『経済研究』（明治学院大学）第 161 号

2021 年

Realized jump beta: Evidence from high-frequency data on
Tokyo Stock Exchange
Masato Ubukata＊†
Abstract
This study measures the realized jump beta of sector portfolios constructed from Japanese
high-frequency intraday data to assess the dynamics in jump beta aggregated over fixed intervalsof time. When we test the null hypothesis that jump beta remains constant over time, the result
strongly rejects the constancy of annually aggregated jump beta, but does not frequently rejectthat of monthly jump beta. Given the worldwide evidence of fractional integration in realized variance and covariance, the estimation result under the assumption of a pure fractional noise processof the monthly jump beta indicates a smaller average degree of integration, namely ARFIMA（0, 0.2,0）, than that of a total realized beta largely including diffusive risk component of asset returns.
I
Moreover, the jump beta might be naturally modeled as a stationary （0）process.
JEL Classification Numbers: G12, G15.
Keywords: Jump beta, Market jump variation, Jump covariation, High-frequency data.

1．Introduction
In the simple one-factor capital asset pricing model（CAPM）, the systematic risk factor loading is market beta determined by the covariance of asset and market portfolio returns. A key question in this framework is whether or not market beta is constant over fixed intervals of time. From
theoretical and empirical perspectives, studies such as Hansen and Richard（1987）, Ferson et al.
（1987）and Jagannathan and Wang（1996）suggest that market beta is likely to vary with condition＊Department of Economics, Meiji Gakuin University, Shirokanedai, Minato-ku, Tokyo, 108-0071 Japan, e-mail:ubukata@eco.meijigakuin.ac.jp†

We would like to thank participants at the 3rd International Conference on Econometrics and Statistics fortheir helpful comments. The financial support from the Ministry of Education, Culture, Sports, Science andTechnology of the Japanese Government through Grant-in-Aid for Scientific Research（No. 15K03397）is gratefully acknowledged.
155

『経済研究』（明治学院大学）第 161 号

ing variables. Within the framework of realized variances and covariances based on the high-frequency intraday data first popularized by Andersen and Bollerslev（1998）,Andersen et al.（2006）analyze
the dynamics of realized market beta, or equivalently nonparametric period-by-period beta, definedas the realization of the ratio of integrated asset and market covariation to integrated market variation of the underlying price processes.
The development in financial econometrics literature based on the high-frequency data sheds
light on large jumps of asset and market return processes（e.g., Andersen et al. 2007; BarndorffNielsen and Shephard 2004; Huang and Tauchen 2005; Lee and Mykland 2008）.A classical asset pricing model assumes that jump risk for individual stock is non-systematic, so that the corresponding
jump beta is zero. However, the observed cross-correlations for extreme asset returns in the financial
market imply that the jump beta is not negligible and the market beta could be decomposed into
systematic diffusive and jump risk components. The presence of substantial jump risks is also supported by the existing positive price against the expected variation of the underlying asset jump return in derivatives markets; in fact, the corresponding option-implied downside jump variation could
be a significant predictor for future equity risk premium（e.g., Pan 2002; Eraker 2004; Bollerslev andTodorov 2011; Bollerslev et al. 2015; Andersen et al. 2020）.
To my knowledge, there is limited research on jump beta based on high-frequency data.
Todorov and Bollerslev（2010）propose an estimator of jump CAPM beta based on higher-order
power variations. They find that the estimates differ statistically from diffusive CAPM beta for someof the U.S. stocks. Mancini and Gobbi（2012）propose an estimator of jump beta based on the ratio oftruncated jump covariation to jump variation. Li et al.（2017）propose a test statistic for time-varying
jump beta. The empirical evidence has been provided only in the U.S. financial market such as byBollerslev et al.（2016）and Alexeev et al.（2016, 2017）. For instance, Bollerslev et al.（2016）show
that the jump betas of 1,000 U.S. stocks entail significant risk premiums and remain significant evenafter controlling for firm characteristics and other explanatory variables. Overall, they suggest theusefulness of separately considering the two different types of components in market beta.
The main purpose of our study is to understand the dynamics of the realized jump beta constructed based on the high-frequency intraday data and to provide new evidence from the Japanesefinancial market. First, we investigate how much sensitivity of major sector portfolios is captured by
the jump beta at the moment when large jumps happen to the market portfolio. To measure thejump beta, we detect the 10-minute intraday market jump returns based on the truncation method
first popularized by Mancini（2001, 2009）, and construct the ratio of jump covariation between market jump and corresponding sector portfolio returns to market jump variation aggregated over fixedintervals of time. We find that annually aggregated jump beta obtained by pooling the 10 -minute
jump returns every year is nonzero and that the dynamics is very smooth, but allowing for the jump156

Realized jump beta: Evidence from high-frequency data on Tokyo Stock Exchangebeta to change on monthly frequency captures more clearly the large changes in the sensitivity of
sector portfolios to the market jumps.
We then examine the time-varying jump beta over years and months using the test statistics
of Li et al.（2017）. If the test statistic rejects the null hypothesis that jump beta remains constantover fixed intervals of time, we investigate the time-series property of jump beta, such as the degreeof persistence. The test shows that the constancy of annually aggregated jump beta is always rejected at the 1% significance level for all sector portfolios used in our study, while that of monthly aggregated jump beta is not rejected for about half of the monthly samples. This result suggests that it
would be more reasonable to investigate the time-series property of realized jump betas at a monthly frequency. Given the worldwide evidence of fractional integration in realized variance and covariances, we apply a pure fractional noise process to monthly jump betas. The estimation result showsa small average degree of fractional integration such that the estimates of the long-memory parameter is around 0.2. It is less persistent than the total realized betas and covariances, typically with thedegree of integration around 0.4 in our samples. Furthermore, we fit the autoregressive integrated
moving average（ARIMA）process to the realized jump beta, and the estimation result suggests that

I
process.
the realized jump beta might be naturally modeled as a stationary （0）The remainder of this article is organized as follows. Section 2 provides a brief explanation of
the procedure for calculating realized jump beta used throughout the study. Section 3 describes thedata that we use in the empirical analysis and presents the empirical results for time-varying realized jump beta. Section 4 concludes.

2．Measuring Realized Jump Beta
Realized jump beta is defined as the realization of covariation between market jump returnand the contemporaneous asset return divided by market jump return variation. To calculate thequantity, we will first detect the high-frequency market jump returns based on the truncation method proposed in Mancini（2001, 2009）. For the simplicity of notation, the daily interval［t－1, t］is divided into non-trading［t－1 , t－1+π］and trading［t－1+π, t］intervals. We then define equallyspaced high-frequency market returns as ri,t＝ft－1+π+iΔm－ft－1+π+（i－1）Δm for i＝1, . . . , m, t＝1, . . .
, T, where f., m, and T are the log market price, the number of intraday market returns in each day,and the number of days, respectively. The interval length of market returns is denoted as（1－π） （1－π）T

Δ m＝

m

＝

mT

（1－π）T＝n＝Δn ，where n is the number of market returns within the fixed

time interval［0, T］. We examine high-frequency data on equally spaced series at 10-minute intervalsof returns in our calculation of the realized jump beta.
157

『経済研究』（明治学院大学）第 161 号

We consider the time-varying threshold value for the identification of market jump returns as
introduced in Bollerslev and Todorov（2011）, Todorov and Tauchen（2012）, Andersen et al.（2015）,
and Li et al.（2017）. The idea is that if an absolute value of intraday market return exceeds thethreshold value, the corresponding market jump return is identified. The computation of the timevarying threshold level considers the intraday periodicity in the return volatility because the deterministic diurnal pattern in volatility is well observed in financial markets. One of the statistics measuring the periodicity is the time-of-day（TOD）factor,∑t＝ r i，t ｛1 |r |≤–νΔ ｝ ，for i＝1，…，m．
TODi＝NOIi× Tm∑t＝ ∑j＝ r j，t ｛1 |r |≤ν–Δ ｝
T

2

1



1

ω
m

i，
t

1

2

ω
m

j，
t

⑴

Theoretically, ω∈（0, 0.5）will work, and we fix ω＝0.49 as with the previous studies. The constant――――truncation level ν is calculated as 3×BV 1/2, where BV is the sample average of open-to-close bipower variation measure BVtopen over［0, T］proposed in Barndorff-Nielsen and Shephard（2004）, that is,––
BV＝1/T∑t＝ BVtopen．The numerator and denominator in ⑴ estimate the continuous or diffusiveT

1

return variation for each i-th intraday interval over［0, T］and the total continuous return variation,∑t＝ ∑j＝ ｛1 |r |≤ –νΔ ｝ and adjusts the different number ofT∑t＝ ｛1 |r |≤ –νΔ ｝
Trespectively. NOIi is expressed as

m

1

1

1

ω
m

j，
t

ω
m

i，
t

continuous returns for each interval i.

We use the time-varying threshold value, denoted as νi,t, which depends on the intraday periodicity and time-varying volatility across days,


νi，
t＝c×

min
（RVtopen，
BVtopen）

ω，
×TODi×Δm1－π̂⑵

where min（RVtopen, BVtopen）is a function that selects the smaller value from open-to-close realized

π is the ratio of close-to-open return variation on close-tovariance and bipower variation measure. ＾∑t＝ （ ft－ ＋π－ft－ ）．In the Japanese stock market,
close return variation over［0, T］expressed asT∑t＝ （ ft－ft－ ）
T

1

1

1

1

1

2

2

π is calculated as the summation of squared overnight and lunchtime returns overthe numerator of ＾［0, T］. The constant scaling factor c is set as 4 in our calculation. The time-varying threshold value

νi,t increases as daily market return volatility increases. Therefore, an absolute value of intraday
market returns｜ri,t｜needs to exceed the large threshold value to be judged as jump. The collection, where L（of high-frequency market jump returns is denoted as（Δnj Z）j∈L（n D）denotes indices of
n D）
158

Realized jump beta: Evidence from high-frequency data on Tokyo Stock Exchangeintervals that contain the market jumps in the region D. Let（Δnj Y）j∈L（n D）be a collection of contemporaneous returns of an asset over the times of market jumps and Δnj X be the 2 by 1 vector of Δnj Y
and Δnj Z, Li et al.（2017）introduce a simple sample analogue estimator of jump covariation matrix

Q（
,
n D）

（

Q（
＝
n D）



）∑

QYY，
（
n D）QYZ，（n D）

Δnj XΔnj X T.
＝
j
L
（D）
QZY，
Q
∈
n
（
n D） ZZ，（n D）⑶

Finally, realized jump beta can be calculated as the ratio of realized jump covariation estimator between Δni Y and Δni Z to realized market jump variation estimator,

βnJ＝



QYZ，
（
n D）

．

QZZ，
（
n D）

⑷

We should note that the estimator in ⑷ corresponds to the no weighting version of the optimallyweighted estimator of the jump beta proposed by Li et al.（2017）.The calculation of weights requires
the pre-jump and the post-jump spot covariance matrices. We do not use the weighted estimator in
our calculation of jump beta because of the shortage of the corresponding pre- and post-jump continuous returns while calculating the weight for the jump return identified at the start- and end-time
intervals in open-to-close markets.

3．Empirical Results
3-1．Data
In our empirical analysis, we construct jump betas of four sector portfolios for electrical appliances（EA）,transportation equipment（TE）, banks（BA）and real estate investment trust（RE）.We
proxy market portfolio in Japan with Tokyo stock price index（TOPIX）which is a value-weighted
index based on all domestic common stocks in the first section of the Tokyo Stock Exchange. To
avoid a market microstructure effect due to ultra high-frequency return series, which is induced byvarious market frictions such as the discreteness of price changes and bid-ask bounces, inter alia, werely on the 10-minute high-frequency returns of the sector and market portfolios obtained by Nikkei
NEEDS-TICK data from January 2012 to December 2016. The equity market timings are 9:00-11:30
and 12:30-15:00 in our sample period, and thus the number of intraday intervals in a day is m＝30.
Figure 1 plots annually and monthly aggregated realized jump betas of the four sector portfolios. The annually estimates in solid lines, which are obtained by pooling the 10-minute market jumpand corresponding sector portfolio returns every year, are very smooth. The jump betas over annual
frequencies tend to be a little higher than 1 for EA, TE, and BA, but much lower than 1 for RE. The
monthly estimates obtained by pooling the 10-minute jump returns every month are reported in dot159

『経済研究』（明治学院大学）第 161 号
Figure 1: Time-series plot of annually and monthly aggregated realized jump betas
2

EA-y

EA-m

2

1

1

0

0

2012
2

2013
BA-y

2014

2015

2016

BA-m

2017 2012
2

1

1

0

0

2012

2013

2014

2015

2016

2017 2012

TE-y

TE-m

2013
RE-y

2014

2015

2016

2017

2015

2016

2017

RE-m

2013

2014

Note: The solid and dotted lines represent the estimates of annually andmonthly aggregated realized jump betas of four sector portfolios constructed from the 10-minute high-frequency returns of market jump andthe contemporaneous sector portfolios from 2012 to 2016.

ted lines. Allowing for jump beta to change monthly, we can capture noisier and higher fluctuationsof jump betas relative to annual jump betas. The estimates for EA, TE, and BA occasionally take below one, and negative estimates for RE occur very rarely.
As noted in Li et al.（2017）, the realized jump beta is consistent with a slope coefficient when
we conduct linear regression without a constant term of sector portfolio returns on market jump returns aggregated over fixed intervals of time. To illustrate the jump regressions, Figure 2 displays
the scatter plots of 10-minute market jumps and the respective sector portfolio returns, together
with their linear fits. We find that the monthly aggregated jump regression provides a reasonably
better fit relative to the annually aggregated jump regression. The results motivate us to explore thetest of constant jump beta over different frequencies.
3-2．Test for Constant Jump Betas
We apply a test statistic for constant jump betas proposed by Li et al.（2017）. The test isbased on the fact that the constant linear jump regression model over fixed intervals of time isequivalent to the singularity of the realized jump covariation matrix. The null hypothesis that thejump beta remains constant over time is rejected when the determinant of a sample analogue estimator of the jump covariation matrix is larger than a critical value,

1Δ－］>C V αn.
［Q（
n det
n D）

⑸

C V αn is the critical value at the significant level α∈（0, 1）. The test statistics can be also written interms of realized jump correlation,
160

Realized jump beta: Evidence from high-frequency data on Tokyo Stock Exchange
Figure 2: An illustration of linear jump regressions with scatter plots and linear fits
1

EA 20120-10
EA 201201
1

1
1

0

1

0

-1
0
EA 201205
1

1

0
1

0

-1
0
EA 201206
1

1

1

-1
0
EA 201210
1

-1

1

1

-1
0
EA 201207
1

1

-1
0
EA 201211
1

1

-1

-1
0
EA 201208
1

1

0
1

0
0

EA 201204

0

0

0
0

EA 201203

0

0

-1
0
EA 201209
1-1

EA 201202-10
EA 201212
1

1

0
0

1

-1

0

1

Note: The x-and y-axes represent the 10-minute returns of market jump andthe contemporaneous electric appliance sector portfolio in 2012 . The
first and the other panels show linear jump regressions over a year
and every month, respectively.



2
1－ρ（
＞
n D）

ΔnCV αn
，
QZZ，
QYY，
（
n D）
（
n D）

where ρ（n D）is the realized jump correlation expressed as

QYZ，
（
n D）

⑹
．The computation

QYY，
QZZ，
（
n D）
（
n D）

of the critical value C V αn is based on Monte Carlo simulations because the null asymptotic distribution is nonstandard, and its conditional（1－α）quantiles cannot be expressed in closed form. We simulate a large number of Monte Carlo simulations to calculate the（1－α）quantiles of copies of thelimiting variable using pre-jump and post-jump spot covariance matrices, realized jump beta, anduniformly distributed on the unit interval and bivariate standard normal variables（See Algorithm 1
and Theorem 1 in Li et al.（2017）for more details）. Then, we compute the p-value for inference.
Table 1 reports annually aggregated realized jump betas of four sector portfolios and the pvalues of the test statistics ⑸ or ⑹ in parentheses. We also report the numbers of 10-minute market（TOPIX）jump returns in Table 1, which range from 107 in 2012 to 248 in 2016. In 2012 and 2016,
the jump betas deviate upward from 1 for the EA, TE, and BA sectors and deviate downward from
1 for the RE sector. Despite the apparent variation of year-by-year jump betas, the null hypothesisthat annually aggregated jump beta remains constant is strongly rejected in all cases. The resultshows the deviations from linearity in the annually aggregated jump regressions and the possibility
that the jump beta changes over a shorter period of time. This might be common with the implications of the conditional capital asset pricing models. As opposed to the annual basis, we test the constancy of monthly aggregated jump betas. Figure 3 plots time-series of p-values of the test statistics161

『経済研究』（明治学院大学）第 161 号
Table 1: Annually aggregated jump beta and test for the constancy
2012

2013

2014

EA-y
p-value

1.19
1.05
1.10
（0.000）（0.000）（0.000）
TE-y
1.19
1.15
1.21
p-value（0.000）（0.000）（0.000）
BA-y
1.27
1.06
1.06
p-value（0.000）（0.000）（0.000）
RE-y
0.14
0.53
0.47
p-value（0.000）（0.000）（0.000）
Number of 10-minute jump returns of TOPIX within a year
107
207
150

2015

2016

1.04
（0.002）0.99（0.002）1.12（0.000）0.30（0.000）1.11（0.000）1.25（0.000）1.37（0.000）0.26（0.000）160

248

Note: T he table reports the estimates of annual jump betas for four sector portfolios and the p-values for the constancy testof the jump beta in parenthesis. The last row reports the number of 10-minute market（TOPIX）jump returns every
year.

Figure 3: P-values for the constancy test of monthly aggregated jump beta
1.00

EA

1.00

0.75

0.75

0.50

0.50

0.25

0.25

2012
2013
BA
1.00

2014

2015

2016

2017 2012
RE
1.00

0.75

0.75

0.50

0.50

0.25

0.25

2012

2013

2014

2015

2016

TE

2017 2012

2013

2014

2015

2016

2017

2013

2014

2015

2016

2017

Note: The circle and cross symbols represent p-values for the null hypothesis of
the constant monthly aggregated jump beta, which are more than andless than 0.01 , respectively. The sample period covers the period fromJanuary 2012 to December 2016.

⑸ or ⑹ from January 2012 to December 2016（60 months）, where the circle（cross）symbol meansthat the p-values exceed（fall below）the 1% significance level. We find that the null hypothesis thatmonthly jump beta remains constant is not rejected even at the 1% significance level for about halfof the monthly samples. These results suggest the linearity of the jump regression model over periods of months and the stability of monthly realized jump beta.
3-3．Time-Series Property of Monthly Jump Betas
We will investigate the time-series property of monthly aggregated jump beta based on the162

Realized jump beta: Evidence from high-frequency data on Tokyo Stock Exchange

results in the previous subsection. Rows two to five in Table 2 report the augmented Dickey-Fuller
unit root test with intercept and different augmentation lags up to four. The test rejects the null hypothesis of unit root for the jump betas of EA, TE, and RE sectors with all augmentation lags and for
BA sector with the first augmentation lag. We also represent the Ljung-Box portmanteau test statistics for up to 12th-order autocorrelation in the last row of Table 2. The null hypothesis of no autocorrelation up to 12 lags is rejected at 5 or 10% significance levels only for jump betas in two out of four
sector portfolios. In fact, the values of LB（12）seem to be quite low, compared with the stylized factof highly significant asset return volatility clustering. This result indicates that the realized jump betas are totally less persistent. This is confirmed by the quite low and moderately declining sample
autocorrelation and partial autocorrelations reported in Figure 4.
Many authors suggest that asset return volatilities are well-described by a fractional noise
process with the degree of fractional integration such that the long-memory parameter d is around
Table 2: Test for the dynamics of monthly aggregated jump beta
EA
ADF
（1）
ADF（2）
ADF（3）
ADF（4）
LB（12）

－4.28

TE
＊＊＊

BA

＊＊＊

－4.13
－2.88＊－3.00＊＊－3.09＊＊＊＊＊－3.58－3.58＊＊＊－2.96＊＊
5.74

RE
＊＊

－4.76＊＊＊－3.63＊＊＊－3.67＊＊＊－2.90＊
10.48

－2.96
－2.26
－2.28
－1.76
19.73＊

21.91＊＊

Note: A DF（i）denotes the augmented Dickey-Fuller unit root test with intercept and with i augmentation lags. LB（12）denotes the Ljung-Box portmanteau statistic for up to the 12th-order autocorrelation. The superscripts ＊＊＊ , ＊＊ , and ＊ indicate significance at the 1, 5, and 10% levels, respectively.

Figure 4: Sample autocorrelation and partial autocorrelation of monthly jump beta
1.0

EA
ACF

1.0

PACF

0.5

0.5

0.0

0.0

-0.5

-0.5

0
BA
1.0

5
ACF

10

15

20

PACF

0
RE
1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

0

5

10

15

TE

20

0

ACF

PACF

5
ACF

10

15

20

10

15

20

PACF

5

Note: T he black and gray bars represent sample autocorrelations and partial
autocorrelations, respectively, for monthly jump betas of four sector
portfolios.

163『経済研究』（明治学院大学）第 161 号

0.4 , indicated by Andersen et al.（2001）for spot exchange rate markets and by Andersen et al.
（2003）for the U.S. equity market, and around 0.5, indicated by Ubukata and Watanabe（2014）forthe Japanese equity market. Andersen et al.（2006）propose the total realized beta including systematic diffusive and jump risk components, which is defined as realized covariance between asset andmarket returns divided by realized market variance, and find that the total realized betas are lesspersistent in the U.S. equity market. Despite the empirical evidence of realized covariances and beta,
very little research exists on the presence of long-memory in realized jump beta. We apply the simplest autoregressive fractionally integrated moving average model, ARFIMA（0, d, 0）, for the time-series of monthly aggregated jump beta to focus on the long-memory parameter d. Table 3 shows theestimation results of the ARFIMA（0, d, 0）model for jump betas of the four sector portfolios. The estimates of the long-memory parameter are lower than 0.4 and the average degree of fractional integration is 0.2. The results indicate very weak long-memory property in the monthly aggregated jump
beta series. We will compare the jump beta and covariation with the total beta and covariation in thenext subsection.
As motivated by the empirical finding of low-order fractional integration, we conduct a test of
short or long-range dependence in Table 4 , which reports the classical Hurst-Mandelbrot rescaled
range statistics with q＝0 and Lo’s（1991）modified rescaled range statistics with q≥1. Judging from

Table 3: Estimation result of ARFIMA（0, d, 0）model for monthly aggregated jump beta

d
s.e.

EA

TE

BA

RE

Mean

0.13
0.00

0.33
0.00

0.27
0.00

0.08
0.00

0.20

Note: The parameter d and s.e. represent the degree of fractional integration and standard error of the estimates. The lastcolumn reports the average of the parameter estimates for four sector portfolios.

Table 4: Rescaled range test statistics of monthly aggregated jump beta
EA
Vq=0
Vq=1
Vq=2
Vq=3
Vq=4
Vq=5
Vq=6
ACF
（1）
q＊

1.659
1.541
1.453
1.402
1.394
1.383
1.383
0.159
2

TE

BA
＊＊

2.206
1.822＊＊
1.677＊
1.575
1.504
1.469
1.442
0.465
5

RE
＊＊

2.278
2.002＊＊
1.797＊＊
1.657＊
1.578
1.516
1.475
0.295
3

1.396
1.310
1.268
1.225
1.227
1.220
1.243
0.135
1

Note: V
 q=0 and Vq≥1 are the classical and modified rescaled range statistics. The superscripts ＊＊ and ＊ indicate significanceat the 5 and 10% levels. The 5 and 10% critical values are 1.747 and 1.620, respectively. q＊ is the lag chosen by Andrews’（1991）data-dependent formula.

164

Realized jump beta: Evidence from high-frequency data on Tokyo Stock Exchange

the statistics with the optimal number of lags q＊ chosen by Andrews’（1991）data-dependent formula, the null hypothesis of no long-range dependence in monthly realized jump betas of EA, TE, andRE sector portfolios is not rejected even at the 10% significance level. In light of the result that thejump beta does not exhibit long-range dependence, we are interested in the alternative where theIprocesses. To explore this possibility, we fit the ARIMA model to therealized jump betas are （0）realized jump betas, with lag selected by the Akaike information criterion（AIC）. Table 5 reports
the ARIMA model selection and estimation results for the monthly jump beta in each sector portfolio. The selected models are ARIMA（0, 0, 0）for EA and RE, ARIMA（1, 0, 0）for TE, and ARIMA（2, 0, 0）for BA. The estimation results suggest that the time series of jump beta might be within the context

Iprocesses.
of stationary （0）3-4．Comparison between Monthly Jump Betas and Total Betas
In this subsection, we compare the time-series properties of the realized jump betas and thetotal realized beta before being decomposed into systematic diffusive and jump risk components. The
sample correlation coefficients between the two betas of four sector portfolios are around 0.8. Despite
the relatively high correlation, the time-series plot of monthly aggregated jump and total betas reported in Figure 5 reveals the difference between the two more clearly. In particular, the jump risk
component of the market beta（dotted line）looks more volatile than the total beta. Furthermore, themore prominent difference is the degrees of fractional integration. Table 6 reports the estimation results of the long-memory parameter d in the ARFIMA（0, d, 0）model for monthly jump beta（secondrow）, market jump variation and covariation for sector portfolios（third row）, and the total ones
（fourth and fifth rows）. We find that the estimates of the long-memory parameter d for total beta
and（co）variation are on average 0 . 43 and 0 . 36 larger than those of jump beta and（co）variation,
which are 0.20 and 0.24, respectively. The results suggest that the total betas and（co）variation arehighly persistent relative to the jump betas and（co）variation. Thus, we conclude that separately
Table 5: ARIMA model selection and estimation results
Constant
EA
TE
BA
RE

1.08
（0.02）
1.13（0.04）
1.22（0.06）
0.25（0.04）

AR（1）

AR（2）

AIC

Inverted Roots－53.450.46（0.11）
0.23（0.13）－28.640.28（0.13）

12.98

0.66

－0.43

22.76

Note: T he table reports estimation results of the fitted ARIMA model selected by AIC for monthly jump beta in each sector
portfolio. The sample covers the period from January 2012 to December 2016 for a total 60 observations.

165

『経済研究』（明治学院大学）第 161 号
Figure 5: Time-series plot of monthly jump and total betas
EA
2

TE
Total beta

Jump beta

2

1

1

0

0

2012
2013
BA
2

Total beta

2014

2015

2016

2017 2012
RE

2013

2

Total beta

Jump beta

1

Jump beta

2014

2015

2016

2017

2015

2016

2017

Jump beta

1

0
2012

Total beta

0
2013

2014

2015

2016

2017 2012

2013

2014

Note: The dotted and solid lines represent monthly jump and total betas fromJanuary 2012 to December 2016, respectively.

Table 6: The degree of fractional integration for monthly jump and total beta and covariation
Jump beta
Jump covariation
Total beta
Total covariation

EA

TE

BA

RE

MarketMeanMedian

0.13
0.31
0.44
0.35

0.33
0.31
0.45
0.36

0.27
0.26
0.46
0.36

0.08
0.02
0.36
0.36

―
0.31
―
0.37

0.20
0.24
0.43
0.36

0.20
0.31
0.44
0.36

Note: This table reports the estimates of long-memory parameter d in ARFIMA（0, d, 0）model for jump beta（second row）,
market jump variation and covariation for sector portfolios（third row）
, and the total ones（fourth and fifth rows）, respectively. The last two columns report the average and median of the estimated d in each row.

considering the jump risk component in the market beta is meaningful to capture the statistical difference between the two betas.

4．Conclusion
Several recent studies investigate the fact that the premia associated with jump risks sometimes appears to be different from the premia associated with continuous risks. This study measuresmarket jump risk factor loading determined by the covariance between market jump and contemporaneous asset high-frequency returns. The estimates show that annually aggregated jump beta issmooth over time, while monthly aggregated jump beta captures large changes in the sensitivity of
sector portfolios to the market jumps. This study also investigates the time-series properties of realized jump beta, such as the constancy over fixed intervals of time and the degree of fractional integration. We have new evidence in the Japanese financial market that in contrast to monthly aggre166

Realized jump beta: Evidence from high-frequency data on Tokyo Stock Exchange

gated jump beta, the constancy of annual jump beta is always rejected, and the time-varying
monthly jump beta has a smaller degree of fractional integration and less persistence compared withthe total realized beta largely including the continuous risk component of asset returns. We note thatthe empirical analysis in this study is limited to the statistical tests and estimation of time-seriesmodels for jump beta. Therefore, it would be worthwhile to investigate whether or not separating
the continuous and discontinuous betas also has important implications for practical portfolio and risk
management.

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