Absolute regularity of semi-contractive GARCH-type processes

TL;DR

This paper establishes the existence, uniqueness, and absolute regularity of stationary distributions for nonlinear GARCH and INGARCH models under semi-contractive conditions, allowing for broader model inclusion and revealing subgeometric mixing rates.

Contribution

It introduces a semi-contractive condition for GARCH-type models, expanding the class of models with proven stationarity and mixing properties beyond fully contractive assumptions.

Findings

Proves existence and uniqueness of stationary distribution.

Establishes absolute regularity with subgeometric decay.

Extends results to non-stationary time series.

Abstract

We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p,q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p,q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes. An extension of our results to non-stationary time series is also provided and we discuss some applications.