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Test for tail index constancy of GARCH innovations based on conditional volatility

  • Moosup Kim
  • Sangyeol Lee
Article
  • 17 Downloads

Abstract

This study considers the problem of testing whether the tail index of the GARCH innovations undergoes a change according to the values of conditional volatilities. Special attention is paid to power-transformed and threshold generalized autoregressive conditional heteroscedasticity processes that can accommodate the GARCH family. We show that the proposed test asymptotically follows a functional of a standard Brownian motion under some regularity conditions. To evaluate our method, we carry out a simulation study and real data analysis using the return series of the Google stock price and DowJones index.

Keywords

Constancy test for tail index Heavy-tailed distribution Conditional volatility GARCH model PTTGARCH model 

Notes

Acknowledgements

We thank the Editor, an AE and referee for their careful reading and valuable comments and suggestions. This research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (No. 2018R1A2A2A05019433).

References

  1. Berkes, I., Horváth, L. (2004). The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics, 32, 633–655.Google Scholar
  2. Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.Google Scholar
  3. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economic Statistics, 31, 542–547.CrossRefGoogle Scholar
  5. Bougerol, P., Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics, 52, 115–127.Google Scholar
  6. Carrasco, M., Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory, 18, 17–39.Google Scholar
  7. Chan, N. H., Deng, S.-J., Peng, L., Xia, Z. (2007). Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations. Journal of Econometrics, 137, 556–576.Google Scholar
  8. Csörgö, M., Horváth, L. (1997). Limit theorems in change-point analysis (Vol. 18). New York: Wiley.Google Scholar
  9. Deo, C. M. (1973). A note on empirical processes of strong-mixing sequences. The Annals of Probability, 1, 870–875.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Drees, H. (2003). Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617–657.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Drees, H., de Haan, L., Resnick, S. (2000). How to make a Hill plot. The Annals of Statistics, 28, 254–274.Google Scholar
  12. Eberlein, E. (1984). Weak convergence of partial sums of absolutely regular sequences. Statistics & Probability Letters, 2, 291–293.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Embrechts, P., Frey, R., McNeil, A. (2005). Quantitative risk management. Princeton series in Finance. Princeton: Princeton University Press.Google Scholar
  14. Engle, R. F., Manganelli, S. (2004). CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business & Economic Statistics, 22, 367–381.Google Scholar
  15. Engle, R. F., Ng, V. K. (1993). Measuring and testing the impact of news on volatility. Journal of Finance, 48, 1749–1778.Google Scholar
  16. Glosten, L. R., Jagannathan, R., Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779–1801.Google Scholar
  17. Hall, P. (1982). On some simple estimates of an exponent of regular variation. Journal of Royal Statistical Society Series B, 44, 37–42.MathSciNetzbMATHGoogle Scholar
  18. Hansen, B. E. (1994). Autoregressive conditional density estimation. International Economic Review, 35, 705–730.CrossRefzbMATHGoogle Scholar
  19. Higgins, M. L., Bera, A. K. (1992). A class of nonlinear ARCH models. International Economic Review, 33, 137–158.Google Scholar
  20. Hsing, T. (1991). On tail index estimation using dependent data. The Annals of Statistics, 19, 1547–1569.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hwang, S., Basawa, I. (2004). Stationarity and moment structure for box-cox transformed threshold GARCH(1,1) processes. Statistics & Probability Letters, 68, 209–220.Google Scholar
  22. Hwang, S., Kim, T. Y. (2004). Power transformation and threshold modeling for ARCH innovations with applications to tests for ARCH structure. Stochastic Processes & Their Applications, 110, 295–314.Google Scholar
  23. Kim, M., Lee, S. (2009). Test for tail index change in stationary time series with Pareto-type marginal distribution. Bernoulli, 15, 325–356.Google Scholar
  24. Kim, M., Lee, S. (2011). Change point test for tail index for dependent data. Metrika, 74, 297–311.Google Scholar
  25. Kim, M., Lee, S. (2016a). Nonlinear expectile regression with application to value-at-risk and expected shortfall estimation. Computational Statistics & Data Analysis, 94, 1–19.Google Scholar
  26. Kim, M., Lee, S. (2016b). On the tail index inference for heavy-tailed GARCH-type innovations. Annals of Institute of Statistical Mathematics, 68, 237–267.Google Scholar
  27. Kim, M., Lee, S. (2016c). Inference for location-scale time series models with ASTD and AEPD innovations. Unpublished manuscript.Google Scholar
  28. Lee, S., Lee, T. (2011). Value-at-risk forecasting based on Gaussian mixture ARMA–GARCH model. Journal of Statistical Computation & Simulation, 81, 1131–1144.Google Scholar
  29. Lee, S., Noh, J. (2013). Quantile regression estimator for GARCH models. Scandinavian Journal of Statistics, 40, 2–20.Google Scholar
  30. Li, C. W., Li, W. K. (1996). On a double-threshold autoregressive heteroscedastic time series model. Journal of Applied Econometrics, 11, 253–274.Google Scholar
  31. Móricz, F. (1983). A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Mathematica Hungarica, 41, 337–346.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347–370.MathSciNetCrossRefzbMATHGoogle Scholar
  33. Pan, J., Wang, H., Tong, H. (2008). Estimation and tests for power-transformed and threshold GARCH models. Journal of Econometrics, 142, 352–378.Google Scholar
  34. Peng, L., Yao, Q. (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrika, 90, 967–975.Google Scholar
  35. Quintos, C., Fan, Z., Phillips, P. C. B. (2001). Structural change tests in tail behaviour and the Asian crisis. Review of Economic Studies, 68, 633–663.Google Scholar
  36. Tweedie, R. L. (1975). Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stochastic Processes & Their Applications, 3, 385–403.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Zakoian, J.-M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics Control, 18, 931–955.CrossRefzbMATHGoogle Scholar
  38. Zhu, D., Galbraith, J. W. (2010). A generalized asymmetric Student-t distribution with application to financial econometrics. Journal of Econometrics, 157, 297–305.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  1. 1.APEC Climate CenterBusanKorea
  2. 2.Department of StatisticsSeoul National UniversitySeoulKorea

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