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Realized Tail-Risk Conditional Autoregressive Expectile Models: Accurate and Stable Forecasting Performance
\spacingset{1}
\maketitle
\begin{abstract}
A realized tail-risk conditional autoregressive expectile (RTCARE) model is proposed for joint
forecasting of value-at-risk (VaR) and expected shortfall (ES). Realized tail measures such as the realized quantile and the realized ES, constructed from intraday high-frequency returns, are incorporated into the
dynamics and measurement equations of the conditional expectile, allowing information about lower-tail risk to enter the forecasts. The parameters are estimated by Markov chain Monte Carlo, and we evaluate the resulting VaR--ES forecasts in numerical examples and in an empirical application using U.S. stock-return data. Across the considered tail levels, the RTCARE models deliver competitive and often superior forecasting performance relative to existing alternatives, with stable results across Monte Carlo replications and rolling forecast evaluations.
\end{abstract}
\begin{keyword}
Expected shortfall, Expectiles, Markov chain Monte Carlo, Realized quantile, Tail risk, Value-at-risk
\end{keyword}
\spacingset{1}
\section{Introduction}
Value-at-risk (VaR) and expected shortfall (ES) are central tools for measuring financial tail risk. VaR, at probability level $\alpha$, is the lower $\alpha$-quantile of the
conditional return distribution and therefore specifies a return threshold
that is breached with conditional probability $\alpha$. Lower-tail levels
such as $\alpha=0.01, 0.025$ and $0.05$ are commonly considered in empirical risk
management.
ES instead measures the conditional mean return in the tail below VaR and thus quantifies the average severity of losses once the VaR
threshold has been breached. Unlike VaR, ES accounts for the entire tail and
is a coherent risk measure as shown in \citet{ArtznerEtAl1999}. Its
regulatory importance has also increased through the Basel Committee's
revised market-risk framework, which uses ES in place of VaR for determining
market-risk capital in Basel I\hspace{-1.2pt}I\hspace{-1.2pt}I \citep[e.g.,][]{BaselCommittee2019}. Joint VaR--ES forecasts are consequently
of practical and methodological importance. Accurate prediction remains
challenging, however, since information about extreme
tails is limited and changes over time with both conditional volatility and
the shape of the return distribution.
A common parametric approach is to model conditional volatility using, for
example, a GARCH model \citep{Engle1982,Bollerslev1986} or a stochastic
volatility model \citep{Taylor1986}, and then derive VaR and ES under an
assumed innovation distribution, such as the Gaussian or Student's
$t$ distribution. The resulting tail forecasts can be sensitive to this
distributional choice, especially at low probability levels \citep[see, e.g.,][]{TakahashiYamauchiWatanabeOmori26}.
Semi-parametric alternatives instead model a
conditional tail functional directly. For example, the conditional autoregressive VaR (CAViaR) model was proposed by \citet{EngleManganelli04} to model the dynamics of conditional quantiles. CAViaR models have been extended to a family of nonlinear dynamic models for estimating time-varying VaR \citep[e.g.,][]{GerlachChenChan11}.
Alternatively, expectiles can be used to model the dynamics of VaR and ES. Expectiles are
defined as minimizers of the expected asymmetric squared loss
\citep{NewyPowell87}. For a zero conditional mean, the expectile associated
with a given quantile level has an analytical mapping to both VaR and ES
\citep{Taylor08}.
Building on this relationship, \citet{Taylor08} proposed the
conditional autoregressive expectile (CARE) model, in which the expectile
evolves dynamically and delivers joint estimation and forecasting of VaR
and ES without specifying a parametric distribution for returns.
Markov chain Monte Carlo (MCMC) methods have also been used in related dynamic VaR--ES models to represent parameter uncertainty in tail-risk forecasts \citep[e.g.,][]{ChenGerlachLu12,GerlachChen14}.
Daily returns alone need not exhaust the information available for forecasting
a daily tail event. High-frequency prices provide realized measures that
summarize intraday variation. Realized variance is commonly constructed
as the sum of squared intraday returns and used to estimate daily
integrated variance
\citep{AndersenEtAl2001,AndersenEtAl2003}.
Related realized-volatility models combine daily returns with
high-frequency volatility measures. Realized GARCH models use a
measurement equation linking realized volatility to the latent volatility
state \citep{HansenRGARCH12}, while realized stochastic-volatility models
incorporate realized volatility into the estimation of a latent
stochastic-volatility process \citep{Takahashi09,Takahashi23}.
\citet{GerlachWang22} extended the CARE framework by
incorporating intraday realized information through a measurement equation
and lagged realized variables in the expectile dynamics.
However, realized-volatility measures
summarize the overall scale of intraday variation and are not designed
specifically for the lower tail. More recently, \citet{Dimitriadis22}
proposed realized quantiles (RQ) and realized expected shortfall (RES) as tail-specific,
daily-scale measures constructed from high-frequency intraday returns. RQ estimates a
daily return quantile by scaling its intraday counterpart, while RES is
constructed analogously for ES. They therefore provide direct estimates of
daily VaR and ES and contain information about the relevant part of the return
distribution rather than only its overall scale. This motivates using RQ and RES as
realized tail measures in a dynamic model for joint VaR and ES forecasting.
This paper proposes realized tail-risk conditional autoregressive expectile
(RTCARE) models, which extend the CARE framework by incorporating these realized
tail measures, RQ and RES, into the expectile dynamics and measurement equations
for joint VaR and ES forecasting.
Our use of realized measures as observed inputs follows the approach of
\citet{Takahashi09} and \citet{HansenRGARCH12}, who use realized volatility
in volatility estimation.
Furthermore, the proposed RTCARE models can be viewed as
a tail-risk extension of the realized CARE model of \citet{GerlachWang22},
which incorporates realized volatility into CARE.
The proposed models are estimated by MCMC, which
also allows parameter uncertainty to be incorporated into inference and
forecasting.
We evaluate the forecasting performance of our proposed models using several standard diagnostics. First, we
use the strictly consistent Fissler--Ziegel loss class \citep{FisslerZiegel16},
specifically the FZ0 loss proposed by \citet{PattonZiegelChen19}, to
assess VaR and ES jointly. We then report the coverage and ES diagnostics and
apply the regression-based ES backtests proposed by
\citet{BayerDimitriadis22}. Finally, we use the model confidence set (MCS) procedure \citep{HansenLundeNason11} to compare
predictive accuracy across models and identify models that are not significantly worse than the best model.
The proposed models are compared with volatility-based, score-driven,
CARE-type, and realized tail-risk benchmarks in the numerical examples and
in both the in-sample and one-day-ahead out-of-sample analyses of IBM daily returns.
The remainder of the paper is organized as follows. Section~2 introduces the
expectile--VaR--ES relationship and the realized tail-risk measures, and then proposes the RTCARE models. Section~3 describes the MCMC estimation scheme.
Section~4 presents the numerical examples. Section~5 presents the empirical application using IBM
data and reports posterior estimates together with in-sample and out-of-sample performance. Section~6 concludes.
\section{Realized Tail-Risk Conditional Autoregressive Expectile Models}
\subsection{Relationships between Expectiles, VaR, and ES}
{\it Definition of an expectile}.
\citet{NewyPowell87} introduced expectiles as generalized quantiles defined
by minimizing an asymmetric quadratic loss and proposed the asymmetric
least squares (ALS) method for estimation. For a random variable $Y$ and $\tau\in(0,1)$, the $\tau$-level
expectile is defined by the following optimization problem.
\begin{align}
\min_{\mu} \mathbb{E}\left[ |\tau - I(Y < \mu)| (Y - \mu)^2 \right],
\end{align}
where $I(\cdot)$ is the indicator function. Let $\mu_t(\tau)$ denote the conditional $\tau$-level expectile of $y_t$ given the information set $\mathcal{F}_{t-1}$. The ALS criterion for $n$ observations is
\begin{equation}
\sum_{t=1}^{n} |\tau - I(y_t < \mu_t(\tau))| (y_t - \mu_t(\tau))^2.
\label{eq:ALSsample}
\end{equation}
\noindent
{\it Definitions of VaR and ES}. The value-at-risk at level $\alpha$ ($\alpha\in(0,1)$) is defined as $\mathrm{VaR}(\alpha) := \inf\{y\in\mathbb{R} \mid F(y) \geq \alpha\}$,
where $F$ denotes the cumulative distribution function of $Y$. The
expected shortfall at level $\alpha$ ($\alpha\in(0,1)$) is defined as $\mathrm{ES}(\alpha) := \mathbb{E}\big[Y \mid Y \leq \mathrm{VaR}(\alpha)\big]$.
\noindent
{\it Relationships between expectiles, VaR, and ES}.
For the expectile level $\tau$ corresponding to quantile level $\alpha$, $\mu_t(\tau)=q_t(\alpha)$ \citep{Jones1994}, and \citet{Taylor08} gives
\begin{align}
\label{eq:expectile-es-map}
ES_{t, \alpha} = \left(1 + \frac{\tau}{(1 - 2\tau)\alpha}\right)\mu_t(\tau)
-\frac{\tau}{(1 - 2\tau)\alpha}\mathbb{E}(y_t\mid\mathcal{F}_{t-1}).
\end{align}
For continuous conditional return distributions, $\alpha=\Pr(y_t<\mu_t(\tau)\mid\mathcal{F}_{t-1})$. We impose a zero conditional mean, eliminating the expected-return term, and select $\tau$ by matching empirical coverage to $\alpha$ using the grid search in Section~\ref{sec:model}.
\subsection{Realized Tail-Risk Measures for VaR and ES}
\label{sec:realized-measures}
High-frequency intraday data provide realized measures that add information
about daily tail risk. \citet{Dimitriadis22} introduced realized quantiles
(RQ) and realized expected shortfall (RES) as daily tail-risk measures
constructed from intraday returns. RQ is an intraday-based estimate of a
lower-tail VaR, while RES is the corresponding estimate of ES. In this
paper, we refer to RQ and RES jointly as realized tail-risk measures.
Because intraday volatility varies over the trading day, a fixed
calendar-time grid does not allocate observations evenly across periods of
high and low activity. We therefore use business-time sampling (BTS),
following the business-time construction of \citet{DongTse17}, to account
for this intraday variation. The detailed construction is given in
the Supplementary Material, Section~A.
\subsection{Realized Tail-Risk CARE Models}
\label{sec:model}
The realized tail-risk conditional autoregressive expectile (RTCARE) models extend the CARE model introduced by \citet{Taylor08}. We propose two RTCARE specifications, RTCARE and RTCARE-RES, which utilize the realized tail-risk measures as described below.\\
\noindent
{\bf RTCARE model}.
First, for a fixed $\tau\in(0,1)$, the RTCARE model is based on RQ and defined as
\begin{align}
y_t &= \mu_t+\varepsilon_t, \qquad \varepsilon_t\sim\mathrm{AN}(0,\sigma_r^2,\tau),
\label{eq:rcare-rq-return}\\
\mu_t &= \beta_0+\beta_1\mu_{t-1}+\beta_2 RQ_{t-1},
\label{eq:rcare-rq-recursion}\\
RQ_t &= \alpha_q+\beta_q\mu_t+u_t, \qquad u_t\sim\mathrm{N}(0,\sigma_u^2),
\label{eq:rcare-rq-measurement}
\end{align}
where AN denotes the asymmetric normal distribution.
Following \citet{GerlachWang22}, we use the asymmetric normal distribution as a pseudo-likelihood device.
Rather than asserting that returns literally follow this distribution, it is chosen so that
the likelihood contribution takes the form of the expectile loss. Under
this pseudo-likelihood, the contribution of (\ref{eq:rcare-rq-return}) has the kernel
\begin{equation}
p(y_t\mid\mu_t,\sigma_r^2,\tau)\propto(\sigma_r^2)^{-1/2}
\exp\left\{-\frac{|\tau-I(y_t<\mu_t)|(y_t-\mu_t)^2}{\sigma_r^2}\right\}.
\label{eq:an-kernel}
\end{equation}
For fixed $\tau$ and $\sigma_r^2$, maximizing this return pseudo-likelihood is equivalent to minimizing the sample analogue
of the expectile loss in \eqref{eq:ALSsample}.
Equation~\eqref{eq:rcare-rq-recursion} specifies the transition dynamics
of the latent expectile through its own lag and lagged RQ.
Moreover, the measurement equation (\ref{eq:rcare-rq-measurement}) links the latent daily expectile to the observed RQ through measurement error. Although we estimate $\alpha_q$ to take account of the possible bias in the RQ, we could set $\alpha_q=0$. We shall investigate whether such a restriction improves tail-risk forecasting performance in numerical examples and empirical studies.
Substituting the measurement equation into the expectile recursion yields
the implied feedback coefficient $\beta_1+\beta_2\beta_q$. We therefore
impose the following restriction on this coefficient.
\begin{equation}
\left|\beta_1+\beta_2\beta_q\right|<1.
\label{eq:restriction}
\end{equation}
\noindent
To select $\tau$, we use a grid search. For each candidate
value, we estimate the proposed model by MCMC and use the
posterior median of $\mu_t$. We choose the candidate for which the empirical
violation rate
$n^{-1}\sum_{t=1}^n I(y_t < \mu_t)$ is closest to the target coverage
level $\alpha$. \vspace{3mm}\\
\noindent
{\bf RTCARE-RES model}. Second, we extend the RTCARE model to incorporate RES, using the relationship in (\ref{eq:expectile-es-map}). For this specification, the expectile recursion in
\eqref{eq:rcare-rq-recursion} is augmented with lagged RES:
\begin{equation}
\mu_t=\beta_0+\beta_1\mu_{t-1}+\beta_2RQ_{t-1}+\beta_3RES_{t-1}.
\label{eq:rcare-res-recursion}
\end{equation}
The return and RQ measurement equations are unchanged, and the RES measurement equation is
\begin{align}
RES_t &= \alpha_e+\beta_e\kappa \mu_t+w_t, \qquad w_t\sim\mathrm{N}(0,\sigma_w^2),
\label{eq:rcare-res-measurement}\\
& \hspace{5mm}\kappa = 1 + \frac{\tau}{(1-2\tau)\alpha}. \nonumber
\end{align}
Substitution of the two measurement equations gives the implied feedback coefficient
$\beta_1+\beta_2\beta_q+\beta_3\beta_e\kappa$; we impose the corresponding coefficient restriction
$\left|\beta_1+\beta_2\beta_q+\beta_3\beta_e\kappa\right|<1$. Using RES computed from intraday returns may improve tail-risk forecasting performance. We also compare this specification with RTCARE models that do not include RES in the numerical examples and empirical application. The selection of $\tau$ follows the procedure above.
\section{MCMC Algorithm for the RTCARE Model}
\label{sec:bayes}
Define $\theta=(\beta_0,\beta_1,\beta_2, \alpha_q,\beta_q,\sigma_r^2,\sigma_u^2)'$, $y=(y_1,\ldots,y_n)'$, and $x=(x_1,\ldots,x_n)'$, where $x_t=RQ_t$.
Before imposing the coefficient restriction, we specify independent proper priors as follows.
$$
\beta_0,\beta_1,\beta_2,\alpha_q,\beta_q
\sim \mathcal{N}(0,10),
\qquad
\sigma_r^2,\sigma_u^2
\sim \mathcal{IG}(0.02,0.02).
$$
The coefficient restriction in \eqref{eq:restriction} is imposed on the
coefficient draws.
For fixed $\tau$, posterior sampling uses a Metropolis-within-Gibbs
algorithm. The MCMC algorithm proceeds as follows.\\
\begin{enumerate}
\item Initialize
$(\beta_0,\beta_1,\beta_2,\alpha_q,\beta_q,\sigma_r^2,\sigma_u^2)$
within the parameter support and compute the corresponding expectile
path.
\item Sample
$(\alpha_q,\beta_0,\beta_1,\beta_2,\beta_q)\mid
\sigma_r^2,\sigma_u^2,y,x,\tau$ as follows.
\begin{enumerate}
\item Integrate out $\alpha_q$ and sample $(\beta_0,\beta_1,\beta_2,\beta_q)\mid
\sigma_r^2,\sigma_u^2,y,x,\tau$ by a joint random-walk
Metropolis--Hastings step.
\item Sample $\alpha_q\mid\beta_0,\beta_1,\beta_2,\beta_q,
\sigma_u^2,y,x,\tau$ from its normal full conditional
distribution.
\end{enumerate}
\item Sample $\sigma_r^2$ from its inverse-gamma full conditional
distribution.
\item Sample $\sigma_u^2$ from its inverse-gamma full conditional
distribution.
\item Repeat Steps 2--4 for a sufficiently large number of iterations.
\end{enumerate}
The details of the sampling algorithm are given in the Supplementary Material,
Section~B.
\section{Illustrative Numerical Examples}
\label{sec:numerical-examples}
\subsection{Data-Generating Process for the Numerical Examples}
To assess finite-sample performance, we generate 50 independent
replications from the data-generating process described below. Each
replication uses a 1000-day rolling estimation window and produces 1000
one-step-ahead forecasts. Every day contains $c=130$ intraday returns,
corresponding to three-minute sampling over a 390-minute trading day.
We use this frequency because the three-minute interval produced the most
stable RQ and RES series among the sampling intervals considered for the
U.S. stock returns in the empirical application.
We evaluate the resulting forecasts using the criteria defined below.
The daily return $y_t$ is assumed to have zero conditional mean and conditional variance
following the GJR--GARCH recursion \citep{GJR93}
\begin{equation}
y_t=\sqrt{h_t}\,\varepsilon_t,\qquad
h_t=\omega+\{0.04+0.10I(y_{t-1}<0)\}y_{t-1}^2
+0.88h_{t-1},
\label{eq:numerical-gjr}
\end{equation}
where $\omega$ is selected to set the unconditional variance to the
prescribed target level. The standardized innovation is a skewed
Student-$t$ variate with six degrees of freedom. The standardized
innovation is defined by
\begin{equation}
\varepsilon_t=\frac{X_t-\mathbb E(X_t)}
{\sqrt{\operatorname{Var}(X_t)}},
\qquad
X_t=\{1.35I(T_t<0)+0.85I(T_t\geq0)\}T_t,
\qquad T_t\sim t_6.
\label{eq:numerical-innovation}
\end{equation}
This specification introduces a leverage effect and a heavier left tail.
Each simulated day is represented by $c=130$ three-minute returns.
Provisional intraday returns are first generated from
$r^{\star}_{t,j}\mid h_t \mathrel{\stackrel{\mathrm{iid}}{\sim}}
\mathcal{N}\left(0,\frac{h_t}{c}\right)$ for $j=1,\ldots,c$.
Because the provisional returns will not generally sum to the generated
daily return $y_t$, the discrepancy is distributed equally across the
$c$ intraday returns. The recentered returns
are $r_{t,j}=r^{\star}_{t,j}
+\frac{y_t-\sum_{\ell=1}^{c}r^{\star}_{t,\ell}}{c}$, with
$\sum_{j=1}^{c}r_{t,j}=y_t$. The realized-variance measure $RV_t$ is
then computed from the recentered returns as
$RV_t=\sum_{j=1}^{c}r_{t,j}^{\,2}$.
The RQ and RES are computed from the intraday returns
$\{r_{t,j}\}_{j=1}^{c}$ using the empirical lower-tail quantile and
tail-average definitions described in the Supplementary Material, Section A.
\subsection{Three RTCARE Models and Ten Competing Models}
\label{sec:competing-models}
In the numerical example, we compare three variants of the proposed RTCARE models with ten competing models in terms of VaR and ES forecasting performance. The proposed RTCARE models consist of the baseline RTCARE specification, RTCARE NO INT, which sets the intercept in the RQ measurement equation to zero ($\alpha_q=0$), and RTCARE-RES, which adds lagged RES to the expectile recursion and a second measurement equation for RES in parallel with the RQ equation. The models included in the comparison are listed below.\\
\noindent
{\bf RTCARE models}:
\begin{enumerate}
\item RTCARE
\item RTCARE NO INT (RTCARE with no intercept in the RQ measurement equation, {\it i.e.}, $\alpha_q \equiv 0$)
\item RTCARE-RES
\end{enumerate}
\noindent
The ten competing models fall into four classes. They are the CARE RV family, the generalized autoregressive score (GAS) model, the GARCH family, and the HAR model. Details are given below. \\
\noindent
{\bf Competing models}:\\
\noindent
(1) {\it CARE RV family models.} We consider three CARE models combined with various realized measures: (i) RV, (ii) RV and RQ, and (iii) RV, RQ and RES. The first CARE model with RV (denoted as CARE RV) is considered in \citet{GerlachWang22}, and we extend it to include lagged realized tail-risk measures in the expectile transition in other specifications. That is, one specification uses RV and RQ (denoted as CARE RV-RQ), whereas the other adds RES as an additional realized tail-risk regressor (denoted as CARE RV-RQ-RES), without a separate measurement equation.
For the realized CARE class, we consider the Realized CARE-SAV specification proposed in \citet{GerlachWang22}. It combines the expectile dynamics with the realized volatility information and a separate equation for the corresponding realized measure. The detailed specification is given in the Supplementary Material, Section~C.\\
\noindent
(2) {\it One-factor GAS model.} We consider the one-factor generalized autoregressive score (GAS) model proposed by \citet{PattonZiegelChen19}, which dynamically updates the parameter governing conditional VaR and ES using the score of the FZ0 loss. FZ0 is the particular formulation proposed by \citet{PattonZiegelChen19} within the strictly consistent Fissler--Ziegel loss class introduced by \citet{FisslerZiegel16}. The parameters are estimated by frequentist M-estimation using the FZ0 loss. The definition of FZ0 loss is given in Section~\ref{sec:evaluation-criteria}.\\
\noindent
(3) {\it GARCH family models.} The four volatility-based competitors are GARCH(1,1), GARCH-t(1,1) \citep{Bollerslev1986,TimtGARCH87}, Realized GARCH(1,1), and Realized GARCH-t(1,1) \citep{HansenRGARCH12}. Their VaR and ES forecasts are obtained from the estimated conditional return volatility and the assumed innovation distribution, with parameters estimated by MCMC.\\
\noindent
(4) {\it HAR models for RQ and RES}. The heterogeneous autoregressive (HAR) model was originally introduced as a parsimonious model for realized-volatility
dynamics and is widely used for volatility forecasting \citep{Corsi2009}. Following \citet{Dimitriadis22}, we use HAR-type dynamics to construct one-step-ahead forecasts of RQ and RES. We refer to this specification as the HAR-RQ-RES model. It forecasts the RQ and the RES using daily, weekly, and monthly autoregressive components. \\
\noindent
Table~\ref{tab:comparison_models} summarizes the ten competing models by the model class and acronym.
The full equations, distributional assumptions, and VaR--ES mappings for all competing models are provided in the Supplementary Material, Section~C.
\vspace{10mm}
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt
}
\caption{Ten competing models.}
\label{tab:comparison_models}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}ll@{}}
\toprule
Model class & Model acronym \\
\midrule
CARE RV family
& CARE RV \\
& CARE RV-RQ \\
& CARE RV-RQ-RES \\
& Realized CARE-SAV \\
\midrule
GAS & One-factor GAS \\
\midrule
GARCH family & GARCH \\
& GARCH-t \\
& Realized GARCH \\
& Realized GARCH-t \\
\midrule
HAR & HAR-RQ-RES \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Evaluation Criteria for Backtesting}
\label{sec:evaluation-criteria}
We evaluate VaR and ES estimates and forecasts using several quantitative
criteria. The definitions and implementation of each criterion are given below.\\
\noindent
{\it FZ0 loss function}. VaR and ES are jointly elicitable, meaning that their
true pair uniquely minimizes the expected value of a single scoring function.
The FZ loss functions form a class of strictly consistent
joint loss functions for these two quantities \citep{FisslerZiegel16}. Under
the convention that both VaR and ES are negative, \citet{PattonZiegelChen19}
define FZ0 as the particular member of this class whose loss differences are
homogeneous of degree zero. This homogeneity makes loss comparisons
invariant to a common positive rescaling of returns and the corresponding VaR and ES forecasts. For an
observed return $y_t$, let $q_t$ and $e_t<0$ denote the one-step-ahead VaR
and ES forecasts for date $t$. The FZ0 loss is
\begin{equation}
L_{FZ0}(y_t,q_t,e_t, \alpha)=
-\frac{I(y_t\leq q_t)(q_t-y_t)}{\alpha e_t}
+\frac{q_t}{e_t}+\log(-e_t)-1.
\label{eq:fz0}
\end{equation}
The sample mean of this loss is computed over the relevant evaluation
period. Smaller values of the FZ0 loss are considered to indicate more
accurate joint VaR--ES forecasts.\\
\noindent
{\it $D$ diagnostic.} The $D$ diagnostic of
\citet{EmbrechtsKaufmannPatie05} evaluates the accuracy of ES forecasts by
measuring the discrepancy between realized lower-tail returns and their
corresponding ES forecasts. For $H$ one-step-ahead forecasts, the
diagnostic is computed from $\delta_t=y_t-e_t$.
Let $c_\alpha$ be the empirical $\alpha$-quantile of
$\{\delta_t\}_{t=1}^H$. Then
\begin{equation}
D_1=
\frac{\sum_{t=1}^{H}I(y_t<q_t)\delta_t}
{\sum_{t=1}^{H}I(y_t<q_t)},\qquad
D_2=
\frac{\sum_{t=1}^{H}I(\delta_t<c_\alpha)\delta_t}
{\sum_{t=1}^{H}I(\delta_t<c_\alpha)},\qquad
D=\frac{|D_1|+|D_2|}{2}.
\label{eq:d-diagnostic}
\end{equation}
Smaller $D$ indicates a smaller average discrepancy between realized tail
returns and the corresponding ES values.\\
\noindent
{\it Violation rate}. The violation rate evaluates the calibration of the VaR values alone,
\begin{equation}
\widehat v=H^{-1}\sum_{t=1}^H I(y_t\leq q_t),
\label{eq:violation-rate}
\end{equation}
and is compared with the nominal level $\alpha$.\\
\noindent
{\it Root mean squared error (RMSE).} In the numerical example, the data-generating process
provides the true conditional VaR and ES. The RMSE is therefore computed
relative to these true values.
\subsection{Comparison of Predictive Performance}
Tables~\ref{tab:sim-gjr-fz0}, \ref{tab:sim-gjr-d}, and
\ref{tab:sim-gjr-violation} report the replication means and standard
deviations for the three main evaluation criteria at all four quantile levels, $\alpha = 0.05, 0.025, 0.01$ and 0.005.
The VaR and ES RMSE values are reported separately in
the Supplementary Material, Section~D.
\vspace{3mm}
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.85\textwidth
}
\caption{Numerical example. Mean FZ0 losses across the four tail levels. Entries are replication means with standard deviations in parentheses. * denotes the best entry in each column.}
\label{tab:sim-gjr-fz0}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{2pt}
\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 1.374 (0.184) & 1.574 (0.195) & 1.821 (0.227) & 2.013 (0.276) \\
RTCARE NO INT & *1.363 (0.184) & *1.556 (0.192) & *1.792 (0.213) & *1.962 (0.246) \\
RTCARE-RES & 1.375 (0.184) & 1.574 (0.193) & 1.819 (0.222) & 2.010 (0.266) \\
\midrule
CARE RV & 1.385 (0.185) & 1.588 (0.200) & 1.833 (0.234) & 2.005 (0.276) \\
CARE RV-RQ & 1.405 (0.204) & 1.622 (0.223) & 1.956 (0.346) & 2.337 (0.803) \\
CARE RV-RQ-RES & 1.419 (0.205) & 1.653 (0.250) & 2.014 (0.394) & 2.438 (1.701) \\
Realized CARE-SAV & 1.377 (0.185) & 1.577 (0.197) & 1.822 (0.230) & 1.998 (0.274) \\
\midrule
One-factor GAS & 1.387 (0.193) & 1.614 (0.221) & 1.944 (0.345) & 2.257 (0.393) \\
\midrule
GARCH & 1.387 (0.197) & 1.623 (0.226) & 1.973 (0.297) & 2.317 (0.400) \\
GARCH-t & 1.380 (0.196) & 1.581 (0.209) & 1.815 (0.231) & 1.976 (0.271) \\
Realized GARCH & 1.388 (0.200) & 1.625 (0.227) & 1.979 (0.296) & 2.323 (0.397) \\
Realized GARCH-t & 1.379 (0.197) & 1.581 (0.207) & 1.816 (0.232) & 1.967 (0.269) \\
\midrule
HAR-RQ-RES & 1.393 (0.208) & 1.656 (0.250) & 2.087 (0.353) & 2.413 (0.470) \\
\bottomrule
\end{tabular*}
\end{table}
\vspace{3mm}
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.85\textwidth
}
\caption{Numerical example. $D$ diagnostics across the four tail levels. Entries are replication means with standard deviations in parentheses. Undefined $D$ values are omitted from the corresponding mean and standard deviation. * denotes the best entry in each column.}
\label{tab:sim-gjr-d}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{2pt}
\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 0.203 (0.137) & 0.288 (0.215) & 0.535 (0.353) & 0.816 (0.609) \\
RTCARE NO INT & *0.180 (0.133) & 0.289 (0.216) & *0.443 (0.293) & 0.684 (0.477) \\
RTCARE-RES & 0.194 (0.125) & *0.283 (0.202) & 0.509 (0.350) & 0.746 (0.575) \\
\midrule
CARE RV & 0.186 (0.117) & 0.284 (0.216) & 0.495 (0.371) & 0.743 (0.621) \\
CARE RV-RQ & 0.250 (0.231) & 0.383 (0.350) & 0.803 (0.524) & 1.470 (1.293) \\
CARE RV-RQ-RES & 0.275 (0.248) & 0.440 (0.382) & 0.930 (0.816) & 1.234 (1.125) \\
Realized CARE-SAV & 0.199 (0.129) & 0.291 (0.213) & 0.519 (0.371) & 0.757 (0.608) \\
\midrule
One-factor GAS & 0.221 (0.184) & 0.350 (0.285) & 0.946 (1.261) & 1.520 (1.240) \\
\midrule
GARCH & 0.674 (0.321) & 0.905 (0.429) & 1.304 (0.640) & 1.598 (0.917) \\
GARCH-t & 0.378 (0.233) & 0.426 (0.301) & 0.575 (0.479) & 0.642 (0.667) \\
Realized GARCH & 0.677 (0.308) & 0.920 (0.420) & 1.328 (0.640) & 1.628 (0.906) \\
Realized GARCH-t & 0.349 (0.206) & 0.415 (0.286) & 0.524 (0.494) & *0.631 (0.661) \\
\midrule
HAR-RQ-RES & 0.763 (0.356) & 1.090 (0.470) & 1.669 (0.752) & 2.199 (1.142) \\
\bottomrule
\end{tabular*}
\end{table}
\clearpage
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.85\textwidth
}
\caption{Numerical example. Violation rates (in percent) across the four tail levels. Entries are replication means with standard deviations in parentheses. * denotes the best entry in each column.}
\label{tab:sim-gjr-violation}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{2pt}
\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
Nominal rate & 5.000 & 2.500 & 1.000 & 0.500 \\
\midrule
RTCARE & 5.058 (0.661) & *2.502 (0.441) & 1.052 (0.373) & 0.570 (0.267) \\
RTCARE NO INT & 5.120 (0.533) & 2.530 (0.375) & 1.084 (0.276) & *0.536 (0.235) \\
RTCARE-RES & *4.990 (0.634) & 2.514 (0.445) & 1.064 (0.350) & 0.590 (0.260) \\
\midrule
CARE RV & 5.048 (0.672) & 2.506 (0.443) & *1.032 (0.345) & 0.548 (0.273) \\
CARE RV-RQ & 5.166 (0.738) & 2.646 (0.570) & 1.202 (0.426) & 0.752 (0.422) \\
CARE RV-RQ-RES & 5.208 (0.801) & 2.682 (0.584) & 1.256 (0.461) & 0.832 (0.428) \\
Realized CARE-SAV & 5.038 (0.712) & 2.528 (0.468) & 1.048 (0.374) & 0.590 (0.284) \\
\midrule
One-factor GAS & 5.420 (0.682) & 2.884 (0.551) & 1.266 (0.392) & 0.718 (0.295) \\
\midrule
GARCH & 6.098 (0.792) & 3.982 (0.599) & 2.326 (0.469) & 1.672 (0.377) \\
GARCH-t & 6.608 (0.870) & 3.804 (0.546) & 1.678 (0.333) & 0.976 (0.274) \\
Realized GARCH & 5.950 (0.777) & 3.896 (0.629) & 2.292 (0.452) & 1.672 (0.382) \\
Realized GARCH-t & 6.472 (0.779) & 3.656 (0.576) & 1.630 (0.334) & 0.906 (0.287) \\
\midrule
HAR-RQ-RES & 6.044 (0.805) & 4.140 (0.735) & 2.548 (0.565) & 1.602 (0.397) \\
\bottomrule
\end{tabular*}
\vspace{3mm}
\end{table}
\noindent
In summary, Tables~\ref{tab:sim-gjr-fz0}--\ref{tab:sim-gjr-violation}
show that the RTCARE models perform competitively across the three criteria. They have low FZ0 losses, relatively small $D$ values at all quantile levels, and violation rates close to their nominal values. The generally modest standard deviations across replications support the proposed RTCARE construction as an accurate and stable approach to joint VaR--ES forecasting.\\
\indent
Figures~\ref{fig:numerical-fz0}, \ref{fig:numerical-d}, and
\ref{fig:numerical-violation-rate} present the FZ0 loss, $D$ diagnostic,
and violation-rate boxplots based on the 50 replication results. Moreover,
RMSE distributions for VaR and ES are reported in the Supplementary Material,
Section~D.
\begin{figure}[htbp]
\centering
\makebox[\textwidth][c]{
\includegraphics[width=\textwidth,height=0.78\textheight,keepaspectratio]{figure/boxplots/boxplot_fz0.pdf}}
\caption{Boxplots of mean FZ0 loss across the 50 replications. Boxes show the interquartile range, the center line is the median, and whiskers show the observed range.}
\label{fig:numerical-fz0}
\vspace{6mm}
\end{figure}
\begin{figure}[htbp]
\centering
\makebox[\textwidth][c]{\includegraphics[width=\textwidth,height=0.78\textheight,keepaspectratio]{figure/boxplots/boxplot_D.pdf}}
\caption{Boxplots of the $D$ diagnostic across the 50 replications. Boxes show the interquartile range, the center line is the median, and whiskers show the observed range.}
\label{fig:numerical-d}
\vspace{6mm}
\end{figure}
Figure~\ref{fig:numerical-fz0} shows the mean FZ0 loss across the 50
replications. The RTCARE NO INT box is consistently among the lowest,
while the CARE RV-RQ and CARE RV-RQ-RES boxes widen in the extreme tail.
GARCH-t and Realized GARCH-t remain competitive in the center of the
distributions, but the proposed specifications show a more stable loss profile.
Figure~\ref{fig:numerical-d} provides the boxplots for the $D$ diagnostics.
The RTCARE boxes remain relatively compact, whereas the CARE RV-RQ,
CARE RV-RQ-RES, and HAR-RQ-RES boxes become more dispersed at 1\% and 0.5\%.
Realized GARCH-t is competitive at 0.5\%, consistent with the corresponding
table entry.
Figure~\ref{fig:numerical-violation-rate} compares the violation-rate boxes
with the red nominal-rate lines. The RTCARE boxes remain close to their
respective lines, while the GARCH, Realized GARCH, and HAR-RQ-RES boxes are
generally above the targets. The wider boxes of the CARE RV-RQ and
CARE RV-RQ-RES variants show greater replication sensitivity, especially at
the 0.5\% level where only a few violations are expected in each replication.
Overall, the boxplots show that the proposed RTCARE models generally combine
lower average FZ0 losses and $D$ values with violation rates close to the
nominal targets. Their relatively narrow boxes and whiskers also show less
variation across replications than many competing specifications. Thus, the
proposed class offers favorable average performance together with a stable
profile across the four tail levels.\\
\clearpage
\begin{figure}[H]
\centering
\makebox[\textwidth][c]{
\includegraphics[width=\textwidth,height=0.78\textheight,keepaspectratio]{figure/boxplots/boxplot_violation_rate.pdf}}
\caption{Boxplots of VaR violation rates across the 50 replications. The red horizontal line marks the nominal violation rate for each tail level.}
\label{fig:numerical-violation-rate}
\vspace{7mm}
\end{figure}
\newpage
\section{Application to IBM Stock Returns}
\label{sec:empirical}
\subsection{Data}
The empirical application uses IBM tick data from 4 January 2005 through 31
December 2024, which is obtained from Tick Data, LLC (https://www.tickdata.com). We calculate daily returns, RQ, and RES from these tick data,
giving 5013 trading days.
Time-series plots of the corresponding daily return, RQ, and RES series
are provided in the Supplementary Material, Section~E.
We evaluated one-,
three-, and five-minute sampling intervals for RQ and RES. Among these
choices, the three-minute interval produced the fewest outliers and the
most stable realized measures, and is therefore used throughout the
empirical analysis.
\subsection{Estimation Results}
The proposed RTCARE model\footnotemark is first applied to the full IBM sample
described above. This full-sample estimation provides parameter estimates
and posterior summaries before the rolling out-of-sample forecasts are
considered. We use the four quantile levels
$\alpha \in \{5\%, 2.5\%, 1\%, 0.5\%\}$. The prior distributions used
for the MCMC estimation are those specified in Section~\ref{sec:bayes}.
The resulting posterior summaries are reported in Table~\ref{tab:empirical-param-summary}.
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.85\textwidth
}
\caption{Posterior summary statistics for RTCARE (in-sample). The table reports the posterior mean, standard deviation, and 95\% credible interval (2.5\% and 97.5\% quantiles) for $\alpha=5\%, 2.5\%, 1\%, 0.5\%$.}
\label{tab:empirical-param-summary}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{3pt}
\setlength{\tabcolsep}{3pt}
\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}lrrrrrrrr}
\toprule
& \multicolumn{4}{c}{$\alpha=5\%$} & \multicolumn{4}{c}{$\alpha=2.5\%$} \\
\cmidrule(lr){2-5} \cmidrule(lr){6-9}
Parameter & Mean & SD & 2.5\% & 97.5\% & Mean & SD & 2.5\% & 97.5\% \\
\midrule
$\beta_0$ & -0.273 & 0.033 & -0.336 & -0.211 & -0.377 & 0.036 & -0.459 & -0.305 \\
$\beta_1$ & 0.525 & 0.015 & 0.496 & 0.554 & 0.604 & 0.015 & 0.574 & 0.633 \\
$\beta_2$ & 0.377 & 0.020 & 0.338 & 0.417 & 0.307 & 0.017 & 0.274 & 0.341 \\
$\alpha_q$ & 0.501 & 0.106 & 0.299 & 0.717 & 0.883 & 0.144 & 0.613 & 1.171 \\
$\beta_q$ & 1.143 & 0.049 & 1.051 & 1.249 & 1.157 & 0.051 & 1.060 & 1.258 \\
$\sigma^2_r$ & 0.709 & 0.014 & 0.682 & 0.737 & 0.505 & 0.010 & 0.486 & 0.526 \\
$\sigma^2_u$ & 0.318 & 0.006 & 0.305 & 0.330 & 0.621 & 0.012 & 0.597 & 0.646 \\
\bottomrule
\end{tabular*}
\vspace{0.5em}
\setlength{\tabcolsep}{3pt}
\begin{tabular*}{0.85\textwidth}{@{\extracolsep{\fill}}lrrrrrrrr}
\toprule
& \multicolumn{4}{c}{$\alpha=1\%$} & \multicolumn{4}{c}{$\alpha=0.5\%$} \\
\cmidrule(lr){2-5} \cmidrule(lr){6-9}
Parameter & Mean & SD & 2.5\% & 97.5\% & Mean & SD & 2.5\% & 97.5\% \\
\midrule
$\beta_0$ & -0.426 & 0.032 & -0.490 & -0.369 & -0.442 & 0.032 & -0.507 & -0.382 \\
$\beta_1$ & 0.762 & 0.011 & 0.741 & 0.783 & 0.846 & 0.010 & 0.825 & 0.864 \\
$\beta_2$ & 0.160 & 0.011 & 0.140 & 0.183 & 0.097 & 0.007 & 0.084 & 0.111 \\
$\alpha_q$ & 1.967 & 0.261 & 1.471 & 2.510 & 2.247 & 0.322 & 1.615 & 2.903 \\
$\beta_q$ & 1.287 & 0.070 & 1.157 & 1.433 & 1.115 & 0.064 & 0.991 & 1.250 \\
$\sigma^2_r$ & 0.324 & 0.007 & 0.312 & 0.337 & 0.184 & 0.004 & 0.177 & 0.191 \\
$\sigma^2_u$ & 2.002 & 0.040 & 1.926 & 2.082 & 15.300 & 0.310 & 14.702 & 15.924 \\
\bottomrule
\end{tabular*}
\vspace{3mm}
\end{table}
\footnotetext{RTCARE is defined by $y_t = \mu_t+\varepsilon_t$ ($\varepsilon_t\sim\text{AN}(0,\sigma_r^2,\tau)$), $\mu_t = \beta_0+\beta_1\mu_{t-1}+\beta_2 RQ_{t-1}$, and $RQ_t = \alpha_q+\beta_q\mu_t+u_t$ ($u_t\sim\mathcal{N}(0,\sigma_u^2)$).}
The posterior means indicate that the coefficient on the lagged expectile, $\beta_1$,
rises from 0.525 at $\alpha=5\%$ to 0.846 at $\alpha=0.5\%$, while
the loading on lagged RQ, $\beta_2$, declines from 0.377 to 0.097.
The measurement
slope $\beta_q$ remains positive and is close to one, ranging between 1.115
and 1.287 across the four levels. The return pseudo-likelihood scale parameter, $\sigma_r^2$, declines
from 0.709 to 0.184, whereas the measurement-error variance, $\sigma_u^2$, increases
from 0.318 to 15.300, indicating substantially greater dispersion in RQ at the most extreme level.
In the RTCARE NO INT model, the omission of the intercept from the measurement equation causes the other parameters to adjust during estimation, and the resulting level of $\mu_t$ is similar to that of the other RTCARE variants.
\subsection{Comparison of In-Sample Predictive Performance}
\label{sec:in-sample-forecasting-comparison}
The full-sample estimates describe the fitted tail dynamics. This subsection compares the competing specifications using their in-sample tail-risk estimates and evaluation criteria. Out-of-sample predictive performance is discussed in Section~\ref{sec:empirical-oos}. \\
\noindent
{\it FZ0 loss, $D$ diagnostics, and violation rates.}
Using the same four quantile levels as in the full-sample estimation above, we evaluate the dynamic model specifications considered in the numerical example. For the in-sample comparison, the HAR-RQ-RES benchmark is replaced by the contemporaneous RQ-RES benchmark.
Tables~\ref{tab:empirical-is-fz0} and~\ref{tab:empirical-is-d} report the average FZ0 loss and $D$ diagnostics for each of the four quantile levels, while Table~\ref{tab:empirical-is-violation} reports violation rates against their nominal rates.
\vspace{3mm}
\begin{table}[htbp]
\centering
\captionsetup{font=tablecaption,skip=3pt,width=0.52\textwidth}
\caption{Mean FZ0 loss for all quantile levels (in-sample). * indicates the smallest value.}
\label{tab:empirical-is-fz0}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{7pt}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 1.072 & 1.333 & 1.683 & 1.915 \\
RTCARE NO INT & 1.065 & 1.319 & 1.673 & 1.909 \\
RTCARE-RES & 1.072 & 1.333 & 1.682 & 1.919 \\
\midrule
CARE RV & 1.137 & 1.400 & 1.718 & 1.940 \\
CARE RV-RQ & 1.070 & 1.324 & 1.680 & 1.906 \\
CARE RV-RQ-RES & 1.070 & 1.324 & 1.673 & 1.913 \\
Realized CARE-SAV & 1.140 & 1.395 & 1.711 & 1.933 \\
\midrule
One-factor GAS & 1.136 & 1.422 & 1.768 & 1.970 \\
\midrule
GARCH & 1.153 & 1.434 & 1.896 & 2.408 \\
GARCH-t & 1.135 & 1.405 & 1.774 & 2.061 \\
Realized GARCH & 1.111 & 1.378 & 1.823 & 2.315 \\
Realized GARCH-t & 1.095 & 1.349 & 1.694 & 1.959 \\
\midrule
RQ-RES benchmark & *0.743 & *0.929 & *1.178 & *1.422 \\
\bottomrule
\end{tabular}
\end{table}
\vspace{3mm}
\begin{table}[htbp]
\centering
\captionsetup{font=tablecaption,skip=3pt,width=0.52\textwidth}
\centering
\caption{$D$ diagnostics at all four quantile levels (in-sample). * indicates the smallest value.}
\label{tab:empirical-is-d}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{7pt}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 0.044 & 0.046 & 0.046 & 0.080 \\
RTCARE NO INT & 0.152 & 0.188 & 0.274 & *0.037 \\
RTCARE-RES & *0.024 & 0.059 & 0.066 & 0.041 \\
\midrule
CARE RV & 0.048 & *0.026 & *0.027 & 0.090 \\
CARE RV-RQ & 0.049 & 0.027 & 0.085 & 0.066 \\
CARE RV-RQ-RES & 0.038 & 0.060 & 0.050 & 0.041 \\
Realized CARE-SAV & 0.276 & 0.280 & 0.195 & 0.243 \\
\midrule
One-factor GAS & 0.145 & 0.119 & 0.067 & 0.117 \\
\midrule
GARCH & 0.557 & 0.963 & 1.608 & 2.269 \\
GARCH-t & 0.191 & 0.338 & 0.752 & 0.950 \\
Realized GARCH & 0.523 & 0.925 & 1.662 & 2.264 \\
Realized GARCH-t & 0.150 & 0.367 & 0.737 & 1.072 \\
\midrule
RQ-RES benchmark & 0.114 & 0.244 & 0.500 & 0.845 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htbp]
\centering
\captionsetup{font=tablecaption,skip=3pt,width=0.52\textwidth}
\centering
\caption{Violation rates (in percent) at all four quantile levels (in-sample). * indicates the value closest to the nominal violation rate in each column.}
\label{tab:empirical-is-violation}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{7pt}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 5.067 & 2.474 & *0.997 & *0.499 \\
RTCARE NO INT & *5.007 & *2.494 & 1.017 & *0.499 \\
RTCARE-RES & 4.867 & 2.513 & 1.017 & *0.499 \\
\midrule
CARE RV & 4.987 & 2.474 & *0.997 & 0.519 \\
CARE RV-RQ & 5.107 & 2.454 & *0.997 & *0.499 \\
CARE RV-RQ-RES & 5.047 & 2.573 & *0.997 & 0.539 \\
Realized CARE-SAV & 5.047 & *2.494 & *0.997 & 0.539 \\
\midrule
One-factor GAS & 5.725 & 2.753 & 1.057 & 0.479 \\
\midrule
GARCH & 3.730 & 2.334 & 1.576 & 1.177 \\
GARCH-t & 4.628 & 2.454 & 1.117 & 0.738 \\
Realized GARCH & 3.850 & 2.354 & 1.416 & 1.157 \\
Realized GARCH-t & 4.907 & 2.394 & 1.117 & 0.658 \\
\midrule
RQ-RES benchmark & 4.947 & 2.833 & 1.237 & 0.738 \\
\bottomrule
\end{tabular}
\vspace{3mm}
\end{table}
\indent
For the FZ0 loss in Table \ref{tab:empirical-is-fz0}, the RQ-RES benchmark based on same-day realized tail-risk measures has the smallest loss at every level, ranging from 0.743 at 5\% to 1.422 at 0.5\%. Among the dynamic specifications, RTCARE NO INT has the smallest loss at 5\% and 2.5\%, ties with CARE RV-RQ-RES at 1\% at the displayed precision, and is surpassed by CARE RV-RQ at 0.5\%. The other RTCARE and CARE RV-RQ variants remain close to these values. The GARCH and Realized GARCH specifications become less competitive as the tail becomes more extreme.
\indent
For $D$ diagnostics in Table \ref{tab:empirical-is-d},
the RTCARE and CARE RV-RQ specifications generally have small $D$ values, whereas the $D$ values for the normal GARCH and Realized GARCH specifications increase markedly at the more extreme levels. RTCARE-RES has the smallest $D$ value at 5\%, CARE RV at 2.5\% and 1\%, and RTCARE NO INT at 0.5\%. CARE RV-RQ-RES and the other RTCARE variants remain competitive on this diagnostic.
Finally, the violation rates in Table~\ref{tab:empirical-is-violation} show that the RTCARE variants stay close to their nominal targets across all four levels. The GARCH and Realized GARCH models have violation rates increasingly above the nominal levels as $\alpha$ decreases, whereas their $t$ versions reduce but do not remove this deviation. The same-day RQ-RES benchmark is close at 5\% but becomes less well calibrated at the more extreme levels.
Overall, the three tables show that the proposed RTCARE variants have small FZ0 losses and $D$ diagnostics, and that their violation rates are close to the nominal targets across tail levels. The lower FZ0 loss of the benchmark based on same-day RQ and RES is an in-sample advantage that does not carry over to one-day-ahead forecasting. The corresponding rank plots are reported in the Supplementary Material, Section~F.
\begin{table}[htbp]
\centering
\captionsetup{font=tablecaption,skip=3pt,width=0.90\textwidth}
\caption{Average ranks for $\alpha \in \{5\%, 2.5\%, 1\%, 0.5\%\}$ (in-sample). Boldface indicates the best rank for each column.}
\label{tab:is_avg_ranking}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{3pt}
\begin{tabular*}{0.90\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Model & Avg FZ0 Rank & Avg $D$ Rank & Avg Violation Rank & Overall Avg Rank \\
\midrule
RTCARE & 5.25 & 3.25 & 3.00 & 3.83 \\
RTCARE NO INT & 2.25 & 6.25 & \textbf{2.25} & \textbf{3.58} \\
RTCARE-RES & 5.75 & \textbf{3.00} & 4.75 & 4.50 \\
\midrule
CARE RV & 9.50 & \textbf{3.00} & 3.00 & 5.17 \\
CARE RV-RQ & 3.00 & 4.25 & 4.00 & 3.75 \\
CARE RV-RQ-RES & 3.75 & \textbf{3.00} & 4.75 & 3.83 \\
Realized CARE-SAV & 9.00 & 8.75 & 3.00 & 6.92 \\
\midrule
One-factor GAS & 10.50 & 6.25 & 9.25 & 8.67 \\
\midrule
GARCH & 13.00 & 12.75 & 12.50 & 12.75 \\
GARCH-t & 10.50 & 10.25 & 8.75 & 9.83 \\
Realized GARCH & 10.00 & 12.25 & 11.50 & 11.25 \\
Realized GARCH-t & 7.50 & 10.00 & 8.50 & 8.67 \\
\midrule
RQ-RES benchmark & \textbf{1.00} & 8.00 & 9.75 & 6.25 \\
\bottomrule
\end{tabular*}
\end{table}
\spacingset{1}
Table~\ref{tab:is_avg_ranking} reports ranks averaged over the four tail levels and, overall, over the three criteria. Ranking conventions are given in the Supplementary Material, Section~F. RTCARE NO INT has the best overall in-sample rank, followed by CARE RV-RQ, while RTCARE and CARE RV-RQ-RES tie at the next position. The RQ-RES benchmark ranks first on FZ0 because it uses contemporaneous realized tail measures, but its $D$ and violation-rate ranks are weaker.
\noindent
{\it Regression-based ES backtesting}. To assess whether the in-sample ES estimates are correctly specified, we use the three regression-based tests proposed by \citet{BayerDimitriadis22}: (1) the strict test, (2) the auxiliary test and (3) the intercept test.
The tests use the joint VaR--ES regression framework described in the Supplementary Material, Section~G.
In Table~\ref{tab:empirical-is-backtest-mark-only}, we report whether we fail to reject the null hypothesis, $H_0:$ the in-sample ES estimates are correctly specified, at the 5\% significance level (two-sided tests are considered in the intercept tests).
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.78\textwidth
}
\caption{Results of the ES backtests proposed by \citet{BayerDimitriadis22} for $\alpha=5\%, 2.5\%, 1\%, 0.5\%$ (in-sample). The check mark (\ensuremath{^{*}}) indicates that we fail to reject the null hypothesis, $H_0:$ the in-sample ES estimates are correctly specified, at the 5\% significance level.}
\label{tab:empirical-is-backtest-mark-only}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{3pt}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}lrrrrrr@{}}
\toprule
Model & \multicolumn{3}{c}{$\alpha=5\%$} & \multicolumn{3}{c}{$\alpha=2.5\%$} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
& Strict & Auxiliary & Intercept & Strict & Auxiliary & Intercept \\
\midrule
RTCARE & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE NO INT & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
CARE RV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
Realized CARE-SAV & -- & -- & -- & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
One-factor GAS & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
GARCH & -- & -- & -- & -- & -- & -- \\
GARCH-t & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
Realized GARCH & -- & -- & -- & -- & -- & -- \\
Realized GARCH-t & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
RQ-RES benchmark & -- & \ensuremath{^{*}} & \ensuremath{^{*}} & -- & \ensuremath{^{*}} & -- \\
\bottomrule
\end{tabular}
\vspace{1.5mm} \\
\begin{tabular}{@{}lrrrrrr@{}}
\toprule
Model & \multicolumn{3}{c}{$\alpha=1\%$} & \multicolumn{3}{c}{$\alpha=0.5\%$} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
& Strict & Auxiliary & Intercept & Strict & Auxiliary & Intercept \\
\midrule
RTCARE & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE NO INT & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
CARE RV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
Realized CARE-SAV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
One-factor GAS & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
GARCH & -- & -- & -- & -- & -- & -- \\
GARCH-t & -- & -- & \ensuremath{^{*}} & -- & -- & \ensuremath{^{*}} \\
Realized GARCH & -- & -- & -- & -- & -- & -- \\
Realized GARCH-t & \ensuremath{^{*}} & \ensuremath{^{*}} & -- & -- & -- & -- \\
\midrule
RQ-RES benchmark & -- & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\bottomrule
\end{tabular}
\vspace{5mm}
\end{table}
\spacingset{1}
\noindent
The check marks show no rejection for the RTCARE specifications, the CARE RV variants, or the GAS model at any of the four levels. The normal GARCH and Realized GARCH specifications are rejected by all three procedures. The GARCH-t and Realized GARCH-t specifications are not rejected at the less extreme levels but are rejected by some tests at 1\% and 0.5\%, while the Realized CARE-SAV model is rejected at 5\% but passes at 2.5\%, 1\%, and 0.5\%. The same-day RQ-RES benchmark passes the auxiliary and intercept tests at most levels, but the null hypothesis of the strict test is rejected except at 0.5\%.
\subsection{Comparison of Out-of-Sample Predictive Performance}
\label{sec:empirical-oos}
The empirical analysis focuses primarily on out-of-sample forecasting because it evaluates predictions for observations that were not used for estimation.
We first compare the forecast diagnostics, then summarize the
model rankings, assess calibration with the ES backtests, and finally report the model confidence sets.
Each model is re-estimated in a rolling window of 1500 days and
produces 3513 one-day-ahead forecasts. The reported evaluation statistics are based on predictions formed using only information available before the forecast date.
\\
\noindent
{\it FZ0 loss, $D$ diagnostics, and violation rates.}
Tables~\ref{tab:empirical-oos-fz0} and~\ref{tab:empirical-oos-d}
report the FZ0 and $D$ diagnostics across the four quantile levels, while Table~\ref{tab:empirical-oos-violation} reports violation rates with their nominal target rates.
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.52\textwidth
}
\caption{Mean FZ0 loss for all quantile levels (out-of-sample). * indicates the smallest value.}
\label{tab:empirical-oos-fz0}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{7pt}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 1.100 & 1.379 & 1.781 & 2.057 \\
RTCARE NO INT & *1.097 & *1.372 & *1.769 & 2.057 \\
RTCARE-RES & 1.101 & 1.382 & 1.782 & *2.054 \\
\midrule
CARE RV & 1.184 & 1.482 & 1.833 & 2.075 \\
CARE RV-RQ & 1.108 & 1.393 & 1.867 & 2.274 \\
CARE RV-RQ-RES & 1.124 & 1.411 & 1.848 & 2.150 \\
Realized CARE-SAV & 1.185 & 1.474 & 1.827 & 2.058 \\
\midrule
One-factor GAS & 1.211 & 1.526 & 1.925 & 2.302 \\
\midrule
GARCH & 1.216 & 1.544 & 2.136 & 2.821 \\
GARCH-t & 1.188 & 1.485 & 1.892 & 2.218 \\
Realized GARCH & 1.182 & 1.504 & 2.065 & 2.734 \\
Realized GARCH-t & 1.163 & 1.450 & 1.840 & 2.147 \\
\midrule
HAR-RQ-RES & 1.195 & 1.593 & 2.299 & 2.884 \\
\bottomrule
\end{tabular}
\end{table}
\vspace{3mm}
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.52\textwidth
}
\caption{$D$ diagnostics at all four quantile levels (out-of-sample). * indicates the smallest value.}
\label{tab:empirical-oos-d}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{7pt}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 0.110 & 0.214 & 0.500 & *0.853 \\
RTCARE NO INT & 0.247 & 0.326 & 0.606 & 0.931 \\
RTCARE-RES & 0.125 & 0.207 & *0.484 & 0.962 \\
\midrule
CARE RV & 0.163 & 0.263 & 0.493 & 1.001 \\
CARE RV-RQ & 0.212 & 0.416 & 0.753 & 1.224 \\
CARE RV-RQ-RES & 0.228 & 0.389 & 0.693 & 1.428 \\
Realized CARE-SAV & *0.106 & *0.165 & 0.552 & 0.866 \\
\midrule
One-factor GAS & 0.146 & 0.185 & 0.641 & 1.339 \\
\midrule
GARCH & 0.765 & 1.237 & 1.990 & 2.647 \\
GARCH-t & 0.368 & 0.602 & 0.999 & 1.237 \\
Realized GARCH & 0.786 & 1.190 & 2.055 & 2.705 \\
Realized GARCH-t & 0.450 & 0.654 & 1.121 & 1.121 \\
\midrule
HAR-RQ-RES & 0.675 & 1.054 & 1.680 & 2.251 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.52\textwidth
}
\caption{Violation rates (in percent) at all four quantile levels (out-of-sample). * indicates the value closest to the nominal violation rate in each column.}
\label{tab:empirical-oos-violation}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{7pt}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $5\%$ & $2.5\%$ & $1\%$ & $0.5\%$ \\
\midrule
RTCARE & 4.782 & 2.420 & *1.082 & 0.626 \\
RTCARE NO INT & *5.095 & 2.676 & 1.224 & *0.598 \\
RTCARE-RES & 4.668 & 2.448 & *1.082 & *0.598 \\
\midrule
CARE RV & 4.668 & *2.505 & 1.167 & 0.626 \\
CARE RV-RQ & 4.782 & 2.420 & 1.167 & 0.740 \\
CARE RV-RQ-RES & 4.611 & *2.505 & 1.196 & 0.712 \\
Realized CARE-SAV & 4.839 & 2.676 & 1.110 & 0.626 \\
\midrule
One-factor GAS & 5.465 & 2.932 & 1.196 & 0.655 \\
\midrule
GARCH & 3.985 & 2.647 & 1.879 & 1.423 \\
GARCH-t & 4.811 & 2.619 & 1.309 & 0.911 \\
Realized GARCH & 4.327 & 3.046 & 1.850 & 1.423 \\
Realized GARCH-t & 5.124 & 3.017 & 1.423 & 1.082 \\
\midrule
HAR-RQ-RES & 7.743 & 4.953 & 2.960 & 1.765 \\
\bottomrule
\end{tabular}
\vspace{3mm}
\end{table}
\indent
Tables~\ref{tab:empirical-oos-fz0} and~\ref{tab:empirical-oos-d} show that the proposed RTCARE class performs well in the out-of-sample comparison. The three variants remain in the leading FZ0 group at every tail level, and the differences within the class are small. RTCARE NO INT has the lowest FZ0 loss at 5\%, 2.5\%, and 1\%, while RTCARE-RES is lowest at 0.5\%.
Forecasts from the GARCH, Realized GARCH, and HAR-RQ-RES models deteriorate markedly as the tail becomes more extreme.
The $D$ columns give further information about this result. The RTCARE variants remain in the leading group, especially at the 1\% and 0.5\% levels, while the GARCH family and HAR-RQ-RES model generally have larger $D$ values. Realized CARE-SAV is also competitive on $D$ at several levels, whereas the CARE RV variants are less consistent across the three diagnostics.
In Table~\ref{tab:empirical-oos-violation}, the violation rates for the three RTCARE models stay close to the nominal target rate at all four levels, including the extreme levels where those of the GARCH, Realized GARCH, and HAR-RQ-RES models substantially exceed the nominal levels. This coverage pattern complements the FZ0 and $D$ results. The three RTCARE models combine accurate coverage with competitive performance across all four quantile levels.
The average ranks are summarized in Table~\ref{tab:oos_avg_ranking} (see
the corresponding rank plots reported in the Supplementary Material, Section~F).
It places RTCARE first overall, followed by RTCARE-RES and RTCARE NO INT. This ordering is driven mainly by consistently low FZ0 losses and competitive $D$ diagnostics. Realized CARE-SAV remains competitive on $D$ values and violation rates, while the GARCH and HAR-RQ-RES models rank lower after the three criteria are combined.
\vspace{10mm}
\begin{table}[htbp]
\centering
\captionsetup{font=tablecaption,skip=3pt,width=0.90\textwidth}
\caption{Average ranks for $\alpha \in \{5\%, 2.5\%, 1\%, 0.5\%\}$ (out-of-sample). Boldface indicates the best rank for each column.}
\label{tab:oos_avg_ranking}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{3pt}
\begin{tabular*}{0.90\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Model & Avg FZ0 Rank & Avg $D$ Rank & Avg Violation Rank & Overall Avg Rank \\
\midrule
RTCARE & 2.00 & 2.50 & 3.25 & \textbf{2.58} \\
RTCARE NO INT & \textbf{1.50} & 5.50 & 4.50 & 3.83 \\
RTCARE-RES & 2.50 & 2.75 & \textbf{3.00} & 2.75 \\
\midrule
CARE RV & 6.50 & 4.25 & 3.75 & 4.83 \\
CARE RV-RQ & 6.25 & 7.25 & 5.25 & 6.25 \\
CARE RV-RQ-RES & 6.00 & 7.75 & 5.75 & 6.50 \\
Realized CARE-SAV & 6.00 & \textbf{2.00} & 4.25 & 4.08 \\
\midrule
One-factor GAS & 10.75 & 5.25 & 8.00 & 8.00 \\
\midrule
GARCH & 12.25 & 12.25 & 10.50 & 11.67 \\
GARCH-t & 9.00 & 8.75 & 7.00 & 8.25 \\
Realized GARCH & 9.75 & 12.75 & 11.25 & 11.25 \\
Realized GARCH-t & 6.00 & 9.00 & 8.25 & 7.75 \\
\midrule
HAR-RQ-RES & 12.50 & 11.00 & 13.00 & 12.17 \\
\bottomrule
\end{tabular*}
\vspace{5mm}
\end{table}
\spacingset{1}
\noindent
{\it Regression-based ES backtesting}.
The three regression-based backtesting procedures proposed by \citet{BayerDimitriadis22} are also applied to the rolling one-day-ahead forecasts as in Section \ref{sec:in-sample-forecasting-comparison}. Their results are reported in Table~\ref{tab:empirical-oos-backtest-mark-only}. A check mark indicates that the corresponding calibration null is not rejected at the 5\% significance level.
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.78\textwidth
}
\caption{Results of the ES backtests proposed by \citet{BayerDimitriadis22} for $\alpha=5\%, 2.5\%, 1\%, 0.5\%$ (out-of-sample). The check mark (\ensuremath{^{*}}) indicates that we fail to reject the null hypothesis, $H_0:$ ES forecasts are correctly specified, at the 5\% significance level.}
\label{tab:empirical-oos-backtest-mark-only}
\fontsize{10}{11}\selectfont
\setlength{\tabcolsep}{3pt}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}lrrrrrr@{}}
\toprule
Model & \multicolumn{3}{c}{$\alpha=5\%$} & \multicolumn{3}{c}{$\alpha=2.5\%$} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
& Strict & Auxiliary & Intercept & Strict & Auxiliary & Intercept \\
\midrule
RTCARE & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE NO INT & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
CARE RV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & -- & \ensuremath{^{*}} \\
CARE RV-RQ-RES & \ensuremath{^{*}} & -- & \ensuremath{^{*}} & -- & -- & \ensuremath{^{*}} \\
Realized CARE-SAV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
One-factor GAS & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
GARCH & -- & -- & -- & -- & -- & -- \\
GARCH-t & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & -- & \ensuremath{^{*}} \\
Realized GARCH & -- & -- & -- & -- & -- & -- \\
Realized GARCH-t & -- & -- & -- & -- & -- & -- \\
\midrule
HAR-RQ-RES & -- & -- & -- & -- & -- & -- \\
\bottomrule
\end{tabular}
\vspace{2.5mm}
\begin{tabular}{@{}lrrrrrr@{}}
\toprule
Model & \multicolumn{3}{c}{$\alpha=1\%$} & \multicolumn{3}{c}{$\alpha=0.5\%$} \\
\cmidrule(lr){2-4} \cmidrule(lr){5-7}
& Strict & Auxiliary & Intercept & Strict & Auxiliary & Intercept \\
\midrule
RTCARE & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE NO INT & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
RTCARE-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
CARE RV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
CARE RV-RQ-RES & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
Realized CARE-SAV & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} \\
\midrule
One-factor GAS & \ensuremath{^{*}} & \ensuremath{^{*}} & \ensuremath{^{*}} & -- & -- & \ensuremath{^{*}} \\
\midrule
GARCH & -- & -- & -- & -- & -- & -- \\
GARCH-t & -- & -- & -- & -- & -- & -- \\
Realized GARCH & -- & -- & -- & -- & -- & -- \\
Realized GARCH-t & -- & -- & -- & -- & -- & \ensuremath{^{*}} \\
\midrule
HAR-RQ-RES & -- & -- & -- & -- & -- & -- \\
\bottomrule
\end{tabular}
\vspace{3mm}
\end{table}
\spacingset{1}
Many models, including the proposed specifications, are not rejected at most tail levels. In contrast, the HAR-RQ-RES benchmark model, which independently forecasts RQ and RES using HAR dynamics, is rejected by all three tests at every tail level. This contrasts with the same-day RQ-RES benchmark, which has the smallest in-sample FZ0 loss. Its advantage does not carry over when RQ and RES must be forecast out of sample. The backtests therefore reinforce the ranking evidence while also showing that a low score alone is not sufficient for a satisfactory VaR--ES forecast. \\
\noindent
{\it Model confidence sets}.
To compare predictive ability across all models while allowing for sampling uncertainty, we use the model confidence set (MCS) procedure of \citet{HansenLundeNason11}. The MCS is the set of models that cannot be statistically distinguished from the best model at a specified confidence level. We apply the range statistic $T_R$ to the common out-of-sample daily FZ0-loss series. $T_R$ is based on the largest studentized pairwise loss
difference, and the equal-predictive-ability null is evaluated with the bootstrap procedure implemented in the \texttt{MCS} package of \citet{BernardiCatania18}. We use 5,000 bootstrap replications, the package's automatically selected block length, and a minimum block length of three to preserve dependence in the loss series. The algorithm sequentially removes the model most responsible for rejecting equal predictive ability and stops when the remaining set is no longer rejected.
A model is retained in the 90\% (75\%) MCS when its $p$-value is at least 0.10 (0.25). Both sets are reported in
Table~\ref{tab:empirical-oos-mcs-90-75-mark-only}. One check mark denotes retention only in the 90\% set, whereas two check marks denote retention in the stricter 75\% set as well.
\vspace{3mm}
\begin{table}[htbp]
\centering
\captionsetup{
font=tablecaption,
skip=3pt,
width=0.7\textwidth
}
\caption{IBM out-of-sample 90\% and 75\% Model Confidence Set (MCS) membership for the $T_R$ statistic. One check mark (\ensuremath{\circ}) indicates retention in the 90\% MCS only; two check marks (\ensuremath{\circ}\ensuremath{\circ}) indicate retention in the 75\% MCS (and therefore also the 90\% MCS).}
\label{tab:empirical-oos-mcs-90-75-mark-only}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Model & $\alpha=5\%$ & $\alpha=2.5\%$ & $\alpha=1\%$ & $\alpha=0.5\%$ \\
\midrule
RTCARE & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
RTCARE NO INT & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
RTCARE-RES & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
CARE RV & -- & -- & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
CARE RV-RQ & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
CARE RV-RQ-RES & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
Realized CARE-SAV & -- & -- & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
One-factor GAS & -- & -- & \ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
GARCH & -- & -- & -- & -- \\
GARCH-t & -- & -- & -- & \ensuremath{\circ}\ensuremath{\circ} \\
Realized GARCH & -- & -- & -- & -- \\
Realized GARCH-t & -- & -- & \ensuremath{\circ}\ensuremath{\circ} & \ensuremath{\circ}\ensuremath{\circ} \\
HAR-RQ-RES & -- & -- & -- & -- \\
\bottomrule
\end{tabular}
\vspace{5mm}
\end{table}
\spacingset{1}
Table~\ref{tab:empirical-oos-mcs-90-75-mark-only} gives a compact comparison of the predictive-ability results. All three proposed RTCARE models are retained in both the 90\% and 75\% sets at every tail probability.
The CARE RV variants are retained more selectively. Their membership is strongest at the most extreme tail, while the Realized CARE-SAV and one-factor GAS models are retained mainly at $\alpha=1\%$ and $0.5\%$.
The normal GARCH and Realized GARCH specifications are excluded at every level. GARCH-t and Realized GARCH-t are retained only at the most extreme tail levels, with the latter also retained at $\alpha=1\%$.
The MCS evidence is consistent with and complements the results of the FZ0 loss, $D$ diagnostics, violation rates and ES backtests reported above.\\
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{figure/FORECAST_RTCARE_HAR_RQ_RES_full_sample_alpha025.pdf}
\caption{One-day-ahead VaR and ES forecasts at $\alpha=2.5\%$ (out-of-sample). The upper and lower panels show VaR and ES, respectively. Solid gray: daily return series. Solid red: forecasts from RTCARE. Dashed blue: forecasts from HAR-RQ-RES.}
\label{fig:rcare-rq-forecast}
\vspace{5mm}
\end{figure}
\noindent
{\it Time-series plot of out-of-sample forecasts}.
Finally, Figure~\ref{fig:rcare-rq-forecast} illustrates the rolling forecasts and compares the proposed specification with the
HAR-RQ-RES benchmark model. Both models produce more negative VaR and ES forecasts during periods of elevated market volatility and return toward their usual levels as conditions calm. ES remains below VaR throughout the period, as required by the ordering of the two risk measures. The HAR-RQ-RES forecasts are visibly closer to zero than the corresponding RTCARE forecasts in both panels. Taken together with the lower loss and stronger diagnostic performance of RTCARE in the preceding tables, this pattern is consistent with the HAR-RQ-RES benchmark model assigning too little probability to large negative tail events and hence potentially underestimating tail risk.
RTCARE forecasts show sharper, short-lived responses to large returns and volatility episodes, whereas HAR-RQ-RES forecasts adjust more gradually, reflecting differences in the dynamic specifications and how realized tail measures enter the models.
\newpage
\section{Conclusion}
This paper proposes RTCARE models, a family of joint VaR--ES models that incorporate RQ and RES into autoregressive expectile dynamics.
In the numerical examples, three RTCARE specifications are compared with ten competing models at four quantile levels using the FZ0 loss, the $D$ diagnostic, the violation rate, and forecast-error measures. The RTCARE specifications remain competitive across the tail levels.
In the application to U.S. stock returns, the proposed models and competing benchmarks are evaluated in-sample and in one-day-ahead out-of-sample forecasting, together with ES backtests and model confidence set analysis. The proposed class delivers accurate coverage, strong ES calibration, and competitive out-of-sample forecast performance without relying on a fully parametric return distribution.
Further work can examine the noise and bias of realized quantiles and realized ES measures, and can extend the
expectile dynamics to capture additional effects, such as leverage effects associated with realized volatility measures.
\section*{Acknowledgments}
This work is partially supported by JSPS KAKENHI (Grant number 24H00142).
The computational results are obtained using Rcpp. Generative AI tools were used to assist with coding and proofreading.
\small
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