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Some limiting properties for GARCH(p, q)-X processes
Journal of the Korean Data & Information Science Society 2017;28:697-707
Published online May 31, 2017
© 2017 Korean Data & Information Science Society.

Oesook Lee1

1Department of Statistics, Ewha Womans University
Correspondence to: Oesook Lee
Professor, Department of Statistics, EwhaWomans University, Ewhayeodaegil 52, Seoul 120-750, Korea. E-mail:
Received March 27, 2017; Revised May 19, 2017; Accepted May 24, 2017.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we propose a modified GARCH(p, q)-X model which is obtained by adding the exogenous variables to the modified GARCH(p, q) process. Some limiting properties are shown under various stationary and nonstationary exogenous processes which are generated by another process independent of the noise process. The proposed model extends the GARCH(1, 1)-X model studied by Han (2015) to various GARCH(p, q)-type models such as GJR GARCH, asymptotic power GARCH and VGARCH combined with exogenous process. In comparison with GARCH(1, 1)-X, we expect that many stylized facts including long memory property of the financial time series can be explained effectively by modified GARCH(p, q) model combined with proper additional covariate.
Keywords : Conditional heteroskedasticity, exogenous variable, GARCH-X model, non-stationarity.