2}\right)\right)$

So, the stationary condition for the proposed model is as follows,

$\displaystyle\textbf{$\beta_{1}-(\gamma_{1}^{1}\phi_{1}+\gamma_{2}^{1}\phi_{2}+\gamma
3_{3}^{1}\phi_{3})<1$},$ (16)

###  7.2 Appendix B: Block Structure for adaptive MCMC

We use $\theta_{i}$ to represent the vector of parameters in the $i^{th}$
block where i depends on the number of realized measures (k) considered.

Table 9: Blocks structure Block number |  k=1 (13 parameters) |  k=2 (21 parameters) |  k=3 (30 parameters)
---|---|---|---
$\theta_{1}$ | $\left\\{\beta_{0},\beta_{1},\tau_{1},\tau_{2}\right\\}$ | $\left\\{\beta_{0},\beta_{1},\tau_{1},\tau_{2}\right\\}$ | $\left\\{\beta_{0},\beta_{1},\tau_{1},\tau_{2}\right\\}$
$\theta_{2}$ | $\left\\{\gamma,\Sigma_{11}\right\\}$ | $\left\\{\gamma_{1},\gamma_{2}\right\\}$ | $\left\\{\gamma_{1},\gamma_{2},\gamma_{3}\right\\}$
$\theta_{3}$ | $\left\\{\delta 11,\delta_{12}\right\\}$ | $\left\\{\xi_{1},\xi_{2}\right\\}$ | $\left\\{\xi_{1},\xi_{2},\xi_{3}\right\\}$
$\theta_{4}$ | $\left\\{\nu_{0},\nu_{1}\right\\}$ | $\left\\{\phi_{1},\phi_{2}\right\\}$ | $\left\\{\phi_{1},\phi_{2},\phi_{3}\right\\}$
$\theta_{5}$ | $\left\\{\xi,\phi,\psi\right\\}$ | $\left\\{\delta_{11},\delta_{12},\delta_{21},\delta_{22}\right\\}$ | $\left\\{\delta_{11},\delta_{21},\delta_{31}\right\\}$
$\theta_{6}$ |  | $\left\\{\nu_{0},\nu_{1}\right\\}$ | $\left\\{\delta_{12},\delta_{22},\delta_{32}\right\\}$
$\theta_{7}$ |  | $\left\\{\psi_{1},\psi_{2}\right\\}$ | $\left\\{\nu_{0},\nu_{1}\right\\}$
$\theta_{8}$ |  | $\left\\{\Sigma_{11},\Sigma_{22},\Sigma_{12}\right\\}$ | $\left\\{\psi_{1},\psi_{2},\psi_{3}\right\\}$
$\theta_{9}$ |  |  | $\left\\{\Sigma_{11},\Sigma_{22},\Sigma_{33}\right\\}$
$\theta_{10}$ |  |  | $\left\\{\Sigma_{12},\Sigma_{13},\Sigma_{23}\right\\}$

###  7.3 Appendix C: Simulation Study

A simulation study at 2.5% quantile with different sample sizes is conducted
to compare the performances and properties of the Bayesian and the maximum
likelihood estimation methods for the proposed Realized-ES-CAViaR-M model.
Parameter estimation and one-step-ahead VaR and ES forecast accuracy are
considered here. Mean and Root Mean Square Error (RMSE) is calculated over the
replicated datasets for the MCMC and ML methods, illustrating their bias and
precision.

For one realised measure, 1000 simulated datasets with sample sizes n = 2000,
3000 and 5000 are generated from the Standard deviation REGRCH model in
equation (2). A similar approach is used with two realized measures. Employing
different sample sizes enables us to test the impact of the sample size on the
VaR and ES forecast accuracy and the properties of the MCMC and ML estimator.
Regarding the prediction of risk measures, the Realized-ES-CAViaR-M framework
thus includes the Standard deviation REGRCH model as a special parametric
case. Parameter values in the equation (17) are selected based on the
estimated values of parameters in the empirical study for models with one
realized measure.

$\displaystyle r_{t}$ $\displaystyle=\sigma_{t}z_{t},$
$\displaystyle\sigma_{t}$
$\displaystyle=\mathrm{exp}(0.05+0.85\mathrm{log}(\sigma_{t-1})-0.12\left(z_{t-1}\right)+0.04\left(z_{t-1}^{2}-1\right)+\boldsymbol{0}.2\mathbf{u}_{t-1}),$
$\displaystyle x_{j,t}$
$\displaystyle=\mathrm{exp}(0.1+0.9\mathrm{log}(\sigma_{t})+0.02z_{t}+0.02(z_{t}^{2}-1)+\mathbf{u}_{j,t}),\;\text{
for }j=1,2,...,k,$ $\displaystyle{u}_{t}$
$\displaystyle\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,0.3^{2}\right),$
(17)

We test $z_{t}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,1\right)$
and therefore Gaussian return errors are used in the data generating process
(DGP). To calculate the corresponding Realized-ES-CAViaR-M model true
parameter values, a mapping from the standard deviation REGRCH model to our
Realized-ES-CAViaR-M model must be derived. Using
$z_{t}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}N\left(0,1\right)$ with
$\sigma_{t}=\frac{Q_{t}}{a_{\alpha}}$, where $\Phi^{-1}(\alpha)\ =a_{\alpha}$
and, $z_{t}=\varepsilon_{t}a_{\alpha}$. Then
$ES_{t}=-\sigma_{t}\frac{\phi\left(\Phi^{-1}(\alpha)\right)}{\alpha}=b_{\alpha}\sigma_{t}$
and we have
$\sigma_{t}=\frac{Q_{t}}{\Phi^{-1}(\alpha)}=-\frac{\alpha\textup{ES}_{t}}{\phi\left(\Phi^{-1}(\alpha)\right)}$
here $\Phi$ is the standard Gaussian CDF, $\phi$ is standard Gaussian
Probability Density Function (PDF) and $\sigma_{t}$ is the volatility. Here,
$b_{\alpha}=-\frac{\phi\left(\Phi^{-1}(\alpha)\right)}{\alpha}$, and
$Q_{t}=a_{\alpha}\sigma_{t}$, $ES_{t}=b_{\alpha}\sigma_{t}$ hence
$\omega_{t}=Q_{t}-ES_{t}=(a_{\alpha}-b_{\alpha})\sigma_{t}=g_{\alpha}\sigma_{t}$.

Also, we have
$\varepsilon_{t}=\frac{r_{t}}{Q_{t}}=\frac{\sigma_{t}z_{t}}{\sigma_{t}\Phi^{-1}(\alpha)}=\frac{z_{t}}{\Phi^{-1}(\alpha)}=\frac{z_{t}}{a_{\alpha}}$
which motivates why $E(\varepsilon_{t})=0$ can be a reasonable assumption.
Here we define $E(\varepsilon^{2})=E\left(\frac{r^{2}_{t}}{Q^{2}_{t}}\right)$
to replace $\frac{1}{a_{\alpha}^{2}}$ from the parametric setting, which
preserves the usual zero-mean trait of the leverage term. Substituting these
back into GARCH and measurement equation of the standard deviation REGARCH
model framework in equation (2), the Realized-ES-CAViaR-M model’s quantile and
measurement equation can be derived as follows. Here, with $2.5\%$ quantile
level, $a_{\alpha}=-1.9600$ and $log(a_{\alpha})$ is not a real number. So, we
consider $log(-Q_{t})$ to avoid this issue. So in the mapping we substitute
$\sigma_{t}=\frac{-Q_{t}}{a_{\alpha}}=\frac{Q_{t}}{-a_{\alpha}}=\frac{-Q_{t}}{-a_{\alpha}}$
and obtain the corresponding equations for the proposed Realized-ES-CAViaR-M
model.

Quantile equation:

$\displaystyle\log\left(\frac{-Q_{t}}{-a_{\alpha}}\right)$
$\displaystyle=0.05+0.85\log\left(\frac{-Q_{t-1}}{-a_{\alpha}}\right)-0.12\left(z_{t-1}\right)+0.04\left(z_{t-1}^{2}-1\right)+\boldsymbol{0}.2\mathbf{u}_{t-1},$
$\displaystyle\log(-Q_{t})$
$\displaystyle=\left(\log(-a_{\alpha})+0.05-0.85\log(-a_{\alpha})\right)+0.85\log(-Q_{t-1})-0.12(a_{\alpha})\varepsilon_{t-1}+0.04a_{\alpha}^{2}\left(\varepsilon_{t-1}^{2}-E(\varepsilon^{2})\right)+0.2\mathbf{u}_{t-1},$
$\displaystyle\log(-Q_{t})$
$\displaystyle=0.093+0.85\log(-Q_{t-1})+0.235\varepsilon_{t-1}+0.153\left(\varepsilon_{t-1}^{2}-E(\varepsilon^{2})\right)+0.2\mathbf{u}_{t-1},$
(18)

Measurement equation:

$\displaystyle log(x_{1,t})$
$\displaystyle=0.1+0.9\log\left(\frac{-Q_{t}}{-a_{\alpha}}\right)+0.02(a_{\alpha})\varepsilon_{t}+0.02a_{\alpha}^{2}(\varepsilon_{t-1}^{2}-E(\varepsilon^{2}))+\mathbf{u}_{1,t},$
$\displaystyle log(x_{1,t})$
$\displaystyle=(0.1-0.9log(-a_{\alpha}))+0.9\log(-Q_{t})+0.02(a_{\alpha})\varepsilon_{t}+0.02a_{\alpha}^{2}(\varepsilon_{t-1}^{2}-E(\varepsilon^{2}))+\mathbf{u}_{1,t},$
$\displaystyle log(x_{1,t})$
$\displaystyle=-0.5056+0.9\log(-Q_{t})-0.039\varepsilon_{t}+0.076(\varepsilon_{t-1}^{2}-E(\varepsilon^{2}))+\mathbf{u}_{1,t}$
(19)

For the difference of VaR and ES, we substitute
$\sigma_{t}=\frac{\omega_{t}}{g_{\alpha}}$ to the volatility equation without
log forms and to simplify the model, we ignore the leverage effects for
simplicity. We introduce the absolute value of the error $u_{t}$ after
considering several versions of the proposed model.

$\displaystyle\left(\frac{\omega_{t}}{g_{\alpha}}\right)$
$\displaystyle=0.05+0.85\left(\frac{\omega_{t-1}}{g_{\alpha}}\right)+\boldsymbol{0}.2\left|\mathbf{u}_{t-1}\right|,$
$\displaystyle\omega_{t}$
$\displaystyle=(0.05)g_{\alpha}+0.85\omega_{t-1}+\boldsymbol{0}.2g_{\alpha}\left|\mathbf{u}_{t-1}\right|,$
$\displaystyle\omega_{t}$
$\displaystyle=0.0189+0.85\omega_{t-1}+0.0756\left|\mathbf{u}_{t-1}\right|,$
(20)

Parameter estimates for one realized measure in the MCMC and ML estimators for
the proposed model with simulated datasets from (17) are given in Table 10.
Similarly, Table 11 represents the estimates for the two realized measures.
The above tables present True values in the “True Values” column. Since we
have time-varying ES to VaR ratios in our Realized-ES-CAViaR-M models, the
true values of ($\gamma_{0},\gamma_{1},\psi_{1},\psi_{2},\psi_{3},$) cannot be
exactly solved as the observed inputs separately influence VaR and ES that
breaks the fixed relationship between VaR and ES holds in a parametric
standard deviation REGARCH type model. (wang2023semi). However, by estimating
other parameters and producing the one-step-ahead VaR and ES forecasts, we can
still check the performances of MCMC and MLE.

In the simulation study, we consider models with only one and two realized
measures of the Realized-ES-CAViaR-M model. With the time and space
restrictions, we do not present the simulation results for the three realized
measures case. Both Realized-ES-CAViaR-M models, including one and two,
realized measures are fit to each dataset, once using MLE and once using MCMC.
Mean and Root Mean Square Error (RMSE) values of parameter estimates are
calculated over 1000 replicated datasets for MLE and MCMC methods. Table 10
and Table 11 present the estimation results.

For both one and two realized measures, relatively accurate parameter
estimates are generated by both MLE and MCMC methods; hence, we can consider
both methods valid. Further, MCMC estimators generate lower bias and higher
precision for each one and two realised measure cases more frequently for
parameter estimates for each sample size considered. Further, when considering
the estimated VaR and ES values, MCMC generates lower bias and higher
precision for each sample size in both models, including one and two realized
measures. In one realized measure case, the precision results are improved
when the sample size is increased for both MLE and MCMC estimators. A similar
pattern can be observed regarding lower bias, which is the same for the two
realized measures case. Hence, by considering the simulation results for one
and two realized measures, it is decided that MCMC is the overall preferred
method; in the empirical study, the MCMC approach is used to estimate the
parameters in Realized-ES-CAViaR-M models with one, two, and three realized
measures.

Table 10: Results for 1000 simulations for one realized measure |  |  | n=2000 | n=3000 | n=5000
---|---|---|---|---|---
MLE | MCMC | MLE | MCMC | MLE | MCMC
| BM | True values | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE
$\beta_{0}$ | 0.05 | 0.1509 | 0.1521 | 0.0282 | 0.1545 | 0.0285 | 0.1512 | 0.0230 | 0.1529 | 0.0227 | 0.1503 | 0.0182 | 0.1512 | 0.0181
$\beta_{1}$ | 0.85 | 0.8500 | 0.8484 | 0.0275 | 0.8462 | 0.0278 | 0.8497 | 0.0221 | 0.8481 | 0.0218 | 0.8508 | 0.0175 | 0.8500 | 0.0174
$\tau_{1}$ | -0.12 | 0.2352 | 0.2420 | 0.0400 | 0.2422 | 0.0394 | 0.2410 | 0.0304 | 0.2414 | 0.0302 | 0.2408 | 0.0246 | 0.2409 | 0.0245
$\tau_{2}$ | 0.04 | 0.1537 | 0.1549 | 0.0482 | 0.1569 | 0.0472 | 0.1564 | 0.0374 | 0.1576 | 0.0368 | 0.1570 | 0.0296 | 0.1577 | 0.0294
$\gamma$ | 0.20 | 0.2000 | 0.2062 | 0.0353 | 0.2075 | 0.0348 | 0.2052 | 0.0274 | 0.2062 | 0.0270 | 0.2069 | 0.0217 | 0.2073 | 0.0216
$\xi$ | 0.10 | -0.5056 | -0.4811 | 0.0963 | -0.4829 | 0.0947 | -0.4816 | 0.0744 | -0.4827 | 0.0736 | -0.4831 | 0.0619 | -0.4843 | 0.0614
$\phi$ | 0.90 | 0.9000 | 0.8751 | 0.0904 | 0.8759 | 0.0889 | 0.8764 | 0.0688 | 0.8766 | 0.0681 | 0.8767 | 0.0589 | 0.8773 | 0.0586
$\delta_{11}$ | 0.02 | -0.0392 | -0.0395 | 0.0252 | -0.0395 | 0.0252 | -0.0384 | 0.0187 | -0.0382 | 0.0188 | -0.0403 | 0.0155 | -0.0403 | 0.0156
$\delta_{12}$ | 0.02 | 0.0768 | 0.0777 | 0.0343 | 0.0786 | 0.0343 | 0.0769 | 0.0276 | 0.0774 | 0.0278 | 0.0779 | 0.0223 | 0.0782 | 0.0224
$\nu_{0}$ |  | 0.0189 | 0.2028 | 0.2194 | 0.2417 | 0.2298 | 0.2229 | 0.2377 | 0.2554 | 0.2424 | 0.2359 | 0.2474 | 0.2682 | 0.2541
$\nu_{1}$ |  | 0.8500 | 0.4338 | 0.4938 | 0.3024 | 0.5508 | v0.4126 | 0.5106 | 0.3067 | 0.5473 | 0.4128 | 0.5089 | 0.3127 | 0.5428
$\psi$ |  | 0.0756 | 0.1090 | 0.1333 | 0.1638 | 0.1345 | 0.0969 | 0.1203 | 0.1392 | 0.1139 | 0.0740 | 0.0871 | 0.1084 | 0.0783
$\Sigma_{11}$ | 0.30 | 0.3000 | 0.3004 |  0.0099 | 0.3014 | 0.0100 | 0.3000 |  0.0080 | 0.3007 | 0.0080 | 0.2999 |  0.0061 | 0.3004 | 0.0062
Estimated VaR |  |  | -2.8937 | 0.1892 | -2.8967 | 0.1852 | -2.8787 | 0.1612 | -2.8816 | 0.1571 | -2.8842 | 0.1474 | -2.8858 | 0.1451
Estimated ES |  |  | -3.3389 | 0.2972 | -3.3476 | 0.2928 | -3.3316 | 0.2809 | -3.3384 | 0.2753 | -3.3434 | 0.2570 | -3.3472 | 0.2555
True VaR |  |  | -2.9865 | -2.9705 | -2.9783
True ES |  |  | -3.5094 | -3.4932 | -3.4989
Benchmark VaR |  |  | -2.8710 | -2.8721 | -2.8566
Benchmark ES |  |  | -3.4244 | -3.4258 | -3.4073

Table 11: Results for 1000 simulations for two realized measures |  |  | n=2000 | n=3000 | n=5000
---|---|---|---|---|---
| MLE | MCMC | MLE | MCMC | MLE | MCMC
| Benchmarks | True values | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE
$\beta_{0}$ | 0.0500 | 0.1509 | 0.1529 | 0.0222 | 0.1541 | 0.0224 | 0.1519 | 0.0179 | 0.1526 | 0.0181 | 0.1517 | 0.0138 | 0.1522 | 0.0138
$\beta_{1}$ | 0.8500 | 0.8500 | 0.8483 | 0.0207 | 0.8473 | 0.0207 | 0.8495 | 0.0161 | 0.8489 | 0.0162 | 0.8499 | 0.0123 | 0.8495 | 0.0122
$\tau_{1}$ | -0.1200 | 0.2352 | 0.2428 | 0.0369 | 0.2429 | 0.0362 | 0.2411 | 0.0302 | 0.2412 | 0.0301 | 0.2428 | 0.0237 | 0.2429 | 0.0236
$\tau_{2}$ | 0.0400 | 0.1537 | 0.1548 | 0.0469 | 0.1562 | 0.0463 | 0.1576 | 0.0378 | 0.1585 | 0.0371 | 0.1547 | 0.0285 | 0.1552 | 0.0283
$\gamma_{1}$ | 0.2000 | 0.2000 | 0.2076 | 0.0407 | 0.2084 | 0.0400 | 0.2042 | 0.0341 | 0.2047 | 0.0335 | 0.2067 | 0.0256 | 0.2070 | 0.0253
$\gamma_{2}$ | 0.3000 | 0.3000 | 0.3074 | 0.0349 | 0.3075 | 0.0341 | 0.3083 | 0.0290 | 0.3083 | 0.0285 | 0.3074 | 0.0226 | 0.3076 | 0.0224
$\xi_{1}$ | 0.1000 | -0.5056 | -0.4856 | 0.0718 | -0.4878 | 0.0707 | -0.4877 | 0.0561 | -0.4895 | 0.0549 | -0.4885 | 0.0450 | -0.4898 | 0.0443
$\xi_{2}$ | 0.1500 | -0.4220 | -0.4029 | 0.0669 | -0.4052 | 00.0662 | -0.4054 | 0.0543 | -0.4073 | 00.0533 | -0.4050 | 0.0435 | -0.4064 | 00.0429
$\phi_{1}$ | 0.9000 | 0.9000 | 0.8778 | 0.0630 | 0.8785 | 0.0624 | 0.8795 | 0.0499 | 0.8803 | 0.0490 | 0.8793 | 0.0410 | 0.8799 | 0.0406
$\phi_{2}$ | 0.8500 | 0.8500 | 0.8293 | 0.0583 | 0.8301 | 0.0581 | 0.8312 | 0.0481 | 0.8319 | 0.0473 | 0.8300 | 0.0397 | 0.8307 | 0.0392
$\delta_{11}$ | 0.0200 | -0.0392 | -0.0396 | 0.0206 | -0.0396 | 0.0207 | -0.0392 | 0.0157 | -0.0392 | 0.0158 | -0.0392 | 0.0129 | -0.0391 | 0.0129
$\delta_{21}$ | 0.0300 | -0.0588 | -0.0583 |  0.0244 | -0.0583 | 0.0244 | -0.0586 |  0.0200 | -0.0586 | 0.0201 | -0.0589 |  0.0155 | -0.0588 | 0.0155
$\delta_{12}$ | 0.0200 | 0.0768 | 0.0776 | 0.0275 | 0.0784 | 0.0278 | 0.0787 | 0.0224 | 0.0792 | 0.0226 | 0.0769 | 0.0172 | 0.0772 | 0.0174
$\delta_{22}$ | 0.0300 | 0.1152 | 0.1154 | 0.0332 | 0.1162 | 0.0335 | 0.1165 | 0.0282 | 0.1171 | 0.0282 | 0.1151 | 0.0214 | 0.1153 | 0.0216
$\nu_{0}$ |  | 0.0189 | 0.1774 | 0.1892 | 0.1712 | 0.1596 | 0.1977 | 0.2086 | 0.1928 | 0.1805 | 0.2187 | 0.2257 | 0.2126 | 0.1985
$\nu_{1}$ |  | 0.8500 | 0.3352 | 0.5725 | 0.2547 | 0.5980 | 0.3431 | 0.5711 | 0.2689 | 0.5848 | 0.3314 | 0.5810 | 0.2735 | 0.5810
$\psi_{1}$ |  | 0.0756 | 0.1070 | 0.1429 | 0.1664 | 0.1317 | 0.0904 | 0.1262 | 0.1413 | 0.1139 | 0.0727 | 0.0944 | 0.1124 | 0.0812
$\psi_{2}$ |  | 0.1134 | 0.0777 | 0.1172 | 0.1277 | 0.0787 | 0.0595 | 0.1035 | 0.1031 | 0.0668 | 0.0397 | 0.0956 | 0.0753 | 0.0603
$\Sigma_{11}$ |  | 0.2000 | 0.2000 | 0.0065 | 0.2009 | 0.0066 | 0.2001 | 0.0051 | 0.2007 | 0.0052 | 0.2001 | 0.0040 | 0.2005 | 0.0040
$\Sigma_{22}$ |  | 0.3000 | 0.3000 | 0.0093 | 0.3013 | 0.0095 | 0.2998 |  0.0077 | 0.3006 | 0.0078 | 0.2999 |  0.0061 | 0.3004 | 0.0061
$\Sigma_{12}$ |  | 0.1000 | 0.1004 |  0.0059 | 0.1008 | 0.0060 | 0.1003 |  0.0047 | 0.1005 | 0.0048 | 0.1002 |  0.0039 | 0.1003 | 0.0039
Estimated VaR |  |  | -3.1667 | 0.2602 | -3.1704 | 0.2500 | -3.0656 | 0.1849 | -3.0689 | 0.1796 | -3.0725 | 0.1482 | -3.0743 | 0.1447
Estimated ES |  |  | -3.5388 | 0.4183 | -3.5544 | 0.4078 | -3.4545 | 0.4444 | -3.4619 | 0.4383 | -3.4654 | 0.3813 | -3.4705 | 0.3774
True VaR |  |  | -3.2129 | -3.1348 | -3.1234
True ES |  |  | -3.8174 | -3.7417 | -3.7164
Benchmark VaR |  |  | -3.1068 | -3.1359 | -3.0141
Benchmark ES |  |  | -3.7057 | -3.7404 | -3.5951

##  8 Acknowledgement

The authors acknowledge the technical assistance of the Sydney Informatics Hub
and the Artemis HPC of the University of Sydney.