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Modeling US Stock Market Volatility-Return Dependence Using Conditional Copula and Quantile Regression

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The Economics of the Global Environment

Part of the book series: Studies in Economic Theory ((ECON.THEORY,volume 29))

Abstract

In this chapter, we examine the return-volatility relationship for some indices reported on exchanges in the United States of America. We utilize both linear quantile regression and copula quantile regression to evaluate the asymmetric volatility-return relationship between changes in the volatility index (VXD, VIX, VXO and VXN) and the corresponding stock index return series (DJIA, S&P 500, the S&P 100 and NASDAQ). The data period is from February 2, 2001 through December 31, 2012. The quantile copula models allow for inference at different quantiles of interest. We find, first, that the relationship between stock return and implied volatility depends on the quartile at which the relationship is being investigated. Second, we obtain results similar to those reported for European exchanges showing the existence of an inverted U-shaped relationship between stock return and implied volatility. This result was obtained even after controlling for changes in volatility of return using a GARCH(1, 1) filter.

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References

  • Abdi, H. (2007). Method of least squares. In Neil Salkind (Ed.), Encyclopedia of measurement and statistics. Thousand Oaks (CA): Sage.

    Google Scholar 

  • Alexander, C. (2008). Market risk analysis: practical financial econometrics (Vol. 2). Chichester: Wiley.

    Google Scholar 

  • Allen, D. E., Gerrans, P., Singh, A. K., & Powell, R. (2009). Quantile regression and its application in investment analysis. The Finsia Journal of Applied Finance (JASSA), 4, 7–12.

    Google Scholar 

  • Allen, D., Singh, A. K., Powell, R. J., McAleer, M., Taylor, J., & Thomas, L. (2012). The volatility-return relationship: insights from linear and non-linear quantile regression. Working Paper, School of Accounting Finance & Economics, Edith Cowan University, Retrieved from Complutense. http://eprints.ucm.es/16688/.

  • Ang, A., & Chen, J. (2002). Asymmetric correlations of equity portfolios. Journal of Financial Economics, 63(3), 443–494.

    Article  Google Scholar 

  • Badshah, I. U. (2012). Quantile regression analysis of the asymmetric return-volatility relation. Journal of Futures Markets, 33(3), 235–265.

    Article  Google Scholar 

  • Barnes, M. L., & Hughes, W. A. (2002). Quantile regression analysis of the cross section of stock market returns (Working Paper). Retrieved from Social Science Research. http://ssrn.com/abstract=458522.

  • Black, F. (1976). Studies of stock market volatility changes, Proceedings of the American Statistical Association, Business and Economic Statistics Section (pp. 177–181)

    Google Scholar 

  • Bouyé, E., & Salmon, M. (2009). Dynamic copula quantile regressions and tail area dynamic dependence in Forex markets. The European Journal of Finance, 15(7–8), 721–750.

    Article  Google Scholar 

  • Breymann, W., Dias, A., & Embrechts, P. (2003). Dependence structures for multivariate high-frequency data in finance. Quantitative Finance, 3(1), 1–16.

    Article  Google Scholar 

  • Buchinsky, M. (1998). Recent advances in quantile regression models: A practical guideline for empirical research. The Journal of Human Resources, 33(1), 88–126.

    Article  Google Scholar 

  • Buchinsky, M., & Leslie, P. (1997). Educational attainment and the changing U.S. wage structure: Some dynamic implications (Working Paper No. 97–13). Department of Economics, Brown University.

    Google Scholar 

  • Buchinsky, M., & Hunt, J. (1999). Wage mobility in the united state. The Review of Economics and Statistics, 8(3), 351–368.

    Article  Google Scholar 

  • Campbell, J. Y., & Hentschel, L. (1992). No news is good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics, 31(3), 281–318.

    Article  Google Scholar 

  • Chen, X., Fan, Y., & Tsyrennikov, V. (2006). Efficient estimation of semiparametric multivariate copula models. Journal of the American Statistical Association, 101, 1228–1240.

    Article  Google Scholar 

  • Cherubini, U., Luciano, E., & Vecchiato, W. (2004). Copula methods in finance. New York: Wiley.

    Book  Google Scholar 

  • Chichilnisky, G. (2009). The topology of fear. Journal of Mathematical Economics, 45(11–12), 807–816.

    Article  Google Scholar 

  • Christie, A. (1982). The stochastic behaviour common stock variances: Value, leverage and interest rate. Journal of Financial Economics, 10, 407–432.

    Article  Google Scholar 

  • Clayton, D. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–151.

    Article  Google Scholar 

  • Cook, R. D., & Johnson, M. E. (1981). A family of distributions for modelling non-elliptically symmetric multivariate data. Journal of Royal Statistical Society B, 43(2), 210–218.

    Google Scholar 

  • Cox, J., & Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166.

    Article  Google Scholar 

  • Dennis, P., Mayhew, S., & Stivers, C. (2006). Stock returns, implied volatility innovations, and the asymmetric volatility phenomenon. Journal of Financial and Quantitative Analysis, 41(2), 381–406.

    Article  Google Scholar 

  • Eide, E., & Showalter, M. H. (1998). The effect of school quality on student performance: A quantile regression approach. Economics Letters, 58(3), 345–350.

    Article  Google Scholar 

  • Eide, E., & Showalter, M. H. (1999). Factors affecting the transmission of earnings across generations. A quantile regression approach. Journal of Human Resources, 34(2), 253–267.

    Article  Google Scholar 

  • Embrechts, P., Höing, A., & Juri, A. (2003). Using copulae to bound the value-at- risk for functions of dependent risks. Finance & Stochastics, 7, 145–167.

    Article  Google Scholar 

  • Embrechts, P., McNeil, A., & Straumann, D. (2001). Correlation and dependence in risk management: Properties and pitfalls. In M. A. H. Dempster (Ed.), Risk management: Value at risk and beyond (pp. 176–223). Cambridge: Cambridge University Press.

    Google Scholar 

  • Engle, R. F., & Manganelli, S. (2004). CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economics Statistics, 22(4), 367–381.

    Article  Google Scholar 

  • Fang, H., Fang, K., & Kotz, S. (2002). The metaelliptical distributions with given marginals. Journal of Multivariate Analalysis, 82, 1–16.

    Article  Google Scholar 

  • Fleming, J., Ostdiek, B., & Whaley, R. E. (1995). Predicting stock market volatility: A new measure. Journal of Futures Markets, 15(3), 265–302.

    Article  Google Scholar 

  • Frank, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y \(-\) F(x, y). Aequationes Math, 19, 194–226.

    Article  Google Scholar 

  • French, Kenneth R., William Schwert, G., & Stambaugh, Robert F. (1987). Expected stock returns and volatility. Journal of Financial Economics, 19, 3–29.

    Article  Google Scholar 

  • Galton, F. (1886). Regression towards mediocrity in hereditary stature. The Journal of the Anthropological Institute of Great Britain and Ireland, 15, 246–263.

    Article  Google Scholar 

  • Genest, C., & Mackay, J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. The American Statistician, 40, 280–283.

    Google Scholar 

  • Genest, C., & Rivest, L. P. (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88, 1034–1043.

    Article  Google Scholar 

  • Ghalanos, A. (2013). rugarch: A garch r package (Version 1.2-3). Retrieved from r-project.org. http://www.r-project.org.

  • Giot, P. (2005). Relationships between implied volatility indices and stock index returns. Journal of Portfolio Management, 31, 92–100.

    Article  Google Scholar 

  • Gumbel, E. J. (1960) Distributions des Valeurs Extremes en Plusieurs Dimensions. Publications de 1Institute de Statistique de 1Universite de Paris 9:171–173.

    Google Scholar 

  • Harper, H. L. (1974–1976). The method of least squares and some alternatives. Part I, II, II, IV, V, VI. International Statistical Review 42, 147–174; 42, 235–264; 43, 1–44; 43, 125–190; 43, 269–272; 44, 113–159.

    Google Scholar 

  • Hibbert, A., Daigler, R., & Dupoyet, B. (2008). A behavioural explanation for the negative asymmetric return-volatility relation. Journal of Banking and Finance, 32, 2254–2266.

    Article  Google Scholar 

  • Joe, H. (1997). Multivariate models and dependence concepts. New York: Chapman and Hall.

    Book  Google Scholar 

  • Kim, T. H., & White, H. (2003). Estimation, inference, and specification testing for possibly misspecified quantile regressions. In T. Fomby & R. C. Hill (Eds.), Maximum Likelihood Estimation of Misspecified Models: Twenty Years Later (pp. 107–132). New York: Elsevier.

    Chapter  Google Scholar 

  • Kimeldorf, G., & Sampson, A. R. (1975). Uniform representations of bivariate distributions. Communications in Statistics, 4, 617–627.

    Article  Google Scholar 

  • Koenker, R. (2005). Quantile regression., Econometric society monograph series New York: Cambridge University Press.

    Google Scholar 

  • Koenker, R. (2012). Quantreg: A quantile regression R package (Version 5.05). Retrieved from r-project.org. http://www.r-project.org.

  • Koenker, R. W., & Bassett, G, Jr. (1978). Regression quantiles. Econometrica, 46(1), 33–50.

    Article  Google Scholar 

  • Koenker, R. W., & Bassett, G, Jr. (1982). Robust tests for heteroscedasticity based on regression quantiles. Econometrica, 50(1), 1577–1584.

    Article  Google Scholar 

  • Koenker, R. W., & Hallock, K. F. (2001). Quantile regression. Journal of Economic Perspectives, 15(4), 143–156.

    Article  Google Scholar 

  • Koenker, R. W., & Xiao, Z. (2006). Quantile autoregression. Journal of the American Statistical Association, 101(475), 980–990.

    Article  Google Scholar 

  • Kumar, S. (2012). A first look at the properties of India’s volatility index. International Journal of Emerging Markets, 7(2), 160–176.

    Article  Google Scholar 

  • Longin, F., & Solnik, B. (2001). Extreme correlation of international equity markets. Journal of Finance, 56, 649–676.

    Article  Google Scholar 

  • Low, C. (2004). The fear and exuberance from implied volatility of S&P 100 index options. Journal of Business, 77, 527–546.

    Article  Google Scholar 

  • MacKinnon, J. G. (1996). Numerical distribution functions for unit root and cointegration tests. Journal of Applied Econometrics, 11, 601–618.

    Article  Google Scholar 

  • Neftci, S. (2000). Value at risk calculations, extreme events, and tail estimation. Journal of Derivatives, 7(3), 23–37.

    Article  Google Scholar 

  • Nelsen, R. B. (2006). Introduction to copulas. New York: Springer Verlag.

    Google Scholar 

  • Patton, A. J. (2004). On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. Journal of Financial Econometrics, 2(1), 130–168.

    Article  Google Scholar 

  • Patton, A. J. (2006a). Estimation of multivariate models for time series of possibly different lengths. Journal of Applied Econometrics, 21(2), 147–173.

    Article  Google Scholar 

  • Patton, A. J. (2006b). Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527–556.

    Article  Google Scholar 

  • Patton, A. J. (2009). Copula-based models for financial time series. In T. G. Andersen, R. A. Davis, J. P. Kreiss, & T. Mikosch (Eds.), Handbook of financial time series. New York: Springer Verlag.

    Google Scholar 

  • Powell, J. L. (1986). Censored regression quantiles. Journal of Econometrics, 32, 143–155.

    Article  Google Scholar 

  • Sklar, A. (1959). Fonctions de Riépartition \(\acute{a}\)n Dimensions et Leurs Marges. Publications de lInstitut de Statistique de lUniversité de Paris, 8, 229–231.

    Google Scholar 

  • Taylor, J. (1999). A quantile regression returns. Journal of Derivatives, 7(1), 64–78.

    Article  Google Scholar 

  • Trivedi, P. K., & Zimmer, D. M. (2005). Copula modeling: An introduction for practitioners. Foundations and Trends in Econometrics, 1(1), 1–111.

    Article  Google Scholar 

  • Whaley, R. (2000). The investor fear gauge. Journal of Portfolio Management, 26, 12–17.

    Article  Google Scholar 

  • White, H. (1994). Estimation inference and specification analysis. New York: Cambridge University Press.

    Book  Google Scholar 

  • Wu, G. (2001). The determinants of asymmetric volatility. The Review of Financial Studies, 14(3), 837–859.

    Article  Google Scholar 

  • Xiao, Z. (2009). Quantile cointegrating regression. Journal of Econometrics, 150(2), 248–260.

    Article  Google Scholar 

  • Yu, K., Lu, Z., & Stander, J. (2003). Quantile regression: Applications and current research areas. Journal of the Royal Statistical Society, Series D. (The Statistician) 52(3), 331–350.

    Google Scholar 

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Acknowledgements

I like to thank Professor David Allen, Professor Graciela Chichilnisky, Professor Roger Koenker, Professor Randall Filer, a referee and the editor, for comments and suggestions. I also like to thank Lucas Dowiak for comments, suggestions and very useful computing assistance. I alone am responsible for any remaining errors.

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Authors and Affiliations

  1. Department of Economics, Hunter College and the Graduate Center, City University of New York, New York City, NY, 10021, USA

    Terence D. Agbeyegbe

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  1. Terence D. Agbeyegbe
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Correspondence to Terence D. Agbeyegbe .

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Editors and Affiliations

  1. Columbia University, New York, New York, USA

    Graciela Chichilnisky

  2. Department of Socio-Economics, Vienna University of Economics and Business, Vienna, Austria

    Armon Rezai

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Agbeyegbe, T.D. (2016). Modeling US Stock Market Volatility-Return Dependence Using Conditional Copula and Quantile Regression. In: Chichilnisky, G., Rezai, A. (eds) The Economics of the Global Environment. Studies in Economic Theory, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-31943-8_26

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