International Research Journal of Finance and Economics
ISSN 1450-2887 Issue 27 (2009)
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5 Conclusion
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References
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Threshold and Leverage Effects of Major Asian Stock Markets ...

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Threshold and Leverage Effects of Major Asian Stock Markets ...

  1. 1. International Research Journal of Finance and Economics ISSN 1450-2887 Issue 27 (2009) © EuroJournals Publishing, Inc. 2009 http://www.eurojournals.com/finance.htm Threshold and Leverage Effects of Major Asian Stock Markets Based on Stochastic Volatility Models Shian-Chang Huang Department of Business Administration, National Changhua University of Education, Taiwan E-mail: shhuang@cc.ncue.edu.tw Tel: 886-953092968; Fax: 886-4-7211292 Abstract This study investigates daily stock market volatility on major Asian stock indices using two stochastic volatility (SV) models, the SV model with threshold effects and the SV model with both threshold and leverage effects. Empirical results indicate the following interesting phenomenon: for threshold effects, the volatility of the NK225 index exhibited significant threshold non-linearity for both positive and negative shocks from NASDAQ returns, while the markets of South Korea and Taiwan revealed significant domestic threshold non-linearity for their own positive return shocks. All three markets display significant negative correlations between returns and volatility. However, leverage effects are most significant in the Taiwanese stock market. Comparing the two types of models in terms of DIC, reveals that the SV model with the threshold and leverage effects fits the data better. Keywords: Stochastic volatility model, threshold effect, leverage effect, Bayesian inference, Markov chain Monte Carlo 1 Introduction Market volatility is a measure of financial risks, and plays a crucial role in financial decision-making, option pricing, hedging strategies, portfolio allocation and Value-at-Risk calculations. However, with the expansion of international financial links and the continued liberalization of cross-border cash flows, international capital flows between the United States and East Asia have become increasingly prevalent. In the increasingly globalized environment, disturbances in one financial market immediately affect the entire international financial systems. This study extends stochastic volatility models to consider these complex market responses to local and international return shocks. In recent years it has been widely accepted that almost all high technological investments around the world are moving closer together, especially in East Asia and Silicon Valley in the United States. To manage risks in such international high technological investments, analyzing the risk and return relationship for major Asian market indices (NK225 (Japan), TWSI (Taiwan), KOSPI (South Korea)) and NASDAQ index are important. The relationship between risk and return has been studied extensively in advanced market economies, but has rarely been studied in East Asian financial markets. A common finding of the literature on leverage effects of financial markets is that a negative correlation exists between shocks to stock returns and volatility. Early studies (Black (1976) and Christie (1982)) asserted that stock price declines increase the debt-equity ratio (financial leverage),
  2. 2. International Research Journal of Finance and Economics - Issue 27 (2009) 107 subsequently increasing the risk (volatility) associated with the firm. Black (1976), Christie (1982), Nelson (1991)(EGARCH model), Glosten et al. (1993)(GJR-GARCH model) and Engle and Ng (1982) all found evidence of volatility being negatively correlated to equity returns. Recently, So et al. (2002) and Asai and McAleer (2004, 2005) considered another asymmetry for SV models. So et al. (2002) proposed a threshold SV model, which is analogous to the threshold ARCH model of Li and Li (1996). Moreover, Asai and McAleer (2004) considered an asymmetric SV model using a threshold effects indicator function, as suggested in Glosten et al. (1993)(GJR-GARCH model). All these models try to capture the threshold non-linearity (So et al. (2002)) of the dynamics of volatility to return shocks. Due to its critical nature, volatility estimation and forecasting has received considerable attention and a significant literature has built up. Traditionally, GARCH models (Engle (1982), Bollerslev (1986)) are the major tools for modeling volatility contagion and transmission, but many studies have pointed out that GARCH-type models do not fit empirical data as well as stochastic volatility models. Furthermore, SV models have a closer connection with financial economics theory. Various empirical studies have shown that SV models provide more accurate descriptions of the volatility dynamics than GARCH models. For instance, the superior performance of univariate SV models compared to GARCH-type models has been documented in Danielsson (1994) and Kim, Shephard and Chib (1998) in terms of in-sample fitting, and in Yu (2002) in terms of out-of-sample forecasting. The above results are due to the only one source of uncertainty in GARCH-type models, which makes GARCH-type models lack sufficient flexibility to capture the complex dynamics of a financial system. Thus this study applies SV models to investigate the threshold and leverage effects of these major Asian stock indices. This study extends SV models to a more complex setting. Two types of SV models, the SV model with threshold effects (TSV model) and the SV model with both threshold and leverage effects (TLSV model), are used in this study. These models are estimated using a Markov Chain Monte Carlo (MCMC) method. An overview of the literature on Bayesian MCMC methods for SV models is provided by the collection of articles in the book of Shephard (2005). The empirical results of this study revealed some interesting phenomenon: first, Japanese stock market displayed significant internal threshold non-linearity for both positive and negative NASDAQ return shocks, while the Taiwanese and South Korean stock markets revealed only significant domestic threshold effects for positive domestic return shocks, and the domestic threshold effects are asymmetric and more significant in the Taiwanese stock market. Second, when both threshold and leverage effects are considered, the threshold non-linearity displays similar patterns to international return shocks, but all three markets displayed significant negative correlations between return shocks and volatility. The leverage effects are most significant in the Taiwanese stock market. Third, comparing the two types of models in terms of DIC 1, reveals that the TLSV model has the best fit to the data. The remainder of the study is organized as follows: section 2 introduces the TSV and TLSV model, and demonstrates how to estimate the latent states and parameters of these models by MCMC methods. Section 3 then describes the data used in the study and its basic statistics. Next, section 4 displays and discusses the empirical findings. Finally, conclusions are presented in section 5. 2 Stochastic Volatility Models with Threshold and Leverage Effects The traditional SV model can be formulated as follows: yt = exp(ht /2 )ε t , ε t : N (0,1) (1) ht = μ + φht −1 + σ ηηt , ηt : N (0,1), (2) 1 The deviance information criterion (DIC) of Spiegelhalter, Best, Carlin and van der Linde (2002) is a generalization of AIC (Akaike information criterion) to complex hierarchical models. Like AIC, DIC consists of two components, the goodness-of-fit measure and the penalty for increasing model complexity
  3. 3. 108 International Research Journal of Finance and Economics - Issue 27 (2009) where ht is the log-volatility. The above model is extended to accommodate the threshold non- linearity of volatility dynamics to domestic and international return shocks, the ``TSV'' model, yt = exp(ht /2 )ε t , ε t : N (0,1) (3) ht = μ + φht −1 + β1 yt −11 y > 0 + β 2 yt −11 y ≤0 t −1 t −1 + β 3 xt −1 1 x + β 4 xt −1 1 x + σ ηη t , η t : N (0,1), (4) t −1 > 0 t −1 ≤ 0 where xt −1 is the lagged NASDAQ returns, and 1{⋅} is a indicator function. 1 y indicates whether t −1 > 0 yt −1 is greater than zero. In the model, β1 , β 2 are used to capture the threshold non-linearity induced by domestic return shocks, and β 3 , β 4 are used to capture the threshold non-linearity resulting from NASDAQ return shocks, which are taken to represent international return shocks. Next, the TSV model is extended to consider the leverage effect (Yu, 2005), the ``TLSV'' model, yt = exp(ht /2 )ε t , ε t : N (0,1) (5) ht = μ + φht −1 + β 3 xt −11 x > 0 + β 4 xt −11x ≤0 + σ ηηt , t −1 t −1 ηt : N (0,1) (6) corr (ε t −1 ,ηt ) = ρ . (7) The above model is equivalent to yt = exp(ht /2 )ε t , ε t : N (0,1) (8) ht = μ + φht −1 + β 3 xt −11 x > 0 + β 4 xt −11 x t −1 t −1 ≤ 0 + ρσ η yt −1 exp(− ht −1/2) + σ η 1 − ρ 2 wt , wt : N (0,1). (9) Because of yt −11 y + yt −11 y = yt −1 , the presence of yt −11 y , yt −11 y and t −1 > 0 t −1 ≤ 0 t −1 > 0 t −1 ≤ 0 ρσ η yt −1 exp(−ht −1/2) lead to multi-collinearity (See Asai and McAleer (2005) for details.). These two regressors are excluded from log-volatility regression equation (9) to form the new model. To estimate these SV models, Bayesian Markov chain Monte Carlo (MCMC) methods can be easily employed for inference. 3. Data and Descriptive Statistics The data used in the study are composed of the following daily indices: NASDAQ (US), NK225 (Japan), TWSI (Taiwan) and KOSPI (South Korea). In each market, this study uses the most comprehensive and diversified stock index. All of the index data encompass the period from January 1998 to December 2004. Because the sample markets operate on different holiday schedules, some daily observations are omitted. After matching the daily observations, we have 1517 observations. These stock market indices are then transformed into daily returns. Figures 1 and 2 plot the returns series, and Table 1 lists selected descriptive statistics of daily returns of these indices. These descriptive statistics include sample means, standard deviations, maximums, minimums, skewness, kurtosis, the Jacque-Bera statistics and their p-values.
  4. 4. International Research Journal of Finance and Economics - Issue 27 (2009) 109 Figure 1: NASDAQ and NK225 Index Returns: The data-sampling period ranges from January 1998 to December 2004. Return series of the NASDAQ index 4 Index Return 2 0 −2 −4 0 500 1000 1500 Return series of the NK225 index 4 Index Return 2 0 −2 −4 0 500 1000 1500 Figure 2: TWSI and KOSPI Index Returns: The data-sampling period ranges from January 1998 to December 2004. Return series of the TWSI index 4 Index Return 2 0 −2 −4 0 500 1000 1500 Return series of the KOSPI index 4 Index Return 2 0 −2 −4 0 500 1000 1500
  5. 5. 110 International Research Journal of Finance and Economics - Issue 27 (2009) Among these return series the KOSPI index displays the largest skewness and kurtosis. This phenomenon results from the East Asian crisis of 1997 to 1998, during which South Korea suffered especially heavily. Based on Jarque-Bera statistics, all of these indices strongly reject the null hypothesis that their distributions are normal, particularly the KOSPI index. Thus stochastic volatility models are reasonable models for capturing their complex volatility dynamics. Table 1: Descriptive Statistics of Daily Stock Returns on NASDAQ, NK225, TWSI, and KOSPI The data-sampling period ranges from January 1998 to December 2004. SD denotes the standard deviation; JBstat denotes Jacque-Bera statistics. Index Obs Mean SD Min Max NASDAQ 1517 0.00894 0.95451 -4.41609 4.42743 NK225 1517 -0.00755 0.69422 -3.14168 3.13636 TWSI 1517 -0.00786 0.82689 -5.54929 3.70009 KOSPI 1517 0.01459 0.77055 -6.18535 8.60836 Index Obs Skewness Kurtosis JBstat p-value NASDAQ 1517 0.09203 1.88643 227.07523 4.91179E-50 NK225 1517 0.13042 1.29993 111.11040 7.45904E-25 TWSI 1517 -0.13691 3.16834 639.24832 1.54522E-139 KOSPI 1517 1.13864 21.88563 30603.36426 0.00000E+00 4. Empirical Results The TSV model is estimated first, and β1 , β 2 are used to capture the threshold non-linearity resulting from to domestic return shocks and β 3 , β 4 to capture the threshold non-linearity owing to NASDAQ return shocks. To obtain sufficient mixing of every Markov chain, the MCMC algorithm is run for 100000 iterations, and the initial 50000 samples are discarded. Tables 2 to 4 list the parameter estimates and their respective standard errors for the TSV model. Figure 3 shows the estimated volatility for the three markets. Table 2: Empirical Results of TSV model for the NK225 Index PC denotes the percentile; DIC denotes the deviance information criterion; * indicates the significance at the 95% confidence level. Mean SD 2.5 PC Median 97.5 PC μ -0.02870 0.04658 -0.12650 -0.02670 0.05457 φ *0.96132 0.01990 0.91660 0.96350 0.99360 β1 -0.09923 0.06461 -0.22170 -0.09946 0.02352 β2 -0.05024 0.06477 -0.18485 -0.04862 0.07470 β3 *-0.05428 0.02719 -0.10880 -0.05398 -0.00226 β4 *-0.07386 0.02021 -0.11690 -0.07276 -0.03650 ση *0.11826 0.01972 0.08485 0.11640 0.16230 DIC 3044
  6. 6. International Research Journal of Finance and Economics - Issue 27 (2009) 111 Table 3: Empirical Results of TSV model for the TWSI Index Mean SD 2.5 PC Median 97.5 PC μ 0.03297 0.04008 -0.05225 0.03545 0.10360 φ *0.96737 0.02016 0.92020 0.97030 0.99630 β1 *-0.30874 0.05698 -0.42445 -0.30650 -0.20340 β2 0.01067 0.05539 -0.10405 0.01192 0.11690 β3 0.04994 0.03910 -0.02119 0.04820 0.13295 β4 -0.05641 0.03103 -0.11800 -0.05606 0.00280 ση *0.23757 0.03593 0.17375 0.23500 0.31545 DIC 3376 Table 4: Empirical Results of TSV model for the KOSPI Index Mean SD 2.5 PC Median 97.5 PC μ -0.05347 0.05873 -0.17805 -0.04965 0.04993 φ *0.95323 0.02091 0.90780 0.95500 0.98890 β1 *-0.14127 0.05675 -0.25000 -0.14270 -0.02586 β2 -0.08390 0.06769 -0.22380 -0.08163 0.04337 β3 -0.02868 0.04899 -0.12490 -0.02873 0.06838 β4 -0.02109 0.03935 -0.09893 -0.02075 0.05611 ση *0.40756 0.04890 0.32045 0.40370 0.51250 DIC 2258 Figure 3: Volatilities of the Three Indices Based on the TSV Model Volatility of the NK225 index 4 2 0 0 500 1000 1500 Volaility of the TWSI index 4 2 0 0 500 1000 1500 Volatility of the KOSPI index 4 2 0 0 500 1000 1500
  7. 7. 112 International Research Journal of Finance and Economics - Issue 27 (2009) Tables 2 to 4 showed that these three markets all have highly persistent volatility. Parameter of persistence, φ , ranges from 0.96737 to 0.95323 . Notably, the NK225 and TWSI indices are more persistent than the KOSPI index. As shown in Figure 5, volatility was highest in the KOSPI index around 1998 because of the heavy damage suffered during the East Asian crisis. The volatility of volatility σ η ranges from 0.40756 in South Korea to 0.11826 in Japan. Japan is the most mature among the three markets and has the lowest volatility throughout most of the sample period. As shown in Tables 2 to 4, the Japanese stock market only displayed significant international threshold non-linearity ( β 3 < 0, β 4 < 0 ) for positive and negative return shocks from the NASDAQ index, while the Taiwanese and South Korean stock markets only revealed significant domestic threshold effects ( β1 < 0 ) for their own positive return shocks, and the domestic threshold effect is more significant in the Taiwanese stock market with β1 = −0.30874 . The volatility responses of the NK225 index to positive and negative NASDAQ return shocks are relatively symmetric, while the volatilities of the TWSI and KOSPI indices only displayed an asymmetrical response to their positive domestic return shocks. The TLSV model is estimated using a similar method. The empirical results are illustrated in Tables 5 to 7. Moreover, the trace and density plots of every parameter of the TWSI index are displayed in Figures 4 and 5 for reference, and the estimated volatilities of the three markets are shown in Figure 6. International return shocks display similar patterns of threshold non-linearity, but all three markets exhibited significant negative correlations between returns and volatility, and revealed lower volatility persistent level φ . The leverage effects are most significant in the Taiwanese stock market, with ρ = −0.44668 , followed by the Japanese stock market, with ρ = −0.37360 , and are least significant in the South Korea's stock market, with ρ = −0.22242 . The threshold non-linearity displays similar patterns: the Japanese stock market exhibited significant international threshold effects ( β 3 < 0, β 4 < 0 ) for both positive and negative NASDAQ return shocks, while both the Taiwanese and South Korean stock markets revealed insignificant international threshold non-linearity. As indicated by Berg et al. (2004), AIC (Akaike information criterion) can not be applied to compare SV models. Consequently, the study compared TSV and TLSV models in terms of DIC (Deviance Information Criterion), and it is obvious that DIC prefers the TLSV model. Table 5: Empirical Results of TLSV model for the NK225 Index Mean SD 2.5 PC Median 97.5 PC μ -0.04891 0.01744 -0.08654 -0.04751 -0.01852 φ 0.95433 0.01209 0.92780 0.95550 0.97490 β3 -0.05493 0.02416 -0.10120 -0.05546 -0.00564 β4 -0.07781 0.01975 -0.11900 -0.07677 -0.04207 ση 0.12529 0.01939 0.09247 0.12380 0.16815 ρ -0.37360 0.12267 -0.59400 -0.38095 -0.11680 DIC 2964
  8. 8. International Research Journal of Finance and Economics - Issue 27 (2009) 113 Table 6: Empirical Results of TLSV model for the TWSI Index Mean SD 2.5 PC Median 97.5 PC μ -0.09035 0.03335 -0.16720 -0.08669 -0.03550 φ 0.91541 0.02451 0.85745 0.91840 0.95450 β3 0.04061 0.04043 -0.03539 0.03946 0.12245 β4 -0.05132 0.03446 -0.12445 -0.04972 0.01149 ση 0.26516 0.04389 0.19330 0.25990 0.36710 ρ -0.44668 0.07598 -0.58675 -0.44990 -0.28895 DIC 3344 Table 7: Empirical Results of TLSV model for the KOSPI Index Mean SD 2.5 PC Median 97.5 PC μ -0.07698 0.03593 -0.15280 -0.07532 -0.01197 φ 0.94428 0.01449 0.91310 0.94530 0.96980 β3 -0.03431 0.04853 -0.12835 -0.03489 0.06112 β4 -0.01544 0.03902 -0.09307 -0.01507 0.06016 ση 0.41670 0.04777 0.32555 0.41500 0.51740 ρ -0.22242 0.06131 -0.34035 -0.22280 -0.10020 DIC 2238 Figure 4: Trace and Kernel Density Plots of Parameters of the TWSI Index Based on the TLSV Model μ Kernel Density 20 0.2 0 10 −0.2 −0.4 0 1 2 3 4 5 −0.3 −0.2 −0.1 0 0.1 4 x 10 φ Kernel Density 20 1 10 0.8 0 1 2 3 4 5 0.7 0.8 0.9 1 4 x 10 β3 Kernel Density 0.5 10 0 5 −0.5 0 1 2 3 4 5 −0.2 0 0.2 0.4 0.6 4 x 10
  9. 9. 114 International Research Journal of Finance and Economics - Issue 27 (2009) Figure 5: Trace and Kernel Density Plots of Parameters of the TWSI Index Based on the TLSV Model β Kernel Density 4 0.5 20 0 10 −0.5 0 1 2 3 4 5 −0.3 −0.2 −0.1 0 0.1 4 x 10 σ η Kernel Density 10 0.5 5 0 0 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 4 x 10 ρ Kernel Density 1 10 0 5 −1 0 1 2 3 4 5 −0.8 −0.6 −0.4 −0.2 0 4 x 10 Figure 6: Volatilities of the Three Indices Based on the TLSV model Volatility of the NK225 index 4 2 0 0 500 1000 1500 Volaility of the TWSI index 4 2 0 0 500 1000 1500 Volatility of the KOSPI index 4 2 0 0 500 1000 1500 Among the three markets, Japan is the most mature and has strong linkages to international financial markets. Information transmissions in the Japanese stock market is more efficient and information disclosures is more transparent. Investors in the Japanese market usually have good hedges on their domestic risks. Consequently, these investors do not react significantly to good and bad domestic news, but react strongly to shocks from the NASDAQ. In comparison, in the Taiwanese and South Korean stock markets, information transmission and disclosure are less efficient and transparent. Investors in these markets are sensitive to domestic good news. Consequently, the TWSI and KOSPI indices are more volatile than the NK225 index throughout the study period. (as shown in Figure 6), and have asymmetric responses to bad and good news.
  10. 10. International Research Journal of Finance and Economics - Issue 27 (2009) 115 5 Conclusion This study applied the TSV and TLSV models to analyze the threshold and leverage effects of major Asian stock markets, including the Japanese, South Korean, and Taiwanese markets. This study found that for threshold effects the Japanese stock market only exhibited significant international threshold non-linearity for positive and negative shocks from NASDAQ returns, while the Taiwanese and South Korean stock markets only revealed significant domestic threshold effects for their own positive return shocks. The domestic threshold effect was asymmetric and more significant in the Taiwanese stock market. When both threshold and leverage effects are considered, the patterns of threshold non- linearity for international return shocks are similar, but all three markets displayed significant negative correlations between returns and volatility. The leverage effects are most significant in the Taiwanese stock market. Comparing the two models in terms of DIC, reveals that the TLSV model fits the data more closely. In sum, the results of this study can help investors control and hedge their risks in international investments, especially in East Asian markets. A future challenge will be to combine SV models with methods from extreme value theory to consider the risk of investing in East Asian financial markets. Another challenge will be to combine behavior finance theories to investigate the detailed mechanism of these complex threshold and leverage effects.
  11. 11. 116 International Research Journal of Finance and Economics - Issue 27 (2009) References [1] Asai, M., M. McAleer, 2004, Dynamic leverage and threshold effects in stochastic volatility models. Unpublished manuscript, Faculty of Economics, Tokyo Metropolitan University. [2] Asai, M and M. McAleer, 2005, Dynamic Asymmetric Leverage in Stochastic Volatility Models, Econometric Reviews, 24(3), 317-332. [3] Berg, A., R. Meyer, and J. Yu, 2004, Deviance information criterion for comparing stochastic volatility models. Journal of Business and Economic Statistics, 22, 107-120. [4] Black, F., 1976, Studies in Stock Price Volatility Changes, Proceedings of the 1976 Business Meeting of the Business and Economic Statistics Section. American Statistical Association, 177-181. [5] Bollerslev, T., 1986, Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 31, 307-327. [6] Christie, A. A., 1982, The Stochastic Behavior of Common Stock Variances: Value, Leverage and Interest Rate Effects, Journal of Financial Economics, 10, 407-432. [7] Danielsson, J., 1994 Stochastic volatility in asset prices: Estimation with simulated maximum likelihood. Journal of Econometrics, 64, 375-400. [8] Engle, R. F., 1982, Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50, 987-1007. [9] Engle, R. F., and V. K. Ng, 1993, Measuring and Testing the Impact of News on Volatility, Journal of Finance, 48 1749-1778. [10] Glosten, L. R., R. Jaganathan, and D. E. Runkle, 1993, On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance, 48, 1779- 801. [11] Harvey, A. C., and N. Shephard, 1996, Estimation of an Asymmetric Stochastic Volatility Model for Asset Returns, Journal of Business and Economic Statistics, 14, 429-434. [12] Jacquier, E., N. G. Polson, P. E. Rossi, 2004, Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. Journal of Econometrics, 122, 185-212. [13] Kim, S., N. Shephard, and S. Chib, 1998, Stochastic volatility: likelihood inference and comparison with ARCH models, Review of Economic Studies, 65, 361-393. [14] Li, C. W. and W. K. Li, 1996, On a double-threshold autoregressive heteroscedastic time series model. Journal of Applied Econometrics, 11, 253-274. [15] Nelson, D. B., 1991, Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59 347-370. [16] Selçuk, F., 2005, Asymmetric Stochastic Volatility in Emerging Stock Markets, Applied Financial Economics, 15, 867-874. [17] Shephard, N., 2005, Stochastic Volatility. Oxford: Oxford University Press. [18] So, M. K. P., W. K. Li, and K. Lam, 2002, A Threshold Stochastic Volatility Model, Journal of Forecasting, 21, 473-500. [19] Spiegelhalter, D. J., N. G. Best, B. P. Carlin, and A. van der Linde, 2002, Bayesian measures of model complexity and fit (with discussion), Journal of the Royal Statistical Society, Series B 64, 583-639. [20] Yu, J., 2002, Forecasting volatility in the New Zealand stock market, Applied Financial Economics, 12, 193-202. [21] Yu, J., 2005, On leverage in a stochastic volatility model. Journal of Econometrics 127, 165- 178.

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