The properties of realized correlation: Evidence from the French, German and Greek equity markets

The Quarterly Review of Economics and Finance 50 (2010) 273–290

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The properties of realized correlation: Evidence from the French, German and Greek equity markets Dimitrios I. Vortelinos ∗ Department of Economics, School of Management and Economics, University of Peloponnese, 22 100, Greece

a r t i c l e

i n f o

Article history: Received 12 January 2010 Received in revised form 18 March 2010 Accepted 21 March 2010 Available online 21 April 2010 Keywords: DAX CAC40 Athens Stock Exchange Realized correlation Bipower variation Range Optimal sampling Long memory Asymmetry Jumps Heterogeneous autoregressive models

a b s t r a c t In this paper I examine the properties of four realized correlation estimators and model their jumps. The correlations are between the French, German and Greek equity markets. Using intraday data I first construct four state-of-the-art realized correlation estimators which I then use to testing for normality, long-memory, asymmetries and jumps and also to modeling for jumps. Jumps are detected when the realized correlation is higher than 0.99 and lower than 0.01 in absolute values. Then the realized correlation is modeled with the simple Heterogeneous Autoregressive (HAR) model and the Heterogeneous Autoregressive model with Jumps (HAR − J). © 2010 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.

1. Introduction The correlation between asset returns is important in risk management, portfolio optimization, trading strategies, option pricing and structured finance.1 The non-parametric estimators of volatility can be also applied to the correlation. The use of high frequency data improves the precision of both volatility and correlation estimation. The realized volatility estimator can be also employed in the computation of covariances and correlations between two assets leading to the analogous concept of realized covariance and realized correlation. The properties of both volatility and correlation are such (e.g. normality, persistence, long memory and jumps) that one has to be quite careful both in modeling and forecasting then.

∗ Tel.: +30 2710 230128; fax: +30 2710 230139. E-mail address: [email protected]. 1 Realized correlation is valuable to the issuers of structured equity products, who have short exposure to equity correlation. So, in order to make offset this exposure dealers buy correlation from hedge funds through correlation swaps as well as many variance swaps.

In this paper, I model the realized correlation between (i) the DAX and CAC40 indices, (ii) the General Index of the Athens Stock Exchange (GD) and DAX index, and (iii) the GD and CAC40 indices using high frequency intraday data and four realized volatility estimators. I examine these correlations for normality in their distribution, through descriptive statistics. Then, the long-memory (d-values) of these correlations is estimated together with asymmetries. The correlations are then tested for jumps. Jumps are detected when realized correlations are higher or lower in their absolute values than two thresholds. The realized correlations, as well as their jumps series, are subsequently modeled using the class of the Heterogeneous Autoregressive (HAR) models. The differences between the U.S. and European equity markets are huge. As it concerns the average daily turnover, the NASDAQ and the NYSE Group were first (for 2005 and 2006), the Athens Stock Exchange was last, with the American SE just above it. It comes not as a surprise that, for the years 2005 and 2006, the domestic market capitalization, was higher for the NASDAQ and NYSE Group exchanges, first and second, respectively, whereas the Deutsche Borse was third, the Euronext fourth, the American SE fifth and the Athens Stock Exchange last. The number of listedcompanies in the exchanges reveals the role that they play locally and also internationally: the higher the number of internation-

1062-9769/$ – see front matter © 2010 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.qref.2010.03.002

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Fig. 1. (a) Return series for French market; (b) return series for German market; (c) volatility series for French market; (d) volatility series for German market.

alised companies the more important the international role of the exchange. The most populated – with regard to listed companies and internationationalized exchanges – are the NASDAQ and the NYSE Group followed by the Euronext, the Deutsche Borse, the American SE and the Athens Stock Exchange. Thus, it is apparent that the American exchanges are more internationalized than their European counterparts. These differences between the U.S. and European equity markets made me use the realized correlation estimates between three European (German, French and Greek) equity markets. The realized correlation depends on the theory of realized volatility (Andersen, Bollerslev, Frederiksen, & Nielsen, 2010) and the bootstrapping of realized volatility (Goncalves & Meddahi,

2009). In order to get the realized correlation, it is assumed that the two assets are sampled simultaneously. The sampling frequency that is mostly used in the literature is that of per 5 min. I choose the specific frequency, because it balances between the advantages of the numerous intraday observations and the bias from the market microstructure noise. I focus on four realized correlation estimators: the realized correlation estimator, the realized bipower variation correlation estimator, the realized Parkinson range-based correlation estimator and the realized optimal correlation estimator. The first estimator was proposed by Andersen, Bollerslev, Diebold, and Labys (1999) the first. The naive realized correlation estimator has been analyzed under various frameworks (Christiansen & Ranaldo, 2007; Ferland & Lalancette, 2006; Wang,

Fig. 2. Correlation series for French and German markets.

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275

Fig. 3. Actual vs. fitted values from HAR model – French and German market.

Fig. 4. Actual vs. fitted values from HAR-J model – French and German market.

Fig. 5. (a) Rolling estimates of daily coefficient from HAR-J; (b) rolling estimates of weekly coefficient from HAR-J; (c) rolling estimates of monthly coefficient from HAR-J; (d) rolling estimates of jump coefficient from HAR-J.

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Fig. 6. (a) Rolling estimates of ˛ from asymmetry model; (b) rolling estimates of ˇ from asymmetry model; (c) rolling estimates of  from asymmetry model; (d) rolling estimates of ı from asymmetry model.

Fig. 7. Rolling estimates of  + ı from asymmetry model.

Wu, & Yang, 2008). The second estimator is a multivariate version of its univariate counterpart, as introduced by Barndorff-Nielsen and Shephard (2002, 2004a,b, 2005, 2006). It is extracted by dividing the realized covariance estimate with the multiplication of the square roots of the realized bipower variation volatility estimators of the two assets. The third estimator is extracted by dividing the realized covariance estimate with the multiplication of the square roots of the realized Parkinson range-based volatility estimators of the two assets. The realized Parkinson range-based covariance and correlation estimators were introduced by Brandt and Diebold (2006), but in a foreign exchange framework. Bannouh, Van Dijk, and Martens (2009) also analyze the realized co-range estimator. The last estimator is the realized correlation estimator with its sampling frequency chosen optimally, as proposed by Bandi and Russell (2005). The properties of normality and stationarity for the realized correlations are examined together with the long-memory (dvalues) of the realized correlations. Choi, Yu, and Zivot (2009) study the long-memory properties of the realized volatility. Realized correlations are examined for the presence of asymmetries, like in Thomakos and Wang (2003), whereas, the methodology of

Andersen, Bollerslev, and Diebold (2007) – ABD (2007) hereafter – is employed in modeling jumps. Giot, Laurent, and Petitjean (2010) also examine jumps in the realized volatility framework. For an indepth analysis of the HAR-class of volatility models see Corsi (2009), Corsi, Pirino, and Reno (2008), and also ABD (2007). The paper is organized as follows. Section 2 presents the methodology for the realized correlation estimators, the jump detection and the modeling of realized correlation and jumps. Section 3 describes the data used in this paper. Section 4 presents the empirical results and Section 5 summarizes the results. 2. Empirical methodology 2.1. Realized correlation estimators Though literature has covered sufficiently the realized volatility estimators, however, it is only the simple realized volatility estimator that has been used in estimating the realized correlation instead. This paper introduces four estimators of realized correlation: (i) the simple realized correlation estimator (i.e. realized correlation), (ii) the simple realized covariance divided by the square roots of the

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277

Fig. 8. (a) Return series for Greek market; (b) return series for French market; (c) volatility series for Greek market; (d) volatility series for French market.

realized bipower variations of the two assets (i.e. realized bipower variation correlation), (iii) the simple realized covariance divided by the square roots of the realized Parkinson estimators of the two assets (i.e. realized Parkinson correlation), and (iv) the realized optimal correlation estimator. It is assumed that for each day, with time confined in the unit interval [0, 1], the observed intraday logarithmic asset prices follow the noise contaminated process: pti ,m =

p∗t ,m i

+ uti

(1)

where p denotes the observed logarithmic price, p∗ the unobservable equilibrium logarithmic price and u the unobservable market microstructure noise.2 The time index ti represents the ith observation in the m + 1 intraday observations with a sampling frequency ∗ = p∗ equal to 1/m. The values for ri,m = pti ,m − pti −1,m and ri,m ti ,m − ∗ pt −1,m are the corresponding intraday returns (at the highest frei

quency of observation), whereas the one for ei,m = uti − uti −1 is the difference of the noise component. It is assumed that the equilibrium price evolves as a function of a stochastic volatility process: def



p∗t = i

0

ti

s dWs + jti

(2)

where t is a stochastic volatility process, Wt is the standard Brownian motion and jti is the component that will appear in the price process in the case of discrete jumps. The integrated volatility over

2

The microstructure noise variable has been dealt under various probabilistic assumptions, the simplest of which is that it is an i.i.d. sequence.

the whole day is then: def



1

s2 ds + t

Vt =

0

(3)



where t = 2 is the contribution of the jumps into the 0


Ct =

1

a,b ds

(4)

0

This covariance is latent, hence the most appropriate estimator is the realized volatility estimator in a multivariate level. BarndorffNielsen and Shephard (2004a) introduced the realized covariance and realized correlation estimator. The realized covariance is given by the cross-products of the two 5-min asset returns series during each trading day. (m) def

RCovt

=

m/5 

ra,i,m rb,i,m

(5)

i=1

The probability limit of the correlation is known by the theory of QV. So, as m → ∞ (m)

RCovt

p

→

Ct

(6)

This is also discussed by Andersen, Bollerslev, Diebold, and Labys (m) (2001). In the absence of noise, RCovt is a consistent estimator for Ct as the sampling frequency increases. Clearly one cannot recover

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Fig. 9. Correlation series for Greek and French markets.

Fig. 10. Actual vs. fitted values from HAR model – Greek and French market.

Fig. 11. Actual vs. fitted values from HAR-J model – Greek and French market.

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279

Fig. 12. (a) Rolling estimates of daily coefficient from HAR-J; (b) rolling estimates of weekly coefficient from HAR-J; (c) rolling estimates of monthly coefficient from HAR-J; (d) rolling estimates of jump coefficient from HAR-J.

Ct because of the noise coming from the bid-ask bounce. This observation was used by Bandi and Russell (2006) to construct optimal sampling frequency estimators as discussed below. (m) The realized correlation coefficient comes from the RCovt divided by the square roots of the realized volatility estimators of (m) (m) the two assets (RVt,a and RVt,b ).

(Par) def

RCt

=

(m)

RCovt



(Par)

RCt,a

m 

ra,i,m rb,i,m def

=



(m) def RCt =



(m)

RCovt

(m) RCt,a

def



=

(m) RCt,b

m 

i=1

   m 2  r

   m 2  r

i=1

i=1

(7)

b,i,m

2.1.2. Realized bipower variation correlation Apart from the realized correlation estimator, this paper uses (BV ) (Par) two other estimators (RCt and RCt ), which come from the same realized covariance estimate as the realized correlation. However, they differ from the realized correlation in the denominator. The realized bipower variation correlation is

m 

ra,i,m rb,i,m



(m)

RCovt (BV )

RCt,a



def (BV )

RCt,b

=



i=1

−2 p

m 

i=2

(ha,i,m − la,i,m )

2

m 

(1/4 log(2))

(9) (hb,i,m − lb,i,m )

2

i=1

ra,i,m rb,i,m

i=1

a,i,m

(BV ) def RCt =



i=1

(1/4 log(2)) m 

(Par)

RCt,b

|ra,i,m ||ra,i−1,m |

−2 p

m 

|rb,i,m ||rb,i−1,m |

i=2

(8) where p = E(|Z|p ) is the mean of the p-th absolute moment of a standard normal distribution. See Barndorff-Nielsen and Shephard (2004b). 2.1.3. Realized Parkinson correlation The second estimator is named realized Parkinson range-based correlation estimator:

where hi,m and li,m are within the ith intraday interval high and low logarithmic prices. Martens and van Dijk (2007) introduced the realized Parkinson range-based volatility estimator – at a univariate level – and Brandt and Diebold (2006) introduced the realized Parkinson range-based covariance and realized correlation estimators. The range-based covariance estimate and the realized co-range have also been studied by Bannouh, Van Dijk, and Martens (2009). The realized Parkinson range-based correlation estimator that they introduced has the same denominator as the realized Parkinson range-based correlation estimator featuring above, though the nominator in the present case is the realized covariance whereas in their case is the realized Parkinson rangebased covariance. A disadvantage of their estimator is its limited applicability only to foreign exchange data. 2.1.4. Realized optimal correlation A different realized covariance estimate from the naive one (used above) is the realized optimal covariance estimate, proposed by Bandi and Russell (2006). High sampling frequencies induce accumulation of microstructure noise and, as a consequence, a substantial bias. Low sampling frequencies deliver imprecise estimates. The bias/variance trade-off can be characterized in the form of a conditional – on the underlying covariance (or volatility) paths – mean square error (MSE) for all frequencies. The MSE can be minimized to determine the optimal sampling frequency which is the optimal number of equispaced observations 1/mopt to be used to define – over any fixed time span – the realized optimal covariance (or volatility) estimator.

 mopt ∼

Qˆ t ˆ e4

1/3 (10)

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Fig. 13. (a) Rolling estimates of ˛ from asymmetry model; (b) rolling estimates of ˇ from asymmetry model; (c) rolling estimates of  from asymmetry model; (d) rolling estimates of ı from asymmetry model.

Fig. 14. Rolling estimates of  + ı from asymmetry model.

where the numerator is the estimated daily quarticity, with def

1

4 0 s

Qt = ds, and the denominator is the square of the estimated noise variance. We have that mopt can be made operational by (a) estimating e2 using the properties of (m) m−1 RVt

(m) RVt ,

i.e. using ˆ e2 =

assuming C = 0 and (b) estimating the daily quarticity m 4 . using sparse sampling every, e.g. 15-min as Qˆ t = (m/3) i=1 ri,m Once mopt is available, one can estimate the underlying covariance and correlation as: mopt  def RCt(mopt ) =



RCovt(mopt ) (mopt ) RCt,a



(mopt ) RCt,b

def

=

  mopt   r2

  mopt   r2

i=1

i=1

a,i,mopt

The asymmetries between risk and return can be estimated by the asymmetry regression. The long-memory of realized volatility has recently been analyzed by Choi et al. (2009). Bollerslev and Zhou (2006) examined asymmetries in volatilties and Thomakos and Wang (2003) examined asymmetries in both volatilities and correlations. The relevant equation is def

1 (1 − L)dRC RCt = ˛ + ϑ · zt−1 + ˇ · wt−1 +  · wt−1 · It−1 2 + ı · wt−1 · It−1 + ut

ra,i,mopt rb,i,mopt

i=1

2.2. Asymmetries

b,i,mopt

(11)

(12)

where dRC is the long-memory estimate for the RCt estimator; wt−1 = rt−1,a + rt−1,b ; zt−1 = (1 − L)dRVt,a · RVt−1,a + (1 − L)dRVt,b · 1 2 RVt−1,b ; It−1 = I(rt−1,a · rt−1,b > 0); and It−1 = I(rt−1,a < 0, rt−1,b < 0). Also, dRVt,a and dRVt,a are the long-memory estimates of the RVt estimator for the two assets; and rt−1,a and rt−1,b are the

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281

Fig. 15. (a) Return series for Greek market; (b) return series for German market; (c) volatility series for Greek market; (d) volatility series for German market.

lagged returns of the two assets (indices). Instead of the lagged volatility series of the two assets (RVt−1,a and RVt−1,b ) in levels, their logarithmic standard deviations counterparts have been used (see Thomakos & Wang, 2003). This specification gives better and more significant estimates for the presence of asymmetries in realized correlations. The ı-coefficient gives the degree of asymmetry. A negative ı indicates asymmetric effects of lagged returns on correlations.

(j)

RCt = xTt−1 ˇ + t

(14)

where the regression component xTt−1 ˇ is given as a linear function of normalized past correlations as follows:

2.3. Testing for jumps Once a realized correlation estimator is available we can apply a test for the presence of correlation jumps. The approach used here is similar to the one discribed by Huang and Tauchen (2005) and Andersen, Bollerslev, and Diebold (2007), as well as the one by Giot, Laurent, and Petitjean (2010). The test statistic is: def

Jc,t = I(|RCt | > c1 )RCt + I(|RCt | < c2 )RCt

(13) (m/5)

where RCt is any of the realized correlation estimators (RCt (BV ) (Par) RCt , RCt

ABD (2007). This class of models allows one to account for standard features of realized correlation, such as long-memory, with a relatively parsimonious, economically coherent model which can be extended to account for jumps. The standard form of a HAR model, in regression format, is:

,

and RCtmopt ). The threshold c can take different values.

def

(j)

(j)

(j)

xTt−1 ˇ = ˇ0 + ˇD RCt−1,t + ˇW RCt−5,t + ˇM RCt−20,t (j)

q

(15)

(j)

with RCt,t+q = q−1 s=1 RCt+s being the mean q-period lagged volatility. In this context, the explanatory variables are the “daily” (for q = 1), “weekly” (for q = 5) and “monthly” (for q = 20) realized correlations. The regression error term t may have heteroskedasticity and serial correlation. The implementation of a HAR model in realized correlation was applied by Corsi and Audrino (2007). The above model could also be extended as follows. The jump component of realized correlation can be explicitly included so as to have

The values used throughout the paper are c1 = 0.95 and c2 = 0.05. 2.4. Modeling correlation and jumps

Modeling correlation and jumps resembles modeling volatility and jumps. So, when volatility is mentioned, the correlation is also implied and vice versa. A class of models can be used to model the evolution of realized volatility (and correlation alike) while accounting for the possible presence of jumps. It is the class of Heterogeneous Autoregressive Models (HAR), as originally proposed by Corsi (2009), and also used by Corsi, Pirino, and Reno (2008) and

Table 1 Descriptive statistics – returns. Mean GD DAX CAC40

−0.017 0.047 0.026

SD 0.016 0.010 0.010

Skew. −0.51 −0.52 −0.45

Kurt.

LB-test

8.43 6.97 7.11

∗∗∗

55 27 30∗

LB2 -test 1190∗∗∗ 248∗∗∗ 367∗∗∗

d-values 0.050 −0.028 −0.069

In rows are reported the descriptive statistics and long-memory estimates (dvalues) of returns (Rt ) for the GD, DAX and CAC40 indices. In the LB-test and LB2 -test columns, the values reported are the statistic values of the tests. The levels of significance are given as follows: ∗ 10% and ∗∗∗ 1%.

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Table 2 Descriptive statistics – GD. Mean

SD

Skew.

Kurt.

d-values

8.29E−5 4.75E−5 3.63E−5 1.75E−4

1.57E−4 8.48E−5 6.08E−5 5.33E−4

7.29 6.00 6.64 10.82

81.4 56.5 70.00 151.21

0.336 0.388 0.351 0.232

7.58E−3 5.74E−3 5.16E−3 9.97E−3

5.04E−3 3.82E−3 3.11E−3 8.67E−3

2.62 2.37 2.49 3.92

14.3 11.5 13.18 28.00

0.438 0.460 0.453 0.343

0.65 0.64 0.75 0.78

2.98 2.90 3.10 3.66

0.453 0.455 0.482 0.393

Panel A. Volatilities (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt ) Panel B. Square roots (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt )

Panel C. Logarithmic volatilities (m/5)

−10.165 −10.683 −10.861 −9.625

RVt (BV ) RVt (Par) RVt RVt(mopt )

1.079 1.087 0.966 1.201

In rows are reported the descriptive statistics and long-memory estimates of volatilities, square roots and logarithmic volatilities for the GD index.

Table 3 Descriptive statistics – DAX. Mean

SD

Skew.

Kurt.

d-values

1.32E−4 7.78E−5 1.09E−4 1.25E−4

3.04E−4 1.86E−4 2.61E−4 3.17E−4

9.27 9.50 10.74 11.41

123.31 132.84 174.23 191.64

0.499 0.436 0.448 0.366

9.43E−3 7.16E−3 8.49E−3 8.98E−3

6.58E−3 5.15E−3 6.04E−3 6.69E−3

3.46 3.52 3.55 3.53

22.38 22.86 24.22 25.01

0.499 0.499 0.499 0.449

0.77 0.78 0.85 0.47

3.87 3.93 3.88 3.64

0.411 0.470 0.493 0.352

Panel A. Volatilities (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt ) Panel B. Square roots (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt )

Panel C. Logarithmic volatilities (m/5)

−9.635 −10.260 −9.852 −9.777

RVt (BV ) RVt (Par) RVt RVt(mopt )

1.032 1.056 1.048 1.130

In rows are reported the descriptive statistics and long-memory estimates of volatilities, square roots and logarithmic volatilities for the DAX index.

the HAR − J model with regression function: def

(j)

(j)

(j)

xTt−1 ˇ = ˇ0 + ˇD RCt−1,t + ˇW RCt−5,t + ˇM RCt−20,t + J Jc,t−1,t

(16)

The additional significance of its parameter J can be easily tested. In addition, the linear representations can be converted into non-linear ones by a logarithmic or square-root transformation in all variables included. 3. Data I have considered the General Index of the Athens Stock Exchange (GD), the DAX index of the Deutsche Borse and the CAC40 index of the Paris Bourse (now Euronext Paris). The GD consists of 60 companies with the highest capitalization among all companies traded in the ASE market. The CAC40 index is a narrow-based, modified capitalization-weighted index of 40 companies listed on the Paris Bourse, which is part of the Euronext market. The DAX index is a total return index of 30 selected German blue chip stocks traded on the Frankfurt Stock Exchange. The data used are intraday, 5-min transaction prices in standard format (open-high-low-close within the 5-min interval) and have been obtained from the Reuters. The dataset begins 2 January 2004 and ends 27 July 2009, for a total of 1,403 trading days for the CAC40 and 1,403 for the DAX. For the GD index, the sample period starts 3 January 2005 and ends 27 July

2009 for 1134 trading days. The average number of 5-min intraday observations is 105 for the CAC40 and DAX indices and 65 for the GD index. In March 2005, October 2006 and August 2007, the return volatility of the Greek equity market appears excessively high and equals to multiple times the volatility in other periods (Fig. 8). In October 2006, August 2007, December 2007 and October 2008, the return volatility of both the German and French equity markets appears excessively high and equals to multiple times the volatility in other periods (Fig. 1). This high volatility in certain periods for both Greek and German markets is also evident in Fig. 15.

4. Empirical results and discussion 4.1. Descriptive statistics of daily returns and realized volatility estimators Table 1 features descriptive statistics for the daily returns from the dataset described in Section 3. According to this table, the DAX has the highest average return (4.7%), with the CAC40 second (2.6%) and the GD last (−1.7%). The standard deviation is very low, at 1% for both DAX and CAC40 indices and at 1.6% for the GD. The skewness values are negative close to zero, with the CAC40 having a value closer to zero. The kurtosis values are high, exceeding the normal value 3, ranging from 7 up to 8. This excess kurtosis shows

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283

Table 4 Descriptive statistics – CAC40. Mean

SD

Skew.

Kurt.

d-values

1.27E−4 7.42E−5 7.08E−5 1.20E−4

2.76E−4 1.60E−4 1.37E−4 2.67E−4

8.74 7.67 6.62 8.73

113.56 84.21 672.34 112.14

0.499 0.437 0.467 0.412

9.20E−3 6.99E−3 6.92E−3 8.82E−3

6.53E−3 5.04E−3 4.79E−3 6.48E−3

3.01 2.94 2.61 2.98

18.14 16.52 13.21 17.94

0.499 0.499 0.499 0.453

0.75 0.76 0.84 0.54

3.26 3.28 3.21 3.32

0.418 0.466 0.477 0.347

Panel A. Volatilities (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt ) Panel B. Square roots (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt )

Panel C. Logarithmic volatilities (m/5)

RVt (BV ) RVt (Par) RVt RVt(mopt )

−9.711 −10.306 −10.365 −9.812

1.087 1.100 1.065 1.149

In rows are reported the descriptive statistics and long-memory estimates of volatilities, square roots and logarithmic volatilities for the CAC40 index.

that the tails of the historical distribution of returns are thicker than the ones of the Gaussian distribution.3 Moreover, the hypothesis of no-serial autocorrelation (according to the Q-values of the Ljung-Box test with 10 lags) in returns is rejected at 1% significance level for GD, 10% significance level for CAC40, but it is not rejected at all for DAX, thus implying stale prices4 for the first two indices. There is also strong positive autocorrelation (according to the Q-values of the Ljung-Box test with 10 lags) in squared returns, indicating volatility clustering.5 Volatility clustering in this case means that high returns tend to coincide with high volatility values and low returns tend to coincide with low volatility values. There are no high values for the long-memory (d-values) estimates of returns. Values range from −6.9% up to 5%. The negative d-values for both DAX and CAC40 indices indicate the lack of long-memory (persistence) in returns. This is something to expect because the long-memory is better revealed in the second moment of returns. Tables 2–4 present the descriptive statistics and the longmemory (d-values) estimates of the realized volatility estimators for the three indices. For the GD index (Table 2), the RVt(mopt ) takes the highest skewness and kurtosis values across the board of lev(m/5) els, square roots and logarithms. The RVt presents the highest (BV )

mean values and d-values. The RVt

features the lowest standard (m/5)

takes the highdeviation. For the CAC40 index (Table 4), the RVt est mean, skewness and kurtosis values and also d-values as well. (Par) The RVt takes the lowest standard deviation. Moreover, for both (m/5)

takes the highest means and DAX and CAC40 indices, the RVt d-values. All indices, at both levels and square roots, feature very high skewness and kurtosis values. Only in logarithms, the realized volatility estimators have skewness and kurtosis values near 0 and 3, respectively. Moreover, in levels the d-values are very high and in many cases close to 0.5. Only in square roots series, the mean

is higher than standard deviation, across the board of the realized volatility estimators. The d-values of the realized volatility estimators for the CAC40 index are higher than those for the DAX index – which is ranked second – and those for the GD index – which comes last – across the levels, square roots and logarithms. The mean values for the DAX are ranked first, with those for the CAC40 second and those for the GD last. The standard deviation values for the GD index are the lowest of all indices, with those for the CAC40 index second lower and those for the DAX as third lower. 4.2. Realized correlations The time series of realized correlation, realized bipower variation correlation, realized Parkinson range-based correlation and realized optimal correlation estimators are estimated for: (i) the DAX and CAC40 indices, (ii) the GD and DAX indices and (iii) the GD and CAC40 indices. According to Fig. 2, the CAC40-DAX correlation is practically constant with values over 80%; though, it features frequent drops to even negative values. This means that the European Integration Table 5 Descriptive statistics – correlations. Mean

SD

Skew.

Kurt.

d-values

0.681 1.252 1.060 0.836

0.237 0.444 0.391 0.296

−2.14 −1.39 −1.23 −0.09

7.23 7.32 6.40 9.60

0.278 0.229 0.256 0.216

0.153 0.276 0.249 −0.003

0.200 0.378 0.337 0.179

0.32 0.73 0.55 −0.61

2.66 4.34 3.36 10.70

0.154 0.142 0.160 −0.004

0.200 0.372 0.399 0.174

0.28 0.65 0.54 −0.08

2.52 4.16 3.03 7.25

0.107 0.098 0.120 −0.006

Panel A. DAX − CAC40 (m/5)

RCt (BV ) RCt (Par) RCt RCt(mopt ) Panel B. GD − DAX (m/5)

RCt (BV ) RCt (Par) RCt RCt(mopt )

Panel C. GD − CAC40 3

This fact has many consequences in option pricing as it connotes that extreme daily returns occur more often in real financial markets than in the Black Scholes framework. 4 This means that returns are clustered which it implies, in its turn, that the positive mean returns will have a slightly increasing trend and the negative mean returns will have a decreasing trend. 5 The high-frequency intraday characteristics are revealed in the behavior of their second moment. Volatility clustering in returns is evident in Andersen et al. (2001) and Andersen, Bollerslev, Diebold, and Labys (2005). According to these papers, the standardized returns displayed no evidence of volatility clustering and were close enough to being Gaussian.

(m/5)

RCt (BV ) RCt (Par) RCt RCt(mopt )

0.151 0.270 0.295 0.009

In rows are reported the descriptive statistics and the long-memory estimates (m/5) ), realized bipower variation correlation (d-values) of realized correlation (RCt (BV )

(Par)

(RCt ), realized Parkinson range-based correlation (RCt ) and realized optimal correlation (RCt(mopt ) ). In columns, there are the mean, standard deviation, skewness, kurtosis values and the long-memory estimates (d-values) for each correlation between the French, German and Greek equity markets.

284

D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290 Table 6 Comparison of realized correlations – Kolmogorov–Smirnov test.

(m/5) RCt

(BV ) RCt

(m/5)

(Par)

Panel A. vs. K–S statistics p-values

Panel B. RCt vs. RCt K–S statistics p-values (BV )

DAX − CAC40

GD − DAX

GD − CAC40

0.870 <2.2E−16∗∗∗

0.2304 <2.2E−16∗∗∗

0.224 <2.2E−16∗∗∗

0.733 <2.2E−16∗∗∗

0.187 <2.2E−16∗∗∗

0.250 <2.2E−16∗∗∗

0.345 <2.2E−16∗∗∗

0.0542 0.077∗

0.045 0.210

0.341 <2.2E−16∗∗∗

0.391 <2.2E−16∗∗∗

0.376 <2.2E−16∗∗∗

0.713 <2.2E−16∗∗∗

0.474 <2.2E−16∗∗∗

0.455 <2.2E−16∗∗∗

0.501 <2.2E−16∗∗∗

0.452 <2.2E−16∗∗∗

0.461 <2.2E−16∗∗∗

(Par)

Panel C. RCt vs. RCt K–S statistics p-values

(m/5)

Panel C. RCt(mopt ) vs. RCt K–S statistics p-values

(BV )

Panel C. RCt(mopt ) vs. RCt K–S statistics p-values

(Par)

Panel C. RCt(mopt ) vs. RCt K–S statistics p-values (m/5)

(BV )

(Par)

abbreviates for realized correlation, RCt for realized bipower variation correlation and RCt for realized Parkinson range-based RCt correlation and RCt(mopt ) for the realized optimal correlation. H0: the two estimators come from the same distribution. There are both statisticand p-values of the Kolmogorov–Smirnov test between the three estimators, for the correlations between the French, German and Greek equity markets. The levels of significance are given as follows: ∗ 10% and ∗∗∗ 1%. The alternative hypothesis is two-sized.

Table 7A Asymmetries. DAX − CAC40

GD − DAX

GD − CAC40

Panel A. ˛-coefficient (m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

−0.008 (0.007) −0.021 (0.011) −0.029∗ (0.011) 7.352E−4 (0.006)

0.019∗ (0.006) 0.031∗ (0.012) 0.041∗ (0.012) −0.010∗ (0.004)

0.023∗ (0.006) 0.043∗ (0.012) 0.054∗ (0.015) −0.005 (0.005)

0.768 (5.977) 0.577 (5.028) 1.841 (4.436) 0.756 (1.106)

0.256 (1.286) 0.629 (2.561) 1.060 (2.216) 0.828 (1.169)

−1.161 (1.427) −1.582 (2.906) −2.221 (3.050) 2.047 (1.063)

Panel B. ˇ-coefficient (m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

DAX − CAC40

GD − DAX

GD − CAC40

0.852 (1.324) 1.231 (2.636) 0.922 (2.287) −0.900 (1.197)

2.882∗ (1.456) 4.480 (2.957) 5.334 (3.112) −2.426∗ (1.120)

−2.467∗ (0.431) −4.294∗ (0.799) −4.031∗ (0.739) −0.061 (0.413)

−2.948 (0.465) −4.860∗ (0.849) −5.234∗ (0.985) 0.510 (0.423)

Panel C. -coefficient −0.793 (6.057) 0.603 (5.040) −1.178 (4.445) −0.249 (1.150) Panel D. ı-coefficient −0.777 (0.872) −2.040∗ (0.875) −2.466∗ (0.757) −0.793 (0.609)

Table entries report standard averages for the corresponding variables in each panel with the Newey–West standard errors in parenthesis. The ‘*’ reveals significance in the 5% significance level.

among European equity markets is true between the most important markets in European equity market. On the other hand, the rest (two) correlations (Figs. 9 and 16) present much lower values. Moreover, these two have a nearly similar pattern throughout the sample period. This means that the Greek equity market is affected by the French and German equity markets in the same direction and in an equal magnitude, which is much less than the one for the French and German markets. So, Greece lacks behind the European Integration of the Western European equity markets. According to Table 5, the mean realized correlations between the DAX and CAC40 indices are the highest across the board of correlations, ranging from 68% to 99%, across the board of the estimators. The mean realized correlations between the GD and DAX indices range from −0.3% up to 27.6%. Correspondingly, between the GD and CAC40 indices, they range from 1% up to 30%. The (m/5) RCt features a lower standard deviation than the other esti-

(BV )

mators, across the board of correlations. The RCt (Par) skewness and kurtosis values. The RCt

has the highest

has the highest mean values and long-memory (d-values) estimates. For the DAX − CAC40 correlation, the mean is higher than the standard deviation in all estimators (Figs. 19 and 20). In the rest (two) correlations, there is no estimator for which the mean is higher than the standard deviation. The skewness values are negative in all estimators for the DAX − CAC40 correlation. In the other two correlations, there are negative skewness values only for the RCt(mopt ) estimator. All in all, there are no excess skewness values. Though, the kurtosis values are close to 3, there are excess kurtosis values in all estimators for the DAX − CAC40 correlation and in the RCt(mopt ) for the other two correlations. (Par) takes the highest d-values for both GD − DAX and The RCt GD − CAC40 correlations. For the DAX − CAC40 correlation, the (m/5) RVt takes the highest d-values. The DAX − CAC40 correlation

D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290 Table 8B Estimation results for HAR model.

Table 7B Asymmetries. DAX − CAC40 Panel A. R

GD − DAX

GD − CAC40

2

(m/5)

RCt (BV ) RCt (Par) RCt RCt(mopt ) (m/5)

(BV )

RCt

(Par)

RCt

RCt(mopt )

(m/5)

RCt (BV ) RCt (Par) RCt RCt(mopt )

0.048 0.037 0.048 0.005

0.055 0.045 0.049 0.006

0.588∗∗∗ (<2.2E−16) 1.659 (0.159) 3.052∗∗ (0.016) 0.385 (0.820)

13.214∗∗∗ (1.67E−10) 10.125∗∗∗ (6.04E−8) 12.965∗∗∗ (2.67E−10) 1.428 (0.223)

15.083∗∗∗ (5.46E−12) 12.147∗∗∗ (1.21E−9) 13.444∗∗∗ (1.10E−10) 1.723 (0.143)

Table 8A Estimation results for HAR model. DAX − CAC40

GD − DAX

GD − CAC40

0.008 (0.037) 0.019 (0.038) 0.021 (0.038) 0.003 (0.032)

0.007 (0.035) −0.002 (0.035) 0.020 (0.037) −0.004 (0.032)

0.215∗ (0.083) 0.094 (0.085) 0.203∗ (0.083) 0.025 (0.076)

0.198∗ (0.085) 0.147 (0.085) 0.191∗ (0.088) −0.084 (0.101)

0.698∗ (0.083) 0.799∗ (0.088) 0.700∗ (0.084) −0.036 (0.164)

0.711∗ (0.082) 0.762∗ (0.083) 0.702∗ (0.085) 0.324∗ (0.149)

Panel A. Estimates of ˇD

(BV )

RCt

(Par)

RCt

RCt(mopt )

0.130∗ (0.049) 0.087∗ (0.036) 0.102∗ (0.039) 0.060 (0.033)

Panel B. Estimates of ˇW (m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

0.420∗ (0.088) 0.354∗ (0.073) 0.433∗ (0.070) 0.386∗ (0.073)

Panel C. Estimates of ˇM (m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

0.220∗ (0.088) 0.324∗ (0.085) 0.199∗ (0.077) 0.267∗ (0.099)

Table entries report the regression coefficient estimates from the HAR model def

(j)

(j)

with regression function given by Eq. (15), i.e. xTt−1 ˇ = ˇ0 + ˇD RVt−1,t + ˇW RVt−5,t + (j) ˇM RVt−20,t

GD − CAC40

0.229 0.186 0.191 0.145

0.403 0.353 0.394 0.001

0.376 0.336 0.373 0.005

108.12∗ (<2.2E−16) 105.24∗ (<2.2E−16) 108.27∗ (<2.2E−16) 78.25∗ (<2.2E−16)

243.64∗ (<2.2E−16) 197.82∗ (<2.2E−16) 234.39∗ (<2.2E−16) 0.05 (0.985)

217.55∗ (<2.2E−16) 183.14∗ (<2.2E−16) 215.26∗ (<2.2E−16) 1.72 (0.162)

Panel B. F-test

takes the highest d-values, with the GD − DAX correlation second and the GD − CAC40 correlation third, across the board of estimators. The d-values of the realized correlation estimators are lower than those of the realized volatility estimators and not higher than 28%. Moreover, the RCt(mopt ) estimator takes negative d-values for both GD − DAX and GD − CAC40 correlations. This result triggers the question whether the ARFIMA models can forecast the realized correlations as accurately as the realized volatilities.

(m/5)

GD − DAX

Panel A. R of the regressions 0.002 0.005 0.009 0.001

Table entries report the R2 and the F-test of the asymmetric regressions for the corresponding variables in each panel. The p-values of the F-test there are in parenthesis. The levels of significance are given as follows: ∗∗ 5% and ∗∗∗ 1%.

RCt

DAX − CAC40 2

Panel B. F-test RCt

285

and dependent variable the corresponding volatility series in the columns of the table. The ‘*’ indicate significance in a 5% significance level. Heteroskedasticity and autocorrelation consistent (Newey–West) standard errors used in the calculation of the corresponding significance levels are reported in parenthesis.

(m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

Table entries report the R2 and the F-test of the HAR regressions for the corresponding variables in each panel. The ‘*’ denotes significance in a 1% significance level. The p-values of the F-test there are in parenthesis.

In order to compare the four realized correlation estimators (as in Yeh & Kuan, 2007), it is applied the Kolmogorov–Smirnov test (where, the alternative hypothesis is two-sized). This test aims at testing whether two samples have been drawn from the same population distribution (see Table 6). For all three correlations, there are statistically significant differences across the three estimators (BV ) at 1% significance level. For the GD − DAX correlation, the RCt and (Par)

have significant differences at 10% and for the GD − CAC40 RCt correlation, their differences are not significant at all. The results from the K–S test reveal the importance of each realized correlation estimator and the different properties they acquire. 4.3. Asymmetries in realized correlations The response of realized correlations to news is quantified by the asymmetry regression (Tables 7A and 7B). The asymmetries in realized correlations are revealed by: (i) the F-test of the asymmetry regression (Table 7B, panel B); (ii) the adjusted R2 -values of the asymmetry regression (Table 7B, panel A); and (iii) the value and the significance of ı-coefficient of the asymmetry regression (Table 7A, panel D). In order to examine time evolution of the parameters of the asymmetry model, I also use a rolling window estimation of 200 days. The more efficient the equity markets are the more likely is for the 5-min realized correlations to capture the incremental intraday information. This becomes clear with the significance of the volatility feedback effect (effect of positive returns on realized correlations; g-coefficient) and the leverage effect (total effect of negative returns on realized correlations; ␥+␦-coefficient). According to Figs. 6, 7, 13, 14, 21 and 22, both asymmetry (␥− and ␥+␦−) coefficients are statistically significant for all three correlations because as we can see, their bands are not close to zero for most of the sample period. However, there are estimates with zero values for a short period of time (i.e. a few days) throughout the sample period. For the CAC40-DAX correlation (Figs. 6 and 7), both asymmetry coefficients feature higher and positive values as time evolves. This is supportive of the increasing positive and direct (not lagged) information change between the French and German equity markets. For the GD-DAX correlation (Figs. 21 and 22), both asymmetry coefficients present lower and negative values as time evolves. This supports the declining and lagged information transfer between the Greek and German equity markets. For the GD-CAC40 correlation (Figs. 13 and 14), both asymmetry coefficients increase over time with only pos-

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D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290

Fig. 16. Correlation series for Greek and German markets.

Fig. 17. Actual vs. fitted values from HAR model – Greek and German market.

Fig. 18. Actual vs. fitted values from HAR-J model – Greek and German market.

D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290

287

Fig. 19. Actual vs. fitted values from asymmetry regression – Greek and German market.

Fig. 20. (a) Rolling estimates of daily coefficient from HAR-J; (b) rolling estimates of weekly coefficient from HAR-J; (c) rolling estimates of monthly coefficient from HAR-J; (d) rolling estimates of jump coefficient from HAR-J.

itive values for the first half of the sample period; and, in the other half, they increase over time with only negative values. This shows there was a period in the past when the information between the Greek and French markets flow and without timelags; though, there was another more recent period when the same information was transferred more difficultly and with timelags. According to Table 7B (panel A), the adjusted R2 -values are not very high. They range from 0.1% up to 5.5%, indicating weak asymmetry effects between lagged returns and correlations. The GD − CAC40 correlation has the highest adjusted R2 -values and the (m/5) DAX − CAC40 correlation, the lowest. Moreover, the RCt has the 2 highest R -values, across the board of correlations. According to the F-test (Table 7B, panel B), the asymmetry regressions are not significant only when the RCt(mopt ) is used. The importance and the size of the contemporaneous relationship between return and correlation is given by the ı-coefficient. According to Table 7A (panel D), there are significant and neg-

ative ı-coefficients for: (i) the DAX − CAC40 correlation for the (BV ) (Par) and RCt estimators; (ii) the GD − DAX correlation for the RCt (m/5)

RCt

(BV )

, RCt

(Par)

and RCt

(BV ) theRCt

estimators; and (iii) the GD − CAC40 (Par)

correlation for and RCt estimators. The highest, in absolute-values, significant asymmetry effect (ı) is −5.234 (which (Par) is for the GD − CAC40 correlation and the RCt estimator). The GD − CAC40 correlation has the highest asymmetry effect, across (Par) the board of estimators. The RCt estimator has the highest asymmetry effect, across the board of correlations. The presence of asymmetries in realized correlations indicate that there is a correlation-return premium which has a reverse connection with correlation. Negative lagged returns suggest a stronger negative response to current correlation, when compared to the weaker positive response of current correlation to positive lagged returns. This might be explained as the result of the incremental information captured by the realized correlation estimators.

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D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290

Table 9A Estimation results for HAR-J model. DAX − CAC40

GD − DAX

GD − CAC40

0.132∗ (0.048) 0.041 (0.092) 0.058 (0.063) 0.134∗ (0.047)

0.008 (0.037) 0.035 (0.041) 0.036 (0.041) 0.014 (0.033)

0.007 (0.035) 0.010 (0.040) 0.032 (0.041) −0.010 (0.034)

0.216∗ (0.082) 0.091 (0.084) 0.198∗ (0.082) 0.022 (0.076)

0.198∗ (0.085) 0.147 (0.085) 0.190∗ (0.088) −0.077 (0.102)

DAX − CAC40

Panel A. Estimates of ˇD (m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

(m/5)

(BV )

RCt

(Par)

RCt

RCt(mopt )

GD − CAC40

0.697∗ (0.083) 0.795∗ (0.089) 0.693∗ (0.084) −0.035 (0.164)

0.712∗ (0.0822) 0.757∗ (0.084) 0.700∗ (0.085) 0.318∗ (0.149)

−0.295 (0.418) −0.030 (0.054) −0.043 (0.059) −0.134∗ (0.053)

−0.065 (0.362) −0.029 (0.047) −0.022 (0.047) 0.115 (0.070)

Panel C. Estimates of ˇM 0.215∗ (0.087) 0.325∗ (0.085) 0.201∗ (0.077) 0.249∗ (0.098)

Panel B. Estimates of ˇW RCt

GD − DAX

0.417∗ (0.087) 0.356∗ (0.073) 0.433∗ (0.069) 0.367∗ (0.072)

Panel D. Estimates of J −0.546 (0.469) 0.041 (0.072) 0.034 (0.034) −0.053∗ (0.021) def

(j)

(j)

(j)

Table entries report the regression coefficient estimates from the HAR-J model with regression function given by Eq. (16), i.e. xTt−1 ˇ = ˇ0 + ˇD RVt−1,t + ˇW RVt−5,t + ˇM RVt−20,t + J Jt−1,t and dependent variable the corresponding volatility series in the columns of the table. The ‘*’ denotes significance in a 5% significance level. The heteroskedasticity and autocorrelation consistent (Newey–West) standard errors used in the calculation of the corresponding significance levels are in parenthesis.

4.4. HAR-type models for correlation This section presents the results from the estimation of the HAR-type models given by Eqs. (15) and (16). The results concern the levels of the HAR and HAR − J models. Every regression coefficient together with the values of the F-test and R2 are reported in separate tables, for each regression and across the board of volatility estimators. The significance of the estimated coefficients is assessed using heteroskedasticity and autocorrelation consistent (Newey–West) standard errors. 4.4.1. HAR models Like in the asymmetry regressions, I examine the time evolution of the parameters of the HAR and HAR-J models by a rolling window estimation of 200 days. Both HAR and HAR-J models fit quite well to the actual correlation values for all three correlations (Figs. 3, 10 and 17). All in all, heterogeneity exists in all three correlations and also in all three lagged times (daily, weekly and monthly). Tables 8A and and 8B contains the results for the HAR model. First, I note that the magnitude of the estimated coefficients declines with the horizon, i.e. the daily impact is larger than the weekly impact which, in turn, is larger than the monthly impact: ˆD > ˇ ˆW > ˇ ˆ M . The DAX − CAC40 correlation is the single excepˇ tion. Most estimates are positive and highly significant. These results were also reported by Corsi and Audrino (2007). This is an anticipated sign of strong volatility persistence, as analyzed thoroughly by Corsi and Audrino (2007).6 Also, I note that the magnitude of the estimates is nearly undifferentiated across the board of realized correlation estimators. However, the RCt(mopt ) has the lowest coefficient estimates, which for some correlations are even negative. The overall average daily impact is 0.038, across the correlations and estimators. The overall average weekly impact is 0.215 and the overall average monthly one is 0.473. Because of the highest impact, there is – in average – a monthly (20-day) persistence (long-memory). Corsi and Audrino (2007) deduced a weekly (5-day) persistence. Finally, the HAR models are highly

6

The HAR-class of models was designed to actually capture this characteristic of volatility data.

significant. This is evident by the F-test and the high adjusted R2 values. The F-test statistics reveal significance for the HAR model across the board of correlations and estimators. The adjusted R2 values range from 0.1% up to 40.3%, yet Corsi and Audrino (2007) (m/5) reported extremely high R2 -values up to 81%. The RCt estimator 2 has both the highest adjusted R -values and the highest F-statistics, and the RCt(mopt ) has the lowest adjusted R2 -values and the lowest F-statistics, which indicate no significant HAR regression for both GD − DAX and GD − CAC40 correlations. 4.4.2. HAR − J models As in the HAR model, the HAR-J model fits well to the actual correlations (Figs. 4, 11 and 18). The daily, weekly, monthly and jump coefficient estimates from the HAR-J model change over time and are significant (different from zero), for the CAC40-DAX (Fig. 5) and GD-CAC40 (Fig. 12) correlations. Only for the GD-DAX correlation (Fig. 20), the daily and jump coefficients are close to zero for most of the sample period. The jump component of correlations also explains well the correlations themselves. Tables 9A and 9B include the estimation results from the HAR − J models. In these models, the daily, weekly and monthly correTable 9B Estimation results for HAR-J model. DAX − CAC40

GD − DAX

GD − CAC40

0.240 0.198 0.200 0.150

0.405 0.358 0.396 0.003

0.378 0.340 0.378 0.008

83.05∗ (<2.2E−16) 78.96∗ (<2.2E−16) 81.45∗ (<2.2E−16) 60.74∗ (<2.2E−16)

182.87∗ (<2.2E−16) 147.85∗ (<2.2E−16) 175.91∗ (<2.2E−16) 0.375 (0.827)

163.24∗ (<2.2E−16) 137.44∗ (<2.2E−16) 161.39∗ (<2.2E−16) 1.45 (0.215)

Panel A. R2 of the regressions (m/5)

RCt (BV ) RCt (Par) RCt RCt(mopt ) Panel B. F-test (m/5)

RCt

(BV )

RCt

(Par)

RCt

RCt(mopt )

Table entries report the R2 and the F-test of the HAR-J regressions for the corresponding variables in each panel. The ‘*’ denotes significance in a 1% significance level. The p-values of the F-test there are in parenthesis.

D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290

289

Fig. 21. (a) Rolling estimates of ˛ from asymmetry model; rolling estimates of ˇ from asymmetry model; (c) rolling estimates of  from asymmetry model; (d) rolling estimates of ı from asymmetry model.

Fig. 22. Rolling estimates of  + ı from asymmetries model.

lations, and also the jump component enter separately into the regression and thus we report on the individual and combined effects. These results corroborate the previous ones on the HAR models. As in the case of the HAR model, also in the HAR − J model (Table 9A), the magnitude of the estimated coefficients declines with the horizon, i.e. the daily impact is larger than the weekly impact which, in its turn, is larger than the monthly. Again, the DAX − CAC40 correlation is the only exception. Also, the daily, weekly and monthly estimates are higher than the jump estimates. Moreover, most jump-coefficient estimates are negative. The overall average daily impact is 0.041 while the weekly impact is 0.213 and the monthly is 0.469. The jump impact is −0.086 and the overall average combined impact is 0.021, across the correlations and realized correlation estimators. This confirms the relative importance of the daily and jump impacts. The impact of the jump components is presented in Table 9, Panel D. For most correlations and realized correlation estimators, the estimates of the J -coefficient are insignificant. They are sig-

nificant when the RCt(mopt ) estimator is used for the DAX − CAC40 and GD − DAX correlations. The negative coefficients are in accordance with ABD (2007), where realized volatilities are used. These estimates also indicate that jumps tend to reverse the direction of future correlation and also to temper its persistence. The fact that some jump-coefficients are positive suggests that there is a correlation reversal effect in a daily basis, where jumps are present in the sample path of correlation. As in ABD (2007), I compute the combined effect of the daily, weekly and monthly volatilities ˆD + ˇ ˆ W /5 + ˇ ˆ M /20 + ˆ J . together with the effect of the jump, i.e. ˇ This combined effect is always positive. Finally, note the adjusted R2 -values of Table 9B, Panel A range between 0.3% to 40.5%. The values of the adjusted R2 -values from these regressions are higher than the ones obtained from the HAR model. This further supports the relative importance of the jump component of correlation. The significance of the HAR − J models is also revealed by the highly significant statistics of the F-test. The HAR − J model is not significant only for the RCt(mopt ) estimator in both GD − DAX and GD − CAC40 correlations.

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D.I. Vortelinos / The Quarterly Review of Economics and Finance 50 (2010) 273–290

5. Concluding remarks

References

The daily returns are not distributed normally, with serial autocorrelation in levels and stronger serial autocorrelation in squares, implying volatility clustering. They also have very weak and even negative long-memory (d-values) estimates. Daily returns seem to follow a standard Geometric Brownian motion. The realized volatility estimators of all three indices are not normally distributed. They have excess skewness, excess kurtosis and strong long-memory (persistence). The RCt(mopt ) takes the highest excess skewness and kurtosis values. The highest mean correlations occur for the DAX − CAC40 correlation. All four realized correlations seem to be normally distributed with weak longmemory. According to the Kolmogorov–Smirnov test, there are significant differences between the four realized correlation estimators. The DAX − CAC40 correlation has the highest long-memory per(Par) sistence. The highest long-memory persistence is for the RCt . The asymmetry regressions are highly significant, according to the adjusted R2 -values and the statistic-values of the F-test. The asymmetry effects (ı-coefficient) are also highly significant with negative value. The GD − CAC40 correlation and the (Par) RCt estimator have the highest asymmetry effects, across the board of estimators and across the board of correlations, respectively. When correlations are modeled by both HAR and HAR − J models, the magnitude of the estimated coefficients decreases with the horizon (Figs. 21 and 22) (i.e. from monthly to weekly and then to daily realized correlations), with the single exception for the DAX − CAC40 correlation. The heterogeneity in realized correlations is highly significant and with meaningful coefficients for all correlations and realized correlation estimators. There are quite enough jumps of correlations above and below the two specific threshold values. Most jump-coefficient estimates are negative, indicating their importance and the reverse connection between correlations and their jumps. This negative sign means that by the time the jumps increase, the correlations of the next day decrease. The most important result is that there is no best-of-the-class realized correlation estimator across the ones used, regarding the descriptive statistics, long-memory, detected jumps and modeling of correlations and their jumps. Perhaps, it would be more thorough to use more non-parametric contemporaneous realized volatility estimators for estimating the realized correlations and modeling them, under this framework. The results revealing the lack of normality and the presence of long-memory, asymmetries and jumps in realized correlations direct my research towards forecasting these realized correlations taking into account these properties. Moreover, the results concerning the DAX − CAC40 correlation seem more robust than the other correlations, in terms of descriptive statistics, long-memory and modeling correlations and their jumps. So, the selection of this correlation would have been the most appropriate for both the forecasting exercises and the usage of new non-parametric realized correlation estimators.

Andersen, T. G., Bollerslev, T., & Diebold, F. X. (2007). Roughing it up: Including jump components in the measurement, modeling and forecasting of return volatility. The Review of Economics & Statistics, 89, 701–720. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (1999). Realized volatility and correlation. Working paper, Northwestern University. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2001). The distribution of exchange rate volatility. Journal of the American Statistical Association, 96, 42–55. Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2005). The distribution of exchange rate volatility. In Shephard, N. (Ed.), Stochastic volatility: Selected readings (pp. 451–479, Chapter 15); In Arellano, M., G. Imbens, G. E. Mizon, A. Pagan, & M. Watson (Eds.), Advanced texts in economics. Oxford, UK: Oxford University Press. Andersen, T. G., Bollerslev, T., Frederiksen, P., & Nielsen, M. (2010). Continuous-time models, realized volatilities, and testable distributional implications for daily stock returns. Journal of Applied Econometrics, 25, 233–261. Bandi, F., & Russell, J. R. (2005). Realized covariation, realized beta, and microstructure noise. Working paper, The University of Chicago. Bandi, F., & Russell, J. R. (2006). Separating market microstructure noise from volatility. Journal of Financial Economics, 79, 655–692. Bannouh, K., Van Dijk, D., & Martens, M. (2009). Range-based covariance estimation using high-frequency data: The realized co-range. Journal of Financial Econometrics, 7, 341–372. Barndorff-Nielsen, O. E., & Shephard, N. (2002). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society B, 64, 253–280. Barndorff-Nielsen, O., & Shephard, N. (2004). Econometric analysis of realized covariation: High-frequency covariance, regression and correlation in financial economics. Econometrica, 72, 885–925. Barndorff-Nielsen, O. E., & Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics, 2, 1–48. Barndorff-Nielsen, O. E., & Shephard, N. (2005). Variation, jumps, market frictions and high frequency data in financial econometrics. In Blundell, R., P. Torsten, & W. K. Newey (Eds.), Advances in economics and econometrics: Theory and applications. Ninth World Congress. Barndorff-Nielsen, O. E., & Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics, 4, 1–30. Bollerslev, T., & Zhou, H. (2006). Volatility puzzles: A unified framework for gauging return-volatility regressions. Journal of Econometrics, 131, 123–150. Brandt, M. W., & Diebold, F. X. (2006). A no-arbitrage approach to range-based estimation of return covariances and correlations. Journal of Business, 79, 1–13. Christiansen, C., & Ranaldo, A. (2007). Realized bond-stock correlation: Macroeconomic announcement effects. Journal of Futures Markets, 27, 439–469. Choi, K., Yu, W.-C., & Zivot, E. (2009). Long memory versus structural breaks in modeling and forecasting realized volatility. Journal of International Money and Finance, forthcoming. Corsi, F. (2009). A simple long memory model for realized volatility. Journal of Financial Econometrics, 7, 174–196. Corsi, F., & Audrino, F. (2007). Realized correlation tick-by-tick. Working paper, no. 2007-02, University of St. Gallen. Corsi, F., Pirino, D., & Reno, R. (2008). Volatility forecasting: The jumps do matter. Working paper, Università di Siena. Ferland, R., & Lalancette, S. (2006). Dynamics of realized volatilities and correlations: An empirical study. Journal of Banking and Finance, 30, 2109–2130. Giot, P., Laurent, S., & Petitjean, M. (2010). Trading activity; realized volatility and jumps. Journal of Empirical Finance, 17, 168–175. Goncalves, S., & Meddahi, N. (2009). Bootstrapping realized volatility. Econometrica, 77, 283–306. Huang, X., & Tauchen, G. (2005). The relative contribution of jumps to total price variance. Journal of Financial Econometrics, 3, 456–499. Martens, M., & van Dijk, D. (2007). Measuring volatility with the realized range. Journal of Econometrics, 138, 181–207. Thomakos, D., & Wang, T. (2003). Realized volatility in the futures markets. Journal of Empirical Finance, 10, 321–353. Wang, T., Wu, J., & Yang, J. (2008). Realized volatility and correlation in energy futures markets. Journal of Futures Markets, 28, 993–1011. Yeh, J., & Kuan (2007). Realized volatility and correlation for non-synchronously traded financial assets. Working paper, Yuan Ze University.