Subgeometrically ergodic autoregressions with autoregressive conditional heteroskedasticity

TL;DR

This paper extends the theory of subgeometric ergodicity to nonlinear autoregressive models with ARCH errors, demonstrating polynomial convergence rates and broadening applicability to heteroskedastic time series.

Contribution

It introduces subgeometric ergodicity results for nonlinear autoregressions with ARCH errors, expanding the scope beyond homoskedastic models.

Findings

Models are subgeometrically ergodic at polynomial rates

Extension to ARCH errors widens application scope

Empirical example with energy sector data demonstrates practical relevance

Abstract

In this paper, we consider subgeometric (specifically, polynomial) ergodicity of univariate nonlinear autoregressions with autoregressive conditional heteroskedasticity (ARCH). The notion of subgeometric ergodicity was introduced in the Markov chain literature in 1980s and it means that the transition probability measures converge to the stationary measure at a rate slower than geometric; this rate is also closely related to the convergence rate of $\beta$-mixing coefficients. While the existing literature on subgeometrically ergodic autoregressions assumes a homoskedastic error term, this paper provides an extension to the case of conditionally heteroskedastic ARCH-type errors, considerably widening the scope of potential applications. Specifically, we consider suitably defined higher-order nonlinear autoregressions with possibly nonlinear ARCH errors and show that they are, under appropriate conditions, subgeometrically ergodic at a polynomial rate. An empirical example using energy sector volatility index data illustrates the use of subgeometrically ergodic AR-ARCH models.