Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subgeometrically ergodic autoregressions with autoregressive conditional heteroskedasticity

Mika Meitz, Pentti Saikkonen

TL;DR

This paper extends subgeometric (polynomial) ergodicity results from homoskedastic autoregressions to univariate nonlinear AR models with ARCH-type conditional heteroskedasticity. It establishes a drift-condition-based proof showing that the $(p+q)$-dimensional AR–ARCH process is $(f,r)$-ergodic with a polynomial rate $r(n)=n^{\delta-1}$ and finite moments up to order $2s_0-\rho$, along with corresponding $\beta$-mixing properties. The main contribution, Theorem 1, handles nonlinear mean dynamics via a carefully constructed Lyapunov function $V$ and a novel matrix-norm approach to control ARCH effects, extending prior IID-error results. An empirical application to energy-sector volatility demonstrates practical applicability and supports the relevance of subgeometric AR–ARCH models for highly persistent financial time series.

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 $β$-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.

Subgeometrically ergodic autoregressions with autoregressive conditional heteroskedasticity

TL;DR

This paper extends subgeometric (polynomial) ergodicity results from homoskedastic autoregressions to univariate nonlinear AR models with ARCH-type conditional heteroskedasticity. It establishes a drift-condition-based proof showing that the -dimensional AR–ARCH process is -ergodic with a polynomial rate and finite moments up to order , along with corresponding -mixing properties. The main contribution, Theorem 1, handles nonlinear mean dynamics via a carefully constructed Lyapunov function and a novel matrix-norm approach to control ARCH effects, extending prior IID-error results. An empirical application to energy-sector volatility demonstrates practical applicability and supports the relevance of subgeometric AR–ARCH models for highly persistent financial time series.

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 -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.
Paper Structure (17 sections, 5 theorems, 103 equations, 2 figures)

This paper contains 17 sections, 5 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Conditional mean
  4. Companion form
  5. Conditional variance
  6. Subgeometric ergodicity at a polynomial rate
  7. Main result
  8. Discussion
  9. Geometric ergodicity.
  10. Homoskedastic case.
  11. Proof strategy.
  12. Mixing and moment results.
  13. Subexponential ergodicity.
  14. Potential extensions.
  15. Examples
  16. ...and 2 more sections

Key Result

Lemma 1

There exists a vector norm $\Vert\cdot\Vert_{*}$ and a corresponding induced matrix norm ${\lvert\lvert\lvert \,\cdot\, \rvert\rvert\rvert}_{*}$ such that ${\lvert\lvert\lvert \boldsymbol{\Pi}_{1} \rvert\rvert\rvert}_{*}=\varpi<1$.

Figures (2)

  • Figure 1: Top row: Daily observations on the Chicago Board Options Exchange energy sector volatility index, 16 March 2011 – 31 December 2021 (left); the corresponding autocorrelation function (right). Middle row: The function $I(x)=-\nu L(x;\gamma,a)+\nu(1-L(x;\gamma,a))$ (left) and the corresponding time-varying intercept term $I(y_{t-1})$ (right), based on parameter estimates in (\ref{['EstimationResults']}). Bottom row: The estimated volatility series $\hat{\sigma}_{t}$ (left) and residual series $\hat{\varepsilon}_{t}$ (right), based on parameter estimates in (\ref{['EstimationResults']}).
  • Figure 2: Further analysis of the residuals shown in (the bottom right graph of) Figure 1: autocorrelation function of $\hat{\varepsilon}_{t}$ (top left), autocorrelation function of $\hat{\varepsilon}_{t}^{2}$ (top right), histogram along with the estimated error density (bottom left), and a Q-Q plot (bottom right). The dashed lines in the autocorrelation function graphs show the conventional bounds $\pm1.96/\sqrt{T}\approx\pm0.038$ ($T=2715$; the first four observations are used as initial values).

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • proof : Proof of Lemma 2
  • proof : Details for the finiteness of moments in Section 3.2
  • proof : Proofs of Propositions 1 and 2