HIAS-E-127

High-frequency realized stochastic
volatility model

Toshiaki Watanabe, Jouchi Nakajima
Hitotsubashi University

January 2023

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:https://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

High-frequency realized stochastic volatility model∗
Toshiaki Watanabe† and Jouchi Nakajima‡
January 2023

Abstract
A new high-frequency realized stochastic volatility model is proposed. Apartfrom the standard daily-frequency stochastic volatility model, the high-frequency
stochastic volatility model is ﬁt to intraday returns by extensively incorporating
intraday volatility patterns. The daily realized volatility calculated using intraday
returns is incorporated into the high-frequency stochastic volatility model by considering the bias in the daily realized volatility caused by microstructure noise. The
volatility of intraday returns is assumed to consist of the autoregressive process,the seasonal component of the intraday volatility pattern, and the announcement
component responding to macroeconomic announcements. A Bayesian method via
Markov chain Monte Carlo is developed for the analysis of the proposed model.
The empirical analysis using the 5-minute returns of E-mini S&P 500 futures provides evidence that our high-frequency realized stochastic volatility model improves
in-sample model ﬁt and volatility forecasting over the existing models.
Key words: Bayesian analysis, High-frequency data, Markov chain Monte Carlo,
Realized volatility, Stochastic volatility model, Volatility forecasting.

∗

The authors thank the participants in the 4th International Conference on Econometrics and Statistics (EcoSta 2021), the Workshop “Macro- and Financial Econometrics” in the 7th Hitotsubashi Summer
Institute (HSI 2021), and the 14th International Conference on Computational and Financial Econometrics (CFE 2020) for their valuable comments. The authors acknowledge ﬁnancial support from the
Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government throughGrant-in-Aid for Scientiﬁc Research No. 19H00588 (Watanabe only) and 20H00073, and Hitotsubashi
Institute for Advanced Study.
†
Center for the Promotion of Social Data Science Education and Research and Institute of Economic
Research, Hitotsubashi University (E-mail: t.watanabe@r.hit-u.ac.jp).
‡
Institute of Economic Research, Hitotsubashi University (E-mail: nakajima-j@ier.hit-u.ac.jp).

1

Introduction

Analysis of intraday time series of ﬁnancial market variables has become popular andoften provided richer information about price and volatility dynamics than traditional
daily-frequency time series analysis. Realized volatility (RV) has been the main workhorseof this context and a wide range of methodological improvements and applications havebeen developed in the literature (e.g., Barndorﬀ-Nielsen and Shephard, 2007; Andersen
and Benzoni, 2009; Andersen et al., 2010). Compared to the traditional daily time series
analysis with the stochastic volatility (SV) models (e.g., Ghysels et al., 2002; Shephard,2005), the RV approach provides easily-computed measures of daily volatility based on theintraday price information for understanding price dynamics and forecasting volatility.
A hybrid approach that builds on the SV model with the RV measure is also proposed as the realized SV (RSV) models (e.g., Takahashi et al., 2009, 2016; Koopman andScharth, 2012). The RSV model utilizes the daily RV in the context of daily SV models
by formulating a measurement equation of the daily RV linked to the latent SV andmicrostructure noise. Takahashi et al. (2021) show that this modeling strategy improves
the volatility forecasting performance of the daily SV model.
Stroud and Johannes (2014) shed light on the SV model in the context of highfrequency intraday time series, proposing a new approach that directly models the intraday returns in the form of traditional daily SV models but with practically relevant,intraday volatility patterns. The new model assumes that log-volatility of intraday (suchas 5-min) returns follow a sum of persistent autoregressive processes and other key ingredients: seasonal components of the intraday volatility patterns and news announcementeﬀects. These ingredients are commonly observed in intraday behaviors of stock returns.
The seasonality of the intraday volatility pattern typically forms a U-shape, which characterizes relatively more volatile trading time at the opening and closing of trading hours.
The stock market can be volatile at macroeconomic announcements such as releases of
GDP estimates and of the central bank’s monetary policy decisions. This news-eﬀect
component enriches the high-frequency SV model with respect to temporal hikes of thevolatility following such announcements.
The current paper proposes a new high-frequency RSV model. We incorporate thedaily RV calculated using intraday returns into the high-frequency SV model, considering the bias in the daily RV caused by microstructure noise. In line with the idea of
the RSV models for daily returns, the daily RV in the proposed model is expected to1

play an important role to pin down the latent variables of intraday SV. We apply the
proposed model to the 5-minute returns of E-mini S&P 500 futures, following Stroudand Johannes (2014). We compare the proposed high-frequency RSV models with thehigh-frequency SV models to examine how introducing the daily RV works in the context of the high-frequency SV models. The empirical analysis provides evidence that ourframework performs eﬀectively in terms of model in-sample ﬁt and volatility forecasting.
The empirical results of in-sample ﬁt and forecasting accuracy indicate that the proposedmodel dominates the high-frequency SV model.
As related literature, Bekierman and Gribisch (2021) propose a mixed-frequency intraday SV model, incorporating daily series of long-lived volatility factor and intraday
series of short-lived volatility factors. The long-lived volatility factor is assumed as thelatent variable which follows a simple AR(1) process evolving on a daily basis. In contrast, the current paper directly incorporates the daily RV with its measurement equation
linked to the latent SV process structured in the high-frequency context to exploit the
information in the daily RV for estimating the latent process and parameters in thehigh-frequency SV model. From another perspective, Santos (2019) uses trading volume
to create a time-deformed series of intraday returns in the high-frequency SV models,
based on the idea that increasing trading volume reﬂects more information arriving to
the market, potentially yielding more changes in volatility.
The paper is organized as follows. Section 2 deﬁnes the high-frequency RSV models
and discusses about each component of the proposed model. Section 3 describes theBayesian computation for model ﬁtting and forecasting. Section 4 provides an empiricalanalysis of the proposed models for E-mini S&P 500 futures. Finally, Section 5 concludes.

2

The model

2.1

High-frequency SV modelDeﬁne a univariate time series of high-frequency intraday return, yt = log(Pt /Pt−1 ), fort = 1, 2, . . . , T , where Pt is the price at time t. The sampling frequency is seconds or
minutes, such as 5 sec, 1 min, and 5 min. For the notation to describe both daily andintraday returns, let y[k,τ ] denote the return for the k-th intraday period on the day τ ,for k = 1, . . . , K, and τ = 1, . . . , N , where K is the number of intraday periods, and N
is the number of days. This notation implies y[1,1] = y1 , and y[k,τ ] = yk+(τ −1)K .

2

For instance, as for the dataset of 5-min returns of E-mini S&P 500 futures used in
the empirical analysis below, the time series covers the trading hours from 18:00 (EST)to 17:00 in each day with a short break from 16:15 to 16:30. In our intraday 5-min returnseries, we include two break-time returns from 16:15 to 16:30 and from 17:00 to 18:00. In
sum, the number of intraday periods is K = 275. The data series continues from 18:00
on Sunday and to 17:00 on Friday, with a weekend trading closure from 17:00 on Friday
to 18:00 on Sunday.
The return is modeled as
y t = V t εt + Jt Zt ,εt ∼ standardized t(ν),(1)

where Vt is the total volatility, and Jt Zt is the jump component for returns.
The total volatility consists of three components:
Vt = Xt · St · At ,where Xt is the autoregressive factor of the SV, St is the intraday seasonal eﬀect, and At
is the announcement eﬀect. Taking the log of the squared total volatility leads to
ht ≡ log(Vt2 ) = xt + st + at ,where xt = log(Xt2 ), st = log(St2 ), and at = log(A2t ). We describe the ingredients of themodel in detail as follows.
Heavy-tailed errors
The heavy-tailedness of the error distribution is a well-known, important element inmodeling stock returns with the daily SV models (e.g., Chib et al., 2002; Nakajima andOmori, 2009) and high-frequency intraday returns (e.g., Andersen et al., 2007, 2010). We
incorporate the Student-t distribution with the degree-of-freedom ν, standardized suchthat the variance of εt is equal to one. Then, using the scale factor λt , we represent εt as
εt =p
λt ϵ t ,ν−2∼ χ2 (ν),λtϵt ∼ N (0, 1), and ν > 2.

This formulation makes the posterior computation of the model easy and eﬃcient withthe latent variable λt . The distribution reduces to the normal distribution if ν is inﬁnite.

3For the model with normal errors, we set λt = 1, for all t.
SV autoregressive process
For the autoregressive process {xt }, we specify a form of AR(1) model as
xt+1 = µ + ϕ(xt − µ) + ηt ,

(2)

where
ϵt
ηt

!
∼ N (0, S),

S =

1

ρσ

!

ρσ σ 2,(3)

for t = 1, . . . , T − 1, and x1 ∼ N (µ, σ 2 /(1 − ϕ2 )), with |ϕ| < 1. The leverage eﬀect
between the high-frequency intraday returns and volatility process is described by anegative correlation (ρ) between the shocks (ϵt , ηt ) in Equation (3).
Stroud and Johannes (2014) assume that the SV component is the sum of two AR(1)processes to capture long- and short-lived autoregressive factors, by deﬁning xt = x1t +x2t ,
x1,t+1 = ϕ1 x1t + η1t , and x2,t+1 = µ + ϕ2 (x2t − µ) + η2t , with 0 < ϕ2 < ϕ1 < 1. This type of
additive multi-process SV has been well studied and known as superposition models (e.g.,
Shephard, 1996; Omori et al., 2007). While Stroud and Johannes (2014) argue that thetwo autoregressive processes improve in-sample model ﬁt and volatility forecasting overthe single autoregressive process, Santos (2019) argues that estimating two autoregressive
processes in the high-frequency SV model can be diﬃcult to appropriately estimate evenwith strong priors and restriction on the parameters (ϕ2 < ϕ1 ), because the additivemodel of two similar latent processes could be too ﬂexible to identify. We also encountereda situation that the MCMC algorithm does not safely converge when we formulate thetwo autoregressive models in our intraday SV and RSV models. For these reasons, weemploy the single autoregressive process as above in this paper.
Jump component
Following a standard approach to modeling jumps in the context of daily SV models (Eraker et al., 2003; Chib et al., 2002; Nakajima and Omori, 2009), the jump component
Jt Zt in Equation (1) is governed by the jump indicator variable, Jt ∈ {0, 1}, and the jump
size denoted by Zt . The jump occurs with the probability 0 < κ < 1: Pr[Jt = 1] = κ, for4t = 1, . . . , T . We assume that the jump size follows the normal distribution:
Zt ∼ N (µz , σz2 ),for t = 1, . . . , T .
Stroud and Johannes (2014) formulate another jump component in the volatility
equation (2), and assume that two jumps occur coincidently. We compared the modelwith jumps only in returns and the one with two coincident jumps and found that thejump in volatility has only a marginal gain for in-sample ﬁt and out-of-sample forecasts.
Therefore, we select the formulation without the jump in volatility for parsimonious
modeling.
Intraday seasonal eﬀect
Let βk denote the intraday seasonal eﬀect at intraday period k, for k = 1, . . . , K. Deﬁne
Htk as an intraday-period indicator such that Htk = 1, if time t corresponds to the periodk, and zero otherwise. Then, the seasonal eﬀect is speciﬁed by
st =

K
X

Htk βk ,t = 1, . . . , T,

k=1

βk+1 = βk + ηβ,k ,ηβ,k ∼ N (0, ck vβ2 ),k = 1, . . . , K − 1.

For the dynamics of the βk process, Stroud and Johannes (2014) utilize a cubic smoothing
spline. Alternatively, we propose the simple random-walk process to reduce a computational burden. Note that ck denotes the time-varying part of the innovation variance. We
set a large value for ck for the time period k when we expect abrupt changes, such as an
opening time of a local and a major international market, and set ck = 1 for other k. To
identify the latent processes and parameters, the formulation requires some restrictionP
for {βk }. Although Stroud and Johannes (2014) propose the restriction, K1 K
k=1 βk = 0,
and use an ad-hoc sampling step to satisfy it, this approach makes an exact sampling
scheme intractable. In the current paper, we instead deﬁne β1 = 0.
Announcement eﬀects
We consider the announcement eﬀect for each macroeconomic announcement j = 1, . . . , J,
where J denotes the number of announcements. Let αjℓ denote the eﬀect of announcement j at ℓ periods after the news release, for ℓ = 1, . . . , L. We set L = 24, which5means that each announcement can impact on the volatility up to 2 hours after the announcement in case of 5-min returns. Further, deﬁne Ijtℓ as an announcement-period
indicator such that Ijtℓ = 1, if time t corresponds to the period ℓ after the announcement j, and zero otherwise. Then, the announcement eﬀect is speciﬁed by the ﬁrst-order
autoregressive process, that is,
at =

J X
L
X

Ijtℓ αjℓ ,t = 1, . . . , T,

j=1 ℓ=12
ζjℓ ∼ N (0, vαj),

αj,ℓ+1 = φj αjℓ + ζjℓ ,ℓ = 1, . . . , L − 1,

where |φj | < 1, for j = 1, . . . , J. For modeling the announcement eﬀect, Stroud andJohannes (2014) use the cubic smoothing spline. For the same reason mentioned above,
we instead employ the simple autoregressive process to reduce a computational burden.

2.2

High-frequency RSV model

We introduce the daily series of RV to propose the high-frequency RSV model. A plain
RV is deﬁned as
RVτ =

K
X

2
y[k,τ
],

τ = 1, . . . , N.

(4)

k=1

In our empirical analysis, we compute this RV using 5-min returns only when the market
is open, i.e., excluding break-time returns.
A key equation in our high-frequency RSV model is the measurement of the RV:log(RVτ ) = ξ + log

X

exp(ht ) + uτ ,uτ ∼ N (0, σu2 ),(5)t∈D(τ )for τ = 1, . . . , N , where D(τ ) denotes a set of intraday time points on Day τ . The
parameter ξ addresses the bias in the RV. If ξ = 0, log(RVτ ) is an unbiased estimator ofP
the log of the true daily volatility, log t∈D(τ ) exp(ht ).
This equation connects the RV calculated using intraday returns to the model-implied
true volatility with the bias correction. The RSV model for daily returns (Takahashi et al.,2009; Koopman and Scharth, 2012) links the daily RV with the daily model-implied
volatility. In line with this idea, it could be eﬀective to connect an intraday (say, 5min) RV to intraday model-implied volatility, exp(ht ), in the high-frequency RSV model.

6However, such an intraday RV is diﬃcult to compute properly due to the microstructurenoise. Therefore, we come up with daily RV for the link in the measurement equation of
our high-frequency RSV model. Another frequency (i.e., a half day) to compute the RV
may also work, which we will discuss later as future work.
Assume that p(t), denoting the log of an asset price, follows a jump-diﬀusion process:dp(t) = µ(t)dt + σ(t)dW (t) + κ(t)dN (t),(6)

where t is continuous time, µ(t) is drift, σ(t) is instantaneous volatility, κ(t) is the jumpsize, and N (t) is counting measure. If p(t) does not include noise,Z τ
plim RVτ =K→∞X

σ 2 (t)dt +τ −1

κ2 (t)dt.

(7)

τ −1<t≤τ

The ﬁrst term of the right-hand side is called the integrated volatility or continuousP
component, which corresponds to t∈D(τ ) exp(ht ). The second term is called the jumpcomponent. There are some methods available for separating RV τ into the estimate of
the continuous component, Ĉτ and that of the jump component Jˆτ (Barndorﬀ-Nielsen
and Shephard, 2004, 2006; Corsi et al., 2010). Strictly speaking, equation (5) should be
replaced bylog(Ĉτ ) = ξ1 + log

Xu1,τ ∼ N (0, σ12 ),exp(ht ) + u1,τ ,t∈D(τ )

log(1 + Jˆτ ) = ξ2 + log 1 +X

(Jt Zt )2  + u2,τ ,u2,τ ∼ N (0, σ22 ).

t∈D(τ )

Since it makes the estimation diﬃcult, we simply use equation (5), so that the jumpcomponent is included in ξ and uτ in equation (5).

3

Bayesian analysis and computation

3.1

Model estimation

Based on observation y ≡ (y1 , . . . , yT ), we consider model ﬁtting using the Bayesian
MCMC methods including conditional samplers for each model component. The al-7gorithm is implemented with collections of the samplers for the daily SV models withleverage, jumps, and heavy-tailed errors (Chib et al., 2002; Omori and Watanabe, 2008;
Nakajima and Omori, 2009) and for the state space dynamic models (e.g., Prado andWest, 2010).
The full set of posterior computations consists of the following conditional samplers
for latent processes and model parameters:
1. The latent autoregressive SV process: x ≡ (x1 , . . . , xT ).
2. The jump components: J ≡ (J1 , . . . , JT ) and Z ≡ (Z1 , . . . , ZT ).
3. The intraday seasonal and announcement eﬀects: β ≡ (β1 , . . . , βK ) and α ={α1 , . . . , αJ }, where αj ≡ (αj1 , . . . , αjL ), for j = 1, . . . , J.
4. The mixing latent process: λ ≡ (λ1 , . . . , λT ).
5. The parameters deﬁning the latent SV process: θX ≡ (ϕ, σ, ρ, µ).
6. The jump parameters: θJ ≡ (κ, µz , σz ).
7. The parameters deﬁning the intraday seasonal and announcement eﬀects: v ≡(vβ , vα1 , . . . , vαJ ) and φ ≡ (φ1 , . . . , φJ ).
8. The degree of freedom for the Student-t distribution: ν.
9. The parameters deﬁning the RV measurement equation: (ξ, σu )This algorithm generates samples from the full posterior distribution π(ψ, θ|y), whereψ = {x, J , Z, λ}, and θ = {β, α, θX , θJ , v, φ, ν, ξ, σu }. Details of the MCMC computation are documented in Appendix.
Stroud and Johannes (2014) employ the so-called mixture sampler developed by Kim
et al. (1998), and Omori et al. (2007) in their MCMC algorithm for the high-frequency
SV model. The main challenge to generating the posterior sample from the SV modelis the nonlinearity of the return equation in terms of volatility. Kim et al. (1998) takethe logarithm of both sides of the equation to make the model linear and approximate
the log chi-squared distribution by a normal mixture distribution. Conditional on an
auxiliary variable associated with the mixture distribution, the model reduces to a linear
Gaussian state space representation, and we can apply an eﬃcient sampler such as thesimulation smoother (e.g., de Jong and Shephard, 1995). Although it is well known that8

an approximation error is negligible (Omori et al., 2007), Stroud and Johannes (2014)
mix up the mixture sampler and other conditional samplers based on the exact model
speciﬁcation. Therefore, the current paper proposes the exact algorithm fully based on
the original model representation using the eﬃcient multi-move sampler (Shephard andPitt, 1997; Omori and Watanabe, 2008).

3.2

Model evaluation

For the model comparison exercises, we consider in-sample ﬁt and out-of-sample forecasts.
This subsection describes methods to compute measures for the model comparison.
3.2.1

In-sample ﬁt

In a Bayesian framework, we can compare models based on a posterior probability thatis proportional to the product of the prior probability of the model and the marginal
likelihood. A Bayes factor, which is the ratio of two posterior probabilities, is often used
to measure he relative degree of in-sample ﬁt. However, a computation of the marginal
likelihood requires considerable a computation burden when the model includes many
parameters. Therefore, we instead calculate the BIC, which is a good approximation of
the Bayes factor when the number of observations is quite large as in our analysis.
The likelihood of the proposed high-frequency RSV model is estimated using theauxiliary particle ﬁlter (e.g., Pitt and Shephard, 1999; Chib et al., 2002; Omori et al.,2007; Stroud and Johannes, 2014). Deﬁne Y t ≡ (y1 , . . . , yt ), as the data up to time t.
The likelihood of the high-frequency RSV model for Y T is given byL(Y T | θ) =T−1Yf (yt+1 | θ, Y T ),t=0

where the right-hand side of this equation is the product of the predictive density:Zf (yt+1 | θ, Y t ) =p(yt+1 | θ, Y t , ψ t+1 ) · p(ψ t+1 | θ, Y t )dψ t+1 ,where ψ t = (xt , λt , Jt , Zt ). In the auxiliary particle ﬁlter, we use the posterior mean θ̂estimated by the MCMC algorithm to approximate the likelihood density.
To compute the BIC, let Li and di denote the likelihood and the number of parameters9in Model i. The BIC for Model i is deﬁned by BICi = −2 log Li (Y T |θ̂) + di log(T ). The
second term of this criterion is the penalty for a more ﬂexible model that includes moreparameters. Note that we treat β and α as the parameters in θ, making them ﬁxed
at their posterior means in the auxiliary particle ﬁlter, and also counting them in thenumber of parameters (di ).
3.2.2

Out-of-sample volatility forecasting

To compare the out-of-sample ﬁt among competing models, we conduct a recursive forecasting exercise based on one-day-ahead daily volatility forecast. Given the data up
to day τ , we run the MCMC algorithm and generate ψ [k,τ +1] , y[k,τ +1] , and V[k,τ +1] , fork = 1, . . . , K, at each MCMC iteration. The resulting posterior predictive density ofthese variables reﬂects parameter uncertainty in the speciﬁed model. Let στ2+1 and σbτ2+1
denote the true volatility and the volatility forecast for day τ + 1, respectively. We
compute the quadratic variation as the forecast of one-day-ahead volatility:
dτ(i)
QV
+1 =

K
X

(i)
(y[k,τ +1] )2 ,

and

k=1

1
σ
bτ2+1 =

I
X

I i=1

dτ(i)
QV
+1 ,

where (i) and I denote the i-th iteration of the MCMC algorithm and the total number(i)of iterations, respectively, and y[k,τ +1] denotes a draw from the posterior predictive distribution of y[k,τ +1] at the i-th iteration. We run the MCMC algorithm to estimate model
parameters and latent variables and to obtain the one-day-ahead volatility forecast foreach business day. We recursively conduct this computation every business day for theforecasting period.
Stroud and Johannes (2014) use the particle ﬁltering method to obtain the out-ofsample forecast with all the parameters ﬁxed in their forecasting exercise. In contrast,
we conduct the rolling estimation with MCMC sampling and obtain the posterior predictive mean as a point forecast. Note that our method takes account uncertainty in theparameters.

101(i) 5-min returns, in percent

0

-1

0
4

5000

10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000(ii) Log of daily RV

2

0

-2
0

5000

10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000

Figure 1: 5-min intraday returns (top) and log of daily RV (bottom) of E-mini S&P
500 futures from January 1 to December 31 in 2015. The RV series is plotted in a step
function with the intraday 5-min frequency.

4

Empirical analysis

4.1

Data and setup

The high-frequency RSV model is ﬁt to 5-min intraday returns of E-mini S&P 500 futures,
which are traded on the Chicago Mercantile Exchange’s Globex platform. As mentioned
above, our dataset covers the prices from 18:00 (EST) on Sunday and to 17:00 on Friday,
for 24 hours including two breaks from 16:15 to 16:30 and 17:00 to 18:00. We count eachbusiness day from 18:00 to 18:00, and create 5-min return series including two break-time
returns to obtain K = 275 intraday returns. The sample period spans from January 1,
2015, to December 31, 2019. We divide this sample period into the ﬁrst year and therest four years as the pre-analysis and the forecasting exercise periods, respectively. The
number of business days is N = 249 and 996 for these two periods, respectively. The
whole sample is N = 1,245 and T = 342,375.

11

0.09

(i) By period of day

breaks

0.06

0.03

0
0.25

2

4

6

8

10

12

14

16

18

20

22

24

(ii) By date (from 2015/Jan to 2019/Dec)

0.20
0.15
0.10
0.05
0

100

200

300

400

500

600

700

800

900

1000

1100

1200

Figure 2: Standard deviations of 5-min intraday returns of E-mini S&P 500 futures inpercent, by the period of day (top) and by date (bottom) for the full sample period, fromJanuary 1, 2015, to December 31, 2019. In the top panel, the horizontal axis shows anhour during the day in EST.
Figure 1 plots the 5-min intraday returns for the pre-sample period, which shows
considerably time-varying volatility. It also plots the daily RV calculated using the 5min intraday returns. It markedly hikes in the second half of the sample, reﬂecting ahigh volatility period as in the top panel. To illustrate intraday and interday volatility
patterns, Figure 2 plots the standard deviation of returns (y[k,τ ] ) by the period of day(horizontal for k = 1, . . . , K) and by date (for τ = 1, . . . , N ), which are computed using
the full-sample period. The intraday pattern in the top panel highlights some hikes whichmostly reﬂect openings in international ﬁnancial markets. That is, we ﬁnd small hikes at
3:00, 4:00, 20:00, when Frankfurt, London, and Hong Kong markets open, respectively,
and a relatively larger increase at 9:30, corresponding to the opening of the New York
market. During the New York trading hours that end at 16:00, we observe a U-shaped
pattern where the prices tend to be more volatile around the opening and closing times.
Table 1 reports summary statistics for the intraday returns in the pre-analysis and12Period(a) 2015/Jan – 2015/Dec(b) 2016/Jan – 2019/Dec

Mean Stdev. Skewness Kurtosis Min.
0.000 0.061-0.54384.40-1.8190.000 0.052-1.061
87.48-3.067

Max.
2.160
1.328

Table 1: Summary statistics for 5-min intraday returns of E-mini S&P 500 futures, inpercent, for the (a) pre-analysis period (N = 259, T = 68,475) and (b) forecasting
exercise period (N = 996, T = 273,900).
1
2
3
4
5
6
7
8
9
10
11
12

ADP Employment (monthly)
Consumer Price Index (monthly)
Durable Goods (monthly)
GDP Advance Estimate (quarterly)Monthly Payrolls (monthly)
Empire State Manufacturing (monthly)
Consumer Conﬁdence (monthly)
Philadelphia Fed Manufacturing Index (monthly)
ISM Manufacturing Index (monthly)
ISM Services Index (monthly)
FOMC Minutes (twice a quarter)
FOMC Announcements (twice a quarter)

Table 2: Macroeconomic statistics for announcements eﬀect.

forecasting exercise periods. For both periods, the skewness is negative, and the kurtosis
is remarkably high. This fact implies that the 5-min return distribution has considerably
heavy tails. We examine how the leverage eﬀects, heavy-tailed errors, and jumps capture
these characteristics of high-frequency intraday stock returns.
For the Bayesian inference, the following priors are used: (ϕ+1)/2 ∼ B(20, 1.5), σ 2 ∼IG(40, 0.2), (ρ + 1)/2 ∼ B(1, 1), µ ∼ N (−5, 1), ν ∼ G(16, 0.8)I[ν > 2], κ ∼ B(1, 500),µz ∼ N (0, 1), σz2 ∼ IG(20, 4), ξ ∼ N (0, 1), σu2 ∼ IG(2.5, 0.025), vβ2 ∼ IG(40, 1), β2 ∼2N (0, 100), vαj∼ IG(40, 1), αj1 ∼ N (0, 100), and (φj + 1)/2 ∼ B(10, 1.5), for j =1, . . . , J, where B and G denote the beta and gamma distributions, respectively. From
the observation of the intraday volatility pattern in Figure 2, we set ck = 100 for k =1, 25, 109, 121, 187, 265, 271, which correspond to the opening and closing time of relevantinternational ﬁnancial markets mentioned above, including the times 16:30 and 18:00,
which corresponds to the time after the breaks of S&P E-mini futures trading. For the
daily series of RV, we calculate it using the 5-minute returns only for trading hours as
explained above.

13

For the macroeconomic announcements, we use releases of major economic statistics
and Federal Reserve Boards’ announcements listed in Table 2, which are the same as usedby Stroud and Johannes (2014) with one exception that we exclude Jobless Claims as
its announcement eﬀects are statistically not relevant in our high-frequency SV and RSV
models, through the pre-sample and forecasting exercise periods. We assume J = 24,
which implies the announcement eﬀects can be alive for 2 hours after the announcement.
Following Stroud and Johannes (2014), we include Sunday opens as it tends to be volatile
responding to information gathered during the weekend. Note that we do not take into
account how diﬀerent the released number of the macroeconomic statistics or contentsof the policy announcements is, compared to market expectations. It is reasonable toconsider that the larger the “market surprise” is, the higher the volatility gets. Examining
the relationship between the market surprise and volatility is of interest, which is left as
future work.
The MCMC analysis was run for a burn-in period of 2,000 samples prior to saving thefollowing 20,000 samples. We use every ﬁfth draw of them for the posterior inferences. To
check the convergence of the MCMC algorithm, we compute the convergence diagnostic(CD) of Geweke (1992). With the MCMC sample size speciﬁed here, the resulting CDs
indicate that the null hypothesis that the Markov chain converges is not rejected at a
5 percent statistical signiﬁcance level in our analysis. The computational results are
generated using Ox version 7.0 (Doornik, 2006).
In the following analysis, the following four types of high-frequency RSV models are
considered:• RSV: normal distribution, no jumps (λt = 0 and Jt = 0, for all t);• RSVt: Student t-distribution, no jumps (Jt = 0, for all t);• RSVJ: normal distribution, with jumps in return (λt = 0, for all t);• RSVJt: Student t-distribution, with jumps in return.
All the models described here include the leverage eﬀect, the intraday volatility pattern,
and the macroeconomic announcement eﬀects. For each formulation above, we consider
the high-frequency SV model, which excludes the measurement equation of the daily RV.

14

ϕ

σ

ρ

µ

vβ

vα1

φ1

RSV
0.981 (0.001)[0.979, 0.983]
13.2
0.219 (0.008)[0.198, 0.229]
26.3-0.064 (0.013)[-0.087, -0.040]
2.3-7.641 (0.185)[-7.856, -7.248]
42.4
0.252 (0.016)[0.225, 0.287]
11.6
0.313 (0.052)[0.234, 0.435]
1.2
0.896 (0.104)[0.625, 0.998]
2.3

κ

µz

σz

RSVJ
0.986 (0.001)[0.984, 0.989]
24.0
0.182 (0.009)[0.160, 0.194]
28.2-0.100 (0.014)[-0.129, -0.076]
3.1-6.301 (0.094)[-6.481, -6.138]
23.2
0.244 (0.015)[0.219, 0.274]
1.2
0.305 (0.041)[0.232, 0.393]
2.6
0.891 (0.111)[0.600, 0.996]
1.1
0.0032 (0.0005)[0.0023, 0.0044]
13.9-0.005 (0.022)[-0.050, 0.039]
0.4
0.264 (0.024)[0.223, 0.313]
8.7

ν

ξ

σu

-0.078 (0.012)[-0.101, -0.055]
2.8
0.109 (0.009)[0.093, 0.128]
4.3-0.036 (0.014)[-0.059, -0.004]
2.2
0.139 (0.014)[0.115, 0.166]
8.8

RSVt
0.989 (0.001)[0.987, 0.990]
10.7
0.160 (0.003)[0.154, 0.164]
23.0-0.106 (0.016)[-0.134, -0.079]
9.2-7.287 (0.091)[-7.440, -7.102]
22.3
0.304 (0.028)[0.255, 0.349]
34.9
0.310 (0.045)[0.231, 0.408]
2.5
0.832 (0.146)[0.470, 0.996]
3.2

7.672 (0.588)[6.912, 8.946]
46.4-0.119 (0.016)[-0.163, -0.093]
11.4
0.159 (0.015)[0.134, 0.193]
2.7

RSVJt
0.989 (0.001)[0.988, 0.991]
16.1
0.157 (0.006)[0.143, 0.166]
31.0-0.112 (0.015)[-0.142, -0.082]
3.2-5.566 (0.390)[-6.324, -5.041]
47.3
0.241 (0.013)[0.219, 0.269]
0.4
0.311 (0.050)[0.236, 0.441]
1.00.854 (0.190)[0.280, 1.000]
0.7
0.0005 (0.0002)[0.0002, 0.0009]
25.2-0.072 (0.114)[-0.307, 0.140]
3.3
0.412 (0.044)[0.340, 0.514]
8.0
8.673 (0.654)[7.942, 10.152]
41.2-0.113 (0.015)[-0.139, -0.061]
23.4
0.164 (0.015)[0.137, 0.195]
2.9

The ﬁrst row: posterior mean and standard deviation in parentheses.
The second row: 95% credible interval in square brackets.
The third row: ineﬃciency factor.

Table 3: Posterior estimates of the selected parameters for the high-frequency RSV
models for E-mini S&P 500 futures, obtained from the pre-sample period: January 1 to
December 31 in 2015 (N = 259 and T = 68,475).

15

4.2

Posterior estimates

Table 3 reports the posterior estimates of selected parameters for the high-frequency
RSV models ﬁt to the pre-analysis period. Note that for vαj and ψj , we report their ﬁrst
elements (j = 1). The posterior mean of the AR(1) coeﬃcient ϕ in the latent SV process
indicates a considerable persistence, ranging from 0.98 to 0.99. The posterior means of(ϕ, σ) are (0.981, 0.219) for the RSV model, while (0.989, 0.157) for the RSVJt model.
The latter model includes heavier tails of the error distribution and jump components,
which makes the autoregressive component more persistent with a smaller variance.
The posterior mean of ρ is around −0.1, with the credible intervals excluding zero,
which indicates that the leverage eﬀect exists as commonly observed for the daily stock
return and volatility. The posterior mean of ν, estimated around 8, indicates that theerror distribution requires considerably heavy tails. In the jump models, the posteriormean of κ is about 0.3% for the RSVJ model and 0.05% model for the RSVJt model.
The lower probability of jumps in the RSVJt model implies that some large returns inabsolute value are captured by the heavy-tailed error distribution. For the same reason,
the posterior mean of σz for the RSVJt model is higher than that for the RSVJ model
as estimated jumps are averagely larger in the RSVJt model than the RSVJ model.
For the parameters in the measurement equation of the RV, the posterior mean of ξis from −0.119 to −0.036 with the credible intervals excluding zero, which indicates thatthe bias underlying the measurement equation is negative. The bias in RV caused by
microstructure noise can be positive or negative (Hansen and Lunde, 2005). As mentioned
above, we use the RV calculated using intraday returns only when the market is open,
while the model-implied true daily volatility is a whole day’s (24-hour’s) volatility, whichcauses a negative in RV. The estimation result suggests that the negative bias derived
from non-trading hours dominates.
The table also reports the ineﬃciency factor to check the eﬃciency of the MCMC
P
algorithm. This factor is deﬁned by 1 + 2 ∞s=1 ρs where ρs is the sample autocorrelation
at lag s, estimating the ratio of the numerical variance of the posterior sample mean tothe variance of the sample mean from uncorrelated draws, as a measure of how well the
MCMC chain mixes (see, e.g., Chib, 2001). We compute the ineﬃciency factor using aParzen window with bandwidth bw = 1,000. The resulting ineﬃciency factors are all lessthan 100, which is as modest as for the standard SV models for daily stock returns. This
result indicates that the MCMC algorithm developed in this paper mixes well enough to

16

3

2

1

0

-1

0

2

4

6

8

10

12

14

16

18

20

22

24

Figure 3: Intraday volatility pattern: posterior means (solid) and 95% credible intervals(dotted) for the volatility intraday seasonal eﬀects (β1 , . . . , βK ), obtained from the RSVJt
model for the pre-analysis period. The horizontal axis shows an hour during the day in
EST. Breaks of market trading occur from 16:00 to 16:15 and from 17:00 to 18:00.
be practically applied to the real-data analysis.
Figure 3 shows the posterior estimates for the intraday volatility pattern with theposterior mean and 95% credible intervals of β, obtained from the RSVJt model. The
result shows a marked intraday pattern of volatility, which shifts up from 9:30 to 16:00
reﬂecting the trading hours of the New York market. The credible intervals are quite
narrow, which suggests that this intraday pattern is empirically relevant.
Figure 4 plots the posterior estimates for the macroeconomic announcement eﬀects,
which indicate that the inﬂuence of releases of CPI and Employment are statisticallyimportant for the intraday volatility. While the eﬀects of the other announcements seemto be modest, we ﬁnd that the eﬀects change over time as some of them become signiﬁcant,with the credible intervals excluding zero when the estimation period is changed. For the
impact of Sunday open, the posterior estimate suggests it should be distinguished.

17Sunday Open

ADP

CPI

Durable

0

0

10
20
Employment

0

0

-1

-1

-1

0
10
20
Empire101020Consumer
1

2

0
10
20
PhilFed
0.5
0.0

0

0

10
ISM-S

20

0

0
-1

-1.0

-1
0
10
FOMC
1

20

0
10
20
ISM-M
1

-0.5

0

0

1

1

2.5

0.0
0

GDP

1

5.0

-1
0
10
20
Minutes
1

0

10

20

0

10

20

0
0
-1

-1

-1

-2
0

10

20

0

10

20

0

10

20

Figure 4: Macroeconomic announcement eﬀects: posterior means (solid) and 95% credibleintervals (dashed) of αjℓ , obtained from the RSVJt model for the pre-analysis period.

4.3

Model comparisons

4.3.1

Result of in-sample ﬁt

Table 4 reports the estimated BICs for the high-frequency SV and RSV models obtained
from the full sample period. Due to a computational burden, we ﬁt the competing models
to the data separately in each calendar year from 2015 to 2019 and sum them up. Because
the high-frequency RSV models specify the additional measurement equation of the daily
RV, we use a partial likelihood only for the measurement of intraday returns (yt ) in thehigh-frequency RSV models to compare the in-sample ﬁt between the high-frequency SV
and RSV models.
The estimated BIC indicates that using daily series of RV improves the in-sample ﬁt.
Based on the BIC, the RSVJt model is the best model, and the RSVt model is the second
among the competing models. In the class of the high-frequency RSV model, the modelwith Student t-error distribution (RSVt and RSVJt) ﬁts the data better than that with
the normal error distribution (RSV and RSVJ).

18

Model
BIC
RSV-682037 [6]RSVJ -678130 [7]RSVt-693029 [2]RSVJt -696145 [1]SV-683399 [5]SVJ-677555 [8]SVt-683883 [4]SVJt-692880 [3]

MSE
QLIKE
2.198 [6] 0.748 [3]*2.048 [3]* 0.763 [4]*1.937 [1]* 0.718 [2]*1.951 [2]* 0.702 [1]*2.352 [8] 0.821 [8]2.325 [7] 0.782 [7]*2.089 [4] 0.774 [6]*2.179 [5] 0.774 [5]*Table 4: Result of model comparison: BIC for in-sample model ﬁt, and MSE and QLIKE
for one-day-ahead volatility forecasts. The sample period is 2015/Jan to 2019/Dec (N =1,245) for BIC and 2016/Jan to 2019/Dec (N = 996) for MSE and QLIKE. Rankings
are in square brackets. The model with the “*” mark is in the 90% model conﬁdence set(MCS).

4.3.2

Result of out-of-sample forecast performance

We compare the predictive ability of one-day-ahead volatility among the competing models, based on the mean squared error (MSE) and quasi likelihood (QLIKE) as loss functions, following Patton (2011):
N0 +N1 
2
1 X
d 2 − QV2 ,
MSE =
QV
τ
τ
N1 τ =N +10

N0 +N11 X
QLIKE =
N1 τ =N +1
0

QVτ2
d2
2 + log QVτ
d
QV

!
.

τ

Since the true QV is unobserved, we must use the proxy. Patton (2011) shows that the
MSE and QLIKE are robust loss functions in the sense that they lead to the same ranking
as the one when the true volatility is used if the proxy is the unbiased estimator of thetrue volatility. In our analysis, we calculate the RV using 30-min returns, including the
break-time returns.
In our out-of-sample forecasting exercise, we employ a rolling-window forecasting
strategy. First, we estimate the model using data for N0 = 249 business days, whichcorresponds to the pre-analysis period, January 1 to December 31, 2015, and forecast
the volatility on the next business day, January 3, 2016. Next, we roll the data window
by one business day with the sample size kept the same (N = 249) and forecast the
one-day-ahead volatility. We repeat this procedure until all the data are used in the19forecasting exercise period, which ends on December 31, 2019. In sum, we obtain theforecasted volatility for N1 = 996 business days.
Table 4 reports the MSE and QLIKE obtained from the forecasting exercise. The MSE
and QLIKE suggest that the RSVt and RSJt models provide the best performance ofvolatility forecasting, respectively. The class of high-frequency RSV models is uniformly
better than that of high-frequency SV models based on both measures. The models with
Student t-error distribution perform better than those with the normal error distribution.
The inﬂuence of introducing the jump component on the predictive ability is marginal,
based on the MSE and QLIKE.
To evaluate the MSE and QLIKE further, we compute the model conﬁdence set(MCS) proposed by Hansen et al. (2011). Following Takahashi et al. (2021), the MCS
is estimated with 1, 000 bootstrap replications and 10 blocks, using R package MCS. In
Table 4, the models indicated by the “*” mark are in the 90% MCS. Based on QLIKE,
the SV model is excluded for the best set of models. Based on MSE, all the speciﬁcations
without the RV are excluded, and the MCS consists of the RSVJ, RSVt, and RSVJt
models.
To further assess whether the diﬀerence in QLIKE is critical in the model comparison,
we examine the hypothesis testing proposed by Giacomini and White (2006), which wecall the GW tests hereafter. For two candidate models i and j, we deﬁne dij,t as a loss
diﬀerence on the day t. Giacomini and White (2006) propose two types of hypothesis
tests, unconditional and conditional.
To test equal unconditional predictive ability, the following null hypothesis is considered:H0 :∀i, j ∈ M1 ,E[dij,t ] = 0,

i > j.

The test statistics for this null hypothesis are computed with the Newey-West heteroscedasticity and autocorrelation consistent estimator.
The conditional predictive ability is examined using the following null hypothesis:
H0 :E[dij,t+1 |It ] = 0,

where It is the information set up to the day t. This hypothesis testing bases the expected
loss diﬀerence between two models on the day t + 1 given the information up to the day t.
Following Takahashi et al. (2021), we deﬁne q t = (1, dij,t )′ and arrange the null hypothesis

20Model
SVt
SVJt
RSVt
RSVJt

SVt

SVJt
0.46*

−0.03
−2.00**−2.74**−2.57**−3.70**

RSVt
0.89*
1.00*

RSVJt
0.93**
1.00**
0.68*

−0.83

Table 5: Result of model comparison in the GW tests for the QLIKE loss function. Lower
triangular elements: the test statistics (t-value) of the unconditional GW tests; upper
triangular elements: the proportion for which the row model has higher predicted losses
than the column model on the conditional GW tests. Values with the “**” and “*”marks indicate the statistical signiﬁcance with the 1% and 5% levels, respectively.

as
H0 :

E[q t dij,t+1 ] = 0.

See Takahashi et al. (2021) for the computation of the test statistics for this hypothesis.
Table 5 reports the test statistics of the unconditional GW tests in the lower triangularelements, where a positive value indicates that the row model has a higher loss than thecolumn model. The table also reports, in its upper triangular elements, the proportion forwhich the row model has higher predicted losses than the column model on the conditional
GW tests. Here, we compare only the candidate models with t-distribution because theseare mostly superior to the normal-error models in the analyses above. Evidently, the
RSVt and RSVJt models perform better than the SVt and SVJt models because boththe unconditional and conditional GW tests indicate that the null hypotheses are rejectedwith the 1% signiﬁcance level. The result for the diﬀerence in QLIKE between the RSVt
and RSVJt models is mixed. The conditional GW test indicates that the RSVJt model
is superior to the RSVt model with 5% statistical signiﬁcance, while the unconditional
GW test does not reject the null hypothesis.

5

Conclusion

This paper proposes the high-frequency RSV model for intraday stock returns. The model
consists of the latent autoregressive SV process, seasonal components of the intraday
volatility patterns, and jumps in prices. The key aspect of the proposed model is to21

specify the measurement equation of RV, which links the daily RV and model-implied
true volatility with the bias of microstructure noise taken into account. We apply the
proposed model to the 5-min returns of E-mini S&P 500 futures and show that the daily
RV plays an important role in describing the intraday volatility. The model comparison
exercise indicates that the daily RV improves both the in-sample ﬁt and performance of
one-day-ahead volatility forecasting.
Although we specify the daily frequency for the RV series, another frequency, suchas half-day, may also work. From the ﬁnding that the intraday volatility gets relatively
higher during the trading hours of the New York market, the RV computed using only the
intraday returns during these hours may be more relevant for the measurement equationin the high-frequency RSV model. We leave this point as future work. From anotherperspective, we can extend the high-frequency RSV model by incorporating a skewed
error distribution. It is of interest to investigate how the skewed distribution capture
the heavy tails and asymmetry of the return distribution (e.g., Takahashi et al., 2016;
Nakajima, 2017), which is relevant in volatility forecasting and risk management analysis.

Appendix. MCMC algorithm for the high-frequency
RSV model
In this appendix, we document details of the MCMC computation method for the highfrequency RSV model, which is outlined in Section 3.1.
A.1 Stochastic volatility process
We derive a posterior sampler for the latent SV process x, conditional on other state√processes and parameters. Deﬁning yt∗ = (yt − Jt Zt ) exp{−(µ + st + at )/2}/ λt , andx̃t = xt − µ, we obtain a non-linear state-space model:yt∗ = exp(x̃t /2)ϵt ,x̃t+1 = ϕx̃t + ηt ,X
log(RVτ ) = ξ + log
exp(x̃t + µ + st + at ) + uτ ,t∈D(τ )

22

where (ϵt , ηt )′ ∼ N (0, S), and x̃1 ∼ N (0, σ 2 /(1−ϕ2 )). We assume that ϵt (or ηt ) and uτ are
uncorrelated for any t and τ . To implement the conditional sampler for (x̃1 , . . . , x̃T ), we
arrange the multi-move sampler for the SV model with leverage (Omori and Watanabe,2008).
A.2 Jump component
Deﬁne ŷt = yt −√λt rt eht /2 , rt = ρt (x̃t+1 − ϕx̃t )/σ, wt2 = λt (1 − ρ2t )eht , where ρt =

ρ · I[t < T ], for t = 1, . . . , T ; and I[ · ] denotes the indicator function that takes one, if
the argument is true, and zero otherwise. Conditional on (y, x, θX , θJ , β, α), the joint
conditional posterior density of the jump indicator and size can be written as(ŷt − Jt Zt )2 (Zt − µz )2−× κJt (1 − κ)1−Jt ,π(Jt , Zt | · ) ∝ exp −
2
2
2wt
2σz
for t = 1, . . . , T . Integrating this joint posterior density with respect to Zt , we obtainZ
π(Jt | · ) ∝∝π(Jt , Zt | · )dZt
(wt2 + σz2 Jt )−1/2 exp

(ŷt − Jt µz )2× κJt (1 − κ)1−Jt .
−2(wt2 + σz2 Jt )We generate the sample of Jt , following the posterior jump probability
Pr[Jt = 1 | · ] =κ · ϕN (ŷt | µz , wt2 + σz2 ),(1 − κ) · ϕN (ŷt | 0, wt2 ) + κ · ϕN (ŷt | µz , wt2 + σz2 )for t = 1, . . . , T , where ϕN (ŷt | µ, σ 2 ) denotes the density function of a normal distribution
N (µ, σ 2 ) at ŷt .
Given the jump indicator, the posterior distribution of the jump size is given by
Zt | (Jt = 1) ∼ N (µ̂t , σ̂t2 ),whereσ̂t2 =

1
1
+ 2
2
wt
σz



−1
,

and

for t = 1, . . . , T .

23µ̂t = σ̂t2

ŷt
µz
+ 2
2
wt
σz


,

A.3 Intraday seasonal and announcement eﬀects
Deﬁne h̃t = xt + at . The conditional posterior distribution of β is given by
)
(
√
T
Y
1
(yt − λt rt e(h̃t +st )/2 − Jt Zt )2π(β | · ) ∝ π(β | vβ ) ×
exp −
st /2
e
2λt (1 − ρ2t )eh̃t +st
t=1
(
)
√
K Y
h̃t /2 βk /22Y1(yt − λt rt e e− J t Zt )∝ π(β | vβ ) ×
exp −
β
/2
2
h̃
k
t
e
2λt (1 − ρt )e eβk
k=1 Htk =1
)
( P
K
∗
βk /2 2
Y
(ỹ
−
g
e
)
1
t
t
Htk =1,∝ π(β | vβ ) ×
exp −Nβk /2
βk
e
2e
k=1
where
y t − J t Ztỹt∗ = p,
λt (1 − ρ2t )eh̃t /2rt.
gt = p1 − ρ2t

It leads to the following non-linear state space model:∗β̃k /2ỹ[k,τ
+ eβ̃k /2 ε[k,τ ] ,] = g[k,τ ] e

βk+1 = βk + ηβ,k ,!

ε[k,τ ]
ηβ,k

∼ N (0, V k ),k = 1, . . . , K, τ = 1, . . . , N,

k = 1, . . . , K − 1,
Vk =

1

0

0 ck vβ2!
,

2). We apply the multi-move sampler for the non-linear
where β1 = 0, and ηβ,1 ∼ N (0, vβ0

state space model (Shephard and Pitt, 1997; Watanabe and Omori, 2004) to generate β.
The announcement eﬀect α is sampled by the multi-move sampler in the same mannerby deﬁning h̃t = xt + st .
A.4 Mixing latent process
The conditional posterior density of λt is− ν+1 −1π(λt | · ) ∝ λt 2
exp



24

ν−2
−
2λt


× f (λt ),where√(yt − λt rt eht /2 − Jt Zt )2f (λt ) = exp −.
2λt (1 − ρ2t )eht
The MH algorithm is employed to generate λt . We generate a candidate as λ∗t ∼ IG((ν +1)/2, (ν −2)/2), and accept it the with probability: min{f (λ∗t )/f (λt ), 1}, for t = 1, . . . , T .
A.5 SV parameters
For the SV parameter ϕ, let π(ϕ) denote the prior distribution. The conditional posterior
density of ϕ is(T −1
(1 − ϕ2 )x̃21 X (x̃t+1 − ϕx̃t − ȳt )22−π(ϕ | · ) ∝ π(ϕ) 1 − ϕ exp −2σ 2
2σ 2 (1 − ρ2 )
t=1
)
(
2
p
(ϕ
−
µ
)
ϕ
∝ π(ϕ) 1 − ϕ2 exp −
,
2σϕ2

p

)

√
where ȳt = ρσ(yt − Jt Zt )e−ht /2 / λt ,
PT −1
µϕ =

t=1 (x̃t+1 − ȳt )x̃t,
s2x

and σϕ2 =σ 2 (1 − ρ2 )
,
s2x

PT −1

2
t=2 x̃t . To sample from this conditional posterior distribution, we use
the MH algorithm. We propose a candidate, ϕ∗ ∼ T N(−1,1) (µϕ , σϕ2 ), where T N(a,b) (µ, σ 2 )2with s2x = ρ2 x̃21 +denotes the normal distribution with mean µ and variance σ truncated on the interval(a, b). Then, we accept it with probability
(
min

)
p
π(ϕ∗ ) 1 − ϕ∗2p,1 .
π(ϕ) 1 − ϕ2

For the parameters, ϑ ≡ (σ, ρ), there is no explicit distribution form which we caneasily sample from. The conditional posterior density is given by)
λt rt eht /2 − Jt Zt )2π(ϑ | · ) ∝ (1 − ρ )exp −2λt (1 − ρ2t )eht
t=1
(
)
P −1
(1 − ϕ2 )x̃21 + Tt=1
(x̃t+1 − ϕx̃t )2−T× σ exp −× π(ϑ),
2σ 2(

2 − T −1
2

T
X
(yt −

25

√

where π(ϑ) is the prior density. We generate a candidate from the normal distribution
based on the approximation of the posterior around the mode of the density. As analternative way, we can generate (ϕ, σ, ρ) jointly by the MH algorithm with the posterior
approximation around the mode, while we found that it is slightly less eﬃcient than theseparate generation of ϕ and (σ, ρ), which we recommend as described here.
2
Assuming the prior, µ ∼ N (µ0 , σµ0), we generate µ from its posterior density, µ | · ∼N (µ̂, σ̂µ2 ), where−11
1 − ϕ2 (T − 1)(1 − ϕ)2
=
+
+
,
2
σµ0
σ2
(1 − ρ2 )σ 2
(
)
PT −1
2
(1
−
ϕ)
(x
−
ϕx
−
ρσê
)
µ
(1
−
ϕ
)x
t+1
t
t
0
1
t=1
,
µ̂ = σ̂µ2
+
+
2
2
2
2
σµ0
σ
(1 − ρ )σσ̂µ2√
and êt = (yt − Jt Zt )e−ht /2 / λt .
A.6 Jump parameters
For the jump parameters θ J , the following priors are assumed: κ ∼ B(aκ0 , bκ0 ), µz ∼N (my0 , cy0 ), and σz2 ∼ IG(ny0 /2, Sy0 /2). Then, the sample generation from the conditional posterior is as follows:κ | · ∼ B(aκ0 + TJ , bκ0 + T − TJ ),where TJ =PT

t=1 Jt ; andµz | · ∼ N (m∗y , c∗y ),σz2 | · ∼ IG(n∗y /2, Sy∗ /2),

where

c∗y
n∗y

=

1
TJ
+ 2
cy0 σz= ny0 + TJ ,

−1

m∗y

,

and

Sy∗

= c∗y

= Sy0 +

my0
+
cy0
T
X
t=1

26

PT

t=1 Zt Jt
σz2(Zt − µz )2 Jt .

!
,

A.7 Seasonal- and announcement-eﬀect parameters
We assume a prior, vβ2 ∼ IG(nβ0 /2, Sβ0 /2), and then simply generate the sample fromits posterior: vβ2 | · ∼ IG(n̂β /2, Ŝβ /2), wheren̂β = nβ0 + K − 1,Ŝβ = Sβ0 +K−1
X(βk+1 − βk )2
.
c
k
k=1

2
The posterior generation of vαj
can be implemented in the same manner.

To generate the parameter φj in the announcement eﬀect, we assume a prior denotedby π(φj ). The conditional posterior distribution of φj is given by(φj − µφj )2π(φj | · ) ∝ π(φj ) exp −
2
2σφj


,

where
PL−1
µφj =

ℓ=1 αjℓ αj,ℓ+1,
PL−1 2
α
jℓ
ℓ=1

2
vαj
2
and σφj= PL−12ℓ=1 αjℓ.

2
Using the MH algorithm, we propose a candidate, φ∗j ∼ T N(−1,1) (µφj , σφj), and accept itwith probability: min{π(φ∗j )/π(φj ) , 1}.
A.8 Degree of freedom for Student-t distribution
The conditional posterior density of ν isπ(ν | · ) ∝ π(ν)

TY((ν − 2)/2)ν/2
t=1

Γ (ν/2)

−ν/2
λt
exp



ν−2
−
2λt


,

where π(ν) denotes the prior distribution. Following Watanabe (2001), we use the MH
algorithm, where we generate a candidate from a normal distribution associated with theapproximation around the mode of the posterior.
A.9 Parameters for RV measurement
For the parameters in the measurement equation of RV, (ξ, σu ), the following priors areP
assumed: ξ ∼ N (ξ0 , q02 ), and σu2 ∼ IG(nu0 /2, Su0 /2). Deﬁne Wτ = log t∈D(τ ) exp(ht ).

27

ˆ q̂ 2 ), σ 2 | · ∼We generate ξ and σu from the conditional posterior distribution: ξ | · ∼ N (ξ,uIG(n̂u /2, Ŝu /2), respectively, whereq̂ 2 =

1
N+ 2
2
q0 σ un̂u = nu0 + N,

−1,ξˆ = q̂ 2

and Ŝu = Su0 +

ξ0
+
q02
N
X

PN

τ =1 {log(RVτ ) − Wτ }
σu2

!
,

{log(RVτ ) − ξ − Wτ }2 .

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