Time-varying return-volatility relation in international stock markets

International Review of Economics and Finance 51 (2017) 157–173

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International Review of Economics and Finance journal homepage: www.elsevier.com/locate/iref

Time-varying return-volatility relation in international stock markets Xiaoye Jin * International School of Financial Law, East China University of Political Science and Law, China

A R T I C L E I N F O

A B S T R A C T

JEL Classification: C10 C14 G12 G15

This study examines the time-varying relationship between stock returns and volatility in sixteen stock markets during January 2001 to October 2014. After estimating the volatility process without assuming any specific form of its behavior, we find the volatility to be long-term dependent with the Hurst exponent on a verge of stationarity and nonstationarity. We then apply the detrended cross-correlation coefficient to overcome this complication and find evidence of a significant and negative relationship between current stock market returns and current market volatility. Additionally, we find the strength of the negative return-volatility relation is different for specific scales and is stronger in longer time horizon. We further investigate the presence of volatility feedback and leverage effects in international stock markets by examining the lead-lag relation between stock returns and volatility and confirm that the negative return-volatility relation seems to be return-driven (the leverage effect). Finally, we examine the dynamic behavior of the return-volatility relation by applying a rolling window approach and find that time-varying negative return-volatility relation is more likely to generate an asymmetric response with a greater effect when returns decline, which is the common characteristic of international stock markets.

Keywords: Return-volatility trade-off Leverage effect Volatility feedback effect Time-varying DCCA

1. Introduction Due to its importance and broad implication in financial economics, the underlying link between the return on a financial asset and its volatility as a proxy for risk has been a fundamental issue in financial research. Ghysels, Santa-Clara, and Valkanov (2005) even argue that the risk-return trade-off is so fundamental in financial economics that it could be described as the “first fundamental law of finance”. Although most asset pricing models postulate a positive link between stock portfolio's expected returns and volatility (Baillie & DeGennarro, 1990) under the assumption of investor risk aversion, it is not uncommon in empirical finance to model stock return volatility as negatively correlated with stock returns (Bekaert & Wu, 2000; Whitelaw, 2000). However, there is no consensus on even the most basic theoretical properties of the return-volatility relationship and either a positive or a negative relationship between current stock returns and current volatility is possible (Bollerslev & Zhou, 2006; Glosten, Jagannathan, & Runkle, 1993). In fact, even if numerous empirical studies have been carried out to investigate the return-volatility relation in stock markets, there is still no clear consensus about the empirical evidence. On the one hand, Campbell (1987), Glosten et al. (1993), Whitelaw (1994) and Brandt and Kang (2004) have reported a negative and/or insignificant relation. On the other hand, French, Schwert, and Stambaugh (1987), Campbell and Hentschel (1992), Guo and Whitelaw (2006) and Salvador, Floros, and Arago (2014) have documented a positive and

* Room B118, HuiXian Building, 555 Long Yuan Road, SongJiang District, Shanghai, 201620, China. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.iref.2017.05.015 Received 12 June 2016; Received in revised form 28 April 2017; Accepted 18 May 2017 Available online 26 May 2017 1059-0560/© 2017 Elsevier Inc. All rights reserved.

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International Review of Economics and Finance 51 (2017) 157–173

significant relation. Particularly, Wu and Lee (2015) reveal that the return-volatility relation is significantly positive in bull markets, but significantly negative in bear markets. In the strand of negative return-volatility relation, two competing theories, i.e. the leverage effect and the volatility feedback effect, have been documented to provide an economic justification for this negative relation in stock markets. The leverage effect, which is firstly discussed by Black (1976) and Christie (1982), relies on a corporate finance argument and indicates that a decrease in stock price causes an increase in the debt-to-equity ratio. The increased leverage gear will make the stock riskier and result in a higher volatility of stock returns. On the other hand, the volatility feedback effect relies on the existence of time-varying risk premiums as the link between changes in volatility and returns (Poterba & Summers, 1986). This effect implies that if expected future stock returns increase when volatility increases, then current stock prices (and hence returns) will fall to adjust to this change in future expectations. Therefore, an increase to volatility causes negative returns (Campbell & Hentschel, 1992). While both of these effects could be at work, the empirical results are mixed and inconclusive in terms of which of these effects is the main determinant of the negative return-volatility relation. Black (1976), Christie (1982), Duffee (1995) and Li, Yang, Hsiao, and Chang (2005) find a negative leverage effect, while, Glosten et al. (1993), Hatemi-J and Irandoust (2011) and Smith and Yamagata (2011) find a negative volatility feedback effect. The inconclusive finding in terms of the nature of the relation as well as the prevailing theory explaining it appears puzzling. Hence, many of the previous studies attribute this to the appropriateness of using the conditional variance to proxy for risk (see Li et al., 2005; Li, 2011). As stock return volatility is a latent variable and cannot be directly observed, most of the abovementioned studies apply parametric GARCH-class models to estimate volatility, and thus suffer the problem of model misspecification and could lead to an inaccurate relation.1 Furthermore, the return-volatility relation is very sensitive to the length of the return horizon and the selection of exogenous predictors (Harrison & Zhang, 1999). In view of the above mixed results, we take a new look at the return-volatility relation by proposing a coherent treatment of it from the nonparametric volatility estimation, the long-term memory characteristics of volatility and its potential non-stationarity, then moving to the estimation of the return-volatility relation under borderline (non-)stationarity, and finally investigating its time-varying behavior through applying a rolling window approach. This coherent treatment enables us to bypass the issue of model misspecification and thus provide more robust empirical results. We then apply the proposed test to the empirical data of sixteen stock markets and the empirical findings can be summarized along four dimensions. First, we show some evidence that a significant negative relationship between current stock market returns and current market volatility prevails in international stock markets and the significance of the negative relationship is quite stable, which corroborates Li et al. (2005)’s finding. Considering the fact that the nonparametric variance specification is more robust than a parametric or semiparametric conditional variance specification, the results of this study provide further support to the argument that stock returns and volatility is negatively correlated (Bekaert & Wu, 2000; Black, 1976; Cox & Ross, 1976). Second, we find that the strength of the negative return-volatility relation is different for specific scales and is stronger in longer time horizon. This finding has not been reported in stock markets and may potentially open a new topic of research.2 Third, in consistent with the spirit of Hibbert, Daigler, and Dupoyet (2008) and Fleming, Ostdiek, and Whaley (1995), we examine the presence of volatility feedback and leverage effects in international stock markets by investigating the lead-lag relation between stock returns and volatility and empirically confirm that the negative return-volatility relation seems to be return-driven (the leverage effect). This finding is in line with Black (1976), Christie (1982) and Duffee's (1995) argument but contradicts Bekaert and Wu's (2000) conclusion that the volatility feedback effect is the main source of the negative correlation between returns and volatility. Finally, we examine the dynamic behavior of the return-volatility relation by applying a rolling window approach and find that time-varying negative return-volatility relation is more likely to generate an asymmetric response with a greater effect when returns decline, which is the common characteristic of international stock markets, albeit with different fluctuating levels, and the strength of the negative return-volatility relation depends on the state of the market and is stronger in the period of market turbulence. This may partially explain the mixed and inconclusive findings in the return-volatility relation. The empirical aspects of our study include four major differences from previous research. First, the model specification problem is taken into consideration. We use a range-based estimator to measure the volatility process without assuming any specific form of its behavior, with the new measure being better metrics of market expectations since it includes the entire price range as well as bypasses the issue of model misspecification by estimating the volatility outside the returns model. This volatility measure also enables us to investigate the time-varying behavior of the return-volatility relation outside the returns model. Second, we conduct the detrended cross-correlation coefficient analysis using a rolling window approach, which enables us to incorporate the possibility of the smooth and continuous process of international stock markets, and to obtain time-varying return-volatility relation that is robust to possible structural changes. We thus extend Li et al. (2005)’s empirical work, albeit within a different framework. Third, we provide a coherent treatment of the return-volatility relation without assuming anything about the relationship between returns and volatility which distinguishes our study from the other studies which are majorly built around assuming some kind of asymmetric volatility model. Indeed, it is well known to financial researchers that the problem that inferences drawn on the basis of GARCH-class model may be highly attributed to model specification. Fourth, we investigate the lead-lag relationship between stock returns and volatility using the detrended cross-correlation coefficient. This approach is consistent with the spirit of Hibbert et al. (2008) and Fleming et al. (1995) that run OLS regressions to investigate the daily lead-lag relation between stock returns and volatility, and thus allows us to disentangle the leverage effect and the volatility feedback effect which have been competing to provide theoretical explanation to the negative nature of the return-volatility relation. Overall, to the best of our knowledge, this is the first time that the detrended cross-correlation coefficient

1 2

Harrison and Zhang (1999) show that GARCH models may be misspecified and could lead to an inaccurate relation. Kristoufek (2014) reports this effect in energy futures markets. 158

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has been applied into examining the time-varying return-volatility relation in international stock markets, and thus provides further empirical evidence to the debate on this issue. The rest of this paper is organized as follows. Section 2 discusses the empirical methodology, including volatility estimation, longterm memory and its tests and estimators, and estimation of correlations under borderline (non-)stationarity. Section 3 presents the analyzed dataset and the empirical findings. Finally, Section 4 contains some concluding remarks. 2. Econometric methodology This section presents a brief review of empirical methodology used in this study to examine the relationship between stock market returns and volatility. We attempt to provide a coherent treatment of the time-varying return-volatility relation in a three-step framework. Firstly, we apply the Garman-Klass (GK) estimator (1980) as a highly efficient estimator of daily variance which circumvents the burden of obtaining high-frequency data. The use of the flexible functional form for variance is appealing because estimation of a parametric GARCH-class model is sensitive to model misspecification. In fact, many of the aforementioned studies attribute the finding of the weak return-volatility relation to the lack of a proper measure of risk. Secondly, as the volatility process is not only standardly considered as a long-term memory process but also usually on the edge of stationarity, we employ four complementary tests to thoroughly examine the long-term memory characteristics of the volatility series to ensure the results are robust. And finally, as it is shown in the Empirical results section that the volatility processes are strongly persistent and on the edge of non-stationarity, we introduce the detrended cross-correlation coefficient (DCCA) proposed by Zebende (2011) that is able to efficiently estimate the return-volatility relation under such circumstance. 2.1. Volatility estimation In majority of the return-volatility relation studies covered in the Introduction section, the volatility process has usually been estimated through a parametric way, for example GARCH-class models or stochastic volatility models, under various assumptions and restrictions. By doing so, the volatility series and its features are strongly dependent on the model choice and specifications such that the return-volatility relation can simply be detected during the estimation procedure due to the model misspecification, even if it does not present in reality. In our study, we circumvent this issue by estimating the volatility outside the returns model through a nonparametric approach. Historically, when we only have daily closing prices and need to estimate volatility on a daily basis, the only estimate we have is squared daily return.3 Although this estimate is very noisy, it is very often the only one we have and has been commonly used. In a sense, most of the volatility models, i.e. GARCH-class models or stochastic volatility models, follow the same logic that daily volatility is first estimated as squared returns and then processed by applying time series techniques. However, Chou, Chou, and Liu (2010) indicate that these measures turn out to be very poor estimators of the true volatility. When intraday high-frequency data are available, the daily volatility can be estimated more precisely. However, it is usually unavailable to obtain and in many cases it is very costly to work with. Particularly, estimating volatility from high-frequency data is rather a complex issue due to market microstructure effects (see Dacorogna, Gençay, Müller, Olsen, & Pictet, 2001). A compromise may be reached when daily open, high and low prices are available too. Range, the difference between high and low prices, is a natural candidate for the volatility estimation. Parkinson (1980) indicates that the estimator based on range is much less noisy than squared returns. According to Molnar (2012), range-based volatility estimators provide significant increase in accuracy compared to simple squared returns and they stay close to the most efficient realized variance family measures. Among several options, we select the Garman-Klass (GK) estimator as a highly efficient estimator of daily variance.4 The estimator is defined as:

b σ 2GK;t ¼ 0:5ðlnðHt =Lt ÞÞ  ð2ln2  1ÞðlnðCt =Ot ÞÞ 2

2

(1)

where Ht and Lt are daily highs and lows, respectively, and Ct and Ot are daily closing and opening prices, respectively (Garman & Klass, 1980). Eq. (1) can be simply explained as an optimal combination of simple and Parkinson volatility estimator. As the estimator is only related to the trading day, i.e. the period from the open to the close of the market, we further work with the trading day return, i.e. rt ¼ lnðCt =Ot Þ. We choose to work with open-to-close returns because overnight returns have been shown to follow a very different dynamic. In addition, including overnight returns may alter our analysis when standardizing returns as we work with volatility computed with intraday transactions data.5 2.2. Long-term memory Long-term memory (long-range dependence) is the specific feature of a time series in which its past values influence its present and future values. More formally, the long-term memory processes are defined in both time and frequency domains. In the time domain, they

3

Alternatively, the volatility series can be estimated as absolute returns of the series. Molnar (2012) finds that for most purposes, the best volatility estimators is the Garman-Klass volatility estimator among range-based volatility estimators. Another convenience of this treatment is that we can examine the lead-lag relation between stock returns and volatility such that the leverage effect and feedback effect can be clearly distinguished. 4 5

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are connected to a power-law decaying auto-correlation function. In the frequency domain, they are connected to the self-affine feature of the series. The self-affinity indicates that each part comprising a fractal resembles the whole. The weaker concept of self-affinity is usually employed in the case of a financial asset (logarithmic) returns series. In both definitions, long-term memory is conveniently represented by a parameter known as the Hurst exponent. For stationary series, the Hurst exponent ranges from 0 to 1 with an exponent ¼ 0.5 indicating a random walk. A Hurst exponent larger than 0.5 indicates a persistent, trend-reinforcing series (positive longrange dependence); a Hurst exponent smaller than 0.5 suggests anti-persistence, and therefore the past trends of a series tend to reverse in the future (negative long-range dependence) (Da Silva, Figueiredo, Gleria, & Matsushita, 2007). However, one important aspect related to the Hurst exponent calculation is that the estimators of Hurst exponent might report values different from 0.5 that is the condition for a time series to reject random walk, and thus hinting long-term memory, even if the series are not long-range dependent (Couillard & Davison, 2005; Lennartz & Bunde, 2009). To deal with this issue, the series should be tested for actually being long-range dependent before estimating the Hurst exponent itself. In this study, we choose the modified R/S and rescaled V/S analysis to test the presence of long-range dependence in the series. The modified R/S test is an improved version of the classical R/S test proposed by Hurst (1951). Lo (1991) shows that the results of R/S analysis are easily affected by the existence of short memory and might lead to a biased estimation of long-term memory. To overcome the drawback of R/S analysis, Lo (1991) refines the classical R/S test by defining the test statistic as

ðR=SÞ VT ¼ pffiffiffiffi T T

(2)

where the range R is the difference between the maximum and the minimum of the cumulative demeaned original series, S is the standard deviation of the series and T is the time series length. Therefore, the modified R/S test is based on an examination of the average rescaled range of the cumulative deviation of a time series from its mean within each of a number of subperiods. We use the standard deviation S in its heteroskedasticity and autocorrelation consistent (HAC) version to control for the potential short-term memory bias, for which the following specification is utilized:  q  X jkj b S x;y ðqÞ ¼ b γ ðkÞ 1 (3) q þ 1 x;y k¼q where bγ x;y is a sample cross-covariance at lag k, q is a number of lags taken into consideration and the cross-covariances are weighted with the Barlett-kernel weights. For the modified R/S test, we set S≡b S x;y ðqÞ as the autocovariance function is symmetric. The optimal bandwidth number q* is chosen according to the data dependent procedure suggested by Lo (1991), which can be described as: 2=3   1=3  3T 2b ρ q* ¼ int (4) 2 1b ρ2 where b ρ is the first-order autocorrelation coefficient and int ½⋅ is the lower integer operator. Under the null hypothesis of no long-range dependence, the limiting distribution of the VT is given by the distribution function of the difference between the maximum and minimum of the Brownian bridge on a unit interval, which is given as ∞ X   2 FV ðxÞ ¼ 1 þ 2 (5) 1  4k2 x2 e2ðkxÞ k¼1

The idea behind the V/S analysis proposed by Giraitis, Kokoszka, Leipus, and Teyssiere (2003) is quite similar to the modified R/S analysis as the only difference is that the range given by the difference between the maximum and the minimum of the cumulative demeaned original series in the modified R/S analysis is replaced by the sample variance. The testing statistic M is then defined as

MT ¼

varðXÞ TS2

(6)

where X is the profile of the original series, S is the standard deviation of the series and T is the time series length. Although the definition of the modified R/S statistic and the V/S statistic seem to be very close, Giraitis et al. (2003) argue that the rescaled V/S test is a very nice improvement in relation to the R/S statistic.6 Under the null hypothesis of no long-range dependence, the statistic is distributed as ∞ X 2 FM ðxÞ ¼ 1 þ 2 ð  1Þk e2ðkπÞ x (7) k¼1

After testing the presence of long-term memory, we utilize four different methods for estimating the Hurst exponent itself based on two definitions of long memory. They are the local Whittle estimator and the GPH estimator (based on the frequency domain definition) and the detrended fluctuation analysis (DFA) and the detrending moving average (DMA) methods (based on the time domain definition). The frequency domain estimators are well suited even for non-stationary or boundary series and the time series estimators are used as a further robustness check as they do not assume any specific functional form of the analyzed process. The local Whittle estimator (Robinson, 1995) is a semi-parametric maximum likelihood estimator using a band of frequencies that degenerate to zero. As it focuses only on a part of the spectrum f ðλÞ near the origin, the full parametric specification is thus not necessary and one does not need to assume any specific underlying long-term memory model but only a model with a spectrum divergent at origin.

6

This argument is also supported by previous results of Lee and Schmidt (1996) and Lee and Amsler (1997). 160

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Therefore, the estimator does not take into account high frequencies and it is in turn resistant to the short-term memory bias. By defining Iðλj Þ as the periodogram, m  T=2, and λj ¼ 2πj=T, the estimate of long memory parameter H can be defined as m X   b ¼ argminH log 1 H λ2H1 I λj m j¼1 j

!

m 2H  1 X logλj  m j¼1

! (8)

Under finiteness of the fourth moment and other mild conditions, the local Whittle estimator is consistent and asymptotically normal, specifically

 pffiffiffiffi b  H 0 →d Nð0; 1=4Þ m H

(9)

This estimator is robust to a certain degree of conditional heteroskedasticity (Robinson & Henry, 1999) and is more efficient than other semi-parametric competitors (Abadir, Distaso, & Giraitis, 2007; Phillips & Shimotsu, 2004). The GPH estimator, named after the authors Geweke and Porter-Hudak (1983), utilizes the full functional specification of the underlying process as the fractional Gaussian noise. Under the specific spectral density assumption, the Hurst exponent is estimated using the ordinary least squares on

    log I λj ∝  ðH  0:5Þlog 4 sin2 λj 2

(10)

where Ið⋅Þ is the periodogram of the data calculated at the harmonic frequencies. The GPH estimator is consistent and asymptotically normal (Beran,p1994), ffiffiffiffi specifically  

 b  H 0 →d N 0; π 2 6 T H (11) The GPH estimator is thus asymptotically infinitely more efficient than the local Whittle estimator as long as the underlying process is in fact the fractional Gaussian noise. However, in financial and economic series this is frequently not the case due to short-term memory. Therefore, the GPH estimator is only based on a part of the spectrum close to the origin similar to the local Whittle estimator. In such, both the local Whittle and GPH estimators are estimated only on parts of the estimated periodogram that are close to the origin.7 The main problem of the local Whittle and GPH estimators was the assumed infinite memory process that may cause the spurious detection of apparent long-range autocorrelations that are an artifact of patchiness. To overcome this issue, the detrended fluctuation (DFA) analysis and the detrending moving average (DMA) analysis were developed by Peng et al. (1994) and Alessio, Carbone, Castelli, and Frappietro (2002) respectively to investigate the possible long-term dependence in the recurrence intervals. The DFA and DMA analysis are regarded as “The Methods of Choice” in determining the Hurst exponent of time series (Shao, Gu, Jiang, Zhou, & Sornette, 2012). Both methods are based on the power-law decaying autocorrelation of the long-term correlated series as the decay implies a power-law behavior of the detrended sums of the long-term correlated series. In the procedure, a profile of the series is constructed as a cumulative sum of a demeaned original series. By removing the local trend series from the profile, the residual sequence series is constructed and then split into N boxes of length s. As the box size s varies, one can determine the power-law relationship between the overall fluctuation function FðsÞ and the box size s

Fq ðsÞ∝sHðqÞ

(12)

where HðqÞ is the generalized Hurst exponent and q is the multifractal order. Especially, when q ¼ 2, Hð2Þ is the Hurst exponent and can be obtained by detecting the slope of plot between Fq ðsÞ and lnðsÞ through the method of ordinary least squares (OLS). The main difference between the DFA and DMA analysis is the determination of the local trend series. The DFA analysis removes the possible trends in time series based on the polynomial fitting. Different from DFA, the DMA analysis removes the local trends by subtracting the local means.8 In the DFA analysis, we opt to vary the box size s between 10 and the largest integer not greater than 10% of series length, and use non-overlapping windows. In the DMA analysis, we opt for the centered moving average resulting in the centered DMA analysis which has been shown to outperform the backward and forward moving average (Carbone & Castelli, 2003; Shao et al., 2012). Again, the range of the moving average window varies from 11 to the odd number closest to the 10% of series length.9 Furthermore, the jackknife procedure is utilized to obtain the standard errors of the estimate. We adopt the parameter values recommended by Kristoufek (2014) with the minimal box size (window length) varying from 10 to 100 and from 11 to 101 for the DFA and DMA analysis, respectively. Therefore, we calculate the actual estimate as the simple average of the jackknife estimates with the standard error as their standard deviation. 2.3. The nonlinear correlation coefficient In order to measure the return-volatility relation, which can be seen as a correlation between returns and volatility, we propose to employ Zebende's (2011) detrended cross-correlation coefficient. Zebende (2011) proposes the detrended cross-correlation coefficient

7 8 9

In our analysis, we apply m ¼ T 0:6 (Phillips & Shimotsu, 2004). For more details about DFA and DMA analysis, please refer to Shao et al. (2012) and Gu and Zhou (2010) respectively. Odd numbers are required for the centered moving average and the specific numbers are chosen to be comparable with the DFA analysis. 161

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as a combination of the detrended cross-correlation analysis (DCCA) (Jiang & Zhou, 2011) and the detrended fluctuation analysis (DFA) (Peng et al., 1994). The nonlinear detrended cross-correlation coefficient ρDCCA ðsÞ, which can be used to measure the correlations between two nonstationary and non-Gaussian time series at different time scales, is defined as

ρDCCA ðsÞ ¼

2 FDCCA ðsÞ FDFAfxðtÞg ðsÞ⋅FDFAfyðtÞg ðsÞ

(13)

2 2 2 where FDCCA ðsÞ is a detrended covariance between profiles of the two series based on a window of size s, and FDFAfxðtÞg and FDFAfyðtÞg are

detrended variances of profiles of the separate series, respectively, for a window size s.10 Obviously, this nonlinear correlation coefficient depends on the length of each part, s, just the time scale in the scaling analysis. ρDCCA ðsÞ is a dimensionless coefficient that ranges between 1  ρDCCA ðsÞ  1. If ρDCCA ðsÞ is equal to zero, for which the two series have no-cross correlation, then it splits the level of crosscorrelation between the positive and the negative case. Contrary to the traditional methods, i.e. Pearson correlation coefficient, the DCCA coefficient is able to estimate the true correlation coefficient between series precisely regardless of the non-stationarity strength (Kristoufek, 2014).11 3. Empirical results We consider the stock price indices for the markets from the following sixteen countries: Australia (AUS), Brazil (BRA), Canada (CAN), China (CHN), France (FRA), Germany (GER), India (IND), Indonesia (IDN), Japan (JAP), Korea (KOR), Mexico (MEX), Russia (RUS), South Africa (SAF), Turkey (TUR), the United Kingdom (UK) and the United States of America (USA).12 The data cover the period from 4 January 1999 to 31 October 201413, observed on a daily basis and expressed in local currency. We acquire the data series from Wind Financial Terminal.14 Table 1 reports the standard descriptive statistics and tests. All raw return series show standard characteristics such as excess kurtosis and negative skewness (apart from Brazil in this case) such that the Jarque and Bera (1980) test statistics clearly confirm the rejection of the null hypothesis of normality for all series at the significance level of 1%. The Ljung and Box (1978) Q statistic on the first thirty lags of the sample autocorrelation function is significant at the 1% significance level suggesting the presence of serial autocorrelations for all raw return series. Moreover, the unit root and stationary tests confirm all raw return series follow asymptotic stationarity. As Andersen, Bollerslev, Diebold, and Ebens (2001) find that the returns normalized by the standard deviations are close to normal and thus are more suitable for statistical analysis, from this point onward, we focus solely on the relationship between standardized returns and volatility, which we examine in its logarithmic form. Although the standardized returns series still reject the null hypothesis of Gaussian distribution, the level skewness and excess kurtosis have been significantly reduced. Moreover, the level of autocorrelations has also been reduced. For the logarithmic volatility, the null hypothesis of normality of the distribution is strongly rejected and very strong autocorrelation has been observed. Moreover, both unit root and stationary hypotheses of the series have been rejected, which leads us to examine potential long-term memory for the series under consideration. Table 2 reports estimation results for the modified R/S and the rescaled V/S tests for the raw returns, the standardized returns as well as the logarithmic volatility series with optimal lag chosen according to Eq. (4). The VT and MT statistics are not significant at the 10% significance level suggesting that the raw returns (apart from China and South Africa) are not long-range correlated. Surprisingly, the VT and MT statistics indicate that the majority of the standardized returns (apart from Australia, France, Korea, and South Africa) are longrange correlated at the 10% significance level. As expected, long-term memory is identified for all logarithmic volatility series at the 10% significance level even high number of lags (between 21 and 34) have been controlled. As the VT and MT tests clearly confirm the existence of long-term dependence in the standardized returns (apart from Australia, France, Korea, and South Africa) as well as the logarithmic volatility series, we therefore estimate the Hurst exponent of the corresponding series using the local Whittle, GPH, DFA and DMA estimators. The estimates are summarized in Table 3. We observe that for all the estimators the Hurst exponent for the standardized returns is drifted away from H ¼ 0:5. Based on the reported standard errors, we can confirm that the majority of the standardized series are long-term correlated. In terms of the logarithmic volatility series, although all the estimators give similar results that are different from one, after considering the reported standard errors, we cannot distinguish whether the Hurst exponents are statistically different from the unity value that is the critical point separating long-term memory and stationarity. 3.1. Tracking the evolution of return-volatility relation As the standardized returns series are long-term correlated and the logarithmic volatility series are either long-term correlated or non-stationary, the standard correlation measures are not able to deal with such series. Therefore, we apply the detrended crosscorrelation coefficient which is capable of working under long-term memory or non-stationarity as well as filtering out well-defined trends (Kristoufek, 2014). It has been pointed out by Zebende (2011) that the value of ρDCCA varies with the timeframe set ðsÞ for

10

For more technical details about the methods, please refer to Podobnik and Stanley (2008). We are indebted to Ladislav Kristoufek for the programming code. 12 Appendix I provides the index descriptions. 13 Due to data availability, the sample starts from 3 April, 2000, 1 August 2002, and 8 May 2001 for Australia, South Africa and Turkey, respectively. 14 We prefer daily return data to lower frequency data, such as weekly and monthly returns, because longer horizon returns can obscure transient responses to innovations that may last for a few days only. 11

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Table 1 Descriptive statistics.

Raw returns

163

AUS

BRA

CAN

CHN

FRA

GER

IND

IDN

JAP

KOR

MEX

RUS

SAF

TUR

UK

USA

0.0001 0.0092 0.5219 7.2600 7709*** 59:74*** 61:98*** 0.1022 0.1004 1.3557 0.0950 4.9723 3548*** 52:08*** 61:38*** 0.1048 10.489 1.0808 0.4022 0.4652 123:7*** 21836*** 6:455*** 1:3273***

0.0005 0.0201 0.6169 17.3958 46279*** 60:49*** 60:61*** 0.1827 0.1424 1.5271 1.7791 32.1127 158844*** 30:04 60:95*** 0:4295* 8.869 0.9515 0.2718 0.8624 158:1*** 10549*** 9:435*** 1:5878***

0.0001 0.0106 1.1606 22.7162 80145*** 91:81*** 59:32*** 0.2106 0.0849 1.3016 0.1818 5.4863 4647*** 48:84** 58:60*** 0.3166 10.179 1.1857 0.4927 3.5307 192:5*** 26086*** 6:674*** 0:5022**

0.0005 0.0147 0.1772 3.2495 1702*** 78:46*** 67:03*** 0:6144** 0.0879 1.2449 0.0858 2.6179 27:941*** 134:47*** 32:97*** 0.3088 9.393 1.0504 0.2825 0.1502 54:5*** 21180*** 7:077*** 1:0594***

0.0004 0.0132 0.1714 6.1625 5976*** 92:12*** 64:53*** 0.0786 0.0246 1.1929 0.1337 2.0529 672*** 42:22* 65:16*** 0.1274 9.538 1.1069 0.1324 0.1453 14:3*** 30894*** 6:301*** 0:6441**

0.0001 0.0148 0.0147 6.4561 6508*** 82:09*** 61:43*** 0.0798 0.0778 1.2185 0.0657 2.7823 1211*** 28.489 62:55*** 0.2816 9.383 1.1625 0.1685 0.1365 20:6*** 34051*** 6:012*** 1:1649***

0.0009 0.0150 0.3531 4.9429 3794*** 49:15*** 62:89*** 0.3416 0.0155 1.2918 0.6985 9.7267 14697*** 40:61* 60:03*** 0.2788 9.361 1.0519 0.4142 0.3085 118:9*** 17942*** 8:962*** 1:4934***

0.0008 0.0147 0.1332 7.6625 8803*** 111:12*** 54:58*** 0.1433 0.3274 1.4131 0.4657 2.0976 789*** 56:55*** 55:91*** 0:6534** 9.669 1.0912 0.2723 0.3214 59:9*** 9259*** 10:351*** 1:9572***

0.0004 0.0128 0.6312 9.9197 21790*** 85:20*** 64:07*** 0.1137 0.0041 1.3407 0.0586 3.8130 2207*** 61:85*** 65:01*** 0.2675 9.789 1.0091 0.1096 0.4213 34:2*** 15561*** 7:888*** 1:7816***

0.0003 0.0141 0.3731 4.9058 3794*** 58:73*** 46:16*** 0.0461 0.0271 1.2426 0.0913 2.2683 798*** 44:19** 63:28*** 0.3250 9.455 1.1409 0.2019 0.0825 26:2*** 36917*** 5:656*** 4:4989***

0.0006 0.0147 0.0436 9.1829 13026*** 85:13*** 55:79*** 0.1313 0.1509 1.5068 0.3208 6.4794 6548*** 37.70 58:51*** 0.2306 9.743 1.0952 0.1572 0.0722 16:1*** 16391*** 7:347*** 0:8596***

0.0003 0.0221 1.3965 32.6388 195096*** 109:63*** 58:07*** 0.1536 0.1983 2.0091 2.0928 47.4811 346855*** 38.53 58:49*** 0:5292** 9.524 1.2838 0.2437 0.8007 134:1*** 16189*** 9:509*** 2:3035***

0.0006 0.0129 0.5765 9.4304 10763*** 93:35*** 52:17*** 0.0958 0.1876 1.5675 0.6661 9.0203 9915*** 40:95* 53:14*** 0.2150 9.868 0.9994 0.2076 0.2796 29:9*** 16146*** 7:108*** 1:2062***

0.0003 0.0188 0.1079 5.6442 4198*** 54:23*** 58:09*** 0:4238* 0.1455 1.2772 0.6589 8.5769 9908*** 33.01 56:86*** 0:9751*** 8.751 1.0181 0.3151 0.3700 70:3*** 16463*** 7:378*** 3:7112***

0.0001 0.0127 0.2219 6.7327 7060*** 141:58*** 27:91*** 0.0791 0.1022 1.3628 0.1704 1.8938 574*** 36.15 63:58*** 0.1771 9.827 1.1177 0.2113 0.0877 28:9*** 30229*** 6:214*** 0:6997**

0.0002 0.0128 0.3814 9.5169 14033*** 100:34*** 46:57*** 0.1823 0.1292 1.4291 0.2755 3.3906 1817*** 42:41* 63:42*** 1:2857*** 9.975 1.1528 0.2188 0.2936 42:8*** 26977*** 6:245*** 1:0179***

Note: ***, ** and * indicate significance at the 1%, 5% and 10% level, respectively.

International Review of Economics and Finance 51 (2017) 157–173

Mean SD Skewness Ex.kurtosis JB Q(30) ADF KPSS Std. returns Mean SD Skewness Ex.kurtosis JB Q(30) ADF KPSS Log volatility Mean SD Skewness Ex.kurtosis JB Q(30) ADF KPSS

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Table 2 Long-term memory tests. Raw returns VT AUS BRA CAN CHN FRA GER IND IDN JAP KOR MEX RUS SAF TUR UK USA

1.4044 0.7044 1.4070 2:0242* 1.1998 0.7961 1.3124 1.4917 1.2167 1.0029 1.1160 1.2379 1.2008 1.5815 1.0379 1.4259

Standardized returns MT 0.0733 0.0198 0.1296 0:2533* 0.0748 0.0214 0.1326 0.1316 0.0548 0.0325 0.0579 0.0583 0:0664* 0.1508 0.0459 0.0624

q

*

VT

3 0 2 5 3 0 3 5 4 3 5 3 2 2 2 3

Logarithmic volatility

MT

1.5878 1.8334 1:9746** 2:9009** 1.5657 2:2286** 1:2691*** 2:7178*** 1:6009** 1.3946 2:0896** 2:1115** 0.8197 2:2656*** 1:5414* 2:2463***

q

0.1044 0:2386** 0.2425 0.3566 0.1223 0:2849** 0:0993** 0:6132*** 0.1073 0.1377 0:2165* 0:2859** 0.0375 0:4414*** 0.1326 0:3126***

*

3 1 3 2 4 2 0 4 4 3 3 2 0 1 3 3

VT

q*

MT ***

3:8528 2:4299** 2:7788** 3:5913*** 2:9827** 2:6655** 2:6754** 2:4308** 2:9753** 3:1428*** 2.2789 4:2719*** 3:2647*** 2:9715** 2:8258* 2:4632**

***

1:5974 0:4028*** 0:6952*** 1:5603*** 0:5398** 0:5071** 0.3963 0.3268 0:6478** 0:6742** 0:5131* 1:7202*** 1:1889*** 0:6871*** 0:5121** 0:4517*

23 21 26 25 30 31 27 23 22 34 23 24 22 25 28 27

Note: ***, ** and * mean significance at 1%, 5% and 10% levels, respectively. Table 3 Estimated Hurst exponents for standardized returns and logarithmic volatility. Local Whittle Value

GPH

DFA

DMA

St. error

Value

St. error

Value

St. error

Value

St. error

Standardized returns series AUS 0.5307 BRA 0.5781 CAN 0.6170 CHN 0.7412 FRA 0.5843 GER 0.5953 IND 0.5547 IDN 0.6626 JAP 0.4886 KOR 0.5323 MEX 0.5919 RUS 0.5708 SAF 0.5875 TUR 0.6083 UK 0.4887 USA 0.6015

0.0437 0.0429 0.0427 0.0423 0.0425 0.0426 0.0429 0.0432 0.0430 0.0427 0.0427 0.0429 0.0462 0.0449 0.0427 0.0427

0.4942 0.5782 0.5974 0.7315 0.5585 0.5882 0.5863 0.6413 0.4691 0.5574 0.5720 0.5989 0.6034 0.6267 0.4685 0.6311

0.0645 0.0501 0.0610 0.0550 0.0623 0.0595 0.0552 0.0725 0.0640 0.0616 0.0637 0.0605 0.0597 0.0594 0.0492 0.0545

0.4917 0.5382 0.5889 0.7319 0.5104 0.5387 0.5284 0.5656 0.4291 0.5289 0.5764 0.6073 0.5039 0.5245 0.4059 0.5364

0.0106 0.0136 0.0161 0.0379 0.0062 0.0167 0.0139 0.0111 0.0088 0.0134 0.0175 0.0165 0.0182 0.0200 0.0036 0.0253

0.5041 0.5624 0.6269 0.7242 0.5102 0.5478 0.5384 0.5815 0.4520 0.5306 0.6001 0.6497 0.5145 0.5390 0.4038 0.5493

0.0114 0.0187 0.0263 0.0307 0.0032 0.0164 0.0176 0.0115 0.0168 0.0107 0.0225 0.0282 0.0238 0.0257 0.0039 0.0263

Logarithmic volatility series AUS 1.0288 BRA 1.0152 CAN 1.0832 CHN 1.0783 FRA 1.0973 GER 1.1051 IND 1.0002 IDN 0.9518 JAP 1.0614 KOR 1.1260 MEX 1.0662 RUS 0.9605 SAF 1.0347 TUR 1.0269 UK 1.1103 USA 1.1094

0.0437 0.0429 0.0427 0.0423 0.0426 0.0425 0.0429 0.0432 0.0430 0.0427 0.0427 0.0429 0.0462 0.0449 0.0427 0.0427

1.0470 0.9936 1.0792 1.1178 1.0859 1.1020 0.9803 0.9081 1.0380 1.1486 1.0441 0.9472 1.0822 0.9918 1.0803 1.0386

0.0560 0.0625 0.0548 0.0595 0.0552 0.0558 0.0614 0.0727 0.0570 0.0589 0.0544 0.0502 0.0657 0.0633 0.0555 0.0609

1.0695 0.9665 1.1652 0.8966 1.0939 1.1161 0.9706 0.8387 0.9952 1.0824 1.0306 0.9588 1.0431 0.9335 1.0837 1.0648

0.0369 0.0041 0.0437 0.0266 0.0146 0.0135 0.0038 0.0022 0.0132 0.0315 0.0190 0.0152 0.0304 0.0237 0.0119 0.0072

1.1003 0.9808 1.1768 0.9099 1.1211 1.1343 0.9806 0.8217 0.9759 1.0979 1.0221 0.9856 1.0703 0.9334 1.1121 1.0903

0.0408 0.0042 0.0392 0.0245 0.0166 0.0136 0.0034 0.0113 0.0034 0.0323 0.0103 0.0179 0.0321 0.0212 0.0126 0.0077

detrending and the cross-correlation effect is stronger in longer time horizon. In order to examine this effect, we vary s between 10 and 100 with a step of 10. In order to generate the corresponding p-values, we adopt Kristoufek's (2014) approach through constructing 1000 series generated using Fourier randomization.15 Results summarized in Fig. 1 suggest that negative relationship between returns and

15

The Fourier randomization ensures that the cross-correlations are shuffled away while the autocorrelation structure remains untouched. 164

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Fig. 1. Correlation coefficients between current returns and current volatility are based on detrended cross-correlation coefficient analysis. Solid lines represent the estimated correlation coefficients (left y-axes) and the dashed lines show the p-values (right y-axes) for varying window lengths s (x-axes).

volatility prevails in all stock markets and the strength of the negative relationship varies with s. This dynamic characteristic indicates that the relationship between return and volatility is sensitive to the time horizon. Particularly, the negative relationship is stronger in longer time horizon for all stock markets. Moreover, the significance of the negative relationship is quite stable as the p-values curves are lower than the 5% threshold for most of these markets. This may be interpreted as evidence for a significant relationship between market return and volatility in international stock markets. Such a finding is generally consistent with many previous works, including Whitelaw (2000), Li et al. (2005) and Bae, Kim, and Nelson (2006), and thus lend some support to the claim that stock returns are negatively correlated with stock return volatility (Bekaert & Wu, 2000; Black, 1976; Cox & Ross, 1976). However, it contradicts the prediction of a positive relation made by many asset pricing models (e.g., Linter, 1965; Merton, 1973; Sharpe, 1964) and empirical findings of an insignificant relation consistently reported in the previous literature (Lee, Chen, & Rui, 2001; Nelson, 1991). Given the fact that the static cross-correlation coefficient analysis for the whole series cannot reflect the local situation, we further

165

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Fig. 2. Time-varying correlation coefficients between current returns and current volatility. Correlation coefficients are based on detrended cross-correlation coefficient analysis.

investigate whether such a negative relationship is time-varying, which we did using a dynamic version of the Zebende (2011) algorithm. In fact, Whitelaw (1994) and Lettau and Ludvigson (2009), among others, advocate the importance of considering the possible time-varying behavior in modeling the risk-return trade-off. To capture the dynamics of the return-volatility relation, we employ the 166

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International Review of Economics and Finance 51 (2017) 157–173

Fig. 3. Correlation coefficients between lagged returns and current volatility (black lines) and between current returns and lagged volatility (grey lines) are based on detrended cross-correlation coefficient analysis. Solid lines represent the estimated correlation coefficients (left y-axes) and the dashed lines show the p-values (right yaxes) for varying window lengths s (x-axes).

method of rolling windows to investigate the evolution of cross-correlation coefficients over time. The length of the rolling windows is fixed to 512 business days, about two years. A two year timeframe strikes a good balance between data sufficiency and avoiding suffering unduly from the time series recency effect.16 Liu and Wan (2011) indicate that care should be taken when deciding the size of rolling window such that on the one hand, the general trend of long-term market dynamics can be captured and on the other hand, the influences of exogenous events on market short-term dynamics will not be missed. In each moving window, the original series is analyzed by the Zebende (2011) algorithm to compute the local cross-correlation coefficient. Since it is impossible to report the computed cross-correlation coefficient for each rolling time window, the results are best

16

We can confirm that the length of rolling window does not have significant impact on the following empirical findings. 167

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communicated through graphical depiction. As the estimated correlation coefficient varies with the time frame set for detrending, in order to overcome this issue and for the convenience of graphing, Fig. 2 presents the evolution of the mean value of cross-correlation coefficients with time frame equal to 20, 40, 60 and 80 business days. The vertical axis shows the value of the cross-correlation

Fig. 4. Time-varying correlation coefficients between lagged returns and current volatility (black lines) and between current returns and lagged volatility (grey lines) are based on detrended cross-correlation coefficient analysis. 168

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Table 4 Estimated leverage and feedback effects.

Leverage effect Feedback effect

Leverage effect Feedback effect

AUS

BRA

CAN

CHN

FRA

GER

IND

IDN

0.2542 0.0438

0.2607 0.1356

0.2607 0.0889

0.1906 0.0138

0.2225 0.0167

0.2989 0.0340

0.3240 0.1359

0.3636 0.2457

JAP

KOR

MEX

RUS

SAF

TUR

UK

USA

0.1864 0.0121

0.2597 0.0734

0.1831 0.0320

0.1754 0.0783

0.2471 0.0849

0.2566 0.1266

0.2858 0.0165

0.3331 0.0315

coefficient in each rolling time window, while the horizontal axis is labelled with the last date of the observations in each moving window.17 It has been observed from Fig. 2 that although the curve of the cross-correlation coefficients exhibits a strongly fluctuating behavior in each panel, most of the fluctuations are within the area defined by the horizontal zero line, which indicates that time-varying negative return-volatility relation is the common characteristic of international stock markets, albeit with different fluctuating levels. We further calculate the percentage of time windows that the estimated cross-correlation are negative, which indicates that the majority of international stock markets have over 90 percentage of negative correlation windows during the whole sample period.18 Only two exceptions are China and Japan with 70% and 80% negative correlation windows, respectively. On a country-by-country basis, the findings clearly demonstrate that the negative return-volatility relation prevails in international stock markets. Moreover, as the evolution of the return-volatility relation is plotted as a function of time in the rolling window approach, it is possible to relate the changing degree of return-volatility relation at different stages of development to major political and economic events. As observed graphically, the majority of the lowest period of the cross-correlation coefficients coincide with the stock market crash of 2008, which indicates that the negative relationship is more significant in the period of market turmoil.19 Generally speaking, this may suggest that the strength of the negative return-volatility relation depends on the state of the market. However, this finding is indicative rather than definitive. Therefore, a more rigorous analysis into this coincidence is warranted in future research due to the interesting policy insight. 3.2. A closer look at the negative return-volatility relation The negative return-volatility relation may be explained by two popular theories: the leverage effect and the volatility feedback effect. Although both theories may explain the negative relationship between stock returns and volatility, they carry different implications for causality between returns and volatility (Bekaert & Wu, 2000). The leverage effect is characterized by a surge in the volatility, subsequent to a drop in the stock price (see e.g., Black, 1976; Christie, 1982). The volatility feedback effect means that positive shocks to volatility cause negative returns. If volatility is priced by market participants, an anticipated increase in volatility would raise the required rate of return on equity, which leads to a current stock price decline (see e.g., Bekaert & Wu, 2000; Campbell & Hentschel, 1992). In terms of which effect is the dominant factor explaining the negative return-volatility relation, most empirical studies produced mixed results. For example, Giot (2005) and Masset and Wallmeier (2008) support the leverage effect while Bekaert and Wu (2000) and Dennis, Mayhew, and Stivers (2006) reveal evidence of volatility feedback effect. In our study, we investigate the presence of volatility feedback and leverage effects in international stock markets by examining the lead-lag relation between stock returns and volatility. Specifically, for the leverage effect, we examine the cross-correlation between stock lagged returns and current volatility; for the volatility feedback effect, we examine the cross-correlation between stock current returns and lagged volatility. Both tests are implemented by the detrended cross-correlation coefficient proposed by Zebende (2011). This approach is consistent with the spirit of Hibbert et al. (2008) and Fleming et al. (1995) that run OLS regressions to investigate the daily lead-lag relation between stock returns and volatility. Same as previous analysis, the timeframe set ðsÞ varies between 10 and 100 with a step of 10 and the corresponding p-values are generated through constructing 1000 series generated using Fourier randomization. Several conclusions can be drawn from results summarized in Fig. 3. Firstly, we find quite stable leverage effect and volatility feedback effect. The strength of both effects varies with s which indicates that the dynamics of both effects are different for specific scales. In fact, both effects are stronger in longer time horizon for the majority of the stock markets. Secondly, the leverage effect is stronger than the volatility feedback effect for all stock markets, which has been illustrated by the fact that the curve of the leverage effect is always below the curve of the volatility feedback effect in all stock markets. Particularly, the curves of the leverage effect and the volatility feedback effect become closer at higher scales. This may indicate that the leverage effect and the volatility feedback effect become indistinguishable in longer time horizon. For example, a volatility movement that is initially induced by a price change could amplify the price change with yet another impulse on volatility. In efficient financial markets, the steps will evolve almost simultaneously and this makes it difficult to identify the different stages of the process in longer time horizon (Hatemi-J & Irandoust, 2011). Thirdly, the leverage effect is statistically significant at the 5% level for the majority of the stock markets. For the volatility feedback effect, the statistical significance is much weaker and erratic than for the leverage effect. Overall, we may conclude that the leverage effect is the more appropriate theory explaining the negative nature of

17 That is to say, for a date Mar-2014, the cross-correlation coefficient was evaluated for the sample beginning with 512 observations earlier and ending in Mar-2014 and so forth. 18 The percentage of time windows that the estimated cross-correlation is negative is 90.20% (AUS), 98.60% (BRA), 99.78% (CAN), 69.96% (CHN), 90.57% (FRA), 99.07% (GER), 99.97% (IDN), 99.97% (IND), 82.58% (JAP), 99.59% (KOR), 94.74% (MEX), 97.78% (RUS), 96.43% (SAF), 99.89% (TUR), 99.84% (UK) and 99.62% (USA). 19 The only exceptions are China, France, South Africa and the USA.

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Fig. 5. Time-varying correlation coefficients between lagged negative returns and current volatility (black lines) and between lagged positive returns and current volatility (grey lines) are based on detrended cross-correlation coefficient analysis.

return-volatility relation in international stock markets. We further investigate the leverage effect and the volatility feedback effect in a time-varying manner by applying the rolling window approach. In consistent with previous analysis, the length of the rolling windows is fixed to 512 business days. Fig. 4 illustrates the 170

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Table 5 Estimated asymmetric leverage effect.

Negative return Positive return

Negative return Positive return

AUS

BRA

CAN

CHN

FRA

GER

IND

IDN

0.2705 0.2738

0.2724 0.2277

0.2405 0.2723

0.2210 0.1137

0.2916 0.1709

0.3047 0.2556

0.3361 0.2546

0.3576 0.2217

JAP

KOR

MEX

RUS

SAF

TUR

UK

USA

0.2251 0.1363

0.2541 0.2405

0.2469 0.1887

0.2077 0.1615

0.2068 0.2467

0.2955 0.1517

0.3077 0.2388

0.3249 0.3199

dynamics of the leverage effect and the volatility feedback effect through presenting the evolution of the mean value of cross-correlation coefficients with time frame equal to 20, 40, 60 and 80 business days. The vertical axis shows the value of the cross-correlation coefficient in each rolling time window, while the horizontal axis is labelled with the last date of the observations in each moving window. It has been observed that the leverage effect is always stronger than the volatility feedback effect in all stock markets as the line of the leverage effect is always lower than the line of the volatility feedback effect. Same as previous observations, most of the strongly fluctuating curve of the leverage effect is below the horizontal zero line in all stock markets. By contrast, significant part of the fluctuating curve of the volatility feedback effect is above the horizontal zero line in the majority of the stock markets. To further corroborate these findings, we also calculate the percentage of time windows that the estimated leverage and volatility feedback effect are negative. For the leverage effect, almost all stock markets have over 90 percentage negative correlation windows during the whole sample period.20 The only exception is Japan with over 80 percentage negative correlation windows. For the volatility feedback effect, only three markets have over 90 percentage negative correlation windows during the whole sample period and almost half markets have less than 70 percentage negative correlation windows.21 Moreover, Table 4 displays the mean value of the leverage effect as well as the volatility feedback effect during the whole sample period. On the one hand, the leverage effect is much stronger than the volatility feedback effect on a country-by-country basis. On the other hand, with the exception of Brazil, India, Indonesia and South Africa, the volatility feedback effect is not far away from zero, which may reveal evidence of the insignificance of it. This strengthens previous finding that the statistical significance is weak and erratic for the volatility feedback effect. As a whole, these empirical findings may support the hypothesis that the negative return-volatility relation seems to be return-driven (the leverage effect). As empirical evidence shows the negatively related relation between return and volatility is more prominent for negative returns (see Badshah, 2013; Black, 1976; Christie, 1982; Low, 2004), we further examine the return-driven (the leverage effect) relation between return and volatility in an asymmetric way.22 For assessing asymmetry we define positive and negative returns as

Rþ it ¼

Rit if Rit > 0 and R it ¼ 0 if Rit < 0



Rit if Rit < 0 0 if Rit > 0

(14)

Consistent with previous analysis, we examine the asymmetric aspect of the leverage effect in a time-varying manner by applying the rolling window approach. The length of the rolling windows is also fixed to 512 business days. Fig. 5 illustrates the dynamics of the asymmetric leverage effect through presenting the evolution of the mean value of cross-correlation coefficients with time frame equal to 20, 40, 60 and 80 business days. The vertical axis shows the value of the cross-correlation coefficient in each rolling time window, while the horizontal axis is labelled with the last date of the observations in each moving window. The graphical evidence shows that the negatively related relation between returns and volatility is more prominent when returns decline as the line of the negative leverage effect is lower than the line of the positive leverage effect in most of the stock markets. Particularly, most of the strongly fluctuating curve of the asymmetric leverage effect is below the horizontal zero line in all stock markets no matter whether the negatively related relation between returns and volatility is positively return-driven or negatively return-driven. Moreover, Table 5 displays the mean value of the asymmetric leverage effect during the whole sample period. In twelve out of sixteen stock markets, with the exception of Australia, Brazil, Canada and South Africa, the negative leverage effect is stronger than the positive leverage effect, which suggests that the leverage effect is asymmetric and more pronounced for negative returns. Overall, our empirical findings are consistent with earlier studies that there is a significant negative and asymmetric correlation between innovations in return and volatility for stock indexes and the leverage effect is more likely to generate an asymmetric response with a greater effect when returns decline.

4. Conclusions This paper presents an empirical analysis of time-varying return-volatility relation in sixteen stock markets. Distinguishing from other studies which are majorly built around assuming some kind of volatility model, we estimate the volatility process without

20 The percentage of time windows that the estimated cross-correlation is negative is 96.72% (AUS), 96.50% (BRA), 99.97% (CAN), 90.28% (CHN), 99.97%% (FRA), 99.97% (GER), 99.97% (IDN), 99.97% (IND), 88.24% (JAP), 99.97% (KOR), 99.34% (MEX), 95.46% (RUS), 97.79% (SAF), 99.96% (TUR), 99.97% (UK) and 99.97% (USA). 21 The percentage of time windows that the estimated cross-correlation is negative is 74.11% (AUS), 90.77% (BRA), 82.47% (CAN), 53.23% (CHN), 43.82%% (FRA), 64.37% (GER), 89.02% (IDN), 99.97% (IND), 67.37% (JAP), 78.16% (KOR), 61.76% (MEX), 79.00% (RUS), 91.49% (SAF), 89.20% (TUR), 54.44% (UK) and 51.48% (USA). 22 We are really grateful to an anonymous referee who brings this issue to our attention.

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assuming any specific form of its behavior and find that the volatility process is strongly persistent and on the edge of non-stationarity. Circumventing this issue by taking the detrended cross-correlation coefficient that is able to efficiently estimate the return-volatility relation under such circumstance, we find negative relationship between returns and volatility prevails in international stock markets and the significance of the negative relationship is quite stable. Interestingly, the strength of the negative effect is different for specific scales and is stronger in longer time horizon. This finding is of great importance given that the empirical relationship between stock returns and volatility remains unsettled. Moreover, we investigate the presence of volatility feedback and leverage effects in international stock markets by examining the lead-lag relation between stock returns and volatility and empirically confirm that the leverage effect is the main source of the negative correlation between returns and volatility. Finally, by employing a rolling window approach, we show that time-varying negative return-volatility relation is more likely to generate an asymmetric response with a greater effect when returns decline, which is the common characteristic of international stock markets, albeit with different fluctuating levels, and the strength of the negative return-volatility relation is stronger in the period of market turbulence. This finding sheds new light on the return-volatility relation in international stock markets and may provide a partial explanation for mixed empirical findings in the return-volatility relation. The findings of this study may have interesting implications for policymakers, financial institutions and investors. For instance, it further corroborates Bekaert and Wu's argument (2000) in a time-varying manner that the negative return-volatility relation implies that the time-varying risk premium theory cannot be valid to explain the stock market behavior. The relationship between stock returns and volatility plays a vital role in assessing investment and leverage decisions, proposing portfolio selection, calculating value at risk, and developing dynamic hedging strategies. In particular, our results should be useful to investors as their investment performance greatly depends on the ability to forecast volatility movements and the related expected returns and, accordingly, to construct their investment portfolios based on these predictions. Further, our approach can clearly be extended to the analysis of sector stock market indices rather than focusing solely on the overall market index, while it would also be of interest to examine the forecasting ability of the proposed model. Future research may also employ this approach to explore other topics concerning the relationship between returns and volatility, such as commodity market, bond market as well as maritime market.

Acknowledgements I would like to thank Carl R. Chen (the Editor) and an anonymous referee for their valuable comments. Appendix I Details of the stock price indices used Markets

Index name

Wind code

Australia (AUS) Brazil (BRA) Canada (CAN) China (CHN) France (FRA) Germany (GER) India (IND) Indonesia (IDN) Japan (JAP) Korea (KOR) Mexico (MEX) Russia (RUS) South Africa (SAF) Turkey (TUR) United Kingdom (UK) United States of America (USA)

S&P/ASX 200 Bovespa S&P/TSX Composite SSE Composite CAC 40 DAX S&P BSE SENSEX Jakarta SE Composite Nikkei 225 Stock Average Korea SE Composite Mexican Bolsa IPC Russian Trading System FTSE/JSE Africa All Share Borsa Istanbul 100 FTSE 100 S&P 500

AS51 IBOVESPA GSPTSE 000001 FCHI GDAXI SENSEX JKSE N225 KS11 MXX RTS JALSH XU100 FTSE SPX

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