Explaining international stock correlations with CPI fluctuations and market volatility

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Explaining international stock correlations with CPI ﬂuctuations

and market volatility

Yijie Cai

a

, Ray Yeutien Chou

b,*

, Dan Li

a

Jinhe Center for Economic Research, Xi’an Jiaotong University, China

b_{Institute of Economics, Academia Sinica and National Chiao-Tung University, Taiwan}

a r t i c l e

i n f o

Article history: Received 6 October 2008 Accepted 18 May 2009 Available online 22 May 2009 JEL classiﬁcation:

C32 E44 G15 Keywords:

International stock markets CPI rates

Global volatility Smooth transition CARR

a b s t r a c t

This paper investigates the dynamic correlations among six international stock market indices and their relationship to inflation fluctuation and market volatility. The current research uses a newly developed time series model, the Double Smooth Transition Conditional Correlation with Conditional Auto Regres-sive Range (DSTCC-CARR) model. Findings reveal that international stock correlations are significantly time-varying and the evolution among them is related to cyclical fluctuations of inflation rates and stock volatility. The higher/lower correlations emerge between countries when both countries experience a contractionary/expansionary phase or higher/lower volatilities.

1. Introduction

International stock market correlations have attracted more attention with the integration and globalization of financial mar-kets. A wealth of qualitative literatures devoted to the intriguing connection between financial markets and economic fundamentals provide sufficient evidences that co-movement of business-cycle fluctuations impact international financial market correlations. However, the controversy continues. Debates on whether eco-nomic fundamentals such as business cycle indicators significantly affect international financial correlations, surfaced in the early 1990s, and have not yet reached a consistent agreement.

Erb et al. (1994)found that correlations between two equity markets vary according to both countries’ economic cycles that economic fundamentals signiﬁcantly affect stock market correla-tions. They show that among the G-7 countries, the highest corre-lations appear when both countries stand in the contractionary phase and lowest correlations appear when both countries are in the expansionary phase. Correlations vary between these two

ex-treme states when they are out of phases. Dumas et al. (2003)

highlighted the statistical evidence that output correlations and stock market correlations are positively related.Forbes and Chinn (2004)showed that direct trade is the predominant factor of the world’s largest markets that affect ﬁnancial markets.Yang et al.

(2009) investigated dynamic interdependence between

interna-tional stock and bond markets affected by real economy (repre-sented as the business cycle, the inﬂation environment and monetary policy stance). Furthermore, they supplied evidence that higher stock-bond correlation coincides with higher short rates and higher inﬂation rates.

On the contrary, other literatures maintain skeptic upon such association between real economic linkages and ﬁnancial-market linkages.King et al. (1994)suggested that co-variances between international stock markets are difﬁcult to interpret by observable economic variables, and can reverse by unobservable variables.

Ammer and Mei (1996) discovered that contemporaneous

co-movement in macroeconomic variables inﬂuence co-variances be-tween international stock markets. However, they ignore this rela-tionship because the real linkages are much stronger in the

long-run than a short-long-run perspective. Kizys and Pierdzioch (2006)

supported Ammer and Mei, showing that the linkage between

monthly conditional international equity correlations and

co-movement of business-cycle ﬂuctuations is not signiﬁcant enough.

*Corresponding author. Tel.: +886 2 27822791; fax: +886 2 27853946.

E-mail addresses: [email protected] (Y. Cai), [email protected]

(R.Y. Chou),[email protected](D. Li).

Contents lists available atScienceDirect

Journal of Banking & Finance

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Recent researches have also focused on the linkages between international stock correlations and market volatility.Longin and Solnik (2001) found that correlation increased in bear markets, but not in bull markets and international integration tightens the ﬁnancial linkage progressively.Connolly et al. (2007)offered plen-tiful evidence that international stock linkages are likely higher/ lower when the level of implied volatility (as a measure of stock

uncertainty) stays higher and its variation is larger. Aydemir

(2008)indicated that the higher the risk aversion periods, the high-er the tendency for market correlations and high market volatility to emerge at the same time.

Besides,Ferreira and Gama (2007)showed that sovereign debt

ratings news tends to increase the international stock market cor-relations. Another literature focuses on the factors explaining the

stock-bond correlations. For example, see Kim et al. (2006), Li

and Zou (2008) and Panchenko and Wu (2009).

Motivated by earlier conﬂicting reports, this research restudies the relationship between economic fundamentals as well as global stock volatility and international stock market interdependence. The current work employs a range-based multivariate volatility

model byChou and Cai (2009). The smooth transition in

condi-tional correlation is controlled by some exogenous variables. The model maintains a parsimonious structure while allowing

ﬂexibil-ity in specifying the dynamic evolutions of conditional

correlations.

This paper is organized as follows. Section 2 introduces the

model including model speciﬁcations, dynamics and tests. Section 3discusses the data set used for the empirical research. Section4 provides empirical results. Section5concludes this paper. 2. The model

Following Chou and Cai (2009), consider the Double Smooth

Transition Conditional Correlation-Conditional Autoregressive

Range (DSTCC-CARR) model. It is an extension of the Dynamic Con-ditional Correlation (DCC) model ofEngle (2002). Two main fea-tures of this model are the additional efﬁciency in using range data (seeChou, 2005; Chou et al., 2009) and the consideration of a ﬂexible mechanism in the correlation dynamics.

2.1. The DSTCC-CARR model

Specifically, the DSTCC-CARR model is constructed with two steps: the CARR specification for estimating volatilities and the smooth transition structure of the conditional correlation allowing more than one explanatory (or transition) variable. For the bivari-ate case, the CARR specification is defined as Eq.(1):

Ri;t¼ ki;t

e

i;t;

e

i;tjIt1 f ð1; Þ; t ¼ 1; 2; . . . ; T; i ¼ 1; 2;

ki;t¼

-

iþ

a

iRi;t1þ biki;t1;

ð1Þ

where the high/low range in logarithm type, of the ith asset during time t is denoted as Ri;t, with a conditional mean of the range ki;t. The distribution of the disturbance term

e

i;t is assumed to be dis-tributed with a density function f ðÞ with a unit mean. Next, the unconditional standard deviation

r

iand the sampling mean of the estimated conditional range ^kiare used to construct an adjustment term (adj) as a ratio. The ratio is used to scale the conditional stan-dard deviation k

i;tfrom ki;t, the expected range from the CARR mod-el. In other words, denote the ithasset return as ri;t and let zi;t be deﬁned as the standardized return:

z

i;t¼ ri;t=ki;t; where k

i;t¼ adji ki;t; adji¼

r

i=^ki: ð2Þ In the second stage, the standardized returns are then used to compute the conditional correlations. InEngle (2002)’s DCC model, the conditional correlations are allowed to vary according to a

GARCH type dynamics. In our formulation of DSTCC, however, the correlations are governed to move smoothly among four re-gimes. Speciﬁcally, let sitbe some exogenous variable, the smooth transition structure of the conditional correlation is deﬁned as following:

Pt¼ ð1 FL1ðs1tÞÞPð1Þtþ FL1ðs1tÞPð2Þt;

PðjÞt¼ ð1 FL2ðs2tÞÞPðj1Þþ FL2ðs2tÞPðj2Þ; j ¼ 1; 2;

ð3Þ

where both transition functions are logistic:

FLjðsitÞ ¼ ð1 þ ecjðsjtcjÞÞ1;

c

j>0; j ¼ 1; 2: ð4Þ Two parameters, named as location parameter cjand speed param-eter

c

j, are used to control the transition from one state to the other. The larger

c

_jis, the faster the correlation changes from one state to the other. If

c

j! 1, the transition function becomes a step func-tion. For details, see Chou and Cai (2009).

Therefore, a DSTCC-CARR model supposes that conditional cor-relation has four extreme states, and switches among these four states ðPð11Þ;Pð21Þ;Pð12Þ and Pð22ÞÞ smoothly under the control of two exogenous transition variables.

Once

c

_j¼ 0; j ¼ 1or2, a DCARR model reduces to an STCC-CARR model. Taking

c

1¼ 0 for example, Eq.(3)should be rewritten as Eq.(5): Pt¼ ð1 FL2ðs2tÞÞP1þ FL2ðs2tÞP2; ð5Þ where P1¼ 1 2ðPð11Þþ Pð21ÞÞ; P2¼ 1 2ðPð12Þþ Pð22ÞÞ:

To complete the model, we followSilvennoinen and Teräsvirta (2005, 2007)in assuming a Gaussian distribution for the joint den-sity function of the standardized returns. Quasi-maximum likeli-hood methods are used for estimation of the parameters and covariance matrices. The Gaussian assumption may be relaxed to allow more fat-tailed conditional density functions. Further more, more ﬂexibility can be obtained by using the copula density func-tions. We do not pursue these approaches in the current study to maintain the tractability of our model.

2.2. Model speciﬁcation tests

Since estimating a model with unnecessary parameters causes inefficiency, specification tests are useful before estimating the DSTCC-CARR model. The tests may help determine whether the exogenous variables are useful as transition variables. Note that some of the model parameters are not identified under the null hypothesis.Luukkonen et al. (1988)adopt a linearization by first-order Taylor expansion around speed parameters to construct the test statistics. Their strategy is followed here. The detailed specifi-cation shows as Eq.(6):

FLiﬃ 1=2 þ 1=4ð

c

iðsit ciÞÞ þ oðÞ; ð6Þ oðÞ is the error term above the second-order.

2.2.1. Tests for CCC against a STCC-CARR model

Based on the structure of the STCC-CARR model as in(5), this

work performs a ﬁrst-order Taylor approximation around

c

₂¼ 0

to the transition function FL2. The dynamic conditional correlations could be written as(7):

P t ¼ P

1þ stP2þ oðÞ: ð7Þ Under the hypothesis: H0:

c

2¼ 0, the STCC-CARR model be-comes a CCC-CARR model. The current study constructs an LM test for conditional correlation constancy against an STCC-CARR model, and the LM statistics are shown as(8):

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LMCCC1¼ T1 XT t¼1 @ltð^hÞ @q0 2 ! ½^ITð^hÞ1ðq 2;q 2Þ XT t¼1 @ltð^hÞ @q0 2 ! : ð8Þ

The derivation of(8)is given inAppendix A. 2.2.2. Tests for CCC against a DSTCC-CARR model

Based on the structure of the DSTCC-CARR model represented as

(3), this study carries out a ﬁrst-order Taylor approximation

around

c

1¼ 0 and

c

2¼ 0 to the transition function FL1 and FL2 respectively. The dynamic conditional correlations could be shown as(9):

P t ¼ P

ð1Þþ s1tPð2Þþ s2tPð3Þ þ s1ts2tPð4Þþ oðÞ: ð9Þ Under the hypothesis:H0:

c

1¼

c

2¼ 0, the DSTCC-CARR model simpliﬁes as a CCC-CARR model. The LM test for constant condi-tional correlations against a DSTCC-CARR model is listed as(10):

LMCCC2¼ T1 XT t¼1 @ltð^hÞ @ðq0 ð2Þ;q0ð3Þ;q0ð4ÞÞ ! ½^ITð^hÞ1ðq ð24Þ;q ð24ÞÞ X T t¼1 @ltð^hÞ @ðq0 ð2Þ;q0ð3Þ;q0ð4ÞÞ ! : ð10Þ

The derivation of(10)is also given inAppendix A. Note that we do not consider the test of the DSTCC-CARR against a DCC model. Unlike the CCC-CARR model which is a special case of the CARR model, the DCC model is not nested by the DSTCC-CARR model. Comparisons of the two models will rely on other types of test statistics and are not pursued here. In this paper, we purposefully preclude the conditional correlations to ﬂuctuate too wildly (as DCC would allow). A smoother and more tractable dynamic structure is given by our DSTCC speciﬁcation, although its null of CCC may be ‘‘too” simple.

3. Data

The current study chooses six international stock markets to cover primary ﬁnancial markets in the world: the US, UK, France and Germany, as representatives of developed western countries in this study, and Hong Kong and Japan, who play irreplaceable roles as Asian ﬁnancial centers.1_{The data consists of three groups:}

a series of stock market indices, consumer price index rates and a CBOE volatility index (VIX).

3.1. Stock indices, returns and ranges

Our stock market data include daily ‘‘high, low and close” price of six stock indices. The original data was extracted from the

web-site Yahoo, China.Table 1presents the original dataset speciﬁca-tion, ranging from February 2, 1991 to May 31, 2007.

Stock returns and ranges are computed by 100 logðpclose t =pcloset1Þ and 100 logðphigh

t =plowt Þ respectively. To compare the correlations among different indices, this work revises the dataset by the fol-lowing rules. (1) Delete daily data when some markets have miss-ing values; (2) cut off the outliers to avoid probable estimation problem; and (3) set daily range to the mean value when there is no change during the day. Details are given in Panel A and B of Ta-ble 2and Figs. B.1–B.3.2 _{All returns and ranges exhibiting excess}

kurtosis and Jarque-Bera tests clearly reject the null of a Gaussian distribution in all cases, so it is appropriate to use the CARR model proposed by Chou (2005).

3.2. CPI rates

Movements in Consumer Price Index (CPI) imply whether the economy goes through inflation or not. As financial markets are absolutely influenced by macro economy trends, CPI may be a meaningful variable to build up correlations between two stock markets.

Six countries’ (regions) CPI’s are downloaded from the IFS3

data-base. Annualized CPI rates are calculated by the formula rateCPI

t ¼ 100 ðCPIt CPIt12Þ=CPIt12. The sample range is from 1991.1 to 2007.4. Panel C ofTable 2andFig. B.4give the details of the sample. At ﬁrst glance, all six countries (regions) have been in expansion phase since the early 1990s, except the USA, and all have gone through a contraction phase since the end of the 20th century. This suggests that the economy is receding in most developed coun-tries. Hong Kong appears as having the largest volatility in the past 17 years, while France holds the most stable state among them.

For solving mismatch between monthly CPI data and daily stock market indices data, monthly CPI data are converted into daily data. We simply allow the CPI to remain constant across the whole monthly days.

3.3. Stock volatility

Since international investors are always reacting to information (including market volatility) obtained in open markets, linkages among international markets are connected with market volatility. Out of variables from past observations such as lagged returns, lagged absolute returns and so on, VIX outperforms other variables for measuring market risk. VIX is the ticker symbol for the Chicago Board Options Exchange (CBOE) volatility index, which represents market volatility expectations over the next thirty days, as well as the popular measure of implied volatility for the S&P 500 index op-tion. Since its introduction in 1993, VIX has become the world’s

1

An older version of the study employs three other markets, including Singapore, Taiwan, and China. As two members of four little dragons in Asia, Singapore and Taiwan are selected for their contributions to the world’s economy. China is involved due to its remarkable journey of becoming an open ﬁnancial market since joining the WTO in 2001. For brevity, the results are not reported here but can be obtained from the authors upon requests.

Table 1

Speciﬁcation of original six stock markets indices dataset.

Country/region Data name Time zone Opening hours

France/FR CAC 40 (CAC) GMT + 02:00 07:30 a.m.–15:00 a.m.

Germany/GER DAX (GDAXI) GMT + 02:00 07:30 a.m.–15:00 a.m.

Hong Kong/HK Hang Seng (HSI) GMT + 08:00 02:00 a.m.–04:30 a.m., 06:30 a.m.–08:00 a.m.

Japan/JP Nikkei 225 (N225) GMT + 09:00 00:00 a.m.–02:00 a.m., 03:00 a.m.–06:30 a.m.

UK FTSE 100 (FTSE) GMT + 01:00 07:30 a.m.–15:00 a.m.

USA 500 Index (GSPC) GMT-04:00 Daylight saving time: 01:30 p.m.–07:30 p.m. Winter time: 02:30 p.m.–08:30 p.m.

Notes: Time zone distribution and opening hours of stock markets are in Greenwich Mean Time.

2

For the sake of brevity, we put ﬁgures of indices, returns and ranges inAppendix

B.

3_{Unfortunately, CPI data of Germany in 1991 is missing, and we make use of the}

values from ‘‘Wind Information database, China” and generate a series of CPI rates in the same way.

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premier barometer of investor sentiment and market volatility, and is often referred to as the ‘‘investor fear gauge”. Index values exceeding 30 usually relate to a large amount of volatility, attrib-uted to investor fear or uncertainty. Contrarily, the index falling below 20 indicates less stressful, even complacent times in the markets. Panel D ofTable 2andFig. B.5show the details. 4. Model testing and estimation results

According to modeling speciﬁcation discussed in Section2, the

current work applies DSTCC-CARR models with ‘‘Aver_CPI”4 and

‘‘VIX” as transition variables for estimation. 4.1. Transition variables

This work mainly focuses on how correlations between two stock markets vary with different inﬂation cycles and worldwide stock volatility. Therefore, two transition variables are adopted: (1) ‘‘Aver_CPI”: average value of both countries’ CPI rates, is de-ﬁned by Aver_CPI = (CPI1 + CPI2)/2; and (2) ‘‘VIX”: is used as a common indicator of the worldwide stock volatility.

Empirically, varying stock interdependence may be decom-posed into two parts. One is high frequency changes related to the micro level of stock market movements, e.g., investor decision in terms of adverse selection, inventory costs, market power, and transaction costs. The other is low frequency (medium frequency) movements dominated by global macroeconomic shocks e.g., shifts in fundamentals, economic trends or preference changes, etc. To accommodate both of these two types of changes, the current re-search employs the above mentioned two transition variables. The variable VIX is a forward looking indicator of market risk. This index updates nearly every day as feedback for investors’

expecta-tions of future market uncertainty, so that it is a response to the high frequency variation of international stock correlations. On the other hand, CPI reflects both countries’ recent inflation and contains the information spread from real economy to financial markets. Macroeconomic influences usually penetrate gradually and evolve over time. As a result, it is more likely to capture low (medium) frequency changes, sometimes lasting several months or more.5_{Other variables such as GDP and interest rates may also}

be considered. However, the CPI variable turns out to be more useful empirically. To maintain the model tractability, this study does not pursue cases with more than two transition variables.

Table 2

Summary statistic of the data.

FR GER HK JP UK USA

Panel A: return series of the six stocks (1991.2.2-2007.5.31)

Mean 0.009 0.010 0.027 0.020 0.003 0.022 Median 0.016 0.057 0.041 0.022 0.023 0.041 Maximum 7.002 7.553 17.247 7.655 5.904 5.574 Minimum 7.575 9.871 10.000 7.234 5.589 7.113 Std. Dev. 1.296 1.371 1.559 1.428 1.005 0.982 Skewness 0.081 0.243 0.064 0.116 0.104 0.116 Kurtosis 5.839 6.841 11.461 5.244 6.366 6.806 Jarq-Bera 1111.048 2060.346 9839.907 699.267 1562.606 1997.487

Panel B: range series of the six stocks (1991.2.2-2007.5.31)

Panel C: the six monthly annual CPI rates (1991.1-2007.4)

Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis J-Bera

Panel D: VIX-volatility implied index (1991.2.2-2007.5.31)

18.555 17.090 45.740 9.820 6.364 1.087 4.082 809.773

Table 3

Results for CCC tests against STCC-CARR (DSTCC-CARR) models.

Aver_CPI (p-value) VIX (p-value) VIX and Aver_CPI (p-value)

FR_GER 0.000 0.000 0.000 FR_HK 0.013 0.006 0.000 FR_JP 0.001 0.333 0.001 FR_UK 0.001 0.000 0.000 FR_USA 0.022 0.098 0.001 GER_HK 0.682 0.278 0.001 GER_JP 0.000 0.001 0.000 GER_UK 0.000 0.000 0.000 GER_USA 0.000 0.000 0.000 HK_JP 0.000 0.000 0.000 HK_UK 0.001 0.000 0.000 HK_USA 0.006 0.258 0.039 JP_UK 0.006 0.354 0.020 JP_USA 0.019 0.013 0.000 UK_USA 0.223 0.909 0.634 4

We also try the transition variables forming of multiplying CPI rates, and the similar results are obtained. Results are available upon request.

5

In a previous version of this paper, a time trend is considered as a transition variable. However, combined with the speciﬁcation of the logistic transition function, such a model implies that the correlation between two stock markets should increase or decrease with calendar time monotonously. Hence it is not realistic in describing the real markets.

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4.2. Time zone and CCC tests

Obviously, stock markets in the current model locate in

differ-ent regions and time zones.Table 1shows Greenwich Mean Time

as the benchmark for measuring stock market opening hours. Stock market opening hours in Asia do not overlap with NYSE’s, while those in Europe overlap with NYSE’s one hour and a half. Asian and European markets present a similar situation.

This work accounts for time zone effect and makes the adjust-ment6 described above for cases without overlap. Reversely, this

work neglects time zone effect as long as two markets have concur-rent opening hours.7

4.3. Model speciﬁcation test

Are ‘‘Aver_CPI” and ‘‘VIX” competent for this research? Does the DSTCC-CARR model with inﬂation cycle and global volatility indi-cators as transition variables outperform the STCC-CARR model with these two indicators separately? Answering these questions requires some preliminary tests.

This study introduces the CCC tests for the constant conditional correlation null hypothesis in the STCC-CARR model and DSTCC-Table 4

LR test for STCC-CARR model against DSTCC-CARR model.

Log likelihood of the models LR statistics

STCC CARR (CPI) STCC CARR (VIX) DSTCC CARR LRCPI PCPI LRVIX PVIX

8014.186 8017.458 7979.180 76.557 0.000 70.012 0.000 9030.749 9073.163 9012.655 121.017 0.000 36.188 0.000 8973.813 8974.783 8964.619 20.329 0.000 18.388 0.000 10604.381 10599.835 10590.678 18.314 0.000 27.406 0.000 10659.538 10661.627 10651.779 19.697 0.000 15.518 0.000 8524.653 8564.267 8519.952 88.631 0.000 9.404 0.002 8039.480 8039.222 8029.962 18.520 0.000 19.037 0.000 9664.087 9666.138 9652.363 27.551 0.000 23.448 0.000 9721.289 9723.546 9710.277 26.539 0.000 22.024 0.000 9037.163 9048.484 9015.496 65.975 0.000 43.334 0.000 10614.404 10614.447 10591.438 46.018 0.000 45.932 0.000 10711.365 10722.819 10704.922 35.794 0.000 12.886 0.000 9505.333 9508.165 9495.682 24.966 0.000 19.303 0.000 9528.893 9531.991 9525.459 13.063 0.000 6.867 0.009 10844.897 10879.522 10828.210 102.623 0.000 33.374 0.000 Table 5

Estimation results of DSTCC-CARR with Aver_CPI and VIX as transition variables.

Part I Range parameters (former) Range parameters (latter)

-1 a1 b1 -2 a2 b2 q1 q2 q3 q4 c1 c2 c1 c2 FR_GER 0.120 0.126 0.778 0.166 0.095 0.765 0.989 0.783 0.426 0.728 0.884 11.451 0.612 0.101 (0.024) (0.020) (0.037) (0.028) (0.015) (0.038) (0.035) (0.012) (0.079) (0.024) (0.215) (0.457) (0.267) (0.043) FR_HK 0.075 0.140 0.797 0.020 0.108 0.867 0.217 0.090 0.476 0.341 5.499 16.071 0.685 0.314 (0.017) (0.024) (0.036) (0.008) (0.014) (0.018) (0.098) (0.050) (0.047) (0.022) (0.269) (0.659) (0.321) (0.211) FR_JP 0.095 0.171 0.750 0.043 0.142 0.824 0.630 0.102 0.473 0.199 1.119 20.557 384.195 1.208 (0.021) (0.029) (0.044) (0.013) (0.017) (0.022) (0.247) (0.095) (0.083) (0.019) (0.011) (0.235) (26.659) (0.775) FR_UK 0.061 0.092 0.857 0.064 0.105 0.826 0.838 0.662 0.598 0.916 2.558 17.929 3.190 0.016 (0.015) (0.016) (0.027) (0.011) (0.013) (0.023) (0.018) (0.036) (0.029) (0.024) (0.048) (0.392) (1.030) (0.004) FR_USA 0.092 0.153 0.770 0.026 0.114 0.852 0.249 0.397 0.785 0.412 2.329 11.423 7.420 0.612 (0.020) (0.025) (0.039) (0.007) (0.014) (0.019) (0.070) (0.037) (0.065) (0.022) (0.063) (0.285) (2.826) (0.329) GER_HK 0.125 0.107 0.788 0.012 0.105 0.875 0.251 0.402 0.866 0.221 2.353 16.014 0.077 500.000 (0.022) (0.018) (0.035) (0.006) (0.012) (0.016) (0.431) (0.030) (0.126) (0.079) (0.661) (0.021) (0.068) (23.472) GER_JP 0.126 0.103 0.790 0.045 0.141 0.825 0.430 0.011 0.989 0.317 4.067 31.905 0.113 3.863 (0.023) (0.018) (0.036) (0.013) (0.017) (0.022) (0.779) (0.046) (0.223) (0.109) (0.707) (0.106) (0.097) (1.182) GER_UK 0.142 0.092 0.790 0.076 0.120 0.799 0.988 0.713 0.047 0.514 1.621 10.631 0.479 0.145 (0.024) (0.015) (0.034) (0.014) (0.016) (0.028) (0.076) (0.014) (0.248) (0.035) (0.159) (0.983) (0.175) (0.083) GER_USA 0.134 0.101 0.786 0.026 0.119 0.847 0.792 0.741 0.244 0.443 1.689 15.815 32.344 0.042 (0.024) (0.017) (0.037) (0.007) (0.015) (0.020) (0.079) (0.047) (0.036) (0.032) (0.016) (0.652) (8.623) (0.025) HK_JP 0.013 0.108 0.872 0.039 0.124 0.846 0.594 0.151 0.968 0.518 5.234 20.209 0.276 75.587 (0.006) (0.013) (0.017) (0.011) (0.014) (0.019) (0.211) (0.041) (0.020) (0.017) (0.206) (0.023) (0.094) (1.063) HK_UK 0.023 0.110 0.863 0.045 0.123 0.825 0.178 0.430 0.572 0.114 1.453 15.933 0.069 7.094 (0.008) (0.014) (0.019) (0.009) (0.016) (0.023) (0.183) (0.028) (0.088) (0.062) (0.318) (0.074) (0.026) (1.057) HK_USA 0.017 0.104 0.873 0.020 0.111 0.862 0.758 0.803 0.487 0.346 7.822 16.759 2.837 0.271 (0.007) (0.014) (0.018) (0.006) (0.015) (0.020) (0.214) (0.183) (0.041) (0.018) (0.110) (0.225) (1.274) (0.092) JP_UK 0.042 0.140 0.827 0.047 0.126 0.820 0.873 0.453 0.259 0.199 1.265 17.544 396.768 4.952 (0.012) (0.016) (0.021) (0.010) (0.017) (0.025) (0.172) (0.068) (0.136) (0.018) (0.009) (0.133) (3.141) (2.785) JP_USA 0.037 0.136 0.834 0.018 0.105 0.870 0.761 0.398 0.109 0.329 1.551 11.057 496.311 0.234 (0.011) (0.016) (0.020) (0.006) (0.014) (0.018) (0.145) (0.083) (0.059) (0.028) (0.009) (3.458) (5.911) (1.135) UK_USA 0.052 0.126 0.815 0.022 0.116 0.854 0.184 0.852 0.384 0.416 3.622 16.441 14.242 0.070 (0.010) (0.016) (0.024) (0.006) (0.015) (0.020) (0.091) (0.172) (0.033) (0.022) (0.098) (0.384) (5.346) (0.034)

6 _{Taking the case ‘‘US_HK” for example, the series of stock markets in Hong Kong}

open 12 h earlier than American markets, so returns and ranges are lagged one period in estimation.

7_{Via this channel, for the cases of ‘‘FR_JP”, ‘‘UK_JP”, ‘‘GER_JP”, ‘‘USA_HK” and}

‘‘USA_JP”, both returns and ranges of the latter stock markets are lagged one period,

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CARR model and gives the LM statistics in Eqs.(8) and (10). Results reported inTable 3indicate that 13 out of 15 cases reject the CCC null hypothesis against the STCC-CARR model with ‘‘Aver_CPI” as the transition variable at a remarkable level of 5%,8_{and nine out}

of 15 cases reject the constant conditional correlation null hypothe-sis against the STCC-CARR model with ‘‘VIX” as the transition vari-able at a remarkvari-able level of 5%. After introducing both transition variables to construct a DSTCC-CARR model, all the components ex-cept for the ‘‘UK_USA” case reject the CCC null hypothesis at the sig-niﬁcant level of 5%. As examined above, the correlations between two stock markets truly have changed with inﬂation cycle and mar-ket volatility.

To testify whether the DSTCC-CARR model outperforms the

STCC-CARR model, this work applies the LR tests. Table 4 gives

LR statistics. Looking over 15 pair wise instances, all cases prefer the DSTCC-CARR models, implying that ‘‘Aver_CPI” and ‘‘VIX” both have indispensable effects on stock correlations. Ignoring either of their inﬂuence may make the model less convictive.

To sum up, modeling speciﬁcations should mention three points. Firstly, it’s important to account for time zone effect for those components without overlapping opening hours. Secondly, ‘‘Aver_CPI” and ‘‘VIX” are useful in explaining the variations in con-ditional correlations between international stock market indices.

Finally, compared to the STCC-CARR models, the DSTCC-CARR model proves to be more appropriate for the current application. 4.4. Model estimation

Table 5reports the estimation results andFig. 1provides rele-vant conditional correlations.

The DSTCC-CARR model’s estimation results show that coeffi-cients in conditional range equations are significant, and the re-vealed characteristics are consistent with what earlier literatures report concerning the range-based volatility model. The detailed content is described inChou (2005). As expected, most coefficients of the four extreme correlation states are significant at a 95% level. The states are different from each other, suggesting that stock cor-relations could be attributed to changing inflation cycles and mar-ket volatility. This paper uses speed coefficients to suggest how fast correlations transit from one state to the other. With ‘‘Aver_CPI” and ‘‘VIX” as transition variables, speed coefficients are small but significant, implying that transitions are smooth. This research cites location coefficients to indicate the sensitivity of asymmetries to inflation cycle and volatility phases. If the coefficients are larger than mean values of transition variables, the correlations between stock markets will remain stable in the state with low inflations or low volatilities. Both countries experiencing large inflations or suf-fering from a strong fluctuation would make stock correlations move towards high inflations or high volatilities. By contrast, cor-relations would stay steady with high inflation or high volatilities for a longer time. Combined with estimating results, this research

0.4 0.5 0.6 0.7 0.8 0.9 1.0 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 FR_GER .0 .1 .2 .3 .4 .5 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 FR_HK -.6 -.4 -.2 .0 .2 .4 .6 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 FR_JP 0.5 0.6 0.7 0.8 0.9 1.0 1991 1992 1993 1994 1996 1997 1998 2000 2001 2002 2003 2005 2006 FR_UK .2 .3 .4 .5 .6 .7 .8 1991 1992 1993 1994 1996 1997 1998 2000 2001 2002 2003 2005 2006 FR_USA 0.2 0.4 0.6 0.8 1.0 1991 1992 1993 1994 1996 1997 1998 2000 2001 2002 2003 2005 2006 GER_HK -0.2 0.0 0.2 0.4 0.6 0.8 1.0 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 GER_JP 0.0 0.2 0.4 0.6 0.8 1.0 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 GER_UK .2 .3 .4 .5 .6 .7 .8 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 GER_USA 0.0 0.2 0.4 0.6 0.8 1.0 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 HK_JP -.2 .0 .2 .4 .6 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 HK_UK -.8 -.4 .0 .4 .8 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 HK_USA -0.4 0.0 0.4 0.8 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 JP_UK .0 .2 .4 .6 .8 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 JP_USA 0.0 0.2 0.4 0.6 0.8 1.0 19 91 19 92 19 93 19 94 19 96 19 97 19 98 20 00 20 01 20 02 20 03 20 05 20 06 UK_USA

Fig. 1. Conditional correlations under DSTCC-CARR model with ‘‘Aver_CPI” and ‘‘VIX” as transition variables.

8

By multiplying CPI rates instead of their mean value, this study constructs another transition variable to replace the ‘‘Aver_CPI”. The tests are repeated and similar results are obtained. For the sake of brevity, the results are not reported but can be obtained from the authors.

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divides the 15 pair-wise cases into several groups. The values of c1 demonstrate that linkages of components ‘‘FR_GER”, ‘‘GER_UK”, ‘‘GER_USA” and ‘‘HK_UK” tend to adhere to states with high inﬂa-tions, while linkages of components ‘‘FR_HK”, ‘‘HK_JP”, ‘‘HK_USA” and ‘‘UK_ USA” are likely to stay at the state with low inﬂations. The remaining cases are not sensitive to either of the states. The values of c2indicate that only ‘‘GER_JP” is sensitive to high volatil-ities while ‘‘FR_GER”, ‘‘FR_USA”, ‘‘GER_UK” and ‘‘JP_USA” are sensi-tive to low volatilities. The remaining cases stay neutral.

To investigate how the correlations vary with different inﬂation cycle phases and volatilities, the current work distinguishes con-traction from expansion by comparing daily CPI rates with their mean CPI. If daily CPI rate is above its mean, it is in expansionary phase, otherwise, it is in contractionary phase. Three symbols are deﬁned to represent these phases. ‘‘Up–up” means both countries are in expansionary phase and ‘‘Down–down” means both coun-tries are in contractionary phase. If one country is in expansionary phase and the other is in contractionary phase, the symbol ‘‘Out of phases” is denoted. Average correlations in these three phases are computed and saved inTable 6and this study analyzes the results. Observations show that 11 out of 15 cases appear in highest corre-lation if both countries are in contractionary phase, while 12 out of 15 cases present the lowest correlation if they are in expansionary phase. Ten out of 15 cases show moderate correlations when two countries are out of phases. Although measured by a different eco-nomic fundamental indicator, the results are consistent withErb et al. (1994).

Moreover, this work calculates the average correlation in the periods with high and low volatilities according to whether the va-lue of ‘‘VIX” is larger than its mean or not.Table 6shows strong support for the fact that higher correlation goes with higher vola-tility, which is identical with the literature mentioned above. Twelve out of 15 cases show higher correlation when volatility ex-ceeds its average level.

Estimation results affirm that interdependence between inter-national stock markets is related to the inflation cycle as well as stock volatility. In the literature, several other factors are proposed in explaining the stock market interdependence. First, low correla-tions across international stock markets may attribute to global portfolio diversification. To reduce their total portfolio risk, inves-tors are willing to diversify across national markets with low cor-relation of returns. Solnik et al. (1996)proves that the linkage occurring between correlation and market volatility is bad news for global money managers. Investors may insist on diversifying

whenever both countries are in the booming period. Aydemir

(2008) ﬁnds counter–cyclical variation between international

ﬁnancial and fundamental linkages for risk sharing. Secondly, a

constant relative risk aversion (CRRA) investor may make out-of-sample portfolio decisions with skewness and asymmetric depen-dence effects.Patton (2004)has found evidence that these charac-teristics may impact on portfolio decisions. As a result, investors may make different decisions on international stock portfolios from downturns to upturns, which would induce international stock correlations to vary according to real sector adjustments. Fi-nally, emerging equity markets could impact on real economy. Bekaert and Harvey (2003)in related literature cited a quantity of cases approving that liberalization in ﬁnancial markets brings on real economic growth. Interdependence of international ﬁnan-cial markets from the world economic relationship cannot be separated.

5. Conclusions

This paper investigates the relationship between real and finan-cial linkages. We use average CPI rates and VIX as transition vari-ables in our model. Empirical results prove the DSTCC-CARR model to be effective. CCC models are rejected in favor of STCC-CARR and DSTCC-STCC-CARR formulations. The tests also indicate that the DSTCC-CARR model with both transition variables to outper-form the STCC-CARR model with either of the two variables alone. By analyzing the estimated results, this study collects ample evidence on varying correlations among different inflation cycle phases. Our results are consistent with those ofErb et al. (1994) that highest correlations appear when both countries are in the contractionary phase and lowest correlations emerge when both countries are in the expansionary phase. Correlations are also vio-lent during periods with different volatilities, coinciding with Con-nolly et al. (2007). Future research could employ other indicators of economic fundamentals such as output and interest rates in our model. Other extensions like considering a richer specification with both countries’ inflation rates as transition variables would also be useful.

Acknowledgements

We thank an anonymous referee and the editor Fariborz Moshirian for very valuable comments and suggestions that helped us to improve the paper, and managing editor Ike Mathur for his help. The first and the third authors acknowledge the financial support of the 3rd phase of Project ‘‘211” and ‘‘985” of Xi’an Jiaotong University. Taiwan’s National Science Commission NSC 96-2415-H-001-019 provides partial financial support for the second author. Of course, we are responsible for any remain-ing errors.

Table 6

Average correlations grouped by inﬂation cycle and stock volatility.

Inﬂation cycle Stock volatility

Expansion Out of phases Contraction High volatility Low volatility

FR_GER 0.633 0.742 0.789 0.769 0.713 FR_HK 0.226 0.318 0.349 0.333 0.280 FR_JP 0.210 0.204 0.217 0.236 0.190 FR_UK 0.730 0.762 0.743 0.754 0.744 FR_USA 0.421 0.382 0.429 0.407 0.412 GER_HK 0.272 0.364 0.335 0.324 0.328 GER_JP 0.048 0.189 0.244 0.227 0.156 GER_UK 0.500 0.639 0.707 0.676 0.621 GER_USA 0.303 0.327 0.468 0.459 0.317 HK_JP 0.270 0.363 0.521 0.482 0.358 HK_USA 0.348 0.369 0.366 0.354 0.368 HK_UK 0.218 0.319 0.353 0.333 0.283 JP_UK 0.169 0.202 0.225 0.212 0.196 JP_USA 0.277 0.306 0.307 0.331 0.276 UK_USA 0.414 0.410 0.405 0.421 0.401

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Appendix A. Hypothesis testing

A.1. Tests for CCC against a STCC-CARR model

Suppose structure of the STCC-CARR model is described as Eq. (5). It is a CCC-CARR model under the hypothesis: H0:

c

2¼ 0. We make a ﬁrst-order Taylor approximation around

c

2¼ 0 to the tran-sition function FL2. After the linearization, the dynamic correlations matrix can be given as(A.1):

P t¼ P 1þ stP2; P 1¼ 1 2ðP1þ P2Þ þ 1 4cðP1 P2Þ

c

; P 2¼ 1 4ðP1 P2Þ

c

: ðA:1Þ

Thus we construct an auxiliary null hypothesis: Haux 0 :q2¼ 0, which stands for the constant correlation. This null hypothesis can be tested by an LM test.

Let q_{¼ ðq}0

1;q02Þ be the vectors holding unique off-diagonal

ele-ments in the two matrices P

1;P 2, where qi ¼ veclðP iÞ; i ¼ 1; 2. Therefore, h ¼ ðx0

1; . . . ;x0N;q0Þ0 is denoted as the full parameter vector and h0the corresponding vector of true parameters under the null hypothesis.

After the linearization, the log-likelihood function could be rewritten as: ltðhÞ ¼ N 2logð2

p

Þ 1 2 XN i¼1 log kit 1 2log jP tj 1 2z 0 tP 1 t z t: Therefore, we construct the LM statistics based on the partial derivatives of log-likelihood function with respect to xi and q. One could ﬁnd the details inChou and Cai (2009)andSilvennoinen and Teräsvirta (2007). The LM statistic is listed as(A.2):

LMCCC1¼ T1 XT t¼1 @ltð^hÞ @q0 2 ! ½^ITð^hÞ1ðq 2;q2Þ XT t¼1 @ltð^hÞ @q0 2 ! : ðA:2Þ

^_I_T_ð^_{hÞ is a consistent estimator of the asymptotic information matrix,} and ½^ITð^hÞ1ðq

2;q2Þis the south-east NðN1Þ

2

NðN1Þ

2 block of the inverse of ^_I

T. The LM statistic has an asymptotic

v

2distribution withNðN1Þ2 de-grees of freedom. For the bivariate case, N ¼ 1.

A.2. Tests for CCC against a DSTCC-CARR model

As the same way, suppose structure of the STCC-CARR model is described as Eq.(3). It is a CCC-CARR model under the hypothesis: H0:

c

1¼

c

2¼ 0. We make the ﬁrst-order Taylor approximation around

c

₁¼ 0 and

c

₂¼ 0 to the transition function FL1 and FL2

respectively. After the linearization, the dynamic correlations ma-trix can be given as(A.3):

P t ¼ P

ð1Þþ s1tPð2Þþ s2tPð3Þ þ s1ts2tPð4Þ; ðA:3Þ where the four correlation states can be illustrated as follows:

P ð1Þ¼ 1=4ðPð11Þþ Pð12Þþ Pð21Þþ Pð22ÞÞ þ 1=8c1

c

1ðPð11Þþ Pð12Þ Pð21Þ Pð22ÞÞ þ 1=8c2

c

2ðPð11Þ Pð12Þþ Pð21Þ Pð22ÞÞ þ 1=16c1

c

1c2

c

2ðPð11Þ Pð12Þ Pð21Þþ Pð22ÞÞ; P ð2Þ¼ 1=8

c

1ðPð11Þþ Pð12Þ Pð21Þ Pð22ÞÞ 1=16c2

c

1

c

2ðPð11Þ Pð12Þ Pð21Þþ Pð22ÞÞ; P ð3Þ¼ 1=8

c

2ðPð11Þ Pð12Þþ Pð21Þ Pð22ÞÞ 1=16c1

c

1

c

2ðPð11Þ Pð12Þ Pð21Þþ Pð22ÞÞ; Pð4Þ¼ 1=16

c

1

c

2ðPð11Þ Pð12Þ Pð21Þþ Pð22ÞÞ:

Under the null hypothesis there are: P

ð1Þ¼ 1=4ðPð11Þþ Pð12Þþ Pð21Þþ Pð22ÞÞ; Pð2Þ¼ 0NN;Pð3Þ¼ 0NN and Pð4Þ¼ 0NN. Thus we construct the auxiliary null hypothesis: Haux0 :qð2Þ¼ qð3Þ¼ qð4Þ¼ 0. The null hypothesis can be tested by an LM test. Let q_{¼ ðq}0

ð1Þ;q0ð2Þ;q0ð3Þ;q0ð4ÞÞ

0 _{be the vectors holding unique off-diagonal}

elements in the four matrices P

ð1Þ;P ð2Þ;P ð3Þ and P ð4Þ, where q ðiÞ¼ veclðP

ðiÞÞ;i ¼ 1;...;4. Therefore, h ¼ ðx01;...;x0N;q0Þ 0

is denoted as the full parameter vector and h0the corresponding vector of true parameters under the null hypothesis.

Therefore, we construct the LM statistics based on the partial derivatives of log-likelihood function with respect to xi and q.

One could checkChou and Cai (2009) and Silvennoinen and

Terä-svirta (2007) for the details. The LM statistic is represented as (A.4): LMCCC2¼ T1 XT t¼1 @ltð^hÞ @ðq0 ð2Þ;q0ð3Þ;q0ð4ÞÞ ! ½^ITð^hÞ1ðq ð24Þ;qð24ÞÞ X T t¼1 @ltð^hÞ @ðq0 ð2Þ;q0ð3Þ;q0ð4ÞÞ ! : ðA:4Þ ^

ITð^hÞ is a consistent estimator of the asymptotic information matrix, and ½^ITð^hÞ1ðq

ð24Þ;qð24ÞÞis the south-east 3NðN1Þ

2

3NðN1Þ

2 block of the in-verse of ^IT. The LM statistic has an asymptotic

v

2distribution with 3NðN1Þ

2 degrees of freedom.

Appendix B. Data description (SeeFigs. B.1–B.5).

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Appendix C. Time zone effects

We divide six stock markets into three groups according to the continent they locate in. Respect to the inner-group cases, time zone effect is supposed to be ignored, because the markets almost open at the same time. Correspondingly, the cross-group cases are likely to be affected by time zone, as we purpose, of which there are totally 11 cases probably involved in for analysis.

The results are reported inTable C.1. We make a comparison be-tween the results with and without time zone effect taken into ac-count, and the answer is straightforward. Correlations between American and Asian markets are found to increase largely if we take time zone effect into account, and the statistical value of CCC tests in both cases are improved intensely. Contrarily, after being adjusted for sake of time zone effect, correlations between American and European countries decline signiﬁcantly, with no

Fig. B.2. Six return (1991.2.2-2007.5.31).

Fig. B.3. Six ranges (1991.2.2-2007.5.31).

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improvement but even worse result in CCC tests. We also become conscious of the indeed increasing correlations between Japanese and European markets and the statistics of CCC test become signif-icant with considering the effect. But the cases of Hong Kong and European countries tell the almost reversed results as shown in Ta-ble C.1.

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Fig. B.5. Index of VIX (1991.2.1-2007.5.31).

Table C.1

Constant Conditional Correlations with and without time zone taken into account.

Overlap time Without time zone

considered

With time zone considered

Correlation Test Correlation Test

FR_USA 1:30* 0.411 0.001 0.238 0.260 FR_HK 0:30 0.308 0.000 0.186 0.252 FR_JP 0:00 0.282 0.077 0.206 0.001 UK_USA 1:30* 0.408 0.634 0.242 0.444 UK_HK 0:30 0.304 0.000 0.223 0.050 UK_JP 0:00 0.278 0.405 0.203 0.020 GER_USA 1:30* 0.391 0.000 0.251 0.000 GER_HK 0:30 0.281 0.001 0.151 0.000 GER_JP 0:00 0.280 0.000 0.177 0.000 USA_HK 0:00 0.118 0.785 0.357 0.039 USA_JP 0:00 0.118 0.037 0.304 0.000

Notes: For the cases between USA and European countries emphasized by ‘‘*”, overlapping time is one and half an hour in daylight saving time while half an hour in winter time.