DOI QR코드

DOI QR Code

Functional central limit theorems for ARCH(∞) models

  • Choi, Seunghee (Department of Statistics, Ewha Womans University) ;
  • Lee, Oesook (Department of Statistics, Ewha Womans University)
  • Received : 2017.03.28
  • Accepted : 2017.09.14
  • Published : 2017.09.30

Abstract

In this paper, we study ARCH(${\infty}$) models with either geometrically decaying coefficients or hyperbolically decaying coefficients. Most popular autoregressive conditional heteroscedasticity (ARCH)-type models such as various modified generalized ARCH (GARCH) (p, q), fractionally integrated GARCH (FIGARCH), and hyperbolic GARCH (HYGARCH). can be expressed as one of these cases. Sufficient conditions for $L_2$-near-epoch dependent (NED) property to hold are established and the functional central limit theorems for ARCH(${\infty}$) models are proved.

Keywords

References

  1. Berkes I, Hormann S, and Horv'ath L (2008). The functional central limit theorem for a family of GARCH observation with applications. Statistics and Probability Letters, 78, 2725-2730. https://doi.org/10.1016/j.spl.2008.03.021
  2. Billingsley P (1968). Convergence of Probability Measures, Wiley, New York.
  3. Bollerslev T (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Economet-rics, 31, 307-327.
  4. Conrad C (2010). Non-negativity conditions for the hyperbolic GARCH model. Journal of Econo-metrics, 157, 441-457.
  5. Conrad C and Haag BR (2006). Inequality constraints in the fractionally integrated GARCH model. Journal of Financial Econometrics, 4, 413-449. https://doi.org/10.1093/jjfinec/nbj015
  6. Csorgo M and Horv'ath L (1997). Limit Theorems in Change-Point Analysis, Wiley, New York.
  7. Davidson J (2002). Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. Journal of Econometrics, 106, 243-269. https://doi.org/10.1016/S0304-4076(01)00100-2
  8. Davidson J (2004). Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business & Economic Statistics, 22, 16-29. https://doi.org/10.1198/073500103288619359
  9. Dedecker J, Doukhan P, Lang G, Leon JR, Louhichi S, and Prieur C (2007). Weak Dependence, Examples and Applications, Springer, New York.
  10. De Jong RM and Davidson J (2000). The functional central limit theorem and weak convergence to stochastic integrals I: weakly dependent processes. Econometric Theory, 16, 643-666. https://doi.org/10.1017/S0266466600165028
  11. Doukhan P and Wintenberger O (2007). An invariance principle for weakly dependent stationary general models. Probability and Mathematical Statistics, 27, 45-73.
  12. Engle RF (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987-1007. https://doi.org/10.2307/1912773
  13. Giraitis L, Kokoszka P, and Leipus R (2000). Stationary ARCH models: dependence structure and central limit theorem. Econometric Theory, 16, 3-22.
  14. Herrndorf N (1984). A functional central limit theorem for weakly dependent sequences of random variables. The Annals of Probability, 12, 141-153. https://doi.org/10.1214/aop/1176993379
  15. Hwang EJ and Shin DW (2013). A CUSUM test for a long memory heterogeneous autoregressive models. Economics Letters, 121, 379-383. https://doi.org/10.1016/j.econlet.2013.09.014
  16. Kazakevicius V and Leipus R (2002). On stationarity in the ARCH($H_{\infty}$) model. Econometric Theory, 18, 1-16. https://doi.org/10.1017/S0266466602181011
  17. Lee O (2014a). Functional central limit theorems for augmented GARCH(p, q) and FIGARCH pro-cesses. Journal of the Korean Statistical Society, 43, 393-401. https://doi.org/10.1016/j.jkss.2013.12.001
  18. Lee O (2014b). The functional central limit theorem and structural change test for the HAR($H_{\infty}$) model. Economic Letters, 124, 370-373. https://doi.org/10.1016/j.econlet.2014.06.029
  19. Li M, Li W, and Li G (2015). A new hyperbolic GARCH model. Journal of Econometrics, 189, 428-436. https://doi.org/10.1016/j.jeconom.2015.03.034
  20. Robinson PM (1991). Testing for strong correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47, 67-84. https://doi.org/10.1016/0304-4076(91)90078-R
  21. Zaffaroni P (2004). Stationarity and memory of ARCH($H_{\infty}$) models, Econometric Theory, 20, 147-160.