Spectral analysis for GARCH processes through a bilinear representation

Karima Kimouche and Abdelouahab Bibi

Communications in Statistics - Theory and Methods, 2023, vol. 52, issue 19, 6840-6856

Abstract: In this paper, we deal with the frequency domain approach to describe some probabilistic properties of the GARCH(p, q) processes using the associated transfer functions which seem more informative. So, in the first part, we propose an appropriate bilinear representation of GARCH(p,q) model governed by a second-order Hermite polynomial with leading coefficient 1, and we give a necessary and sufficient condition for the causal and second-order stationary solution. In a second part, we establish the autocovariance and the associated spectral density function of such a representation, and for its squared version in terms of transfer functions. As a consequence, it is observed that the second-order properties are similar to an ARMA process with uncorrelated innovations and hence the resort to higher-order properties however becomes inevitable. Therefore, the explicit expressions of bispectral densities of the process and its squared version are given and several illustrative examples are emphasized. Our results are related to the work of Terdik (2000, §3) for the GARCH(1, 1) model.

Date: 2023
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DOI: 10.1080/03610926.2022.2034019

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