Discussion Paper No. 991

MODELLING REALIZED VOLATILITYIN ELECTRICITY SPOT PRICES:NEW INSIGHTS AND APPLICATION
TO THE JAPANESE ELECTRICITY MARKET

Aitor Ciarreta
Peru Muniain
Ainhoa Zarraga

February 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Modelling Realized Volatility in Electricity Spot Prices:
New insights and Application to the Japanese Electricity
MarketAitor Ciarreta, Peru Muniainy, Ainhoa Zarragaz
University of the Basque Country, Spain
February 24, 2017

Abstract
The paper analyzes volatility of the electricity prices in the Japanese day-aheadmarket using realized volatility. We use several jump tests to decompose total realized variation into jump and continuous components. Then, we estimate several
HAR models that show the time-dependence structure of the volatility. Our results show that even though that market is narrow, it is relevant to identify jumps
in volatility. Besides, modeling residuals improve estimation results. The timedependent structure of the prices is present in volatility as well.

1

Introduction

Two decades ago, the electricity sector worldwide used to be organized as a vertically integrated industry in which prices were set on a cost-of-service regulation. In this setting,prices used to change on a yearly basis.1 Deregulation has implied the launching of poolmarkets in which selling bids and purchase orders meet to determine the equilibrium
price and quantity for each time interval of the following day. There is a considerable
and increasing amount of trading in both the spot and derivative markets that haveemerged depending on the maturity of the market.
The nature of electricity makes the market price more sensitive to demand and supply
changes. The resulting high level of price volatility and ocasional price jumps experienced
in these markets has allowed the development of instruments to hedge market agentsagainst risk involved. In this respect, empirical evidence shows that volume traded is
Email: aitor.ciarreta@ehu.eus. I would like to thank …nancial support from Ministerio de Economíay Competitividad under research ECO2015-64467-R and from Dpto. de Educación, Universidades e
Investigación del Gobierno Vasco under research grant IT-783-13. I would like to thank the Institute ofSocial and Economic Research, University of Osaka, for …nancial aid.
y
Email: peru.muniain@ehu.eus.
z
Email: ainhoa.zarraga@ehu.eus.
1
Most of the countries updated prices according to the in‡ation rate observed the year before.

1negative correlated with price variation.2 Power suppliers and retailers seek to manage
their portfolios in this volatile environment. Understanding price volatility is important
for agents to design their bidding strategies optimally. Development of intraday markets
and …nancial contracts for future delivery of power is the result of agents seeking toe¤ectively manage risk associated to short-term day-ahead transactions. Thus, electricity markets, as other …nancial markets, are evolving to allow participants to developee¤ective risk management strategies. The more electricity markets evolve the larger theamount of risk that has to be managed and controlled by market agents.
Models that incorporate the stochastic nature of volatility have developed rapidly
following initial works by Hull and White (1987) and Hull (2000). Applications toelectricity prices can be found in Deng (1999), Escribano et al (2011), Lucia and Schwartz(2002), Skantze et al. (2000), Geman and Roncoroni (2006) among others.
Our interest is to use the information contained in within-day prices to analyze andforecast volatility. Variation of asset prices is measured by the sum of squared returns
over de…ned time periods. The mathematics behind is based on the quadratic variation
process (Back, 1991). Increments of the risk premium are associated to increments ofquadratic variation in asset pricing theory. The estimation of quadratic variation using
realized volatility measures was initially discussed in the works of Andersen and Bollerslev (1998), Comte and Renault (1998), and Bandor¤-Nielsen and Shephard (2002).
Since then, the estimation of daily volatility using realized measures relying on highfrequency intraday data has stimulated a growing …eld of volatility modeling and estimation.
The development of nonparametric methods for volatility estimation by Bandor¤Nielsen and Shephard (2004) o¤er the possibility to estimate separately the di¤usive
and the jump components of the price process using realised and bipower variation.
Huang and Tauchen (2005) and Andersen et al. (2007) showed that jumps play animportant role in the movement of prices. Since then, non-parametric jump tests using
realised measures have been proposed attending the particular features of the stochastic
process. All of them consider jumps as rare events and try to extract the jumps fromthe continuous part of the volatility. The literature has grown rapidly since the works byBarndor¤-Nielsen and Shephard (BNS, 2006). Further developments by Andersen et al.
(ABD, 2007), Lee and Mykland (LM, 2008), Ait-Sahalia and Jacod (2008), Jiang andOomen (JO, 2008), Corsi et al (CPR, 2010), Podolskij and Ziggel (2010) and Andersen
et al. (ABD, 2012), among others have been also proposed, simulated and tested.
There are not that many applications of realized volatility to electricity markets. The
are several characteristics of the data generating process to consider when modelling
realized volatility in electricity prices. First, the non-storability nature of electricity(except for hydroelectric generation) which forces prices to be highly dependent on spot
demand and supply conditions. Second, strong seasonal e¤ects, mostly observed yearly,
weekly and daily periodic cycles. Third, presence of frequent price spikes coupled with
2
Examples of the negative and signi…cant correlation between price volatility (measured by the
daily standard deviation) and quantity traded are 0:0494 in the Japanese market, 0:1889 in EPEXGermany, 0:3662 in EPEX-Switzerland, and 0:1709 in the Spanish-Portuguese market.

2non-Gaussian characteristics of the implied probability distribution of the day-ahead
prices. Therefore, proper statistical modelling of spot prices is necessary for optimal
valuation of …nancial products and optimal physical assets management in the electricity
sector.
Although each estimator has its own properties, there is no single method which ispreferred to the rest. Two select the best model we use two approaches: (1) In-sample
goodness of …t and (2) a test for bias in the estimation.
Several studies using Heterogenous Autorregressive (HAR) estimation to di¤erent
speci…cations of realized volatility models in electricity markets are present in the literature. Ciarreta and Zarraga (2016) for the sequential intraday market, Qu at el (2016)
propose a logistic beta smooth transition heterogeneous autoregressive (LSTHAR) modelof realized volatility outperforms other HAR models based on Corsi (2009). To ourknowledge, there are not studies on the Japanese market using statistical models of
price formation and volatility.
Several contributions are in the paper. First, we propose a departure from the existing literature on how seasonality is removed in the returns. We smooth the time series of
prices using a robust non-linear smoother by Velleman (1980). The smooth technique iseasy to implement. Second, once seasonality is removed, we decompose realized volatility
following BNS (2006), ABD (2007), CPR (2010) and ABD-LM (2012). At this point wepropose a modi…ed version of the ABD-LM test in which the local volatility is estimatedusing a kernel approach. We propose a discrete Weibull distribution. Then, we estimate
the bias of the di¤erent decomposition statistics following Fang et al. (2012). And third,a minor contribution which is the study of volatility in the Japanese electricity market
that has not been addressed before, basically because the market only trades 2% of thetotal consumption.
The paper is organized as follows. In Section 2, we review the time series processmodel and the nonparametric tests based on realized measures. Section 3 brie‡y describes the Japanese electricity market structural, characteristics and regulation. Section 4 describes the data. Section 5 reports empirical results and discusses the model
selection criteria. Section 5 proposes new lines of research. Section 6 concludes.

2

Methodology

2.1

Jump tests

Assume the dynamics of the (logarithmic) prices follow a jump-di¤usion process whichcontains a non-stochastic component and a stochastic component of the form
dpt =t dt +

t dWt + dJt

where t is the drift, t is the di¤usion parameter,
Wt is a standard Brownian motion
PNt
and Jt is the jump process such that Jt = j=1 ctj , where ctj is the size of the jumpand Nt is the number of jumps up to time t. Jumps are assumed to be countable rare
events that migh happen along the path of the continuous price process. The quadratic3

variation of the process (QVt ) has a continuous part corresponding to the integrated
volatility or integrated variance (IVt ), which is the continuous component, and the sumof the squares of the jump sizes, which is the discontinuous or jump component. Thus
QVt is the sum of both components,
QVt =

Z t
0

2
s ds +

Nt
X

c2tj .

j=1

Assume we divide the time interval [0; t] into M subintervals with length = 1=M
where M is taken constant for each t (as! 0 frequency of interavals increases).
Overall, there are T trading periods observed with the same time interval. Therefore,
following Andersen and Bollerslev (1998), realized volatility for day t, RVt , is de…ned as
the sum of the squares of intraday returns,
RVt =

M
X

2
rt;j
,

j=1

where rt;j is the intraday log-price di¤erence at j, rt;j = pt;j pt;j 1 , which is the usualde…nition of return in …nancial markets.3 Under the null of absence of jumps, QVt is
consistently estimated by RVt . Therefore, RVt is taken as a measure for the unobserved
volatility of a high-frequency time series.
IVt captures the continuous, predictable component of RVt and it can be estimatedusing di¤erent methods. In this paper, we use several jump test statistics that havebeen proved to converge to a normal distribution when M ! 1. Hence, if the statistic
exceeds the critical value 1 , day t is classi…ed as a jump day. For a chosen signi…cancelevel , the jump component of volatility at day t is obtained as:
JVt = IZt > 1(RVtc t );IVc t the corresponding estimator of IVt obtained
where Zt is the jump test statistic and IV
with each method, and IZt > 1 is 1 if Zt > 1 , and 0 otherwise. The Zt test statitic
is the one proposed by Huang and Tauchen (2005) which is a class of Hausman (1978) c t ; 0 . Once jump
type of test. To ensure JVt > 0 it is de…ned as JVt = max RVt IV
variation (JVt ) is estimated, the continuous component of the total variation, CVt , isgiven by the di¤erence between the two, CVt = RVt JVt .
In this paper, existing non-parametric jump-robust type estimators are grouped into
two categories. The …rst category includes those jump tests that enable us to estimate
JVt and CVt from total RVt , these are BNS (2006), Andersen, Dobrev and Schaumburg(2009), and CPR (2010). The second category includes those tests that do not allow us
3

As we show in the descriptive statistics, in the Japanese market, the minimum pt;j is strictly positive.
Therefore, we can de…ne realized volatility in terms of returns. However, in many electricity markets
pt;j can be zero or negative thus either up-scaling logarithmic transformations of prices are consideredor the price process is taken as such, and realized volatility is de…ned in terms of price di¤erences.

4

to directly estimate JVt and CVt from total RVt but only identify jump-times, these are
ABD, (2007), and LM, (2008) and JO, (2008).
I. Tests based on multipower variation:
BNS (2006) test is based on the use of the realised bipower variation. They proof(Theorem 1) how IVt can be consistently estimated using bipower variation (BVt ). The
1, 1-order BVt process, if it exists, is de…ned as4
M

2
1

BVt =

X
M
jrj jjrj 1 j,
(M 1)
j=2

p p
where 1 = E juj = 2= . Following Huang and Tauchen (2005), the test statistic,
called ZtBN S , is based on the di¤erence between RVt and the jump-robust measure ofvariance, BVt , to detect signi…cant jumps:
ZtBN S =

p

Mr

(RVt
2

4

BVt )=RVt
5 max[1; T Qt =BVt2 ]+The denominator represents the e¤ect of integrated quarticity, which can be estimatedusing the tripower quarticity,
T Qt = 1:74

M2
M 2

M
X
j=3

(jrj jjrj 1 jjrj 2 j)4=3

As the sampling frequency increases, ! 0, the impact of jumps is reduced because
jump-returns are multiplied by adjacent non-jump returns. In …nite samples, it canbe shown that BVt is upward biased in the presence of jumps and as a result the jumpcomponent is underestimated. Besides, BVt is also a¤ected by zero returns, thus reducing
its value and consequently detecting more jumps when returns are zero or close to zero.
ABD, (2012) propose two jump-robust consistent estimators of IV using the nearest
neighbour truncation. These are the M inRVt and M edRVt estimators.
M inRVt = 2:75

M edRVt = 1:42

M
M 2
M
M 2

M
X
j=2

min(jrt;j j; jrt;j 1 j)2

M
X1
j=2

med(jrt;j 1 j; jrt;j j; jrt;j+1 j)2The M inRVt is exposed to zero returns and is less e¢ cient than the M edRVt estimatorwhich is based on the median on blocks of three consecutive returns. These two estimators can be used to construct the jump test statistic following Huang and Tauchen(2005):
4

BNS (2004) extend the properties of BVt for the products of k-lag length absolute returns for k 1.
Because we want to make our results comparable for the di¤erent tests, we restrict to the case of k = 1.
Taking k > 1 allows for correction for serial correlation. For instance, Ullrich (2012) analyzes the casefrom k = 1 to k = 8.

5

ZtM inRV

=

p

=

p

(RVt M inRVt )=RVt
Mr
L N (0; 1)hi !
M inRQt
1:81 max 1; M inRV 2t

ZtM edRV(RVt M edRVt )=RVt
Mr
L N (0; 1)hi !
M edRQt
0:96 max 1; M edRV 2twhere the estimator of the integrated quarticities in the denominators areM

M inRQt = 2:21

M2 X
min(jrt;j j; jrt;j 1 j)4
M 1
j=2

M edRQt = 0:92

M2
M 2

M
X1
j=2

med(jrt;j 1 j; jrt;j j; jrt;j+1 j)4 ,respectively. ABD (2012) argue that increasing the block length to any k > 1 in theestimation of IVt reduces e¢ ciency. Therefore, we restrict all the jump tests for the IV
estimators based on adjacent returns.5
CPR, (2010) propose a consistent and nearly unbiased estimator of IVt based on
threshold bipower variation:
CT BP Vt = 1:57

M
X

Z1 (rj ; #j )Z1 (rj 1 ; #j 1 );j=2

where the function Z1 is de…ned as:
Z1 (rj ; #j ) =

(
jrj j

1=2
1:094#jif rj2#j2
if rj > #j

which depends on the returns and the value of the threshold #j = c2# V^jZ . The local
volatility estimate, V^jZ , is estimated following an iterative process using a Gausssian
kernel as
LX
2 I
K Li rj+i2c2 VbjZ 1 g
frj+ii= L;i6= 1;0;1
for Z = 1; 2; :::VbjZ =LX
K Li Ifr2 c2 Vb Z 1 gj+iji= L;i6= 1;0;1

5
In the appendix we also report results for both estimators taking one block jump although results
should be taken cautiously due to data frequency.

6As they suggest, we consider c# = 3 and the number of iterations needed to converge
depends on the process itself.6 The choice of the threshold does not a¤ect the robustnessof the IVt estimator, and its impact on estimation is marginal. The corresponding jump
test statistic is also based on Huang and Tauchen (2005):
ZtCT BP V =

p

Mr

(RVt
2

4

CT BP Vt )=RVtih
T riP Vt
5 max 1; CT
CT BP V 2

+

t

PM Q3

where CT T riP Vt = 1:74M j=3 k=1 Z4=3 (rj k+1 ; #j k+1 ) and Z4=3 is de…ned in theoriginal paper. The test statistic is more powerful than those based on multipower
variation, but it is also a¤ected by zero returns.
II. Tests based on jump-time identi…cation:
The ABD-LM (The test compares standardized returns to a threshold. The null
hypothesis is that there are no jumps at each time. The test is able to identify the exact
time i and the number of jumps within an intraday trading day t. The local volatility
is estimated computing the bipower variation at each time interval, BVt;i , choosing awindow size K.
i 1
X
BVt;i
1
b2t;i ==
jrt;j jjrt;j 1 j
K 2
2K 2
j=i K+2

Thus, the standardized return is & t;i = rt;j =bt;i .
Since it is constructed using the bipower variation of BNS (2006), the distribution
of & t;i is asymptotically standard normal when there are no jumps. The jump test is
constructed using a threshold based on Gaussian extreme value theory max (& t;i )max (zj )
Sn

Cn

L
!

and P ( ) = exp ( exp ( x)), where
Cn =
Sn =

(2 log n)1=2
0:8
1

log

+ log (log n)

1:6 (2 log n)1=2

0:8 (2 log n)1=2,.

The choice of K usually considers the number of trading days and the number of intervals within the trading day. When a jump occurs the size is rt;j and the jump componentP
2
is de…ned as the sum of intraday squared returns, JLMt = M
j=1 rt;j If& t;i >& g . FurtherP2
more, JLMt is decomposed into positive and negative jumps, JLMt+ = M
j=1 rt;j Ifrt;j >0^& t;i >& g
6

In the empirical application we use several kernel density functions: Normal, Parzen, cosine andEpannechnikov. The choice of c# determines the jump detection. Choosing larger values of c# reduces
the percentage of detected jumps. We have also tried with c# = f4; 5; 6g.

7P2
and JLMt+ = M
j=1 rt;j Ifrt;j <0^& t;i >& g (Patton and Sheppard, 2015). This statistic isonly used for jump detection and cannot be used to decompose RV into JV and CV .
JO (2008) test uses SwVt (M ) which is a function of the di¤erence between arithmetic,
Rt;j = (pt;j+1 pt;j ) =pt;j , and logarithmic, rt;j = log pt;j+1 log pt;j , intraday returns
SwVt (M ) = 2

M
X

(Rt;j

rt;j ) .

j=1

In the absence of jumps the di¤erence between SwVt (M ) and RVt is zero,lim (SwVt

M !1

RVt ) =

0Rt

2 0 (exp (Ju )Ju1) dquRt

2
0 Ju dqu

if no jumps in [0; t]
if there are jumps in [0; t]

Thus, jumps are captured in a exponential way. The corresponding JO test statistic isnBVtqb SwVt

1

RVt
SwVt

M 4 4

3
X Y
b SwVt = 3:05 M
L
N
(0;
1)
where
jrt;i+k j3=2!
M 3
i=0 k+1

The statistic takes very large values in the presence of high returns. Therefore, thestatistic tends to overeject the null hypothesis of absence of jumps, that is jumps areusually over-identi…ed. This statistic is only used for jump detection and cannot be used
to decompose RV into JV and CV .

2.2

Realized volatility models

We propose to estimate several heterogenoeus autoregressive (HAR) models based on theinitial work by Corsi (2009) for the RVt (HAR-RV). The HAR-RV model is a parsimoneous model because it reduces the number of parameters to be estimated. In particular,
we propose the linear regression equation
RVt =

0+

3
X

i RVi;t 1 + "t

i=1

P
1 P30
where RV1;t 1 = RVt 1 , RV2;t 1 = 17 7t=1 RVt 1 and RV3;t 1 = 30
t=1 RVt 1 . Then,with the decomposition into continuous and jump components for the di¤erent tests, webuild the corresponding HAR-CV-JV model,
RVt =

0+

3
X

i CVi;t 1 +

i=1

3
X

i JVi;t 1 + "t

i=1

1 P7

1 P30
where CV1;t 1 = CVt 1 , CV2;t 1 = 7 t=1 CVt 1 , CV3;t 1 = 30
t=1 CVt 1 , JV1;t 1 =
1 P7
1 P30
JVt 1 , JVw;t 1 = 7 t=1 JVt 1 and JVm;t 1 = 30 t=1 JVt 1 . These models are estimated by ordinary least squares with heteroscedasticity and autocorrelation-corrected8consistent standard errors to control for volatility clustering usually observed in timeseries analysis.
The assumption of identically and independently distributed Gaussian errors is tested.
As Corsi et al. (2008) and Ciarreta and Zarraga (2016) show residuals can often show
volatility clustering and probability distributions with excess kurtosis and skewness.
Thus, we test for the presence of GARCH structures using the Ljung-Box (1978) statistic. Out of the several GARCH models proposed in the literature, we propose thefollowing: GARCH(1,1) and EGARCH(1,1). The error term "t is written as "t = t ut
where ut are independet and identically random standard normal distributed variables.
The GARCH(1,1) models volatility as2t =2
1 "t 1 +

0+

2
2 t 1

where 0 > 0, 10, 20 and 1 + 2 < 1 to ensure that the variance is positive
and the process is stationary. The asymmetric EGARCH(1,1) is given by
log

2
t =

0+ 1

j"t 1 j

+

2 log

t 1

"t 1
2
t 1+ 3t 1

where there are no restrictions on the sign of the coe¢ cients. The coe¢ cient 3 captures
the asymmetric response of the volatility due to shocks of di¤erent sign: 3 > 0 indicatesan inverse leverage e¤ect; 3 < 0 indicates a direct leverage e¤ect.

2.3

Bias in the volatility measures

We investigate the unbiasedness of the volatility measures following Jin and Maheu(2010) and Fang et al. (2012). We use daily data and estimate the following equation

where red = rd

PNd

j=1
P
Nd

red =

+

d;V d

ed , which are equal to the returnd;j are the de-jumped returns, r

on day d, rd , and j=1 d;j is the jump component. The jump component is the sumof all jumps within day d, Nd , of magnitude d;j . Finally, is the drift, d;V is thevolatility on day d estimated using volatility measure V (BNS, ABD, ALM and CPR),
is expected to be one in thed is a Gaussian perturbation. The estimated parametercase of no-bias, and greater (less) than one when V incorporates a downward (upward)
bias.

3

Japanese electricity market structure and regulatory framework

The Japanese electricity industry used to be organized around 10 local monopolies.
These local monopolies are vertically integrated Electricity Power Companies (EPCOs)
that comprise generation, transmission and distribution. In addition, Japan has two
9

types of frequency zones, 50Hz and 60Hz, that are linked. EPCOs controll the bulkof the generation capacity. The process of liberation from these formerly verticallyintegrated fully-regulated electricity …rms started in Japan in 1995 when the Electricity
Utility Industry Law (EUIL) was passed that promoted the entrance of Independent
Power Producers (IPPs). The target of the EUIL was to promote competition. Besides,
it opened the possiblity of competitive bidding system creating a wholesale market. At
the same time, the electricity rate system was revised, with the introduction of theyardstick assessment method for rate revision approval7 , optional contract provisions,fuel cost adjustment system, and management e¢ ciency review.
Further liberalization steps were taken in 2005 when the Japanese Electric Power
Exchange (JEPX) was launched in April. JEPX operates as a pool and calculates thesystem marginal price for each of the areas.8 In the absence of congestions in thetransmission grid, the price is the same. This is quite often the case in the areas that
operate with the same frequency. The bidding is done by a uniform price auction system.
Under this system, a bid is made for the combination of price and quantity of eachproduct. A point of intersection where the buying and selling conditions comply witheach other is sought, and the price and contract quantity are decided at this point. The
purpose of a power exchange is to increase liquidity and ‡exibility in electricity trading,allowing market participants to …t their positions closer to delivery. The amount ofelectricity traded through JEPX has signi…cantly increased although it remains at low
levels (below 2%).
There exists still a prominent position of these incumbents in the market. Table 1
illustrates this fact by showing the capacity mix of the EPCOs in 2013.
7

Under yardstick regulation the performance a regulated utility is compared against of a group ofcomparable utilities (Shleifer, 1985). Thus, the price cap of each utility is determined in a way that canendup in losses. This system is expected to promote e¢ ciency through cost cutting.
8
The exchange operator does not open a real price trading. It opens a system marginal price for theoperating regions and a virtual one for the whole country (asumming no restrictions in the network andno di¤erences in the frequencies).

10

TABLE 1. Capacity mix of EPCOs in 2013 (GW).
EPCO
Frq.
TH
NU
HY RES Total
1. Hokkaido 50
4:21
2:07
1:65 0:03
7:96
2. Tohoku
50
12:03
3:27
2:43 0:23 17:96
3. Tokyo
50
44:28
1:26
9:86 0:05 55:45
4. Chubu
60
24:02
3:62
5:50 0:03 33:17
5. Hokuriku 60
4:40
1:75
1:92 0:01
8:08
6. Kansai
60
19:41
8:93
8:23 0:11 36:68
7. Chugoku
60
7:80
0:82
2:91 0:01 11:54
8. Shikoku
60
3:45
2:02
1:15 0:01
6:63
9. Kyusyu
60
10:21
4:70
3:58 0:21 18:70
Total
129:81 28:44 37:23 0:69 207:38
Frq stands for frequency (Hz). Source FEPC (2014). TH, Thermal
NU, Nuclear; HY, Hydroelectric; RES, Renewable.

The total installed capacity of EPCOs is 208 GW which represents around two thirdsof the total installed capacity in the country. Limited transmission capacity between
frequency zones (1200 MW) prevents prices to be exactly the same between zones. Generation mix has been diversi…ed with a twofold target: (1) Reduction of carbon emissions
to comply with international agreements, and (2) reduce dependence from international
imports and fuel sources. Table 2 reports generation from di¤erent energy sources tototal generation mix.
TABLE 2. Generation mix in 2013 (%)
NU CO LNG OIL HY RES
2009 27:0 28:0 27:0
9:0 8:0
1:0
2010 28:6 25:0 29:3
7:5 8:5
1:1
2011 10:7 25:0 39:5 14:4 9:0
1:4
2012 1:7 27:6 42:5 18:3 8:4
1:6
2013 1:0 30:3 43:2 14:9 8:5
2:2
Source FEPC (2014). NU, Nuclear; CO, Coal;
LNG, Liqui…ed Natural Gas; OIL, Oil-…red,
HY, Hydroelectric; RES, Renewable.

Generation mix is dominated by thermal plants which makes the country highly
dependent on imports of fossil fuels. Only around 10% of total generation is fromrenewable sources. The perspectives to increase the share from hydroelectric sources
is quite limited and up to the limit the country can reach, so it is not expected tosigni…cantly increase in the future. Renewable generation has not received the support
as in other OECD countries such as Spain and Germany. The Fukushima Daiichi nuclear
power plant accident in March of 2011 evidenced many problems of the electricity supply
11system that had to be corrected urgently. There exists little competition and strong pricecontrol. Little ‡exibility in changing the existing energy mix; hard to increase the ratioof renewable energy. Therefore, a deep electricity reform started to be promotting ina short-term horizon. Three regulatory advances in this line should be remarked: (1)Legal unbundling of the network by 2020, (2) retail liberalization by April 1st, 2016,
and (3) renewable generation and distributed power generation promotion. Thus, smart
metering should be fully accomplished in order to be successful to achieve (1) and (2)goals.

4

Data and descriptive statistics

Our data consists of 30 minute day-ahead price intervals for each day (48 intra-day
observations). Data covers the period from April 2, 2005 until December 31, 2015.
Therefore, we have 188; 448 half-hourly observations. Table 3 reports basic descriptive
statistics of the price as well as by season and day of the week.9

TABLE 3: Descriptive statistics of prices per season and day of the week
Min.
Max.
Mean
MedianSt. Dv.
Sk
Kt (Ex.)
J-B

Wtr
4
42
11.90
11.50
4.67
1.07
5.60
5 106

Spr
3.30
35
10.89
10.22
4.07
0.59
2.84
7 105

Sum
2.95
60
10.93
10.10
5.02
1.68
9.98
2 107

Aut
4.34
39.23
11.09
10.24
4.08
0.91
4.62
7 106

Mo
2.95
39.67
11.50
10.91
4.56
0.89
4.73
5 106

Tu
4.00
44.92
11.66
11.00
4.72
1.06
5.57
6 106

Wd
3.99
43.67
11.64
11.00
4.72
1.09
5.63
5 106

Th
3.78
55.00
11.72
11.00
4.86
1.44
8.74
9 106

Fr
3.83
60.00
11.67
11.00
4.87
1.41
8.73
1 107

All
2.95
60.00
11.20
10.52
4.49
1.18
6.84
8 107

Min, minimum; Max., maximum; St. Dv., Standard deviation; Sk, Skewness; Kt (Ex), Excess kurtosis
J-B, Jarque-Bera test of normality.

The mean price is 11:20 U/kWh with a standard deviation of 4:49 U/kWh, theminimum price is 2:95 U/kWh and the maximum price is 60 U/kWh. Thus, the rangeof the price is quite large since large prices can be up to …ve times the mean price.
Several seasonal e¤ects are observed. Mean and median prices are higher in winter thanin the rest of the seasons, probably due to house heating. Although air-conditioning
consumption is high in summer it is not re‡ected in signi…cantly higher prices than inother seasons maybe due to lower economic activity that o¤sets consumption. Day of
the week seasonality is observed with lower mean, median and volatility of prices during
weekends and Monday. The coe¢ cient of skewness is positive and in many cases greater
9

Descriptive statistics are also available for each zone upon request. For example, in the islandof Honshu, the average prices in the Tohoku and Tokyo zones are 11:49 U/kWh and 11:44 U/kWh,respectively, whereas the average prices in Hokuriku, Chobu, Kansai and Chugoku zones are around
11:18 U/kWh. The rest of descriptive statistics follow a similar pattern.

12than 1, thus the distribution is highly right-skewed. In all cases there is signi…cant excess
of kurtosis which indicates that the distribution of prices is leptokurtic. Jarque Beratest con…rms that the distribution of prices is not normal.
Figure 1(a) plots the estimated kernel density against the standard normal ditribution
to con…rm the tests. Figure 1(b) plots the time series of prices.
[Insert Figure 1(a) and 1(b)]Figure 1(a) shows how the distribution of prices is skewed to the left and bimodal.
Figure 1(b) shows important price jumps in 2007/09/20, 2007/09/21, 2013/08/22, 2014/02/14,
2014/07/25 and 2014/08/04 to 07. These larger prices are observed mostly during peakdemand hours.
There exists a positive correlation between larger prices and traded quantities as
demand e¤ect dominates. However, this is not necesarily the case for returns. We
control for return seasonality using two di¤erent …ltering approaches. The …rst approach used by Chan et al. (2008), Ullrich (2012) and Ciarreta and Zarraga (2016),
where the medians of the returns are subtracted from the returns for each month of
the year, m, day of the week, d, and half-hour, hh, of the day to take into accountthe seasonal pattern e¤ect observed in the descriptive analysis of the prices, rt;(m;d;hh) .
Therefore, the de-median returns are de…ned as rt;j1 = rt;j rt;(m;d;hh) . The second
approach (REFERENCES) is using a nonlinear median smooth of size p and obtainrt;j2 = median (rt;j p ; :::; rt;j 1 ; rt;j ; rt;j+1 ; :::; yt;j+p ). For p = 1 is the usual de…nitionof median. Both approaches yield to similar results since both are based on the sameunderlying median smoothing theory. Alternatively we could have …ltered the data toremove the trend, , and cyclical components, c, ret;i = t;i + ct;i . The trend may be nonstationary, contain a deterministic or stochstic trend. We do not obtain large di¤erences
for the di¤erent …ltering methods this is why we stick to the …rst approach.10
Table 4 reports descriptive statistics of the de-median returns rt;j1 as a whole andalso by season and day of the week.
TABLE 4: Descriptive statistics of returns per season and day of week
Min.
Max.
Mean
MedianSt. Dv.
Sk
Kt (Ex.)JB

Wtr-0.7130.613-0.000
0.000
0.053
0.272
18.95
4 105

Spr
-0.551
0.488-0.000
0.000
0.056
0.114
14.92
2 105

Sum
-0.7680.889-0.002
0.000
0.073
0.046
21.84
6 105

Aut
-0.719
0.608
0.000
0.000
0.053
0.278
18.76
5 105

Mo
-0.7490.696-0.000
0.000
0.061
0.246
19.65
3 105

Tu
-0.7190.755-0.0010.000
0.063-0.003
20.34
3 105

Wd
-0.7680.680-0.0000.000
0.064-0.031
21.24
3 105

Th
-0.6530.856-0.000
0.000
0.063
0.344
20.14
3 105

Fr
-0.7150.889-0.000
0.000
0.063
0.274
20.90
3 105

All
-0.7680.889-0.000
0.000
0.060
0.128
21.43
3 105

On average adjusted returns are slightly negative. The largest return is 0:889 observed in summer which corresponds to a ratio of 2:5 between two consecutive prices
10

In Appendix 1 we report summary statistics of the …ltered returns for di¤erent …ltering methods.

13

(Friday 21, September, 2007 at half-hour 27). Thus, spikes are not as severe as those observed in other electricity markets worldwide.11 The percentage of zero returns is larger
during weekends when market transactions are lower. There is no signi…cant correlation
between returns and quantity traded. Figure 2(a) plots the estimated kernel density
against a standard normal ditribution of the returns and Figure 2(b) the time series of
the returns.
[Insert Figure 2(a) and 2(b)]Figure 2(a) clearly shows the excess of kurtosis and ‡atter than the normal distribution whereas Figure 2(b) shows the presence of return spikes above and below zero.
Extreme returns are also associated frequently with price spikes as it is the case for thedays 2007/09/20, 2007/09/21, and 2013/08/22, although not for the rest of the days.
Risk managers do not like volatility because it involves complex hedging strategies
and increases the risk of failures. Indeed, this is shown by the negative correlation
between RVt , de…ned in terms of the adjusted returns is used as our estimator of IVt ,and total quantity sold ( 0:1462).12 Table 5 summarizes the descriptive statistics of
RV .
TABLE 5: Descriptive statistics of RV
Min.
0.0031Max.
2.4296Mean
0.1728Median0.1189
St. Dv.
0.1840Sk
3.6120Kt (Ex.)25.2898
J-B
2.6 106
There exists skewness and excess of kurtosis, besides de Jarque-Bera test con…rms
thar the distribution is not normal. Figure 3(a) plots the estimated kernel density andthe standard normal ditribution and Figure 3(b) plots the time series of the RV .
[Insert Figure 3(a) and 3(b)]Figure 3(a) shows how a visual inspection of the distribution of RV is not normal.
Figure 3(b) shows that there are few days when RV presents sharp spikes. In particular,larger values are computed for 2007/09/20 and 2007/09/21, and also for 2011/08/19,
2011/08/31, 2013/08/21, 2013/08/29, 2015/08/04, and 2015/08/05. Thus, larger RV isfound during summer time days. Thus, a visual inspection justi…es to test the presenceof jumps using the tests proposed in Section 2. In order to restrict jumps, we choose
11
For instance, in the EPEX day-ahead the ratio can be 25 times larger when prices are positive or in
the Iberian market can be up to 7 times larger.
12
In empirical applications two transformations are often used.p One is the square root of the realized variance called properly realized volatility (see BNS, 2004) RV , and another is the logarithmictransformation, log RV . These transformations are used to smooth the time series. However, in ourapplication we restrict to RV because some of the tests are not suitable for any of these transformations.

14

= 0:01 as the signi…cance level to detect jumps. For the BNS, MIN and MED tests,
we choose 1-order processes. For the CPR test, we report results
p for the Gaussian kerneland L = 100. Finally, for the ABD-LM we have that K = 365 48 = 132. Table 6
reports the percentage of days with jumps detected by the di¤erent tests for adjacent
returns and …ltered returns.13
TABLE 6: Percentage of jump days
BNS MIN MED CPR ABD-LM% days 42:61 8:99
2:96 68:37
13:75

JO
33:11

Note that the BNS test detects many jump days, 42:61% of the days are classi…ed
as jump days. By construction of the CPR test detects even more jump days.14 On
the contrary, the MIN and MED tests are more conservative and detect around one …thand one tenth of jump days, respectively. ABD-LM and JO detect jump days inbetween
the other tests. This is probably the result of having 25% of the observations with zero
returns that tends to over-represent zero values in the calculations of BVt and T Qt .15
Figure 4 plots JV for the di¤erent tests.
[Insert Figure 4(a)] [Insert Figure 4(b)][Insert Figure 4(c)] [Insert Figure 4(d)][Insert Figure 4(e)] [Insert Figure 4(f)]Before we turn to the estimation of the models, we test for bias of the di¤erent
intraday volatility measures used in this paper. The estimated equation of the returns
against the di¤erent jump measures provides a test of the biasedness of the di¤erent
approaches to the decomposition between continuous and jump components. Table 7
reports estimation results.
TABLE 7: Test for Bias
BNS
MIN
MED
b
0:0012
0:0123
0:0204b 0:6074
0:551
0:566

CPR
0:01090:5812

ABD-LM
0:00220:6016RV
0:02590:5756Signi…cant at 1% level.

In all cases there is an upward bias in the volatility measures. The closest to no-boas( b = 1) is the BNS test followed by ABD-LM.
13

Appendix 2 reports di¤erent jump detection results considering several speci…cations of the …lteredreturns and products of k-spaced adjacent returns. We have also analyzed other kernel functions for the
CPR test but results do not change signi…cantly.
14
We have implemented the CPR test for values of L = f25; 50; 150g. For instance in the case of
L = 150, the percentage of jump days is 70:45. We have also considered c# = 4 and c# = 5 and thepercentage of jump days reduces to 59:60% and 55:11%, respectively.
15
We have also considered the serial autocorrelation of the returns and using the staggered version of
BNS the percentage of jump days decreases substantially. However, we continue considering this version
in order to make results more comparable.

15

5

Estimation Results

We estimate the di¤erent HAR models with and without the decompositions. Results
are shown in Table 8 below.

TABLE 8: HAR model estimation results
RV
CV-JV
BNS
CPR
MIN
b
b
0:02090:0186
0:0185
0:021000bb0:3417
0:3717
0:37170:33091
1
bb2 0:2919
0:3037
0:2913
0:29692bb3 0:2673
0:2334
0:26730:25453
b1 0:2936
0:2936
0:4122b2 0:3173
0:3172
0:3949b3 0:1971
0:19700:0372
2
adj-R
0:3786
0:3796
0:3796
0:3798AIC
3977:83981:03981:03982:6Signi…cant at 1% level.

MED
0:0212
0:3346
0:30400:23800:4921
0:3103
0:0863
0:37983982:6

ABD-LM
0:0144
0:3418
0:2612
0:2471
0:3011
0:31700:18120:38053981:1Signi…cant at 5% level.

We only report the estimation results of the CPR test when the Gaussian-normal
kernel is considered because results for the other kernel functions are very similar. Notethat all the coe¢ cients are positive and signi…cant at the 1% level for the CV andshowing strong volatility persistance. It is observed that the magnitude of the e¤ect on
RVt is larger from the previous day followed by the weekly and monthly average realized
volatilities. This is due to the presence of volatility clusters decreasing over longer
horizons. Considering the decomposition, it also shows high volatility persistance. In
the case of the jump component, weekly e¤ect is larger in magnitude than the previous
day e¤ect, and the monthly average e¤ect is clearly smaller in accordance with the shortterm jump e¤ect. Overall, based on adjusted R2 and AIC criteria the best model is theone that decomposes the realized volatility based on the approach by ABD (2010).
The error terms are analyzed in each of the estimated models in order to determine
the presence of GARCH structures. In the case of RV, the Ljung-Box statistic on thecorrelations of the residuals and squares of residuals do not reject the hypothesis of theexistence of GARCH e¤ects in the residuals. Thus, we estimate the HAR-GARCH and
HAR-EGARCH models for RV.

16

TABLE 9: HAR-GARCH model estimation results
RV
CV-JV
BNS
CPR
MIN
b
b
0:01050:00640:00590:002100bb0:3351
0:36680:3676
0:34751
1bb2 0:3881
0:3876
0:3886
0:39382bb3 0:2472
0:1416
0:25650:22443
b1 0:2995
0:31080:3125b2 0:4340
0:42730:4956b3
0:1170
0:13160:016
b0
0:0004b 0 0:0004
0:00040:0004b1
0:2088b 1 0:2244
0:00040:2201b2
0:7904b 2 0:7789
0:7982
0:7820AIC
6613:262
6618:556
6620:366
6612:484
Signi…cant at 1% level.

MED
0:00690:34130:4106
0:20980:3626
0:40450:042
0:0004
0:2181
0:7837
6612:484

ABD-LM
0:0061
0:32680:3901
0:2341
0:30050:4245
0:11330:0004
0:2044
0:76896611:511

Signi…cant at 5% level.

Note that b 0 > 0, b 1
0, b 20 and b 1 + b 2 < 1. According to the AIC thebest model is the ABD-LM. Note that estimated coe¢ cients do not much di¤erent inmagnitude with respect to the simple HAR model. The signi…cance of the GARCH
parameters indicates that it is relevant to model volatility in the error term.
Finally, results from the HAR-EGARCH estimation are reported in Table 10.
TABLE 10: HAR-EGARCH model estimation results
RV
CV-JV
BNS
CPR
Min
b
b
0:0016
0:0000
0:0000
0:000200bb10:3517
0:3366
0:3426
0:32761b2
b0:3627
0:35580:3734
0:51122bb0:1267
0:33980:3433
0:32473
3
b1
0:2972
0:3174
0:2834b2
0:40870:4207
0:4570b3
0:184801711
0:0005b0b0:13760:1601
0:15480:16190b1
b1
0:08780:0585
0:0614
0:0633b2
b2
0:2777
0:33490:32780:3368b3
b0:9614
0:9545
0:95590:9542
3AIC
6604:216608:55
6608:566615:468
Signi…cant at 1% level.

Med
0:0005
0:3167
0:3836
0:3103
0:3524
0:3897
0:0721
0:1625
0:0521
0:3494
0:9536
6617:979

ABD-LM
0:00000:3355
0:3427
0:32980:2962
0:4066
0:17480:15000:0577
0:33480:9544
6608:58

Signi…cant at 5% level.

We have estimated 15 non-nested models. To evaluate the in-sample performance ofthe models we select as preferred model the one with the minimum AIC value. In this17

case, the best model is the HAR-GARCH-CV-JV with CPR Gaussian decomposition,
followed by the HAR-GARCH-CV-JV with BNS decomposition. Thus, it is clearly
relevant to decompose RV into continuous and jump component. We are interested
in testing the robustness of our model. For that reason we also consider the Model
Con…dence Set approach by Hansen et al (2011).

6

Conclusions, policy implications and further research

The liberalization of the Japanese electricity market has implied the launching of a halfhourly Day-ahead market. Electricity subject to competitive market transactions is still
marginal compare to the bulk of electricity consumed. However, the government following recommendations from regulatory agencies is trying to push further liberalization.
As it happens in more developed electricity markets worldwide this trend tends to push
prices down but volatility increases. As a response, agents implement hedging strategies
following the experience of …nancial mrkets.
In this paper, we have shown how despite being marginal, there is volatility in themarket that can be modelled using realized volatility. After implementing several test
statistics, we show that there are jumps in volatility that can be best identi…ed using
the BNS approach. Using the HAR estimation, the nature of the price process is suchthat volatility depends on its own past. Including GARCH e¤ects improves estimation
results. Our results call for further research on the liberalization process of the Japaneseelectricity market. Even though we have identi…ed the best model to explain volatility,di¤erences in the jump detection call for further research on this initial part.

References[1] Ait-Sahalia, Y., and Jacod, J., (2008), "Testing for jumps in a discretely observed
process". The Annals of Statistics, 37, 184-222.
[2] Andersen, T. G. and T. Bollerslev (1998). "Answering the skeptics: yes, standard
volatility models do provide accurate forecasts". International Economic Review 39,
885-905.
[3] Andersen, T. G., T. Bollerslev, and D. Dobrev (2007), "No-arbitrage semimartingale restrictions for continuous-time volatility models subject to leveragee¤ects, jumps and i.i.d. noise: Theory and testable distributional implications".
Journal of Econometrics 138, 125-180.
[4] Andersen, T. G., T. Bollerslev, and E. Schaumburg (2012), "Jump-robust volatility
estimation using nearest neighbor truncation". Journal of Econometrics 169, 75-93.
[5] Asano, H., Okada, K, Yokohama, R., and Zhao, M., (2011), "Japanese approachto electricity deregulation policy - Modeling the spatial puchased power pricing".
Discussion paper CRIEPI and TMU Japan.
18

[6] Back, K. (1991). "Asset pricing for general processes". Journal of Mathematical
Economics 20, 371–395.
[7] Bandor¤-Nielsen, O., and Shephard, N., (2002), "Econometric analysis of realized
volatility and its use in estimating stochastic volatility models". Journal of the Royal
Statistical Society, Series B, 64, 253-280.
[8] Bandor¤-Nielsen, O., and Shephard, N., (2004), "Power and Bipower Variation with
Stochastic Volatility and Jumps". Journal of Financial Econometrics, Vol. 2, No. 1,
1-37.
[9] Bandor¤-Nielsen, O., and Shephard, N., (2006), "Econometrics of Testing for Jumps
in Financial Economics Using Bipower Variation". Journal of Financial Econometrics, Vol. 4, No. 1, 1-30.
[10] Chan, K.F., Gray P., and van Campen, B., (2008), "A new approach to characterizing and forecasting electricity volatility". International Journal of Forecasting 24,
728-743.
[11] Ciarreta, A., and Zarraga, A., (2016), "Modeling realized volatility on the Spanish
intra-day electricity market". Energy Economics, Volume 58, 152-163.
[12] Comte, F. and E. Renault (1998). "Long memory in continuous-time stochastic
volatility models". Mathematical Finance 8, 291-323.
[13] Corsi, F., (2009), "A Simple Approximate Long-Memory Model of Realized Volatility". Journal of Financial Econometrics, 7, No. 2, 174-196.
[14] Corsi, F., Mittnik, S., Pigorsch, C., and Pigorsch, U., (2008), "The volatility ofrealized volatility". Econometric Reviews 27, 46-78.
[15] Corsi, F., Pirino, D., and Reno, R., (2010), "Threshold bipower variation and theimpact of jumps on volatility forecasting". Journal of Econometrics 159, 276-288.
[16] Degiannakis, S., and Floros, C., (2016), "Intra-day realized volatility for Europeanand USA stock indices". Working paper.
[17] Deng, S. (1999). "Stochastic models of energy commodity prices and their applications: Mean reversion with jumps and spikes". Unpublished manuscript, Georgia
Institute of Technology.
[18] Dumitru, A-M., and Urga, G., (2012), "Identifying jumps in …nancial assets: A
comparison between nonparametric jump tests". Journal of Business & Economic
Statistics, 30 : 2, 242-255.
[19] Escribano, A., Pena, J.I., and Villaplana, P. (2011). "Modeling electricity prices:
International evidence". Oxford Bulletin of Economics and Statistics, 73, 5.

19

[20] Fang, Y., Ielpo, F., and Sevi, B., (2012), "Empirical bias in intraday volatility
measures". Finance Research Letters 9, 231-237.
[21] Federation of Electric Power Companies of Japan, (2015). Electricity Statistics Information.
[22] Geman, H., and Roncoroni, A., (2006), "Understanding the …ne structure of electricity prices". The Journal of Business, 79, No 3, 1225-1261.
[23] Hosoe, Nobuhiro, (2006), "The deregulation of Japan’s electricity industry". Japanand the World Economy 18, 230-246.
[24] Huang, X., and Tauchen, G. (2005), "The Relative Contribution of Jumps to Total
Price Variance". Journal of Financial Econometrics, Vol. 3, No. 4, 456-499.
[25] Hull, J., and White, A., (1987), "The pricing of options on assets with stochastic
volatilites." Journal of Finance, Vol. 42, 281-300.
[26] Hull, J., (2000), Options, Futures and other Derivatives, Prentice Hall.
[27] Ikeda, Shin, (2013), "An empirical microstructure analysis of the implied spread
cost in the Japanese day-ahead electricity market". National Graduate Institute for
Policy Studies, Tokyo, Japan.
[28] International Energy Agency, (2015), Statistics Energy Japan.
[29] Lee, S.S., and Mykland, P.A., (2008), "Jumps in …nancial markets: A new nonparametric test and jump dynamics". The Review of Financial Studies, 21 (6),2535-2563.
[30] Jiang, G.J., and Oomen, R.C.A., (2008), "Testing for jumps when asset prices areobserved with noise – a “swap variance” approach". Journal of Econometrics 144,352-370.
[31] Liu, Z., Ma, F., Wang, X., and Xia, Z., (2016), "Forecasting the realized volatility:
The role of junps". Applied Economics Letters, 23 : 10, 736-739.
[32] Ljung, G. M., and G. E. P. Box. (1978). On a measure of lack of …t in time seriesmodels. Biometrika 65, 297-303.
[33] Lucia and Schwartz[34] G. M. Ljung; G. E. P. Box (1978). "On a Measure of a Lack of Fit in Time Series
Models". Biometrika. 65 (2): 297-303.
[35] Makiko Nakanoa and Shunsuke Managi, (2008), "Regulatory reforms and productivity: An empirical analysis of the Japanese electricity industry". Energy Policy
36, 201-209.

20

[36] Miyauchi, Hajime, and Misawa, Tetsuya, (2014), "Regression analysis of electric
power market price of JEPX". Journal of Power and Energy Engineering, 2, 483488.
[37] Patton, A.J., and Sheppard, K., (2015), "Good volatility, bad volatility: Signed
jumps and the persistence of volatility". The Review of Economics and Statistics,
97(3), 683-697.
[38] Podolskij, M., and Ziggel, D. (2010), "New tests for jumps in semimartingale models". Statistical Inference for Stochastic Processes, 13, 15-41.
[39] Qu, H., Chen, W., Niua, M., and Li, X., (2016) "Forecasting realized volatility
in electricitymarkets using logistic smooth transition heterogeneous autoregressivemodels". Energy Economics 54, 68-76.
[40] Skantze, P., Ilic, M., and Chapman, J., (2000), "Stochastic modelling of electric
power prices in a multi-market environment", IEEE.
[41] Shleifer, A. (1985), "A Theory of Yardstick Competition," Rand Journal of Economics, Vol. 16, No. 3, 319-327.
[42] Ullrich, C.J., (2012), "Realized volatility and price spikes in electricity markets:
The importance of observation frequency". Energy Economics 34, 1809-1818.
[43] Velleman, P. F. (1980), "De…nition and comparison of robust nonlinear data smoothing algorithms". Journal of the American Statistical Association, 75, 609-615.
[44] Yamazaki, Takuya, (2015), "Japan’s Electricity Market Reform and Beyond", Presentation of Agency for Natural Resources and Energy (ANRE), Ministry of Economy, Trade and Industry.

21

Appendix: Filtering returns, realized volatility and jumps
TABLE #: Descriptive statistics of returns
rt;j1
rt;j2 (p = 1) rt;j2 (p = 2) ret;j
Min.
0:768
0:9851
0:86590:7520Max.
0:889
0:9337
0:97290:7528Mean
0:00080:00030:00050:0000Median0:0000
0:0000
0:0000
0:0000St. Dv.
0:060
0:0605
0:0635
0:0570Sk
0:128
0:41990:49100:4056
Kt (Ex.)
21:43
26:34
24:33
15:75
TABLE #: Descriptive statistics of RV
rt;j1
rt;j2 (p = 1) rt;j2 (p = 2)
ret;j
Min.
0:0031
0:00000:00120:0007
Max.
2:4296
2:5393
2:7758
2:5278Mean
0:17280:17580:1937
0:1561
Median0:11890:1172
0:1325
0:1068
St. Dv.
0:1840
0:1921
0:20820:1645Sk
3:61203:3284
3:527
3:6179Kt (Ex.) 25:2898
22:8928
24:7908
28:6389
TABLE #: Percentage of jump days
BNS MIN MED CPRrt;j1
k = 1 42:61 8:99
2:96 68:37
k=2
rt;j2 (p = 1) k = 1 49:41 3:59
2:54
k=2
rt;j2 (p = 2) k = 1 61:11 7:54
2:17
k=2
ret;j
k = 1 14:70 3:67
3:93
k=2

22

ABD-LM
13:75

JO
33:11

Technical appendix: Kernel functions
A kernel density estimate is constucted using a weighted sum of values calculatedusing a kernel function K (x) such that
fbK =

n
1 X
wi K
q h

x

Xi
h

i=1

p
P
2 ; iqrx . Call z = x Xi then K (z)where q = ni=1 wi , h = 0:9m
with m = min
x 1:349hn1=5can be one of these functions:
CosineEpanechnikov
GaussianParzenWeibull

1 + cos (2 z) if jzj < 10otherwise
(
1 2
p
1
z
(
)
3
p5
if
jzj
<
5
4
5
K (z) =
0otherwise
K (z) =

2z
K (z) = p12 exp
28
4
2 + 8 jzj3>
8z
if jzj 12< 3
38(1 jzj)
K (z) =
if 12 < jzj 1
3>:0otherwise

K (z) = (1

p)z

23

(1

p)(1+z)Technical appendix: Regression with Newey–West standard errors
The Newey-West standard errors for coe¢ cients estimated by OLS regression. The
error structure is assumed to be heteroskedastic and possibly autocorrelated up to some
lag,
b

OLS

Vd
ar b OLS

=

X 0X

1

X 0y

=

X 0X

1

X 0 b X X 0X

The coe¢ cients b are those of OLS linear regression (y =estimates of the estimated coe¢ cients are calculated as
X 0 b X = X 0 b 0X =

n

n

k

n
X
i=1

1

0

X + e). The varianceeb2i x0i xi

where n is the number of observations, k the number of regressors, and ebi = yi
If lags are included in the estimation then
X

0b

X=X

0b

0X +

n

n

k

m
X
l=1

1

l
m+1

24

n
X

t=l+1

ebt ebt l x0t xt l + x0t l xt

xi b OLS .

Technical appendix: Ljung-Box statistic
The Box-Ljung test (1978) is a diagnostic tool used to test the lack of …t of a time series model. The test is applied to the residuals of a time series after …tting an ARMA(p,q)
model to the data. The test examines m autocorrelations of the residuals. If the autocorrelations are very small, we conclude that the model does not exhibit signi…cant lackof …t. In general, the Box-Ljung test test the null hypothesis that the model does notexhibit lack of …t against the alternative hypothesis that the model exhibits lack of …t.
Given a time series Y of length T , the test statistic is de…ned as:
Q = T (T + 2)

m
X
k=1

T

rbk2

k

where rbk is the estimated autocorrelation of the series at lag k and m the number of lags q.
The null hypothesis is rejected if Q > X12 ;h , being X12 ;h the chi-square distribution
with h = m p q degrees of freedom and signi…cance level . The following tablesummarizes the Box-Ljung statistic for all the estimated models for q = 20:

RV

b
"

b
"2

HAR

GARCH

EGARCHHARGARCH

EGARCHq = 20q : 0:01 > Q

216:35
5

117:44
6

131:83
71024:7201097:55
18

75:61
18

q = 20q : 0:01 > Q
q = 20q : 0:01 > Q
q = 15q : 0:01 > Q
q = 15q : 0:01 > Q

217:96
4
217:71
5
214:98
5
211:795

131:66
7
130:207
132:34
7
131:55
7

133:61
7
132:38
7
134:13
7
133:48
71055:7101029:8101005:9601012:82
0

94:44
18
95:51
18
103:5918
100:94
18

81:75
18
81:62
18
87:32
18
85:97
18

CV+JV
BNS
CPR
Min
Med

25

Appendix: Quadratic Variation
First, variation of a real-valued continuous function f de…ned on an interval [a; b]
R is a measure of the arclength of the curve with parametric equation x 7! f (x) forx 2 [a; b]. Thus, total variation of a real valued function f de…ned on an interval[a; b] R is de…ned asnp
X
V (f ) = supjf (xi ) f (xi 1 )jP

i=1

where the supremum runs over the set of all partitions of [a; b], P = p = x0 ; :::; xnp np is a partition of [a;If f is a di¤erentiable function then V (f ) can be expressed as an integral,
V (f ) =Z bajf (x)j dx

Quadratic variation is a kind of variation process used to analyze stochastic
processes such as martingales and semimartingales. Suppose now that yt is a real-valued
continuous stochastic process de…ned on a probability space ( ; F; P ), the quadratic
variation of the process, denoted as [y]t , is de…ned as the sum of the squares of thedi¤erences of two consecutive numbers of the partition of t,[y]t = lim

kpk!0

n
X

yti

yti 1

2

i=1

where kpk is the norm or mesh of the partition. Recall that a partition of an interval
i=nsuch that xt0 < xt1 < ::: < xtn
t on the real line is a …nite sequence xt = (xti )i=1
where the norm is max xti xti 1 : i = 1; :::; n . A process yt has a …nite varation if
prob xti xti 1 < 1 = 1 for every i = 1; :::; n. Note that by the de…nition it implies
that the quadratic variation exists for all continuous …nite variation processes and iszero. Instead suppose that yt is a real-valued non-continuous stochastic process de…ned
on a probability space ( ; F; P ). If yt is a cadlag …nite variation process then yt has left
limit and it is right continuous, the quadratic variation can be expressed as
X[y]t =( ys )20<s t

It is important to relate quadratic variation to Ito processes. An Itô process is de…ned
to be an adapted stochastic process that can be expressed as the sum of an integral withrespect to Brownian motion, W , and an integral with respect to time of a predictablecomponent, ,ZZtyt = y0 +0ts ds + +The quadratic variation of the process is[y]t =

Z t
0

26

s ds

s dWs

0

Write the Ito process in the form of a di¤erential equation:dyt =t yt dt +t yt dWt

then, by Ito’s lemma, the stochastic process of any function G (y; t) isdG (y; t) =yt

@G @G 1
+
+
@y
@t
2

2
2 2@ Gy@y 2

dt + yt

2

@G
dWt
@y

@G
1 2 2@ G
where
yt @G
y @y2 dt is the non-stochastic component and yt @G@y + @t + 2@y dWt is
the stochastic component. Now let us consider the process with jumps,dyt =t yt dt +27

t yt dWt + dJt

60
50

smp

.1

Density

.08

40

.06

30

.04

10

20

.02
0

0

10

20

30

40

50

60
0

smp
Kernel density estimate

Normal density
0

Figure 1(a).- Kernel density of smp and standard normal.

40000

60000

80000

100000120000140000160000180000

80000

1000001200001400001600001800002500

3000

Returns
.8
1

9
8

-.8

-.6

-.4

-.2

0

.2

.4

.6

.8

-.6

0

-.4

1

-.2

2

0

3

4

.2

5

.4

6

.6

7

Density

20000

Figure 1(b).- Time series of prices

1-.8

Returns
Kernel density estimate

Normal density
0

Figure 2(a).- Kernel density of de-median returns and standard normal

40000

60000

2.5

RV

4.5

2

4
3.5

1.5

3
2.5
2

1

1.5
1

.5

.5
0

0

.5

1

1.5

2

2.5
0

Density

20000

Figure 3(b).- Time series of de-median returns

Kernel density estimate

Normal density

0

Figure 3(a). Kernel density estimate of RV

500

1000

Figure 3(b).- Time series of RV

1

1500

2000

3500

4000

1.4

JV

0

.2

.4

.6

.8

1

1.2

1.4

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

3000

3500

4000

2000

2500

3000

3500

4000

.2

.4

.6

.8

1

1.2

JV

1.4

Figure 4(a): Time series of JV using ADS-MED

0

JV

1.2
1
.8
.6
.4
.2
0

0

Figure 4(a): Time series of JV using BNS

0

500

1000

1500

Figure 4(d): Time series of JV using ADS-MIN

2