Browse > Article
http://dx.doi.org/10.5351/CSAM.2017.24.5.443

Functional central limit theorems for ARCH(∞) models  

Choi, Seunghee (Department of Statistics, Ewha Womans University)
Lee, Oesook (Department of Statistics, Ewha Womans University)
Publication Information
Communications for Statistical Applications and Methods / v.24, no.5, 2017 , pp. 443-455 More about this Journal
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
functional central limit theorem; ARCH(${\infty}$) model; $L_2$-NED property;
Citations & Related Records
연도 인용수 순위
  • Reference
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.   DOI
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.   DOI
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.   DOI
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.   DOI
9 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.   DOI
10 Dedecker J, Doukhan P, Lang G, Leon JR, Louhichi S, and Prieur C (2007). Weak Dependence, Examples and Applications, Springer, New York.
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.   DOI
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.   DOI
15 Hwang EJ and Shin DW (2013). A CUSUM test for a long memory heterogeneous autoregressive models. Economics Letters, 121, 379-383.   DOI
16 Kazakevicius V and Leipus R (2002). On stationarity in the ARCH($H_{\infty}$) model. Econometric Theory, 18, 1-16.   DOI
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.   DOI
18 Lee O (2014b). The functional central limit theorem and structural change test for the HAR($H_{\infty}$) model. Economic Letters, 124, 370-373.   DOI
19 Li M, Li W, and Li G (2015). A new hyperbolic GARCH model. Journal of Econometrics, 189, 428-436.   DOI
20 Robinson PM (1991). Testing for strong correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47, 67-84.   DOI
21 Zaffaroni P (2004). Stationarity and memory of ARCH($H_{\infty}$) models, Econometric Theory, 20, 147-160.