Flexible Multivariate GARCH Modeling for Heteroskedastic Asset Returns with an Asymmetric and Leptokurtic Distribution

10.6293/AQAFA.2013.11.10

Pilsun Choi；Kiseok Nam；Augustine C. Arize

S(subscript U)-Normal Distribution ； Asymmetric and Leptokurtic Distribution ； Value-at-Risk (Var) ； Heteroskedastic Asset Returns ； Multivariate GARCH Model

Advances in Quantitative Analysis of Finance and Accounting

11期（2013 / 12 / 01）

261 - 292

英文

We propose the S(subscript U)-normal distribution for the estimation of multivariate GARCH models to describe the nonnormality features, such as asymmetry and fat tails, embedded in heteroskedastic asset returns. We show that the S(subscript U)-normal distribution consistently outperforms the normal, Student-t and skewed-t distributions for describing the conditional distribution and the extreme lower and upper tail shapes of daily returns of individual stocks, industry portfolios, and national equity indexes over the sample period of 1989:01 - 2009:12. The exceeding ratio (ER) test for VaR forecasts suggested by Kupiec (1995) shows that the S(subscript U)-normal consistently outperforms the normal, Student-t, skewed-t distributions in multivariate CCC- and DCC-GARCH models. The results indicate that (a) compared to the S(subscript U)-normal, both the normal and Student-t distributions tend to underestimate the tail-thickness around the lower and upper extreme tails, and (b) even with an improvement relative to the normal and Student-t, the skewed-t distribution is still problematic because it tends to overestimate the extreme tails.

1. Engle, R. F., & Sheppard, K. (2001). Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH. Working Paper, University of California, Santa Cruz, CA.
2. Theodossiou, P. (2000). Skewed Generalized Error Distribution of Financial Assets and Option Pricing. SSRN working paper, Cyprus University of Technology, Cyprus.
3. Johnson, N. L. (1949a). Systems of frequency curves generated by method of translation. Biometrika, 36, 149-176.
4. Johnson, N. L. (1949b). Bivariate distributions based on simple translation systems. Biometrika, 36, 297-304.
5. Bera, A. K., & Premaratne, G. (2000). Modeling asymmetry and excess kurtosis in stock return data. Working Paper, University of Illinois, Illinois, IL.
6. Tse, Y. K., & Tsui, A. K. C. (2000). A multivariate GARCH model with timevarying correlations. Working Paper, National University of Singapore, Singapore.
7. Angelidis, T.,Benos, A.,Degiannakis, S.(2007).Angelidis, T., Benos, A., & Degiannakis, S. (2007). A robust VaR model under different time periods and weighting schemes. Review of Quantitative Finance and Accounting, 28, 187-201..Angelidis, T., Benos, A., & Degiannakis, S. (2007). A robust VaR model under different time periods and weighting schemes. Review of Quantitative Finance and Accounting, 28, 187-201.,28,187-201.
8. Baba, Y.,Engle, R. F.,Kraft, D. F.,Kroner, K. F.(1990).San Diego, CA.,University of California.
9. Baillie, R. T.,Bollerslev, T.(1989).The message in daily exchange rates: A conditional variance tale.Journal of Business and Economic Statistics,7,297-305.
10. Baixauli, J. S.,Alvarez, S.(2006).Evaluating effects of excess kurtosis on VaRestimates: Evidence for international stock indices.Review of Quantitative Finance and Accounting,27,27-46.
11. Bauwens, L.,Laurent, S.(2005).A new class of multivariate skew densities, with application to GARCH models.Journal of Business and Economic Statistics,23,346-354.
12. Bera, A. K.,Kim, S.(2002).Testing constancy of correlation and other specifications of the BGARCH model with an application to international equity returns.Journal of Empirical Finance,9,171-195.
13. Bollerslev, T.(1987).A conditionally heteroskedastic time series model for speculative prices and rates of return.Review of Economics and Statistics,69,542-547.
14. Bollerslev, T.(1990).Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model.Review of Economics and Statistics,72,498-505.
15. Bollerslev, T.,Engle, R. F.,Wooldridge, J. M.(1988).A capital asset pricing model with time-varying covariances.Journal of Political Economy,96,116-131.
16. Choi, P.,Nam, K.(2008).Asymmetric and leptokurtic distribution for heteroscedastic asset Returns: The SU-normal distribution.Journal of Empirical Finance,15,41-63.
17. Cotter, J.(2001).Margin exceedences for European stock index futures using extreme value theory.Journal of Banking and Finance,25,1475-1502.
18. Danielsson, J.,de Vries, C.(1997).Tail index and quantile estimation with very high frequency data.Journal of Empirical Finance,4,241-257.
19. Diebold, F.,Schuermann, T.,Stroughair, J.(2000).Pitfalls and opportunities in the use of extreme value theory in risk management.Journal of Risk Finance,1,30-35.
20. Engle, R. F.(2002).Dynamic conditional correlation: A simple class of multivariate GARCH models.Journal of Business and Economic Statistics,20,339-350.
21. Engle, R. F.,Gonzalez-Rivera, G.(1991).Semiparametric ARCH models.Journal of Business and Economic Statistics,9,345-359.
22. Engle, R. F.,Manganelli, S.(2004).CAViaR: Conditional autoregressive value at risk by regression quantiles.Journal of Business & Economic Statistics,22,367-381.
23. Gencay, R.,Selcuk, F.(2004).Extreme value theory and value-at-risk: Relative performance in emerging markets.International Journal of Forecasting,20,287-303.
24. Ghose, D.,Kroner, K. F.(1995).The relationship between GARCH and symmetric stable processes: Finding the source of fat tails in financial data.Journal of Empirical Finance,2,225-251.
25. Hansen, B. E.(1994).Autoregressive conditional density estimation.International Economic Review,35,705-730.
26. Harvey, C. R.,Siddique, A.(1999).Autoregressive conditional skewness.Journal of Financial and Quantitative Analysis,34,465-487.
27. Hsieh, D. A.(1989).Modeling heteroscedasticity in daily foreign-exchange rates.Journal of Business and Economic Statistics,7,307-317.
28. Hsu, C. P.,Huang, C. W.,Chiou, W. J. P.(2012).Effectiveness of copulaextreme value theory in estimating value-at-risk: Empirical evidence from Asian emerging markets.Review of Quantitative Finance and Accounting,39,447-468.
29. Huang, Y. C.,Lin, B. J(2004).Value-at-risk analysis for Taiwan stock index futures: Fat tails and conditional asymmetries in return innovations.Review of Quantitative Finance and Accounting,22,79-95.
30. Jansen, D. W.,Koedijk, K. G.,de Vries, C. G.(2000).Portfolio selection with limited downside risk.Journal of Empirical Finance,7,247-269.
31. Johnson, N. L.,Kotz, S.,Balakrishnan, N.(2002).Continuous multivariate distributions: Models and applications.New York:Wiley.
32. Jondeau, E.,Rockinger, M.(2003).Conditional volatility, skewness, and kurtosis: Existence, persistence, and comovements.Journal of Economic Dynamics and Control,27,1699-1737.
33. Kupiec, P. H.(1995).Techniques for verifying the accuracy of risk measurement models.Journal of Derivatives,3,73-84.
34. Lee, C. F.,Su, J. B.(2012).Alternative statistical distributions for estimating value-at-risk: Theory and evidence.Review of Quantitative Finance and Accounting,39,309-331.
35. Lee, T. K. Y.,Tse, Y. K.(1991).Term structure of interest rates in Singapore Asian dollar market.Journal of Applied Econometrics,6,143-152.
36. Maddala, G. S.(Ed.),Rao, C.(Ed.)(1997).Handbook of Statistics.Amsterdam, the Netherlands:Elsevier.
37. McDonald, J. B.(1991).Parametric models for partially adaptive estimation with skewed and leptokurtic residuals.Economics Letters,37,273-278.
38. McDonald, J. B.(1996).An application and comparison of some flexible parametric and semi-parametric qualitative response models.Economics Letters,53,145-152.
39. McDonald, J. B.,Newey, W. K.(1988).Partially adaptive estimation of regression models via the generalized t distribution.Econometric Theory,4,428-457.
40. McNeil, A. J.,Frey, R.(2000).Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach.Journal of Empirical Finance,7,271-300.
41. Mittnik, S.,Paolella, M. S.,Rachev, S. T.(2000).Diagnosing and treating the fat tails in financial returns data.Journal of Empirical Finance,7,389-416.
42. Nagahara, Y.(1999).The PDF and CF of Pearson type IV distributions and the ML estimation of the parameters.Statistics and Probability Letters,43,251-264.
43. Nelson, D. B.(1991).Conditional heteroskedasticity in asset returns: A new approach.Econometrica,59,347-370.
44. Newey, W. K.,Steigerwald, D. G.(1997).Asymptotic bias for quasimaximum likelihood estimators in conditional heteroskedasticity models.Econometrica,65,587-599.
45. Pagan, A.(1996).The econometrics of financial markets.Journal of Empirical Finance,3,15-102.
46. Richardson, M. P.,Smith, T.(1993).A test for multivariate normality in stock returns.Journal of Business,66,295-321.
47. Rockinger, M.,Jondeau, E.(2002).Entropy densities with an application to autoregressive conditional skewness and kurtosis.Journal of Econometrics,106,119-142.
48. Tauchen, G.,Pitts, M.(1983).The price variability-volume relationship in speculative markets.Econometrica,51,485-505.
49. Theodossiou, P.(1998).Financial data and the skewed generalized t distribution.Management Science,44,1650-1661.
50. Tse, Y. K.(2000).A test for constant correlations in a multivariate GARCH model.Journal of Econometrics,98,107-127.
51. Wang, K. L.,Fawson, C.,Barrett, C. B.,McDonald, J. B.(2001).A flexible parametric GARCH model with an application to exchange rates.Journal of Applied Econometrics,16,521-536.
52. Weiss, A. A.(1986).Asymptotic theory for ARCH models: Estimation and testing.Econometric Theory,2,107-131.