Modeling and Forecasting the Volatility of the Nikkei 225 Realized
Volatility Using the ARFIMA-GARCH Model
ISAO ISHIDATOSHIAKI WATANABE

Faculty of Economics andInstitute of Economic Research

Graduate School of Public Policy

Hitotsubashi University

University of Tokyo
October, 2008
Preliminary and incomplete

Abstract
In this paper, we apply the ARFIMA-GARCH model to the realized volatility and the continuous
sample path variations constructed from high-frequency Nikkei 225 data.

While the homoskedastic

ARFIMA model performs excellently in predicting the Nikkei 225 realized volatility time series and theirsquare-root and log transformations, the residuals of the model suggest presence of strong conditional
heteroskedasticity similar to the nding of Corsi et al. (2007) for the realized S&P 500 futures volatility.
An ARFIMA model augmented by a GARCH(1,1) speci cation for the error term largely captures thisand substantially improves the t to the data. In a multi-day forecasting setting, we also nd evidence ofpredictable time variation in the volatility of the Nikkei 225 volatility captured by the ARFIMA-GARCH
model. A battery of speci cation tests including the BDS, CCK, and Hong-Li tests for detecting higherorder dependence are run. The results of these tests reveal various forms of misspeci cation remaining
in the ARFIMA-GARCH model, which suggest the model can be further improved upon.

JEL classi cation: C22, C53, G15.

Keywords: ARFIMA-GARCH, Volatility of realized volatility, Realized bipower variation, Jump
detection, BDS test, Hong-Li test, High-frequency Nikkei 225 data

1

Introduction

Volatility plays key roles in the theory and applications of asset pricing, optimal portfolio allocation, andrisk management.

This fact, together with the development of econometric tools for volatility analysis

and empirical evidence for the predictability of the volatility of numerous markets, spurred the phenomenal
growth of the volatility literature as well as the birth of an entire nancial risk management industry during
the last quarter century. It is well-documented by now that time-variation in nancial market volatility is tosome extent predictable but stochastic, hence volatility itself is volatile. This paper empirically investigates
whether the volatility of the Japanese stock market volatility is predictably time-varying. The obtained
empirical results indicate that it is indeed time-varying with some predictable component.
The pace of progress at the frontier of volatility research has had at least four notable upsurges. The
rst followed the invention of the ARCH model by Engle (1982) and led to the development of the rst setof econometric procedures for the empirical analysis of time-varying volatility (see e.g. Bollerslev et al.
1994), and our deeper understanding of the empirical properties, e.g., volatility clustering, leverage effects
in volatility, and fat-tails, of many nancial time series. The second wave centered around the stochastic
volatility (SV) modeling, which capitalized on and often contributed in turn to the concurrent developmentin the Bayesian statistical analysis using the Markov chain Monte Carlo procedure (see e.g. Shephard 2005).
This paper mainly concerns modeling and forecasting of the volatility of the realized volatility, and is partof the currently continuing third upsurge involving realized volatility measures, which was ignited by the
recent availability of intraday nancial data collected near or at the tick-by-tick frequency and the need toharness such high-frequency data fraught with microstructure noise and apparent short-term seasonalities
before the rich information contained can be tapped into. The seminal paper by Andersen and Bollerslev(1998) de ned the sum of squared intraday returns as the realized volatility (RV) for the day, and proposed
to use it as a proxy for the ex post realization of the daily volatility. The squared daily return, typicallyused as a measure of the ex-post daily volatility in earlier volatility prediction studies, is a very noisy, albeit
unbiased, proxy for the conditional variance. On the other hand, under ideal conditions in the absenceof microstructure effects, the RV not only is an unbiased and much less noisy measure of the conditionalvariance but also converges in probability to the integrated variance over the measurement period as the
sampling frequency increases to in nity if the asset price follows a diffusion process. Hence, RV may be1considered an almost observable measure of volatility. A further case for the RV can be made based on thework of Hansen and Lunde (2005a) and Patton (2005), who showed how the use of noisy proxies for the
ex-post volatility such as the squared daily return may lead to a choice of an inferior volatility prediction
model.
In a milestone paper, Andersen et al. (2003) used a Gaussian fractional VAR model, i.e., vectorARFIMA model with MA order zero, for directly modeling and forecasting several exchange rate RV series,
building on earlier empirical investigations that had found long-range dependence and approximate normality of the daily log RV time series constructed from high-frequency intraday data of exchange rates (e.g.
Andersen et al. 2001a) and stock prices (e.g. Andersen et al. 2001b). They provided compelling empirical evidence for the superiority in predictive accuracy of this direct “reduced-form” daily RV modelingapproach over the daily returns approach with short- or long-memory GARCH-type models. Earlier papers
including Andersen et al. (1999), Blair et al. (2001) and Martens (2001) that explored how to take advantage of intraday data within the GARCH framework also underlie the shift of the focus of the volatility
literature to this approach. Ebens (1999) was also among the rst to apply the ARFIMA model directly
to RV time series.

His ARFIMAX model for the RV of the DJIA index portfolio returns incorporated

terms to capture the leverage effect, a well-documented stylized fact about equity returns. Koopman et al.
(2005) conducted an extensive forecast performance comparison study of the ARFIMA model for the RV
series of the S&P 100 stock index and the unobserved components RV (UC-RV) model of Barndorff-Nielsen
and Shephard (2002) as well as more traditional GARCH-type and SV models based on daily returns andtheir implied-volatility-augmented versions, and reported that the ARFIMA model outperformed the othermodels although the performance of the UC-RV model was nearly as good.
Corsi (2004) proposed the heterogeneous autoregressive (HAR) model for the RV as an alternative to the
ARFIMA model. The HAR-RV model employs a few predictor terms that are past daily RVs averaged overdifferent horizons (typically a day, a week, and a month), and is capable of producing slow-decay patterns
in autocorrelations exhibited by many RV series. Because of the ease in estimation and extendability ofthe baseline model, the HAR model has quickly become popular for modeling the dynamics of RV andother related volatility measures. Corsi et al. (2005), after nding strong conditional heteroskedasticity
and non-Gaussianity in the ARFIMA and HAR residuals of the RV of the S&P 500 futures, introduced
the HAR-GARCH-NIG model, which augments the basic HAR model with NIG (normal inverse Gaussian)
distributed standardized innovations having a GARCH volatility dynamic structure.

Subsequent papers

by Bollerslev et al. (2005) and Andersen et al. (2007a) also formulated the HAR errors as a GARCH
process. Note that when the time series being modeled measures volatility, its volatility is related to the2volatility of volatility of the primitive price process. The volatility of volatility of an asset price process
is an important determinant of the tail property of the distribution of the asset's return, precise modeling of
which is crucial, for example, for managing the extreme risk of a portfolio involving the asset, or pricing
and hedging of out-of-the-money options written on the asset. Furthermore, a suite of volatility options
and futures at the Chicago Board of Options Exchange (CBOE) and the CBOE Futures Exchange (CFE)
written on U.S. equity market volatility indices, as well as a variety of over-the-counter volatility derivatives
on major indices around the world, have been traded actively in recent years. For each of these products,
the underlying is itself some measure of market volatility. As far as these volatility derivative products areconcerned, the volatility of volatility is a second-moment property, rather than a fourth-moment property, ofeach of the respective underlying processes, and hence potential payoffs to accurate volatility-of-volatility
modeling may be substantial.
In this paper, we use the ARFIMA model with the GARCH(1,1) speci cation for the error term (ARFIMAGARCH), to empirically investigate the dynamic behavior of the volatility of the daily RV of the Nikkei 225
index. The Nikkei 225 is the most widely watched indicator of the overall moves of the Japanese stock
market, one of the largest in the world in terms of capitalization and trading volume. Shibata (2004, 2008),
Shibata and Watanabe (2004), Watanabe (2005), Watanabe and Sasaki (2006), Watanabe and Yamaguchi(2006) among others studied the RV of the Nikkei 225 index or index futures, and reported empirical ndings similar to those obtained for other major markets1 . Although neither a volatility index calculated anddisseminated on a real-time basis by a major nancial organization nor an exchange-traded volatility derivative has been introduced for any of the Japanese equity indices, Japanese-equity-related volatility derivatives
have recently been traded actively over-the-counter. Several papers applied the ARFIMA-GARCH model
to lower frequency macroeconomic and nancial time series (e.g. Baillie et al. 1996, Ling and Li 1997,
Ooms and Doornik 1999), and a large number of papers in the RV literature employ the ARFIMA model
without a conditionally heteroskedastic error speci cation to t daily RV series (e.g. Oomen 2001, Giotand Laurent 2004 as well as those already referenced above). To the author's knowledge, this paper is the
rst to apply the ARFIMA-GARCH model to RV time series. Although several recent RV studies used the
HAR-GARCH in place of the ARFIMA-GARCH model primarily for ease of estimation, it is not dif cult toestimate a low-order ARFIMA or ARFIMA-GARCH(1,1) model by the conditional sum of squares (CSS)
estimator either. The CSS may be considered an approximate Gaussian maximum likelihood estimatorwith all pre-sample innovations of the series set to zero or the unconditional mean. One advantage of the1

See also Takahashi et al. (2007), which applied their novel Bayesian SV approach that uses both trading-hour RV and daily

returns to the TOPIX, another widely watched index representing the Japanese stock market.

3

ARFIMA model is that it has the fractional integration parameter d explicitly incorporated into the model,
allowing one to estimate it jointly with the other parameters, a feature not shared by the HAR model, which
is not formally a long-memry model.
Another notable recent development in the RV literature is the approach due to Barndorff-Nielsen andShephard (2004, 2006a) of decomposing the RV into the contribution of continuous sample path variations
and that of jumps.

Extending the theory of quadratic variation of semimartingales, Barndorff-Nielsen

et al. (2006a) provided an asymptotic statistical foundation for this decomposition procedure under very
general conditions; See also an exposition paper by Barndorff-Nielsen et al. (2006b).

Andersen et al.

(2007a) used the HAR framework to study the roles of these two distinct components in RV prediction while
Andersen et al. (2007b) documented improvements in the RV forecasting accuracy achieved by modelingthese components separately.

In light of these results, we estimate and remove the jump contributions

from the daily Nikkei 225 RV using the Barndorff-Nielsen procedure modi ed by Andersen et al. (2007),and study the continuous sample path variations as well as the raw realized volatility.

Removal of theestimated jump component did not affect our empirical results very much, but given the recent interest in theliterature in this methodology, we document both results. We nd strong empirical evidence of conditional
heteroskedasticity in the ARFIMA errors, and some evidence of predictability of the time variation in thevolatility of the Nikkei 225 realized volatility.
The pace of progress in realized volatility research is yet to quiet down, but it is worth brie y mentioning
before closing the introductory section that another new sub-area of active research has already emerged,
triggered by the introduction of volatility indices by several major exchanges. This strand of the literature,
which may be called the fourth wave of volatility research, studies the empirical properties of volatility
indices directly, rather than treating volatility as a latent process and attempting to conduct inferences about
it through GARCH-type models or SV models applied to daily asset returns data. By far the most prominentof the volatility indices is the CBOE Volatility Index (VIX). The VIX index is a measure of the market's
expectation of the S&P 500 index volatility over the next 30 days implied by the prices of some of the
S&P 500 options traded at the CBOE. As mentioned above, the CBOE and its futures exchange (CFE)
already offer volatility options and futures based on some of their volatility indices, and obviously thestatistical properties of the levels of these indices are of interest on their own right. Moreover, to the extent
that the stock and options markets are integrated, the VIX as well as other indices formulated in a similar
fashion may be regarded as the latent volatility of the underlying index turned into an essentially observable
quantity. As is the case with the RV and related measures, this observability makes it possible to explore
ner structures of volatility processes. Going beyond the analytically tractable class of af ne jump diffusion

4

models, Bakshi et al. (2006), Dotsis et al. (2007), and Duan and Yeh (2007) among others estimated theconstant-elasticity-of-variance(-of-variance) and other continuous-time volatility models using daily VIXdata and their empirical results are broadly in line with the predictable variation of the volatility of the
Nikkei 225 realized volatility that we document in this paper.
The remainder of the paper is organized as follows. Section 2 brie y reviews the results from the theoryof bipower variation and jump component extraction. Section 3 describes the data and summary statistics.
Section 4 reviews the ARFIMA-GARCH model and reports estimation and forecasting results. Section 5
concludes.

2

Realized variance, realized bipower variation, and jump component extraction

The starting point of the realized volatility research is the recognition of the well-known result in the theoryof continuous-time stochastic processes that the volatility of a process would be completely known if we
observed a continuous record of the sample path of the process.

Although in reality we do not obtaina continuous record and only observe the realized sample path of the process at discrete points in time,
the sum of squared increments of the process approaches the integrated variance as the return measurement
intervals shrink to zero. More precisely, if the process is a continuous semimartingale, under mild regularity
conditions,
RVt :=

1=
X
j=1

2 p

jrt+j ; j !

Z t+1
t

2
s ds as(1)#0

where rt+j ; is the increment over the interval [t + (j1) ; t + j ] (in our context, the process is thelog of the Nikkei 225 index level process so that rt+j ;is the log return),

t is the diffusion coef

cient(instantaneous volatility) of the process, time t has a daily unit so that RVt is the tth day realized variance2 .
We will hereafter use the term realized volatility (RV) to refer both to RVt de ned in (1) and more loosely toother related measures such as the realized bipower variation de ned below3 . If the process is a semimartingale with nite-activity jumps, i.e., only a nite number of jumps occurring in any nite time interval, suchas Poisson jumps, then the realized variance converges to the quadratic variation, which can be decomposed2

Stock exchanges (the Tokyo Stock Exchange in our case) are not open 24 hours a day. Strictly speaking, time t here is used in

two different ways: when we refer to the tth day RV and when we divide the trading hours [t; t + 1] of a particular day into 1=
intervals. Our notation here also glosses over the existence of a lunch break.
3

Some authors preserve the term realized volatility strictly forp
RVt ; and call RVt the realized variance.

5

into the integrated variance (the continuous sample path variation) and the sum of squared jump sizes:
p

RVt !
where

Z t+1
t

X

2
s ds +

2s as

s is the size of the jump occurring at time s.

(2)

#0

t<s t+1

Barndorff-Nielsen and Shephard (2004, 2006c) showedthat even in the presence of jumps the realized bipower variation
BVt :=

1=
X

2

j=2

p

jrt+j ; j rt+(j 1) ;

!

Z t+1
t

2
s ds

(3)

holds under mild conditions, and proposed to use
X

p

RVt

BVt !

2
s

(4)

t<s t+1or
BVt )+

Jt := (RVt
as an estimator for the sum of realized squared jumps(5)Pt<s t+1

2.
s

Jt is known to take non-zero, smallvalues very frequently due to measurement errors and due possibly to the presence of jumps of in niteactivity types. Based on the asymptotic distributional theory for these quantities developed by BarndorffNielsen and Shephard (2004, 2006c) and Barndorff-Nielsen et al. (2006a) and an extensive simulation study
by Huang & Tauchen (2005), Andersen et al. (2007a) introduced what they call a shrinkage estimator forthe jump contribution
Jt := I (Zt >

a)

(RVt

BVt )

(RV

BV )RV

1

(6)

t
t
t
where I is an indicator function, Zt := q
is asymptotically standard nor2(( =2) + 5) max(1;T Qt BVt 2 )pmally distributed, 1 := 2= ; a := (a) is the standard normal distribution function, and the (stan-

dardized) realized tripower variation
T Qt :=

1

4

1 3=2

7
6

1=
3X
j=3

4=3

jrt+j ; jrt+(j 1) ;4=3

rt+(j 2) ;

4=3 p

!

Z t+1
t

4
s ds as#0(7)

The last convergence result holds even in the presence of jumps. a is usually set to values such as .999
so that Jt picks up only "signi cant" jumps.

With Jt ; another estimator of the continuous sample pathvariation:
Ct := RVt
may be used in place of BVt .

6

Jt

(8)

In this paper, we use the microstructure-effects-robust versions of BVt and T Qt ; due also to Andersen
et al. (2007a). In these versions, the summands are respectively (14 ) 1 jrt+j ; j4=3 rt+(j 2) ;

(1

4=3

rt+(j 4) ;4=3

2 ) 1 jrt+j ; j rt+(j 2) ;, skipping an interval of lengthandin sampling

short-period returns. The de nitions of Jt and Ct are modi ed accordingly as well.

3

Data and summary statistics

3.1

Calculation of ve-minute returns from minute-by-minute Nikkei 225 data and RV
measuresNihon Keizai Shinbun, Inc. (Nikkei) computes and disseminates the Nikkei 225 index once every minuteduring the trading hours of the Tokyo Stock Exchange (TSE) (09:00-15:00 with a 11:00-12:30 lunch break)4 .
In this paper, we use minute-by-minute Nikkei 225 index data provided directly by Nikkei and maintainedby the Center for Advanced Research in Finance at the University of Tokyo. The sample period spans March
11, 1996 through August 31, 2007. From the minute-by-minute data, we construct a series of ve-minutelog differences multiplied by one hundred, which we call " ve-minute (percentage) returns." This choice
is made to strike a balance between alleviating the microstructure-related noise and increasing the precisionof volatility measurement, following the standard practice of the RV literature in handling high-frequency
intraday data from highly liquid markets. Andersen et al. (2000) and the other empirical papers dealing
with the RV of the Nikkei 225 cash or futures referenced in the introduction also used ve-minute returns.
For further discussions of the miscrostructure-related issues, see Hansen and Lunde (2005a) and references
therein.
The of cial minutely Nikkei 225 index starts at 09:01 for the morning session, and the rst ve-minutereturn in a given day that we calculate is for the 09:05-09:10 interval as in Andersen and Bollerslev (2000).
Removal of the rst four observations partially alleviates possible effects of the sluggish response of the
Nikkei 225 to the information accumulated overnight (or over a weekend or holiday) on volatility measurement. Given the TSE trading rules, the 9:05 index value is unlikely to have fully impounded all overnight
information in it when overnight domestic and overseas events move the overall Japanese stock market up
or down by a large amount relative to the previous close5 , and again our choice here is an attempt for a4

Most of the major TSE-listed companies are dually listed on the Osaka Securities Exchange (OSE), which closes at 15:10. The

OSE is the only domestic exchange where the Nikkei 225 futures and options are traded. Nikkei, however, calculates the Nikkei
225 index based on electronic data feed from the TSE with generally higher volumes in individual stock trading.
5

The TSE trading rules limit the range of prices at which shares of each individual stock is allowed to trade. For each stock,7

balance between higher precision and lower bias. For the afternoon session, the index starts at 12:31. Since
the effect of the TSE trading rules hampering the Nikkei 225 index from quickly re ecting the information
accrued over a lunch break is likely to be much milder, the 12:31-12:35 interval is retained and used in our
calculation of the rst ve-minute return.
For the end price of the last ve-minute return calculation of each session, we use the closing price ofthe session. Due to such factors as delayed arrival of individual stock price data from the TSE and thereal-time nature of the Nikkei 225 calculation and dissemination, the nal few observations of each session
are occasionally marked by time stamps up to several minutes later than 15:00 (11:00). For sessions withsuch observations as well, we use the last recorded observation of the session for closing the last ve-minuteinterval.

In total, there are 53 ve-minute returns for a typical trading day, 23 from the morning session

and 30 from the afternoon session. We exclude sessions from half trading days including the rst and last
trading days of each year from our sample, retaining 2,802 trading days.
We then calculate RVt for each day using all ve-minute returns from the day. For the other associatedmeasures such as BVt involving lagged returns, we do not treat the morning session's last few and theafternoon session's rst few as consecutive observations since there is an intervening lunch break. This2 +entails a loss of several more ve-minute return observations for these measures. RVt := RVt + Rn;t
2 may be used to de ne the RV for the day, where R
Rl;tn;t is the previous-day-close-to-open (15:00-9:05)

overnight return and Rl;t is the lunch break (11:00-12:30) return6 . Since the TSE is open only 4.5 hours
a day, however, it would be a stretch to treat RVt an observed realization of the volatility for the wholeday.

We therefore concentrate on RVt and related measures calculated from ve-minute returns wheninvestigating the behavior of the volatility of volatility.

3.2

Properties of the realized volatility and related measures

Summary statistics for various returns, RV measures, and their log and square-root transformations arepresented on Tables 1.

In addition to the sample skewness and kurtosis, the Jarque-Bera (JB) statistic is

this range is initially set around the reference price based on the last execution price, and when no trade takes place within it forve minutes due to an order imbalance, it is given a shift of a predetermined size once every ve minutes until a trade takes place.
The TSE announces either the upper or the lower end of the range as the "special quote." The special quotes of the constituent
individual stocks, if there are any, are used in place of the more stale last transaction prices in the calculation of the Nikkei 225
index.
6

In their empirical analysis of the Nikkei 225 futures RV, Shibata and Watanabe (2004) used RVt and several other measureswith more sophisticated weighting schemes that extend Martens (2002).

8

presented for each series.

It is designed to jointly measure the deviations of the sample skewness andkurtosis from their respective population values, zero and three, under normality.

For an i:i:d: normal

series, the JB statistic is asymptotically distributed

The simulation work

2 (two degrees of freedom).

of Thomakos and Wang (2003) has shown, however, that the JB statistic grossly overrejects the null of
normality if the data are a sample path of a long-memory process. Hence, it should be treated here as an
informal descriptive statistic.
For checking temporal dependence, the rst-order sample autocorrelation and the Ljung-Box statisticsof orders 5, 10, and 22 (corresponding to roughly one week, two weeks, and a month) for no serial correlations up to their respective orders are shown for each series. Since the usual Bartlett's standard error,
T

1=2 = 0:01 9; is biased under heteroskedasticity, the heteroskedasticity-adjusted standard error for the

rst-order autocorrelation and Ljung-Box statistics due to Diebold (1988) are also presented. Previous empirical studies have documented that daily volatility measures such as the daily return squared, the absolutedaily return, and various daily RV measures of nancial time series appear to have long-memory properties.
For long memory processes, the in uence of shocks does not last forever unlike in the case of integrated
processes, but decays very slowly relative to the geometrically fast decay for short-memory processes. Formally, there are several de nitions of long memory. A usual de nition of long memory for a covariancestationary time series, which we adopt in this paper, is that
1
X

k= 1

holds wherej (k)j = 1(9)(k) is the kth order autocovariance. Another de nition of long memory for a covariancestationary process is that (k) decays hyperbolically, i.e.,(k)k 2d 1 l (k)(10)

as k ! 1, where l is some slowly varying function and d < 1=2 is called the long-memory parameter.
If the process is a covariance stationary one satisfying some regularity conditions, (10) with d 2 (0; 1=2)
implies (9). For a review of alternative de nitions and their relations to each other, see e.g. Palma (2007).
Before estimating the ARFIMA and the ARFIMA-GARCH models for the RV series, we estimate
the long-memory parameter d for our various series via two popular semiparametric estimators dbGP H ;dbRobinson , due respectively to Geweke-Porter-Hudak (1983) (GPH) and Robinson (1995a), with the band-

width parameter m set at m = T 0:7 where T is the sample size. The asymptotic standard errors for dbGP H
pp
and dbRobinson are respectively = 24m and 1= (2 m), and hence the latter is asymptotically more ef cient

for a given m. d is equal to the d in the fractional differencing parameter d if the process is a fractionally9integrated process, reviewed in the next section, and its de nition can be extended to the nonstationary region d1=2. The region of the true d over which dbGP H and dbRobinson are consistent and asymptoticallynormal extends beyond 1=2.

For these properties, neither dbGP H nor dbRobinson requires Gaussianity, andthe latter does not require conditional homoskedasticity. See Robinson (1995b), Velasco (1999), Deo and
Hurvich (2001), and Robinson and Henry (1999).

Summary statistics for Rt , Rn;t , Ram;t (9:05-11:00 returns), Rl;t , and Rpm;t (12:30-15:00 returns) arepresented on the top part of Table 1. All of these log return series have means insigni cantly different fromzero and exhibit evidence for nonnormal unconditional distributions (in particular Ram;t ) and little evidence
for the presence of autocorrelations except that there is some weak evidence of autocorrelations in Rl;t andRpm;t :
Looking at the summary statistics for RVam;t (the morning RV component), RVpm;t (the afternoon RV
component) and RVt = RVam;t + RVpm;t ; their unconditional distributions all seem to be highly nonnormal
with very large positive values of sample skewness and kurtosis. The LB statistics indicate that each of
the three series is highly signi cantly serially correlated. The values of the rst-order sample autocorrelations, 0.429 (RVam;t ), 0.355 (RVpm;t ), and 0.544 (RVt ), are at medium-persistent levels and well below
one, but the values of dbGP H (.460 for RVam;t ; .364 for RVpm;t ; and .470 for RVt with standard errors.040) and dbRobinson (0.454 for RVam;t ; 0.376 for RVpm;t ; and 0.468 for RVt with standard errors .0311) aresigni cantly positive but below the stationary/nonstationary border of 1/2 for all three, indicating that autocorrelations decay slowly for these series. Note, however, that for RVt and RVam;t , dbGP H and dbRobinson
are within two standard errors from 1/2.

Deviations from normality seem to be vastly reduced by the square-root transformation, but remain
large. The log transformation brings down the sample skewness and kurtosis values for each series evenfurther and close to zero (-.177 for RVam;t ; -.182 for RVpm;t , and -.130 for RVt ) and three (3.272 for
RVam;t ; 3.260 for RVpm;t , and 3.234 for RVt ) respectively. Each of these two transformations increases
the values of the rst-order sample autocorrelation, LB statistics, and the two semiparametric estimates ofpd. For example, for RVt ; the rst-order sample autocorrelation increases to .663 ( RVt ) and .713 (ln RVt ),pand dbGP H ; dbRobinson increase to values in excess of 1/2 (557, .524 for RVt and .584, .533 for ln RVt ).
Other than our point estimates of d being in the nonstationary region, these results are roughly in line withearlier empirical ndings about RV measures constructed from high-frequency intraday exchange rate andstock returns data, which led to the popularity of the Gaussian ARFIMA model; See e.g. Andersen et al.
(2003).
Additionally, summary statistics for RVN W;t ; RVHL;t ; RVN W HL;t and their respective transformed ver-10

sions are shown on Table 1. RVN W;t := RVN W;am;t + RVN W;pm;t where RVN W;am;t and RVN W;pm;t are
respectively the Newey-West-type estimates of the morning-session and afternoon-session RVs. RVHL;t
2 , RV
22
2(RVHLN W;t ) weights the four components Rn;t
am;t ; Rl;t ; RVpm;t ( Rn;t ; RVN W;am;t ; Rl;t ; RVN W;pm;t )more ef ciently. All of these estimators are due to Hansen and Lunde (2005a, 2005b). RVN W;t behaves
similarly to RVt and Ct ; and we do not further report their dynamic properties in this paper.

We do notfurther investigate RVHL;t or RVN W HL;t for the same reason we do not study RVt :Turning next to the Barndorff-Nielsen decomposition of RVt into the contribution of squared jumps andthat of continuous sample path variations, we set a = :999 in (6). Again, we present separate results for themorning, the afternoon, and the whole daily trading hours, but focus on the whole-trading-day statistics inour brief discussion. Summary statistics of the square-root and log (ln (1 + Jt ) for Jt ) transformed series
are also presented on Table 1. The sample mean of I (Zt >a ) ; which is an estimate of the unconditional

jump probability over the trading hours of a day, is .159, implying a little more than one jump occurrences
per week7 . On average, the jump contribution Jt comprises about 4% (= :045=1:087) of the RV overtrading hours. Not surprisingly, given the results of previous studies on the S&P 500 futures and othernancial time series (e.g. Andersen et al. 2007a, Andersen et al. 2007b), Jt is distinctly less persistentalthough the unobserved conditional jump probability series (as opposed to the realized jump series Jt )might be more persistent. And the standard deviation of Jt (.154) is not negligible relative to that of Ct(1.03). Hence, using Ct ; purged of the jump component with a different dynamic behavior, may reveal ahigher-resolution picture of the dependence structure of Ct : For our analysis via the ARFIMA-GARCH
model, we use Ct as well as RVt : It turns out, however, that the ARFIMA-GARCH estimation results for
Ct are similar to those for RVt :4

Modeling and forecasting the conditional mean and the conditional variance of the RV with the ARFIMA-GARCH model

4.1

The ARFIMA-GARCH model and its estimation

The ARFIMA model, introduced by Granger and Joyeux (1980), and Hosking (1981), is a natural extensionof the ARIMA model for parsimoniously modeling time series with long memory. An ARFIMA(p; d; q)
process fYt g may be de ned as a causal solution to(L) (17L)d (Yt)=(L) "t(11)

Using a less stringent value a = :99, the estimated jump probability becomes .296, or an average once per 3.38 trading days.

11

where2 R,(L) = 1

p
pL ;

1L

(L) = 1 +AR and MA operators, sharing no common roots, (1f"t g

W N 0;

2

;0 <

2

< 1.

If all roots of

1L +

+

q
q L are respectively theL)d is the fractional differencing operator, and(z) and(z) lie outside of the unit disk andd 2 ( 1; 1=2) holds, then there is a unique covariance stationary solution which is invertible8 . It can beshown that the autocovariance function of this solution satis es(k)jkj2d 1 c

(12)

as jkj ! 1 where c is a constant (see e.g. Palma 2007, p.48). In particular, it exhibits long memory ifd > 0: Hence, the fractional integration parameter d in (11) corresponds to the long-memory parameter din (12). Also of note is that an ARFIMA process with d

1=2 is nonstationary but still mean-reverting as

long as d < 1 (See Baillie 1996, p.21). If d 2 [1=2; 1) ; we may interpret fYt g as a process, starting fromsome nite past and satisfying (11), which becomes a stationary ARFIMA(p; d1; q) process after being

differenced once.
Following Ding et al. (1993) that found extremely slow decay patterns in the sample autocorrelation
functions of daily volatility measures such as absolute returns, features meant to capture the long-range
dependence in volatility have been incorporated into GARCH-type models (e.g. the FIGARCH model) andstochastic volatility models; see e.g. Baillie et al. (1996a), Bollerslev and Mikkelsen (1996), Breidt et al.
(1998), Deo and Hurvich (2003). Since the daily RV time series appears to have long-memory properties
and is a series of observed quantities, the ARFIMA model is a natural modeling choice.

Baillie et al.

(1996b) extended the ARFIMA model to include a GARCH speci cation for conditional heteroskedasticity
and used it to analyze the in ation rate time series from the G7 and three other high in ation countries. See
also Hauser and Knust (1998, 2001) for applications of the ARFIMA model with ARCH errors. In thispaper, we apply the ARFIMA-GARCH(1,1) model in which ht := Et 1 "2t ; the conditional variance of"t with respect to the sigma- eld(Yt 1 ; Yt 2 ;) ; is given the following formulation:

ht = ! + ht 1 + "2t 1(13)

For the stationary ARFIMA model with Gaussian homoskedastic errors, Sowell's (1992) algorithm for exact
maximum likelihood estimation is available. For the ARFIMA-GARCH model, however, no closed-form
expression for the exact likelihood function is available. Hence, we employ the conditional sum of squares
8

Bondon and Palma (2007) recently proved invertibility for d >literature.

12

1, relaxing the condition d >1=2 often cited in the

(CSS) estimator:

where

:=

0

; 01 ;

1 :=

1;

;

p;

b := argmaxL ( )1;;q ; d; !;

;

(14)
0

and

T

1X
ln ht ( ) + Zt ( )22
t=1
p
Zt ( ) := "t ( ) = ht ( )(15)

L ( ) :="t ( ) :=(L) 1 (L) (1

L)d (Yt)

ht ( ) = ! + ht 1 ( ) + "t 1 ( )2
with the presample values of Yt
If d(1

set to zero and h1 ( ) set to some initial value.

; t = 0; 1; 2;

1=2, Yt does not have an unconditional mean andL)d (Yt) = (1

presample values so thatL)d 1

Yt but note that(16)

cannot be interpreted as such.

Y1 = Y1

Y0 = Y1

For this case,under our choice of thedoes not disappear. We equate h1 ( ) to the value of the sample variance as

is often done in the estimation of GARCH-type models. In estimation, we do not impose d < 1=2 (plus+< 1 for the GARCH speci cation) necessary for covariance stationarity. We estimate the ARFIMA

with a homoskedastic speci cation, i.e., ! =2;== 0; by the CSS estimator as well for comparabilityof results across homoskedastic and GARCH speci cations. The objective function maximized by the CSS
estimator is essentially the Gaussian maximum likelihood (without the constantT
2 ln (2) ) for the AR(1)representation of the ARFIMA-GARCH model with the conditional distribution of "t speci ed as N (0; ht ),conditional on the presample values and h1 . Hence, the CSS estimator for the ARFIMA-GARCH model isa long-memory analogue of what we usually refer to as the (quasi-)maximum likelihood estimator, (Q)MLE,
in the short-memory ARMA-GARCH setting. Note that the exact likelihood is not available in closed form
for the latter case either. Extending the results of Beran (2004) for the ARFIMA model, Ling and Li (1997)
pshowed that the CSS estimator is T -consistent and asymptotically normal for the ARFIMA (p; d; q)GARCH(P; Q) model. Their results are also valid for the nonstationary case in which the true value of d islarger than 1/2. This is a desirable property particularly because d estimates are found to be near or greaterthan 1/2 in our semiparametric estimation as well as other studies of nancial time series data, and is anotherjusti cation for using the CSS estimator. Of course, one could work with a differenced series, but withestimated d near the boundary 1=2, one cannot be con dent about the appropriateness of such a procedurep
Ling and Li (1997) derived T -consistency for b1 assuming that is known. Hence, strictly speaking,0

their results are not applicable to b: However, it seems reasonable to expect that T 1=2 d b; T 1=2b1

0

has

a Gaussian limiting distribution as long as the other conditions of Ling and Li (1997) are satis ed and13d < 1=2 9 . Another caveat is that the asymptotic results of Ling and Li (1997) are based on the assumptionno
1=2
E "4t < 1. If f"t g follows a GARCH(1,1) process and the standardized error sequence Zt := "t ht
is i:i:d: with:= E Zt4 < 1; this condition amounts toindicate that a covariance staionarity condition

2+2

+

2

< 1. Our estimation results< 1; let alone the more stringent condition

+

< 1,

may not be satis ed by some of the series we study. However, this is not likely to invalidate estimationand inference based on Ling and Li (1997). Hansen and Lee (1994) established (local) consistency andasymptotic normality of the Gaussian QMLE for the GARCH(1,1) model when the true values of ;not be in the covariance stationarity region
E ln

+ Zt2

< 0.

+

may

< 1 but are in the strict stationarity region satisfying

Jensen and Rahbek (2004), focusing on the estimation of ( ; ) and working

with a different set of assumptions on fZt g proven that the consistency and asymptotic normality results ofHansen and Lee (1994) extend to the case of conditional variance explosion E ln

+ Zt2

> 0. Unlike

in our case, however, they both assumed Yt =+ "t with either

a constant to be estimated jointly with theother parameters (Hansen and Lee 1994) or= 0 known (Jensen and Rahbek 2004). To our knowledge,

rigorous asymptotic theory for the ARFIMA-GARCH model with

1 is not yet available in the+literature.
Given the strong evidence to be reported shortly against conditional normality of the standardized error
Zt except when Yt is ln RVt or ln Ct , we also present the robust standard errors of Bollerselv and Wooldridge(1992) interpreting our estimator as a QMLE. For the models with the GARCH speci cation, the BollerslevWooldridge standard errors are robust to distributional misspeci cation of Zt under correct speci cation of
the conditional mean and variance and regularity conditions. We use the BIC10 as our model selection criteria and con ne our search for the best model to a total of 64 models: The full ARFIMA(2,d,2)-GARCH(1,1)
model and its 63 restricted versions (at least one of the ARFIMA parameters,
zero and/or no conditional heteroskedasticity ( ; ) = 0).

1;

2;

1;

2 ; d; isxed at; ! are always estimated with the other para-

meters. Let us denote, for example, the ARFIMA(2,d,0) model with the rst-order AR coef cient restricted
to be zero as the ARFIMA({2},d; 0). We denote the other models with second-order terms similarly.
When modeling a time series of daily currency or stock returns, the conditional mean is small relative
9

For the case of d(L) "t withd1=2; Ling and Li (1997) considered estimating the parameters in(L) (1L)d m ((1L)m Yt)=6= 0 unknown and estimated separately from the other paramters, where m is the smallest positive integer such thatm < 1=2. If m = 1; this implies that fYt g has a linear time trend, which is counterintuitive in our case where Yt is a measureof volatility.
10

BIC =

function L

2L

b + N ln T where N is the number of parameters, L

b =L bT ln (2 ) ; and L ( ) is as de ned in (15).

14b is the value of the maximized log likelihood

to the variance so that setting it to zero or a constant instead of tting a more elaborate model usually does
not much affect the estimation of the volatility process. Since the RV is highly persistent, it is essential toadequately model the conditional mean part of the RV even if one's primary interest is in the conditionalvariance of the RV. Otherwise, misspeci cation in the mean may masquerade as conditional heteroskedasticity. See Diebold and Nason (1990) for a study focusing on this issue.
Recently, the HAR model proposed by Corsi (2003) has also been frequently used in the RV literature
because of its exibility in terms of accommodating predetermined variables other than pure lagged dependent variables and ease of estimation; See e.g. Andersen et al. (2007), Bollerslev et al. (2005), Forsberg
and Ghysels (2006), and Shibata (2008). The HAR model is a high-order AR model which restricts theparameters in an intuitively appealing way, and can also be extended to have conditionally non-Gaussian,conditional heteroskedastic error speci cations; See Corsi et al. (2007). Although it can roughly match
slowly decaying autocorrelation patterns exhibited by observations of many RV time series, the HAR model
is formally not a long-memory process. Of course, lack of the long memory property in the formally de ned
sense per se is not a shortcoming of the HAR model. Long memory being an asymptotic property of (k)as jkj ! 1; the concept is strictly speaking not applicable to a nite segment of a time series anyway, anda primary objective of using long-memory-type models is to give a parsimonious approximation to the DGP
that generates RV data with slowly decaying sample autocorrelations. For this purpose, both ARFIMA andHAR models seem to do an admirable job. Nevertheless, it is still of interest to estimate the long-memory
parameter d in a simple uni ed framework provided by the ARFIMA model.

4.2

ARFIMA-GARCH estimation results

Many empirical studies of time series data set aside a hold-out sample for out-of-sample performance evaluation.

In our case, however, we already know, based on the results reported in the literature, that the

ARFIMA model would perform well as a time series model for the conditional mean for our entire sample,
and we are not running a horse race of a myriad of models in out-of-sample forecasting. Rather, our objective is simply to get a rst handle on the time-series behavior of the volatility of the Nikkei 225 RV using the
ARFIMA-GARCH model. We do search for the best order within the ARFIMA(2,d,2)-GARCH(1,1) class,but it is done in an attempt to adequately lter out the conditional mean component before the volatilityof the RV can be analyzed. Remaining serial correlations in the ARFIMA residuals may lead to spurious
predictability of the volatility of the RV. Hence, we rst use the entire sample (2802 daily observations)available to us to estimate the full ARFIMA(2,d,2)-GARCH(1,1) model and 63 restricted versions. For
order selection, we employ the BIC.

15

In applying the ARFIMA-GARCH model, we work with transformed series ln RVt ; ln Ct and
p

p

RVt ;

Ct as well as the original series RVt ; Ct : It should be emphasized that, although the results for all six series

are presented and discussed together and some comparisons in terms of model adequacy are made acrossthe two series RVt and Ct and across different transformations, each series is being investigated on its own
right, and that the BIC values and other speci cation test statistics and forecast performance measures are
not directly comparable across the six series. It is beyond the scope of this article to address the issue of
which series should be used as the LHS variable of the ARFIMA-GARCH model in a forecast exercise witha particular loss function11 although our results contain information relevant to this issue.
Table 3 summarizes the parameter estimation results. For brevity, only the results for the best shortmemory model, the best homoskedastic model and its GARCH(1,1) counterpart, and the best overall modeland its homoskedastic counterpart (if the best overall model is not a homoskedastic one). Table 4 presentssummary statistics and some speci cation test statistics for the residuals of these models.
4.2.1

Estimation results for the ARFIMA model with the constant error variance speci cation

Our estimation results for the ARFIMA model with the constant error variance speci cation and various
order restrictions are similar to those previously reported in the literature for the RVs of the Nikkei 225 cash
index and index futures and other nancial time series. The order selected by the BIC is the ARFIMA(0,d,1)
for all series except ln RVt : For ln RVt , the ARFIMA(2,d,0) is selected. So the effect of ltering out thejump component is not so large as to in uence the ARFIMA order selection for our sample except when
log transformation is applied.

The difference in the BIC values between the ARFIMA(2,d,0) and the

ARFIMA(0,d,1) is rather small (Estimation details of the latter are not reported). Note that for all series the
BIC selects an order (p,d,q) with either p > 0 or q > 0; but not both, and d not restricted to be zero.
The estimated values of d in the BIC-selected homoskedastic ARFIMA models are 0.496 (RVt ), 0.519p(ln RVt ), 0.539 (ln RVt ), 0.494 (Ct ), 0.516
Ct , 0.509 (ln Ct ). At usual levels, all of these are signi cantly higher than zero, but none are signi cantly different from the nonstationarity boundary of 1/2. Hence
it is hard to conclude from db whether these series are nonstationary.

On Table 4, the rst-order sample serial correlations (the Bartlett's standard errors T

1=2 = 0:019; andthe heteroskedasticity-adjusted standard errors given in parentheses) and the LB portmanteau statistics (therst lines) for testing the null of no serial correlations of orders up to 5, 10, and 22 (roughly corresponding
11

Suppose, for example, that our goal is to minimize the mean squared errors in forecasting RVt one-step-ahead.

Since

exp (Et 1 [ln RVt ]) 6= Et 1 [RVt ] by Jensen's inequality, we need to speci y more than the rst two condtional moments of
ln RVt to produce an optimal forecast for RVt if we are to work with fln RVt g instead of directly with fRVt g :16to a week, two weeks, and a month) are presented on Table 4.

The sample autocorrelations are smaller

than 0.01 in absolute magnitude and are hence insigni cant at usual levels even using T

1=2= 0:019,

which is much smaller than the heteroskedasticity-adjusted versions. The values of the LB statistics forthe residuals of the models of the raw series RVt and Ct are apparently large enough, whether or not thedegrees of freedom is reduced by the number of estimated parameters in obtaining the

2 critical values, toindicate that serial correlations are not adequately ltered out by the selected ARFIMA models although forthe log and square-root transformed series they are not signi cant. However, in the presence of conditional
heteroskedasticity, the Bartlett's standard error is not a consistent estimator of the standard deviations of thesample serial correlations, and consequently the LB statistic is not asymptotically distributed

2 (k) under

the null of no serial correlations of orders up to k. In particular, if the squared series is positively autocorrelated, the Bartlett's standard error overestimates the estimation precision of sample autocorrelations andas a result the LB statistics overreject the null. Note that under the null of conditional homoskedasticity,
the squared ARFIMA errors are serially uncorrelated, but that under correct speci cation of the conditional mean only, the squared residuals may be serially correlated while b; b1 ;; bp ; b 1 ;; b q maystill consistently estimate the true values. Therefore, we also report Diebold's (1988) heteroskedasticityadjusted LB statistics (on the second lines), according to which there is no strong evidence of remaining
serial correlations in the residuals of any of the best homoskedastic models.

The reductions in the LB

statistics due to the adjustment are particularly substantial for the residuals of the models of the raw series
RVt and Ct .
Although the estimates of the long-memory parameter d; when not restricted to be zero, are signi cantly
greater than zero and the BIC favors the long-memory models, the heteroskedasticity-adjusted LB statistics
for the residuals of the BIC-selected short-memory models are only marginally signi cant for the raw series
RVt and Ct and insigni cant for the four transformed series. For each series, the best short-memory ARMA
model achieves this by mobilizing more AR and MA terms than the best homoskedastic long-memory
model, but yet are not quite as successful12 .
The very large values of sample skewness and kurtosis of the residuals from the raw series RVt andCt shown on Table 4 suggest that the unconditional distributions of the error term are highly nonnormally
distributed. The square-root transformation of RVt and Ct vastly reduces the values of these measures of
nonnormality, but still far above those of a normal distribution. The log transformation bring the residual
distributions closer to normality, but the sample kurtosis values are still nearly 4.
12

To save space, only the results for the BIC-selected models among the short-memory models with or without the GARCH

speci cation are presented. The same is true, however, when the best models are selected from among the homoskedastic models.

17

In sum, it appears that parsimonious ARFIMA models are reasonably successful in removing serial
correlations although some evidence of remaining serial correlations is found for the residuals of the modelsof the raw series RVt and Ct .
The discrepancies between the values of the unadjusted LB statistics and those of the heteroskedasticityadjusted ones already indicate the presence of conditional heteroskedasticity in the form of serial correlationsin "2t : Next we directly test the null of no serial correlations in "2t : McLeod and Li (1983) showed that,
under the null of homoskedasticity, the asymptotic distribution of the sample autocorrelations in the squared
residuals b"2t is multivariate standard normal, not affected by the replacement of the unobservable errors"t of an ARMA model by the residuals b"t . Hence, parameter estimation does not impact the asymptotic
null distribution of the LB statistic, called the McLeod-Li statistic in this context, which issample serial correlations of k different orders in the squared residuals.

2 (k) using

The McLeod-Li statistics forthe squared residuals from the selected models are all very large, cleanly rejecting the null of no serial
correlations of orders up to 5, 10, and 22 at usual levels. With this evidence, we turn to modeling conditional
heteroskedasticity with the ARFIMA model augmented by the GARCH(1,1) speci cation for the error term.
4.2.2

Estimation results for the ARFIMA-GARCH(1,1) model

We next compare all 64 versions of the ARFIMA(2,d,2)-GARCH(1,1) model including those without theconstant error variance restriction= 0. For each of the six series, the BIC selects a model withp
the GARCH(1,1) speci cation, more speci cally, the ARFIMA(2,d,0)-GARCH(1,1) for RVt ; RVt ; thep
ARFIMA(0,d,1)-GARCH(1,1) for Ct ; ln Ct ; and the ARFIMA(f2g,d,1)-GARCH(1,1) for ln RVt ; Ct :=The estimation results for the selected model are shown at the bottom of the section of Table 3 for eachseries, together with the results for the homoskedastic version of it. As expected from the LB statistics forthe squared residuals from the conditionally homoskedastic models, the improvement in the log likelihood
value achieved by giving the GARCH(1,1) speci cation for the error process is substantial for each series.
Also as expected, the addition of the GARCH(1,1) speci cation to the homoskedastic model of the same
ARFIMA order does not alter the ARFIMA parameter estimates very much in most cases, and consequentlyp
the distributions of the residuals remain similarly nonnormal (Table 4). In spite of this, for RVt ; RVt ;ln RVt ; Ct , the ARFIMA order changes when the GARCH(1,1) speci cation is allowed, which is not surprising since the differences in the BIC values amongst the highest BIC-ranking homoskedastic versions are
small.
As for the estimation of the GARCH equation, we rst discuss the results for the raw series RVt ; Ct :The point estimates of the volatility-of-the-RV persistence measure18+

are in excess of the covariancestationarity threshold of one although not signi cantly so. The Bollerslev-Wooldridge standard errors of
the GARCH parameters are rather large, in fact several times larger than the Hessian-based ones and renders
even b = 0:263 insigni cantly different from zero. The values of the sample skewness and kurtosis and the

bt := "t b =h 1=2 b (shown on Table 4) indicate
JB normality test statistic of the standardized residuals Ztvery high degrees of conditional nonnormality of the error terms of the models for the raw series RVt and
Ct 13 . This at least partially explains the large discrepancies between the Bollerslev-Wooldridge standard

errors and the Hessian-based standard errors and suggest that substantial ef ciency gains may be obtainedby adopting a non-Gaussian QMLE, but non-Gaussian QMLE requires additional conditions for ensuringconsistency under distributional misspeci cation; See Newey and Steigerwald (1997). As a check of the
GARCH(1,1) speci cation, we calculate the LB statistics for no correlations in the squared standardized
residuals (Table 4). Unlike in the case of the McLeod-Li test statistics for no serial correlations in thesquared raw residuals, parameter estimation affects the asymptotic distribution in this case. Bollerslev andMikkelson (1996) suggest a heuristic adjustment of reducing the degrees of freedom of the

2 distribution by

the number of estimated parameters. The values of the LB statistics are greatly reduced by standardization
although they are still high enough to reject the null whether the degrees-of-freedom adjustment is appliedor not (when the number of lag orders is 5, this adjustment obviously cannot be applied)14 except for thestatistic for lag orders up to 22 and Yt = RVt .
As for the four transformed series, both
estimates of+andare estimated to be signi cantly above zero, but the

do not exceed one, although for the log transformed series, it is very close to one.

For the log transformed series,

estimates are rather small, which together with+estimated to benearly one implies a slowly varying conditional mean process as shown on Figures 2a and 2b. Again, thevalues of the LB statistics for the squared standardized residuals are substantially reduced from the versions
calculated for the raw squared residuals, but are still large enough to reject the null of no serial correlations.
pp
For RVt and Ct ; the degrees of nonnormality in the residuals are hugely reduced but not to near normal
levels. For ln RVt and ln Ct ; the raw residuals are not highly nonnormal to begin with, and the effects ofstandardization are much smaller than in the case of the raw or square-root transformed series. Consistent
13

Kulperger and Yu (2005) proved that the asymptotic distribution of the Jarque-Bera-type moment-based distributional test

bt rather than Zt is
statistic based on Z2(2) ; unaffected by parameter estimation if the conditional variance is correctly speci edand fZt g is an i.i.d. sequence. For testing normality, these conditions imply that the Gaussian QMLE is the MLE. One of theiradditional assumptions is that the conditional mean of the observed variable Yt is zero, which is not satis ed by our ARFIMA-GARCH case.
14

Li and Mak (1994) proposed a more elaborate statistic that correctly accounts for this effect.

19

with near normality, the discrepancies between the Bollerlsev-Wooldridge standard errors and the Hessianbased ones are not very large for these log-transformed variables. However, the values of the JB statistic
still reject the null of normality easily.
All in all, the addition of the GARCH(1,1) speci cation for the ARFIMA error term helps to capture much of the serial correlations in the squared residuals though not completely.

We may augment

the ARFIMA model by other GARCH-type speci cations that have proven successful in improving the
GARCH(1,1) t in more traditional volatility (rather than the volatility-of-the-RV) prediction contexts, butsuch attempts are beyond the scope of the current paper.

4.3

Further speci cation tests

The LB statistics are for testing the null of no remaining serial correlations in the residuals and the squared
residuals, and are not designed to detect more general forms of serial dependence. Hence, we run a batteryof other tests capable of detecting nonlinear dependence.

In this paper, we only attempt to model theconditional mean by the homoskedastic ARFIMA model and the rst two conditional moments by the
ARFIMA-GARCH model. By estimating them using the CSS, which is interpreted as the Gaussian QMLE,
we do not take a stand on neither the higher-order dependence structure nor the shape of the conditional
distribution (beyond zero mean and unit variance) of fZt g : Nevertheless, it is of interest for a variety ofreasons to test the independence and (standard) normality of fZt g jointly or one at a time assuming the other.
n obt but not fZt g ; and the in uence of parameter estimation may or may
The dif culty is that we observe Z

not vanish in the distributions of the test statistics even asymptotically as we have discussed above in thecontexts of the LB and JB portmanteau statistics. Unfortunately, for all of the diagnostic statistics that wep
employ, asymptotic theory assumes some form of mixing for fYt g and/or T -consistency of the parameter
estimator for establishing invariance in the presence of parameter estimation (the "nuisance-parameter-free"property) or justifying the adjustment when invariance does not hold whereas an ARFIMA process withpd > 0 is not mixing and b cannot be expected to be T -consistent. Hence, strictly speaking, the diagnostic

tests in this subsection as well as the more traditional ones used in the previous subsections remain somewhat
informal.

The values of the statistics are summarized on Table 5. The rst columns show the values of the BDS
nonlinearity test statistics due to Brock et al. (1996), which test the null of fZt g
to detect higher-order dependence.

i:i:d: and has power

All different pairs of two consecutive segments of a xed length k

are taken from an observed time series and the number of cases in which the distance (using a particularmeasure) between the two segments in a pair is shorter than a preset value

20

is counted. After standard-

ization by its consistent estimator and the asymptotic standard deviation, this number becomes the BDS
statistic with (k; ), which is asymptotically distributed N (0; 1) under the null of fZt g

i:i:d. De Limap

(1996) showed that under some conditions (including a mixing condition on fYt g and T -consistency ofnobt2 , where Z
bt is the standardized residual of an ARCH model, isthe estimator) the BDS applied to ln Z
nuisance-parameter free (Note that if fZt g

i:i:d:; then ln Zt2

i:i:d holds as well). Caporal et al.

(2005) investigated by Monte Carlo simulations the nite-sample size properties of the BSD test statistics
under GARCH(1,1) DGPs with various combinations of true parameter values (estimated by the Gaussian
QMLE) and distributions of Zt not necessarily satisfying De Lima's (1996) suf ciency conditions and reported that they are well-behaved for T

1000. We apply the BDS test to the log standardized residuals,

and following Chen and Kuan (2005), we set equal to 0:75 times the sample standard deviation of thenob2 ); and calculate the sample standard deviation based on theseries under investigation (in our case ln Ztbt and use it for normalization. The results
1000 bootstrap resamples from the empirical distribution of Z

for k = 2;

; 5 are presented (on the rst lines). For RVt and Ct , the null of fZt g

i:i:d: is strongly

rejected using the log squared residuals from the best homoskedastic model, but not rejected using thosepfrom the best overall model, which has the GARCH(1,1) speci cation, except when k = 5: For RVt andp
Ct , the null is strongly rejected for the best homoskedastic model except when k = 5 (but the rejection is
not as strong as in the case of the log transformed variables), and not rejected for any k = 2;

; 5: Finallyfor ln RVt and ln Ct , rejection never occurs, indicating lack of power in light of serial correlations in then obt ;squared residuals detected by the LB statistics. We also present the BDS statistics calculated with Z
which violates one of De Lima's (1996) suf cient conditions for invariance, (on the second lines) because
the impact of parameter estimation on the variance of the statistic is accounted for by the bootstrap although
we still presume that the asymptotic normality of the BSD statistics extend to this case, and use the standard
normal critical values for inference. The pattern is similar to the case of the log squared residuals, but thevalues of the BDS statistic are much larger, rejecting the null in more cases.
We next turn to a test of time reversibility due to Chen et al. (2000) (hereafter the CCK statistic).
fZt g

i:i:d: implies time reversibility, which in turn implies that the unconditional distribution of Zt Zt k

is symmetric around the origin. The CCK statistic tests this symmetry and is calculated as k =bk ; wherePk) 1=2 Tt=k+1 (Zk ) and bk is a consistent estimator for the standard deviation of k : Therek := (Tare a variety of functions that can be chosen as ( ). Following Chen and Kuan (2005), we use (x) :=n obt , and calculate bk by bootstrap similarly to
x= 1 + 2 x2 ; replace the true errors with the residuals Z
the case of the BDS statistics15 , and present the results for k = 1; 2; 3 and15= 0:5; 1: A general tendency

Chen and Kuan (2005) showed that, unlike in the case of the BDS statistic, the impact of parameter estimation on the CCK

21

again is that the values of the CCK statistics decrease when square-root transformation is applied to Yt andturn insigni cantly or marginally negative when log-transformation is applied. The CCK statistics appear
to be less sensitive to the addition of the GARCH speci cation to the model than the BDS statistics are,particularly for ln RVt and ln Ct :For testing the correctness of the speci cation of the joint distribution of fYt g in its entirety, we may
use the nuisance-parameter-free Hong-Li statistics. In our Gaussian ARFIMA-GARCH case, it is equivalent to jointly testing the independence and (standard) normality of fZt g (as opposed to the JB statistic thattests the normality under the maintained hypothesis of independence). The Hong-Li statistics are basedon the observation that, under the null of correct model speci cation in its entirety (as opposed to just the
rst two conditional moments), the probability integral transformed series fUt g implied by the model is asequence of i:i:d: uniform [0; 1] random variables, and in particular the joint density of (Ut ; Ut k ) should
be f (u1 ; u2 ) = 1 over [0; 1][0; 1]. Hong and Li (2005) showed that, under the null, a properly normal-

ized measure of the distance16 (call it the Hong-Li statistic of order k; QHL (k)) between f (u1 ; u2 ) = 1n obt ; is asymptotically distriband fb(u1 ; u2 ), a nonparametric estimate of f (u1 ; u2 ) constructed using U
uted N (0; 1). Furthermore, they showed that the asymptotic distribution of (QHL (1) ;; QHL (K)) isp PK
standard multivariate normal and hence that of a portmanteau statistic WHL (K) := K k=1 QHL (K) is

N (0; 1) under the null. Noting that negative values of

HL (k) occur only under the null if the sample size issuf ciently large, they suggest using the upper-tailed critical values for individual

HL (k)'s and QHL (K).

For this reason, the portmanteau statistic is a scaled sum rather than a scaled sum of squares that would yielda2 (K) statistic.

For the choice of the kernel function and the bandwidth parameter involved in nonpara-

metric density estimation, we follow Hong and Li (2005). QHL (1) ; WHL (5) ; WHL (10), WHL (22) are
shown on Table 4. The overall pattern across variables and models is similar to the case of the JB statistics
for the standardized residuals, and as expected, the null of i:i:d. normality is very strongly rejected for RVt ;pp
Ct ; RVt ; and Ct . However, it is of note that, while all Hong-Li statistics still cleanly reject the null forthe best model for ln RVt , the statistics values are much reduced for ln Ct and QHL (1) only marginally rejects the null for ln Ct at the 5% level (Note that the upper-tailed 5% critical value is 1.645). This seemingly
contradicts the strong rejection of the normality of Zt for ln Ct by the JB statistic. However, recall that thestatistic is of the same stochastic order as the CCK statistic calculated from the true fZt g provided that the parameter estimatorpis T -consistent. Hence, correcting the asymptotic variance based on the assumption of no parameter estimation errors is crucial here, and boostrapped bk serves the purpose. It might turn out to be the case, however, that using the ARFIMA-GARCH
standardized residuals, the effect of the error in estimating
16

asymptotically dominates the other terms.

Their expression of the statistic has a typo, which is corrected in Egorov et al (2006, Footnote 11).

22

JB statistic tests the normality under the maintained hypothesis of fZt g

i:i:d: whereas the Hong-Li sta-tistics in our context test the independence and standard normality jointly. If the i:i:d: assumption does nothold as indicated by the LB statistics for the squared standardized residuals, the JB statistic may overrejectnormality.
Hong and Li (2005) proposed another set of nuisance-parameter-free statistics, which they call the “separate inference” statistics, designed to detect possible sources of misspeci cation when QHL (k) or WHL (K)reject the null of correct speci cation.

To obtain the Hong-Li separate inference statistic M (m; l) for

given m and l; we calculate cross-correlations of all orders j

btm and
1 (up to a truncation point) in U

b j and take a weighted average of the squares of them, which after normalization is asymptotically disUt jtributed N (0; 1) under the null of correct speci cation of the entire joint density of fYt g. M (m; l) for(m; l) = (1; 1) ; (2; 2) ; (3; 3) ; (4; 4) ; (1; 2) ; and (2; 1) are shown on Table 5. M (m; m) for m = 1; 2; 3; 4
are meant to detect autocorrelations in level, volatility, skewness, and kurtosis respectively, and M (1; 2) andM (2; 1) ARCH-in-mean and leverage effects. Note that, in our context, leverage means that the volatilityof the RV responds asymmetrically to positive and negative shocks in the RV, and hence is different from thephenomenon of asymmetric reactions of equity market volatility to positive and negative returns17 . These
M (m; l) statistics are revealing. Substantial reductions in the values of M (2; 2) in all cases are as expected. But it is rather surprising, for example, to nd the very high value of M (4; 4), 28.74, for the best
bt relative to mere 0.74 for the best model of Ct with highly positively
model of ln Ct with nearly normal Z

bt : Failure of the best model for ln Ct at these M (m; l) tests should be kept in mind
skewed and leptokurtic Z
in predicting the entire conditional density for risk management or choosing the variable to directly target
given a loss function.

4.4

RV prediction

For one-step prediction of the RV measures, we use
Ybt+1jt := b +

t
X
s=1

bs (Yt+1 s

b)

(17)

where b is the CSS estimate of the unconditional mean and bs are the coef cients in the AR(1) expansion of
the ARFIMA model, implied by the ARFIMA parameter estimates. While the Durbin-Levinson algorithm
may be applied to calculate the best linear one-step predictor based on the nite past, the formula (17) is
17

As mentioned in the introduction, some studies incorporated a speci cation for the latter effect in time series models for the

RV by using daily returns data.

23

more in line with our CSS estimator. We also evaluate the performance of the selected model for each ofPP
the RV series in forecasting the k-day average RV, k 1 ks=1 Yt+s . For this, we use k 1 ks=1 Ybt+sjt as
our k-day forecast whereYbt+kjt := b +

k 1
X
s=1

bs Ybt+sjt

b +

t 1
X
s=0

bk+s (Yt s

b)

(18)

For evaluating predictive accuracy, we mainly look at R2 from the Mincer-Zarnowiz regression of the realization of the target variable on the prediction (plus an intercept).
p
The results for k = 1; 5; 10; 22 are summarized on Table 5. Again, the results for RVt ; RVt ; ln RVt
pand their respective jump-free versions Ct ; Ct ; ln Ct are similar. In conformance with the previously
reported results for the RVs of the Nikkei 225 index and other nancial time series, the log transformedppseries ln RVt and ln Ct appear to be most predictable, followed by RVt and Ct . For example, R2 isnearly 60% when the target is the one-step-ahead ln RVt or ln Ct : Although R2 tapers off as the horizon
increases, predictability of the average daily RV over the next month is still substantial (nearly 30% forln RVt or ln Ct ). Excellent performance of the ARFIMA-GARCH model can be visually con rmed by thetime series plot (Figure 1) of the ARFIMA-GARCH t together with the target series.

4.5

Prediction of the variance of the RV

We next evaluate the performance of the ARFIMA-GARCH model selected for each of the six series in
predicting the volatility of the RV. For one-step-ahead forecasting, we use b
ht+1 as the volatility-of-the-RV
forecast and b"2t+1 =

Yt+1

Ybt+1

2

as the target, whether the object of our interest is V art (Yt+1 ) =Et "2t+1 or some other measure of the volatility of the RV.

Our ARFIMA-GARCH estimation results

indicate that the variability of b
"2t+1 is much higher than that of Et "2t+1 , which would lead to an apparently

low R2 value of the Mincer-Zarnowitz regression even if the time-variation in Et "2t+1 could be wellapproximated by the ARFIMA-GARCH model.

This parallels the solution by Andersen and Bollerslev(1998) of a puzzle regarding the low R2 phenomenon prevalent in earlier volatility prediction studies basedon data sampled at daily or lower frequencies. The introduction of the RV measures in empirical nance has
rendered nancial volatility nearly observable, but here we are back in the unobservability territory. Whatmakes the problem even more complicated in our context is that the RV is highly predictable unlike daily
asset returns and that V ar (Et [Yt+1 ]) =V ar (Yt+1 ) seems to be large and hence the effect of the deviation of
Ybt+1 from the true Et [Yt+1 ] due to possible model misspeci cation, parameter estimation errors and otherfactors on the Mincer-Zarnowitz R2 is potentially more serious for volatility-of-volatility prediction than forequity or currency return volatility prediction.
24

For forecasting the multi-day volatility of the RV, we use k 1

Pk

b

b

s=1 ht+sjt ; where ht+sjt is the s-period-

ahead single-day conditional variance of the RV implied by the ARFIMA-GARCH, as our forecast, and thedaily squared errors averaged over the horizon,t (k) := k

1

k
X
s=1

as a proxy for the target.

Yt+s

Ybt+sjt

2(19)

The right half of Table 5 summarizes the forecasting performance evaluation results for horizons 1, 5, 10,
and 22 days. As expected, R2 's are in fact low, but not negligible. For brevity, we do not report the details of
the results of our multi-day-ahead single-day volatility-of-the-RV forecasting exercise, but the predictability
does not increase as we attempt to forecast further into the future. In spite of this, R2 values are higher for
multi-day average forecasts than for a one-day ahead forecast. This may happen because our target proxyt (k) ; being squared forecast errors aggregated over forecast horizons of a week to several weeks, is a sortof realized volatility (of the RV). Although the sampling frequency is much coarser here, a noise reduction
effect similar to those observed in the daily volatility forecasting studies using RV measures constructedfrom intraday high frequency asset returns may lead to a higher R2 in Mincer-Zarnowitz regressions withpplonger-horizon targets and predictors. For RVt and Ct , the R2 values are respectively 3.63% and3.55% (one-day-ahead, single-day), 8.23% and 7.60% (5 days), 6.87% and 6.50% (10 days), 5.44% and5.27% (22 days). Presumably, the effect of decreasing predictability outweighs the noise reduction as thehorizon increases beyond a week or so. For ln RVt and ln Ct ; the R2 values are respectively 1.70% and3.31% (one-day-ahead, single-day), 3.68% and 7.29% (5 days), 4.65% and 9.30% (10 days), 6.15% and12.50% (22 days). It is not clear why the squared errors appear to be more predictable for ln Ct than for
ln RVt in spite of the similar behavior of Ct and RVt observed in other respects.

5Concluding remarks

In this paper, we investigated the volatility of the daily Nikkei 225 realized volatility. Although much of therecent advances in volatility research has been due to the recognition that high-frequency intraday data makedaily volatility essentially observable in the form of the realized volatility and related measures, we are backto the condition of unobservability when we move one order higher in terms of the moments from volatility
to volatility of volatility. This makes evaluation of models such as the ARFIMA-GARCH for forecasting
the volatility of the RV a dif cult task. Nevertheless, the GARCH speci cation for the conditional varianceof the RV added to the ARFIMA speci cation for the conditional mean of the RV seems to capture the25persistent time variation in the conditional variance of the Nikkei 225 RV regardless of whether we use theraw, square-root- or log-transformed series. Filtering out the jump component from the RV using a versionof the Barndorff-Nielsen procedure does not seem to have much impact on the estimation of the ARFIMAGARCH model or the degree of predictability in the RV. However, the volatility of the RV seems to be morepredictable for the jump-free RV than the usual RV at the forecast horizon of about a month.

Acknowledgments
The authors are grateful to Yoshihiro Yajima and Yuji Sakurai for helpful discussions.

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32

Table 1: Summary statistics: Nikkei 225 daily returns, realized volatility and related measures

R
Rn-0.007
0.039

Std.
Dev.
1.423
0.631

R am

-0.0250.802-0.2427.700-7.3945.078

2,606.19-0.033 ( 0.022 )Rl-0.026

0.164

0.015

4.905

-0.7360.786

423.620.026 ( 0.024 )

R pm

0.005

0.774

0.103

5.818

-3.6754.881

932.38-0.099 ( 0.025 )53.10733.12160.19437.744

RV am

0.573

0.622

5.970

78.3500.011 12.748

679,502.99

0.429 ( 0.060 )

2,253.238

223.283

3,635.044

442.419

1/2

0.691

0.309

1.587

9.207

0.103

3.570

5,674.29

0.569 ( 0.083 )

3,862.612

214.711

6,603.898

406.563

11,267.971

ln RV am

-0.9250.871-0.177

3.272-4.551

2.545

23.19

0.616 ( 0.043 )4,532.985

949.938

7,921.581

1,761.857

RV pm

Series

RV am

Mean

ρ1

S.E.(ρ 1)

Skew.

Kurt.

Min.

Max.

Jarque-Bera-0.064
0.009

4.962
3.480

-7.234-3.186

7.660
3.849

451.1326.89-0.041 ( 0.022 )0.036 ( 0.020 )LB (5)DLB (5)

LB (10)13.251
10.26523.80514.545

9.678

6.342

16.08310.71219.207
12.112DLB (10)

LB (22)DLB (22)Long-memory d
GPH Robinson

16.59812.33437.58833.34526.73229.016-0.0180.054-0.011
0.044

11.876

7.938

23.16715.856-0.048-0.03122.00715.01757.63540.153

0.071

0.113

80.96053.633-0.107-0.0605,801.812

809.331

0.460

0.454

803.268

0.532

0.495

14,268.434 3,406.661

0.525

0.491
0.376

0.514

0.595

12.096335.0240.015 18.941 12,938,877.08

0.355 ( 0.059 )1,298.273

168.714

1,981.101

390.580

2,873.448

822.065

0.364

1/2

0.655

0.292

1.773

13.810

0.122

4.352

15,111.33

0.535 ( 0.081 )

3,128.317

196.9115,208.527

373.5108,450.013

734.922

0.504

0.472

ln RV pm

-1.0280.866-0.182

3.260-4.202

2.941

23.35

0.600 ( 0.047 )

4,070.792

710.029

7,035.733

1,290.913

12,361.541 2,405.026

0.524

0.477

RV = RV am + RV pm

0.468

RV pm

1.087

1.030

5.336

67.8400.065 20.493

1/2

0.968

0.388

1.459

8.520

0.255

ln RV
2
Rn

-0.214
0.399

0.778
0.628

-0.1308.937

3.234
168.634

RV|R n |
2ln (R n )2

Rl2

|R l |
2ln (R l )2

2

504,141.92

0.540 ( 0.071 )3,237.586

250.406

4,970.638

529.470

7,586.328 1,004.412

0.470

4.527

4,551.03

0.663 ( 0.097 )4,997.214

211.504

8,342.521

402.196

13,887.091

793.872

0.557

0.524

-2.733 3.020
0.000 14.812

14.25
3,240,305.54

0.713 ( 0.030 )0.062 ( 0.017 )5,927.105
88.8762,427.793
81.982

10,322.146
159.7214,474.032
133.651

18,365.815 8,625.484
241.634 221.775

0.584
0.197

0.533
0.202

0.508

0.376

1.192

7.069

0.000

3.849

2,596.51

0.097 ( 0.028 )192.32073.951330.598

131.412515.651

222.424

0.261

0.250

-2.1492.195-1.5406.627-16.1512.695

2,643.12

0.050 ( 0.020 )58.13644.332104.587

79.669175.383

133.484

0.209

0.190

0.028

0.054

4.178

26.776

0.000

0.618

74,150.20

0.145 ( 0.033 )

241.33193.607396.055

161.320793.223308.027

0.278

0.255

0.121

0.114

1.687

6.476

0.000

0.786

2,739.04

0.175 ( 0.035 )397.477122.080713.096233.969

1,358.050

459.849

0.288

0.302

-5.2792.438-1.1395.284-19.501-0.481

1,213.66

0.096 ( 0.030 )157.867

53.404320.240

106.402605.525

213.052

0.210

0.258

143,704.07

0.461 ( 0.059 )2,475.549

257.636

3,928.348

521.401

6,076.282

980.450

0.451

0.458

2,626.47

0.563 ( 0.090 )3701.681

179.1796248.175

337.833

10273.433

651.265

0.542

0.502

12918.003
12918 003 8,479.105
8 479 105

1.514

1.321

4.040

37.1410.082 21.151

2 1/2

1.149

0.440

1.283

6.990

0.286

4.599

2

lln (RV +R n +R l )

0.139
0 139

0.752
0 752

-0.174
0 174

3.327
3 327

-2.506
2 506

3.052
3 052

26.58
26 58

0.603
0 603 ( 0.026
0 026 )

4291.539
4291 539

2415.713
2415 713

7482.022
7482 022

4,479.544
4 479 544

0.571
0 571

0.505
0 505

RV NW

1.153

1.420

5.879

61.9940.041 21.909

422,464.73

0.467 ( 0.077 )2,409.258

160.784

3,494.761

357.302

4,982.016

710.337

0.427

0.445

1/2

0.968

0.466

1.931

10.572

0.201

4.681

8,436.53

0.563 ( 0.087 )

3,652.271

194.0906,029.797

378.271

9,857.148

759.299

0.490

0.504
0.507

RV +R n +R l
2

(RV +R n +R l )
2

RV NW

ln RV NW

-0.2670.894-0.006

3.301-3.204

3.087

10.57

0.606 ( 0.028 )4,327.317

2,030.833

7,633.914

3,797.857

13,702.610 7,391.286

0.527

RV HL

1.514

1.289

3.637

31.7920.083 20.565

102,960.04

0.503 ( 0.063 )

2,835.818

272.087

4,492.889

553.123

7,071.670 1,057.351

0.459

0.464

RV HL 1/2

1 150
1.150

0 438
0.438

1 184
1.184

6 272
6.272

0 288
0.288

4 535
4.535

1 904 83
1,904.83

0 596 ( 00.092
0.596
092 )

4 050 849
4,050.849

189 043
189.043

6 821 396
6,821.396

355 864355.864

11 373 832
11,373.832

690 552690.552

0 554
0.554

0 505
0.505

ln RV HL

0.141

0.753

-0.186

3.261-2.487

3.024

24.14

0.634 ( 0.026 )4,679.843

2,634.268

8,132.032

4,865.625

14,158.498 9,273.210

0.588

0.509

RV HLNW

0.438

1.580

1.512

3.405

23.2290.051 17.863

53,190.72

0.412 ( 0.059 )

1,833.103

220.9022,876.402

452.062

4,335.164

867.892

0.439

1/2

1.156

0.494

1.233

5.956

0.226

4.226

1,730.24

0.476 ( 0.077 )2,578.344

172.929

4,344.404

326.894

7,017.121

622.922

0.505

0.475

ln RV HLNW

0.114

0.849

-0.2173.232-2.978

2.883

28.29

0.495 ( 0.024 )

2,891.338

1,889.880

5,051.168

3,470.476

8,568.755 6,304.374

0.526

0.469

RV HLNWTable 1: Summary Statistics (Continued)

I am

0.054

Std.
Dev.
0.227

3.936

16.492

0.000

1.000

28,485.68

0.082 ( 0.030 )40.00318.27666.50130.686

171.48377.806

I pm

0.105

0.306

2.585

7.680

0.000

1.000

5,676.54,

0.063 ( 0.024 )66.48037.339113.491

65.899191.848114.649

0.137

0.189

I
J am

0.159
0.018

0.390
0.111

2.325
12.3787.663
254.404

0.000
0.000

2.000
3.126

5,063.29
7,450,609.97

0.111 ( 0.030 )0.024 ( 0.022 )154.40323.63470.2074.842

270.145
24.062124.1205.409

515.36435.036245.01245.902

0.214
0.115

0.231
0.094

J am 1/2

0.029

0.133

5.330

36.177

0.000

1.768

141,774.19

0.049 ( 0.026 )19.2348.781

21.62411.18750.31527.145

0.108

0.102
0.088

Series

Mean

Skew.

Kurt.

Min.

Max.

Jarque-Bera

ρ1

S.E.(ρ 1)

LB (5)DLB (5)

LB (10)DLB (10)

LB (22)DLB (22)Long-memory d
GPH Robinson
0.102
0.148

ln (1+J am )

0.015

0.077

7.480

78.774

0.000

1.417

696,469.20

0.035 ( 0.027 )17.3195.526

17.7775.924

36.05321.733

0.094

J pm

0.027

0.108

6.936

72.084

0.000

1.819

579,666.63-0.009 ( 0.008 )22.952

11.62234.80017.74978.48137.759

0.108

0.137

J pm 1/2

1/2

0.049

0.156

3.512

16.148

0.000

1.349

25,944.59

0.017 ( 0.017 )39.21620.93565.20036.154124.11768.731

0.100

0.164

ln (1+J pm )

0.022

0.083

5.197

37.851

0.000

1.036

154,421.82-0.003 ( 0.011 )28.39013.38444.81121.80497.20545.817

0.125

0.149

0.045

0.154

6.750

82.923

0.000

3.126

767,042.11

0.099 ( 0.087 )82.20215.45095.86622.705150.284

53.468

0.152

0.156

J 1/2

0.076

0.198

2.834

11.464

0.000

1.768

12,114.74

0.063 ( 0.030 )80.41336.245114.470

59.428226.672124.609

0.177

0.184

ln (1+J )
C am

0.037
0 554
0.554

0.112
0.617
0 617

4.232
6.094
6 094

26.9398181.196196

0.000 1.417
00.011
011 1212.748748

75,267.00
731,228.51
731 228 51

0.067 ( 0.046 )0.429
0 429 ( 0.060
0 060 )

71.4252,210.926
2 210 926

21.879215
215.305305

90.85033,526.428
526 428

33.287431
431.947947

172.347
55,639.629
639 62975.446789
789.466466

0.165
00.426426

0.168
00.446446

10,979.110

J

C am 1/2

0.677

0.310

1.581

9.214

0.103

3.570

5,675.75

0.567 ( 0.082 )

3,749.278

216.737

6,386.549

411.409ln Cam-0.9770.900-0.225

3.272-4.551

2.545

32.28

0.609 ( 0.043 )4,322.297

887.450

7,584.786

1,642.283

C pm

0.488

0.592

12.327343.571

0.012 18.941 13,612,657.54

0.349 ( 0.059 )

1,247.790

188.381

1,891.862

415.115

C pm 1/2

0.633

0.295

1.779

13.664

0.110

4.352

14,753.15

0.531 ( 0.078 )3,089.449

208.804

5,137.232

396.071

ln Cpm-1 115-1.115

0 910
0.910

-0 210-0.210

3 215
3.215

-4 413-4.413

2 941
2.941

26 08
26.08

0 596 ( 00.048
0.596
048 )

4 006 274
4,006.274

678 345678.345

6 938 593
6,938.593

1 234 260
1,234.260

C

1.042

1.021

5.426

70.1600.056 20.493

540,350.80

0.537 ( 0.070 )3,177.375

250.504

4,859.086

527.560

C 1/2

0.943

0.391

1.449

8.415

0.237

4.527

4,403.37

0.658 ( 0.094 )4,880.797

218.977

8,129.868

416.252

ln C-0.2780.811-0.161

3.183-2.879

3.020

16.02

0.703 ( 0.031 )5,698.077

2,265.783

9,909.682

4,165.279

818.777

0.535

0.485

13,818.262 3,195.502

0.540

0.482

2,714.858

825.181

0.368

0.373

8,194.508

766.741

0.519

0.480

12 007 228 2,271.501
12,007.228
2 271 501

0 515
0.515

0 486
0.486

7,371.936

995.301

0.477

0.464

13,491.335

821.386

0.549

0.518

17,598.452 7,995.639

0.577

0.523

The sample period is from March 11, 1996, through August 31, 2007 (2802 observations). ρ 1 and S.E.(ρ1) in parentheses are respectively the first-order sample autocorrelations and heteroskedasticity-robust
standard errors (the usual Bartlett's standard errors are T-1/2 = 0.018). LB(k ) are the Ljeung-Box statistics of orders up to k and DLB(k ) are their heteroskedasticity adjusted versions. The 5% critical values for χ2(k)are 5.991 (k=2), 11.070 (5), 18.307 (10), 33.924 (22). The standard errors of the GPH and Robinson estimators for the long-memory parameter d are 0.034 and 0.031 respectively.

Table 2: ARFIMA-GARCH estimation results
Var.

Model

μ

d

Best short-memory(2,0,{2})-G

0.7415(0.0915)(0.1667)
Best homoskedastic
0.8843(0,d ,1)(0.2118)(0.0936)GARCH ver. of
0.5751best homosked.
(0.2495)
RV(0,d ,1)-G(0.1917)
Homoskedastic ver.
0.8765of best overall model (0.3450)(2,d ,0)(0.2259)
Best overall
0.6332(2,d ,0)-G(0.5415)(0.6374)
Best short-memory
0.8838(1,0,2)-G(0.0465)(0.0471)
Best homoskedastic
0.8818(0,d ,1)(0.1215)(0.1011)GARCH ver. of
0.7579
1/2
best homosked.
(0.1019)RV(0,d ,1)-G(0.0922)
Homoskedastic ver.
0.8790of best overall model (0.1191)(2,d ,0)(0.0922)Best overall
0.7721(2,d ,0)-G(0.1210)(0.1150)
Best short-memory-0.1962
memory(0.0957)(2,0,1)-G(0.0859)
Best homoskedastic-0.3558(0.2239)(2,d ,0)(0.2146)GARCH ver. of-0.3739(0.1529)lnRV best homosked.
(2,d ,0)-G(0.0953)
Homoskedastic ver.
-0.3569of best overall model (0.1610)({2},d ,1)(0.1080)
Best overall-0.3740({2},d ,1)-G(0.2582)(0.2639)

ARFIMA parameters
φ1
φ2
0.3592(0.0252)(0.0375)

0.4963(0.0320)(0.0865)
0.5604(0.0478)(0.0859)
0.5085(0.0282)(0.0822)0.6218(0.0384)(0.0950)

0.5193(0.0333)(0.0607)0.4979(0.0304)(0.0323)
0.5427(0.0260)(0.0419)0.5346(0.0302)(0.0335)

0.5388(0.0257)(0.0282)
0.5316(0.0257)(0.0263)0.5553(0.0314)(0.0348)0.5470(0.0324)(0.0351)ψ1

0.5887(0.0282)(0.0506)

ψ2-0.4579(0.0290)(0.0438)-0.2262(0.0389)(0.1068)-0.2105(0.0605)(0.0998)-0.2345(0.0330)(0.0928)-0.2554(0.0430)(0.0901)
0.9741(0.0055)(0.0063)-0.2061(0.0306)(0.0460)-0.1831(0.0355)(0.0382)
1.1414(0.0323)(0.0375)-0.1826(0.0309)(0.0349)-0.1688(0.0308)(0.0317)-0.0866(0.0254)(0.0684)-0.1725(0.0330)(0.0774)-0.6178(0.0223)(0.0231)-0.1898(0.0436)(0.0819)-0.1545(0.0388)(0.0414)-0.0911(0.0237)(0.0330)-0.0988(0.0271)(0.0288)-0.1597(0.0299)(0.0345)-0.0913(0.0239)(0.0249)-0.0849(0.0239)(0.0233)-0.0638(0.0210)(0.0214)-0.0623(0.0217)(0.0214)-0.7730(0.0239)(0.0286)-0.1993(0.0368)(0.0417)-0.1846(0.0380)(0.0414)-0.0949(0.0232)(0.0263)

GARCH paramters
σ 2, ω
βα

LL
BIC

0.0136(0.0024)(0.0127)
0.6391(0.0171)(0.1352)0.0106(0.0015)(0.0068)
0.6381(0.0170)(0.1344)0.0115(0.0017)(0.0085)
0.0033(0.0007)(0.0016)0.0700(0.0019)(0.0049)
0.0031(0.0007)(0.0015)0.0698(0.0019)(0.0049)
0.0031(0.0007)(0.0015)0.0006(0.0005)(0.0007)0.2434(0.0065)(0.0078)0.0006(0.0005)(0.0007)0.2434(0.0065)(0.0078)0.0006(0.0005)(0.0007)-2,410.93

0.7873(0.0260)(0.1291)

0.2630(0.0397)(0.1893)

4,877.42-3,348.67

6,729.09
0.8238(0.0141)(0.0577)

0.2119(0.0219)(0.0819)-2,416.28

4,880.19-3,346.53

6,732.75
0.8079(0.0184)(0.0879)
0.8561(0.0209)(0.0436)0.2340(0.0283)(0.1299)
0.0990(0.0142)(0.0263)-2,406.79

4,869.15-17.3090.17-250.25532.260.8590(0.0200)(0.0435)

0.0988(0.0140)(0.0270)-14.1875.99-246.75

533.180.8590(0.0202)(0.0435)
0.9874(0.0045)(0.0056)

0.0991(0.0143)(0.0277)0.0104(0.0032)(0.0039)

-9.68

74.93
-1,970.04

3,995.65-1,996.23

4,032.14
0.9870(0.0048)(0.0061)0.0106(0.0034)(0.0042)-1,963.06

3,981.68-1,996.23

4,032.14
0.9871(0.0048)(0.0060)

0.0105(0.0034)(0.0041)-1,962.95

3,981.46

Table 2 (Cont.): ARFIMA-GARCH estimation results
Var.

Model

μ

Best short-memory
memory(2,0,{2})-G
Best homoskedastic(0,d ,1)C

C 1/2

0.7155(0.0932)(0.1343)0.7893(0.3326)(0.2169)
GARCH ver. of
0.4635best homosked.
(0.2665)(0,d ,1)-G(0.2163)
Homoskedastic ver.
0.7704of best overall model (0.7140)({2},d ,1)(0.8760)
Best overall
0.5185({2},d ,1)-G(0.3330)(0.2393)
Best short-memory
0.6985
memory(0.0872)(2,0,2)-G(0.0661)
Best homoskedastic
0.8151(0,d ,1)(0.1152)(0.0811)GARCH ver. of
0.7145best homosked.
(0.1177)(0,d ,1)-G(0.1107)
Homoskedastic ver.
0.8151of best overall model (0.1152)(0,d ,1)(0.0811)
Best overall
0.7145(0,d ,1)-G(0.1177)(0.1107)
Best short-memory-0.2637
memory(0.1042)(2,0,1)-G(0.0945)

Best homoskedastic(0,d ,1)lnC

d

ARFIMA parameters
φ1
φ20.3404(0.0245)(0.0361)0.4941(0.0322)(0.0869)
0.5661(0.0500)(0.1022)0.5163(0.0342)(0.0978)
0.6251(0.0467)(0.0843)1.8988(0.0175)(0.0188)

0.6116(0.0267)(0.0425)-0.0321(0.0211)(0.0504)-0.0966(0.0267)(0.0450)-0.8990(0.0174)(0.0188)0.5160(0.0305)(0.0515)0.4990(0.0297)(0.0306)
0.5160(0.0305)(0.0515)0.4990(0.0297)(0.0306)
1.1508(0.0322)(0.0368)ψ1-0.1685(0.0298)(0.0337)

ψ2-0.4749(0.0296)(0.0396)-0.2216(0.0394)(0.1086)-0.2375(0.0626)(0.1301)-0.2371(0.0383)(0.1010)-0.2753(0.0527)(0.0879)-1.5908(0.0298)(0.0354)-0.1847(0.0393)(0.0681)-0.1633(0.0373)(0.0397)-0.1847(0.0393)(0.0681)-0.1633(0.0373)(0.0397)-0.7792(0.0240)(0.0289)

0.5965(0.0287)(0.0342)GARCH paramters
2
σ ,ω
β
α

LL
BIC

0.0134(0.0023)(0.0110)0.6329(0.0169)(0.1367)
0.0120(0.0018)(0.0084)
0.6324(0.0168)(0.1344)0.0120(0.0019)(0.0084)
0.0029(0.0007)(0.0015)0.0723(0.0019)(0.0050)
0.0027(0.0006)(0.0013)0.0723(0.0019)(0.0050)
0.0027(0.0006)(0.0013)0.0002(0.0002)(0.0003)-2,365.42-0.5282(0.3443)0.5089(0.0283)-0.1641(0.0374)

0.2731(0.0073)(0.4589)(0.0330)(0.0465)(0.0088)

GARCH ver. of-0.5293
best homosked.
(0.2520)(0.1958)(0,d ,1)-G
Homoskedastic ver.
-0.5282of best overall model (0.3443)(0.4589)(0,d ,1)Best overall-0.5293(0,d ,1)-G(0.2520)(0.1958)

0.7857(0.0223)(0.0932)

0.2595(0.0320)(0.1325)

4,786.40-3,335.09

6,701.93
0.7993(0.0184)(0.0730)

0.2423(0.0274)(0.1083)-2,366.79

4,781.21-3,333.97

6,707.64
0.7966(0.0191)(0.0732)
0.8630(0.0198)(0.0415)

0.2457(0.0284)(0.1099)
0.0982(0.0138)(0.0271)-2,360.72

4,777.00-47.29158.08-295.10621.940.8714(0.0179)(0.0370)

0.0926(0.0126)(0.0248)-50.53148.68-295.10621.940.8714(0.0179)(0.0370)0.9894(0.0032)(0.0040)0.0926(0.0126)(0.0248)0.0102(0.0028)(0.0036)-50.53148.68-2,100.31

4,256.19-2,157.44

4,346.63

0.5030-0.14520.00020.98890.0105-2,095.63(0.0273)(0.0362)(0.0002)(0.0034)(0.0030)

4,238.89(0.0286)(0.0398)(0.0003)(0.0046)(0.0039)0.5089-0.1641

0.2731-2,157.44(0.0283)(0.0374)(0.0073)

4,346.63(0.0330)(0.0465)(0.0088)

0.5030-0.14520.00020.98890.0105-2,095.63(0.0273)(0.0362)(0.0002)(0.0034)(0.0030)

4,238.89(0.0286)(0.0398)(0.0003)(0.0046)(0.0039)

The sample period is from March 11, 1996, through August 31, 2007 (2802 observations). For each of the six series, theparameter estimates for the best short-memory model (top), the best homoskedastic model and its GARCH version with thesame ARFIMA order, the homoskedastic version of the best model, and the best model (bottom), selected by the BIC fromthe 64 restricted versions of the ARFIMA(2,d ,2)-GARCH(1,1) model, are shown with the Hessian-based (on the first line
beneath the parameter estimates) and the Bollerslev-Wooldridge (on the second line) standard errors in parentheses, the log
likelihood (LL) and BIC values. For C 1/2 and lnC , the best homooskedastic model and the best model have a common
ARFIMA order. The resulting two duplicates (the GARCH version of the best homoskedastic model and thehomoskedastic version of the best model) are not omitted from Table 2. The ARFIMA(2,d,1)-GARCH(1,1) model with thefirst-order AR coefficient restricted to be zero, for example, is denoted as ({2},d ,1)-G.

Table 3: Residual Diagnostic Statistics
Var.

RV

Model
Short-mem.
(2,0,{2})-G
Homosked.
(0,d ,1)+ GARCH(0,d ,1)-G
Homosk..ver.of best

RV

1/2

(2,d ,0)Best overall(2,d ,0)-G
Short-mem.
(1,0,2)-G
Homosked.
(0,d ,1)+ GARCH(0,d ,1)-G
Homosk..ver.of bestln RV

(2,d ,0)Best overall(2,d ,0)-G
Short-mem.
(2,0,1)-G
Homosked.
(2,d ,0)+ GARCH(2,d ,0)-G
Homosk..ver.of bestC({2},d ,1)Best overall({2},d ,1)-G
Short-mem.
(2,0,{2})-G
Homosked.
(0,d ,1)+ GARCH(0,d ,1)-G
Homosk..ver.of best

C 1/2

ln C

({2},d ,1)Best overall({2},d ,1)-G
Short-mem.
(2,0,2)-G
Homosked.
(0,d ,1)Best overall(0,d ,1)-G
Short-mem.
(2,0,1)-G
Homosked.
(0,d ,1)Best overall(0,d ,1)-G

mean
0.033

std.
0.806

skew.
6.306

kurt.
117.688

Residuals εt
JB1,554,221.518

0.001

0.800

7.059

126.740

1,810,892.767

0.005

0.803

6.556

122.018

1,673,875.665

0.001

0.799

6.996

125.325

1,769,826.414

0.003

0.805

6.212

115.545

1,496,806.010

0.008

0.265

1.685

14.567

16,946.021

0.000

0.265

1.761

14.91018,008.312

0.003

0.265

1.773

14.973

18,205.261

0.000

0.264

1.729

14.685

17,336.279

0.002

0.264

1.727

14.661

17,267.676-0.002

0.495

0.268

3.905

129.249-0.001

0.493

0.278

3.916

134.075-0.001

0.493

0.280

3.913

133.913-0.001

0.493

0.281

3.922

136.115-0.001

0.493

0.282

3.919

135.664

0.030

0.801

6.472

122.420

1,684,532.807

0.003

0.796

7.167

131.645

1,956,131.830

0.007

0.798

6.740

127.021

1,816,978.170

0.003

0.795

7.098

130.267

1,914,529.203

0.004

0.799

6.483

122.088

1,675,364.830

0.009

0.269

1.726

14.521

16,888.778

0.001

0.269

1.712

14.59017,050.511

0.004

0.269

1.730

14.66017,270.139-0.001

0.524

0.196

3.897

111.742

0.002

0.523

0.219

3.908

118.640

0.002

0.523

0.221

3.908

119.116

ρ1-0.084(0.075)0.004(0.059)-0.071(0.073)-0.001(0.060)-0.095(0.075)-0.015(0.032)0.006(0.032)-0.007(0.032)0.000(0.032)-0.015(0.032)-0.008(0.021)-0.001(0.021)-0.007(0.021)0.000(0.021)-0.007(0.021)-0.061(0.072)0.003(0.060)-0.049(0.070)-0.003(0.061)-0.071(0.073)0.028(0.032)0.005(0.032)0.001(0.032)-0.016(0.021)
0.006(0.022)-0.007(0.022)

LB (5)
53.1767.023
18.1742.476
37.032
4.885
13.1781.473
52.404
6.665
7.272
3.882
9.366
4.726
10.348
4.970
1.697
0.577
3.450
1.396
3.446
2.943
0.318
0.270
0.541
0.447
0.168
0.145
0.468
0.393
32.7075.340
13.6252.237
23.326
3.848
9.912
1.619
24.334
3.563
6.392
3.167
7.183
3.859
7.538
3.882
5.457
4.502
5.708
5.056
6.063
5.367

LB (10) LB (22)82.729 126.167
19.617 33.220
28.554 63.534
7.480 19.087
49.867 88.438
11.312 24.346
25.098 60.567
7.115 18.701
71.944 114.655
15.709 29.049
22.272 41.796
13.162 25.380
17.903 33.033
10.283 19.831
19.734 35.480
11.098 20.941
10.961 26.835
6.592 16.590
13.500 30.073
7.936 18.270
11.024 21.219
9.552 19.226
4.702 13.178
4.211 12.326
5.062 13.636
4.521 12.732
4.497 12.902
4.007 12.044
4.876 13.355
4.337 12.448
61.594 109.406
17.063 30.983
22.272 62.813
6.465 19.133
34.366 77.775
9.357 23.242
20.051 61.243
6.422 19.014
40.853 88.084
11.070 25.013
9.984 31.365
5.574 18.568
11.215 28.938
6.519 17.750
12.025 30.246
6.803 18.233
10.085 19.664
8.199 17.053
8.300 16.803
7.264 15.227
8.751 17.454
7.664 15.808

Squared residuals εt 2
ρ1
LB (5) LB (10) LB (22)0.126 118.963 120.793 133.099

mean
0.075

Standardized residuals Z t
std.
skew.
kurt.
JB
0.997 3.594
31.503 100,878.194

0.070

80.39281.72291.537

0.001

1.000

7.059

126.740 1,810,892.767

0.114 105.383 106.727

117.021

0.023

0.999

3.719

33.8680.072

83.81094.211

0.001

1.000

6.996

125.325 1,769,826.414

0.128 123.181 124.726

138.032

0.027

0.999

3.629

32.2630.138 220.138 251.703

304.449

0.030

1.000

1.385

0.128 203.585 230.174

275.851

0.000

1.000

0.131 202.744 228.330

274.450

0.006

0.133 209.458 236.776

286.246

0.137 211.853 238.638
0.082

51.1560.094

0.000 22.683

23.82626.627106,124.409

0.000 20.808

22.14125.0638.082

3,910.744

0.010 26.629

29.21833.8341.761

14.91018,008.312

1.000

1.400

8.313

4,211.003

0.012 28.392

31.17535.567

0.000

1.000

1.729

14.685

17,336.279

290.040

0.004

1.000

1.388

8.239

4,103.537

0.013 27.560

30.65035.00078.28988.733-0.004

0.995

0.304

3.707

101.4440.038 23.449

27.81133.16057.20482.51092.203-0.002

1.000

0.278

3.916

134.0750.094

57.20382.85992.823-0.004

0.995

0.303

3.735

105.983

0.047 25.559

30.92036.7090.093

56.28881.68491.640-0.002

1.000

0.281

3.922

136.1150.093

56.37882.08292.258-0.004

0.995

0.307

3.741

107.926

0.046 25.143

30.42036.2840.112

83.03185.19798.084

0.071

0.997

3.295

25.34463,358.309-0.002 31.719

33.07036.1020.070

52.80154.18964.745

0.004

1.000

7.167

131.645 1,956,131.830

0.101

70.51872.02883.001

0.026

0.999

3.403

27.339-0.002 32.466

33.76636.7530.074

55.92857.38768.599

0.004

1.000

7.098

130.267 1,914,529.203

0.117

86.03887.770101.463

0.027

0.999

3.360

26.61970,401.910-0.002 32.707

34.05536.9370.135 182.378 217.176

273.101

0.028

1.000

1.310

7.685

3,363.966

0.013 28.953

32.97137.0290.137 179.276 214.392

268.211

0.004

1.000

1.712

14.59017,050.511

0.138 177.706 211.891

265.939

0.009

1.000

1.332

7.801

3,518.869

0.011 34.676

38.19742.3180.099

68.566 133.160

171.932-0.003

0.987

0.250

3.636

76.4440.033 24.887

30.48538.1840.112

78.223 140.330

175.775

0.004

1.000

0.219

3.908

118.6400.112

77.339 140.100

176.198

0.000

0.988

0.250

3.666

80.9570.041 26.919

33.70041.15682.462

117,699.427

Squared standardizd residuals Z t 2
ρ1
LB (5) LB (10) LB (22)-0.001 19.607 20.903 24.063

74,569.180

2JB, ρ 1, and LB (k ) stand respectively for the Jarque-Bera statistic for nonnormality, first-order sample autocorrelation, and the Ljung-Box statistic for no serial correlations of orders up to k (LB for the squared residuals ε t are also called the McLeod-Li

statistics). For the residuals ε t , the heteroskedasticity-consistent standard errors for ρ1 are given in parentheses (the usual standard error is T2

-1/2

= 0.018) and both the usual LB (upper lines) and the heteroskedasticity-adjusted LB (lower lines) are shown. The

5% critical values for χ (k ) are 5.991 (k =2), 7.815 (3), 9.488 (4), 11.070 (5), 12.592 (6), 14.067 (7), 15.507 (8), 16.919 (9), 18.307 (10), 19.675 (11), 21.026 (12), 22.262 (13), 23.685 (14), 24.996 (15), 26.296 (16), 27.587 (17), 28.869 (18), 30.144 (19), 31.410(20), 32.671 (21), 33.924 (22).

Table 4: Further specification tests
Var.

RV

RV 1/2

ln RV

C

C 1/2

ln C

Model
Short-mem.
(2,0,{2})-G
Homosked.
(0,d ,1)+ GARCH(0,d ,1)-G
Homosk.. ver. of best
(2,d ,0)Best overall(2,d ,0)-G
Short-mem.
(1,0,2)-G
Homosked.
(0,d ,1)+ GARCH(0,d ,1)-G
Homosk.. ver. of best
(2,d ,0)Best overall(2,d ,0)-G
Short-mem.
(2,0,1)-G
Homosked.
(2,d ,0)+ GARCH(2,d ,0)-G
Homosk.. ver. of best({2},d ,1)Best overall({2},d ,1)-G
Short-mem.
(2,0,{2})-G
Homosked.
(0,d ,1)+ GARCH(0,d ,1)-G
Homosk.. ver. of best({2},d ,1)Best overall({2},d ,1)-G
Short-mem.
(2,0,2)-G
Homosked.
(0,d ,1)Best overall(0,d ,1)-G
Short-mem.
(2,0,1)-G
Homosked.
(0,d ,1)Best overall(0,d ,1)-G

BDS
k =2
-0.0022.408
12.646
18.6300.820
3.782
12.91318.4881.176
3.363-0.540
1.226
2.822
9.674
0.382
1.904
2.706
9.388
0.480
1.911
0.001
1.677
0.363
3.512
0.302
2.157
0.200
3.407
0.202
2.110
1.296
1.511
12.20118.8801.947
0.913
12.65118.891
1.079
1.861
0.550
0.818
2.712
9.168
0.351
0.967
-0.249
0.802
0.248
3.291
0.376
1.155

3
-0.456
2.557
12.83221.9120.627
4.491
13.31221.731
0.913
3.613-0.7961.341
2.822
12.136-0.0172.136
2.640
11.7330.025
2.054-0.023
1.914
0.112
4.171
0.017
2.564
0.043
4.008
-0.043
2.492
1.085
2.004
13.21722.3702.557
1.159
13.74222.310
1.352
2.245
0.259
0.912
2.868
11.6060.288
1.171-0.338
0.874
0.342
4.015
0.391
1.192

Line 1: ln Z t 2
Line 2: Z t
4
5-0.593-3.241
3.411
4.695
13.249-3.27425.045 28.791
0.431-2.1345.406
6.766
13.865-3.36024.901 28.667
0.688-2.9154.375
5.659-0.779-2.506
1.930
2.942
2.672
-1.85514.470 17.515-0.177-1.761
2.663
3.627
2.549
-2.57014.029 17.040-0.130-2.235
2.505
3.432
0.007
-0.8382.292
3.199-0.075-1.691
4.577
5.629-0.240-1.5872.811
3.620-0.098-1.3794.461
5.478-0.210-1.363
2.747
3.545
0.881
-1.896
3.174
4.439
14.048-2.59225.656 29.452
3.709
5.024
1.304
-1.792
14.481-2.84625.595 29.404
1.333-2.360
3.274
4.589
0.026
-1.182
1.846
2.922
2.915
-1.253
14.520 17.888
0.094-0.8892.278
3.456-0.515-0.154
1.373
2.197
0.382
-0.153
5.097
6.344
0.451
-0.184
1.675
2.476

Hong-Li

Hong-Li separate inference(m ,l ) (1,2)
(2,1)
(2,2)
(3,3)
= (1,1) ARCH-m Leverage-0.2161.038
0.229 1.198 -1.363-2.867730.094 1,002.689 1,435.300

5.420

13.89325.255 14.293-1.095-2.973140.927301.625

611.230-0.213

1.416

0.311

1.790

-1.088-2.7774.664

341.694728.647 1,001.012 1,431.934

4.608

15.16725.546 14.788-1.051-2.959

3.602

3.690

134.742287.149399.346585.305-1.223

0.622

0.475

1.454

-1.190-2.769-0.036

1.196

1.195

36.74579.042110.621161.434-0.880

0.517

2.622

3.203

5.156

4.152

1.748

1.378

2.039

1.664

56.780123.072167.576239.940

1.627

2.240

23.649 48.013

14.567

7.456

2.260

0.939

0.468

0.842

0.932

36.96280.971112.648164.097-0.021

1.011

2.521

3.580

5.668

4.303

2.729

2.633

1.368

1.164

2.438

1.998

56.167120.665

164.562236.100

0.147

2.924

23.733 49.772

14.781

7.845

2.249

2.617

0.287

-0.014

1.275

1.294

37.11678.697109.086

159.376-1.723

0.408

2.728

3.495

5.239

4.127

-1.434 -0.667 -2.216-2.240 -0.673 -0.708

2.815

6.414

8.762

11.689-0.522

3.217

0.339

3.148

7.241

25.083-1.262 -0.603 -2.092-2.083 -0.653 -0.633

4.618

9.634

13.70118.604-1.4495.065

1.187 14.761

5.425

14.710-1.479 -0.635 -2.171-2.176 -0.755 -0.705

3.677

6.522

8.524

11.296-1.754

3.674

0.207

3.904

6.812

23.126-1.254 -0.615 -2.134-2.129 -0.636 -0.654

4.686

9.729

13.76118.607-1.4785.049

1.253 14.583

5.440

14.461-1.471 -0.648 -2.207-2.215 -0.751 -0.726

3.579

6.414

8.387

11.142-1.836

3.754

0.261

3.799

6.632

22.747

4.547

5.571

1.925

2.155

3.095

3.478

135.709296.375

412.433

602.941-0.188

0.690

0.117

3.569

3.368

0.896

5.179

4.923

4.328

4.281

4.953

4.424

342.874740.090 1,011.522 1,441.198

4.639

13.20522.4588.789-2.623-3.358

4.290

5.353

3.055

3.098

2.163

2.664

137.079

297.039605.267-0.619

0.854

0.138

3.689

3.008

0.419

5.242

5.071

3.835

3.833

5.168

4.649

341.469735.879 1,006.099 1,433.161

4.110

14.51622.9979.529-2.598-3.352

4.895

5.854

1.373

1.621

2.758

3.228

134.695

288.186402.269

589.806-1.831

0.251

0.233

3.754

3.202

0.737

1.694

2.362

0.894

0.500

1.425

1.725

30.90668.27695.601139.663-0.358-0.240

1.551

3.768

7.181

8.302

2.498

2.800

1.340

1.038

2.215

2.206

52.669116.046158.849226.107

0.624

1.522

23.293 45.588

5.783

0.825

2.097

2.663

0.828

0.334

1.102

1.423

33.28473.196102.143149.058-1.010

0.334

1.985

4.840

10.672

13.526-1.229 -0.422 -2.242-2.538 -0.341 -0.084

0.489

2.636

3.312

4.350

-0.4942.941-0.128

3.731

8.468

27.987-1.531 -0.737 -2.347-2.481 -0.202 -0.094

3.677

8.651

11.50015.658-0.7555.292

1.415 28.678

6.514

21.892-1.536 -0.754 -2.099-2.407 -0.430 -0.215

1.725

3.017

3.681

5.178

-0.8462.767-0.14410.05628.736CCK
γ = 0.5
γ = 1.0
k =1
2
k =1
2
3
4.716 5.451 1.834
2.002 3.158

3
3.359

139.297300.546

5.577

4.750

4.683

4.469

5.177

4.230

340.963

4.768

5.350

3.181

3.084

2.157

2.408

5.570

4.859

4.260

4.112

5.708

5.047

5.735

1.370

1.594

2.344

2.598

0.322

2.576

2.379

1.996

Q HL (1) W HL (5) W HL (10) W HL (20)418.937

418.049

414.158

613.858

4.459

(4,4)

Table 5: In-sample forecast performance evaluation
Variable /HorizonSelected: Day (s) RMSE
Model

RV
(2,d ,0)-G

RV 1/2(2,d ,0)-G

lnRV({2},d ,1)-G

C({2},d ,1)-G

C 1/2(0,d ,1)-G

lnC(0,d ,1)-G

10.804850.905410

0.949122

1.01001

0.264350.3041100.3244220.35061

0.49345

0.5675100.6065220.65671

0.798650.9006100.945022

1.0052

1

0.268950.309010

0.3294220.35481

0.5226

5

0.5996

100.6386220.6885

RV prediction
Mincer-Zarnowits Regression
Int.
Coef.
R2
χ2
0.1245(0.0459)
0.2291(0.0554)0.2956(0.0527)0.4598(0.0837)0.8878(0.0485)0.7938(0.0560)
0.7345(0.0521)0.5856(0.0711)

0.39590.0162(0.0201)
0.0480(0.0316)0.0822(0.0424)0.1751(0.0694)

0.9856(0.0228)0.9551(0.0351)
0.9215(0.0462)0.8277(0.0701)

0.5347-0.0022(0.0099)-0.0053(0.0160)-0.0092(0.0218)-0.0203(0.0292)

0.9931(0.0159)
0.9818(0.0297)
0.9675(0.0438)
0.9222(0.0694)

0.1063(0.0433)
0.2224(0.0534)
0.2934(0.0523)0.4571(0.0816)0.9014(0.0484)
0.7930(0.0565)0.7273(0.0534)
0.5727(0.0715)

0.39330.0086(0.0198)
0.0271(0.0319)0.0592(0.0437)
0.1491(0.0723)

0.9951(0.0230)0.9800(0.0364)
0.9490(0.0489)0.8573(0.0750)

0.52690.0015(0.0106)
0.0070(0.0175)
0.0072(0.0240)-0.0011(0.0319)0.9979(0.0163)1.0080(0.0311)1.0031(0.0462)
0.9656(0.0750)

0.58430.24480.17650.08850.38560.30330.19420.5982
0.46910.39470.29290.23970.17010.08350.37640.29320.18660.45340.38080.2813RMSE

8.325(0.0156)
17.159(0.0002)31.805(0.0000)
34.744(0.0000)

7.26031.130(0.5684)
2.959(0.2278)
4.279(0.1177)
6.364(0.0415)0.25480.192(0.9084)
0.394(0.8214)0.589(0.7450)1.513(0.4694)
7.074(0.0291)17.408(0.0002)31.865(0.0000)
36.492(0.0000)

4.24083.94773.60340.16280.15570.1421
0.41220.2846
0.27200.2766
7.39714.36434.00763.64971.042(0.5939)1.997(0.3685)
3.004(0.2227)
4.487(0.1061)0.2641

0.063(0.9692)
0.170(0.9184)
0.094(0.9542)0.224(0.8939)0.45850.1706
0.16340.14980.31420.29870.3006

Volatility-of-the RV prediction
Mincer-Zarnowits Regression
Int.
Coef.
R2
χ2
0.3977(0.0987)
0.4836(0.0906)
0.6027(0.1161)
0.7313(0.1392)

0.2943(0.0832)
0.2842(0.0612)0.2240(0.0389)
0.1650(0.0281)0.02010.0190(0.0095)
0.0276(0.0082)
0.0395(0.0088)
0.0524(0.0140)0.7189(0.1700)
0.7678(0.1349)
0.7313(0.1116)0.7261(0.1528)

0.03630.0088(0.0443)0.0485(0.0440)0.0670(0.0493)0.0820(0.0588)

0.9528(0.1914)0.9710(0.1870)
1.0255(0.2056)
1.1473(0.2421)0.4192(0.1091)0.5073(0.0976)
0.6141(0.1195)
0.7351(0.1407)0.2631(0.0676)0.2544(0.0488)0.2062(0.0329)0.1543(0.0277)0.0197(0.0093)0.0306(0.0081)0.0425(0.0096)
0.0561(0.0149)

0.7190(0.1630)
0.7422(0.1272)
0.7081(0.1148)
0.6952(0.1611)0.03550.0176(0.0305)
0.0584(0.0315)0.0761(0.0369)0.0897(0.0468)

0.9062(0.1204)
0.9345(0.1198)
0.9912(0.1354)
1.1124(0.1730)

0.0331

0.06870.05980.05620.08230.06870.05440.01700.03680.0465
0.06150.01590.05470.05170.05030.07600.06500.05270.07290.09300.1250103.968(0.0000)
177.385(0.0000)
421.560(0.0000)
824.011(0.0000)
4.216(0.1215)
15.915(0.0003)21.515(0.0000)
18.512(0.0001)
0.131(0.9366)
25.042(0.0000)
50.321(0.0000)
77.823(0.0000)
156.044(0.0000)
277.402(0.0000)
590.623(0.0000)
856.786(0.0000)
4.728(0.0940)
17.726(0.0001)
20.623(0.0000)
18.422(0.0001)
1.083(0.5819)
21.464(0.0000)
43.871(0.0000)
69.770(0.0000)

In volatility prediction, the target variable is RV , RV 1/2, lnRV , C , C 1/2, or lnC (one-day-ahead, or multi-days-ahead singleday). In volatility-of-RV prediction, the proxy for the target is the squared prediction errors ( one-day-ahead or multi-days
average ) from the selected model for each of the six series. Newey-West standard errors for the OLS estimates of the2
regression intercept and the coefficient are given in parentheses. The χ test statistic with p-value in parentheses is for testing
the joint hypothesis of the intercept being zero and the coefficient being one.

20

RVt

15

10

5

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

RVt

4

3

2

1

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

3

ln RVt
2
1
0
-1
-2
-3
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Figure 1a: The figure plots time series of daily Nikkei 225 RV, square-root RV, and log RV (the dottedlines), along with the corresponding in-sample one-day-ahead ARFIMA-GARCH forecasts (the solid lines).

20

Ct

15

10

5

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Ct

4

3

2

1

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

3

2006

2007

ln Ct

2
1
0
-1
-2
-3
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Figure 1b: The figure plots time series of daily Nikkei 225 C , C 1/2, and lnC (the dotted lines) where C is thecontinuous sample path component of RV , along with the corresponding in-sample one-day-ahead ARFIMAGARCH forecasts (the solid lines).

80

RVt
60

40

20

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

1.5

RVt
1

0.5

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

1

ln RVt

0.5

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Figure 2a: The figure plots time series of squared residuals (dotted lines) and one-day-ahead GARCH
1/2
conditional variance estimate (solid lines) from the ARFIMA-GARCH model for daily Nikkei 225 RV , RV ,and lnRV .

80

Ct
60

40

20

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

1.5

Ct
1

0.5

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

1

ln Ct

1

0
1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

Figure 2b: The figure plots time series of squared residuals (dotted lines) and one-day-ahead GARCH
conditional variance estimate (solid lines) from the ARFIMA-GARCH model for daily Nikkei 225 C , C 1/2,
and lnC (the dotted lines) where C is the continuous sample path component of RV .

RV 1/2

RV 1/2

Horizon = 5 days

Horizon = 22 days

1.0
3

2
0.5
1

0
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
4

lnRV

0.0
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Horizon = 5 days

lnRV

Horizon = 22 days

2

3

2

1

1

0
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

01996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 3a: The figure plots time series of squared residuals (dotted lines) and five-day-average and 22-day-average conditional variance estimates (solid lines) from the
ARFIMA-GARCH model for daily Nikkei 225 RV1/2 and lnRV .

C 1/2

C 1/2

Horizon = 5 days

Horizon = 22 days

1.0
3

2
0.5
1

0

0.0
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
lnC

Horizon = 5 days

lnC

Horizon = 22 days

4
2
3

2
1
1

0
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

01996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

Figure 3b: The figure plots time series of squared residuals (dotted lines) and five-day- average and 22-day-average conditional variance estimates (solid lines) from the
ARFIMA-GARCH model for daily Nikkei 225 C1/2 and lnC .