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.

翻译：本文研究具有自回归条件异方差（ARCH）的单变量非线性自回归模型的次几何（具体而言，多项式）遍历性。次几何遍历性概念于20世纪80年代在马尔可夫链文献中提出，意指转移概率测度以慢于几何速率的速度收敛于平稳测度；该速率也与$\beta$-混合系数的收敛速度密切相关。现有关于次几何遍历自回归模型的文献均假设误差项为同方差，而本文将其拓展至条件异方差ARCH型误差的情形，显著扩大了潜在应用范围。具体而言，我们定义了适当的高阶非线性自回归模型（允许具有非线性ARCH误差），并证明在适当条件下这些模型以多项式速率满足次几何遍历性。本文利用能源行业波动率指数数据的实证案例，展示了次几何遍历AR-ARCH模型的应用。