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Remarks on stationary GARCH processes under heavy tail distributions

Marc Taberner-Ortiz, Manfred Denker

Abstract

Let $(X_n)_{n\in \mathbb Z}$ be a GARCH process with $E(X_0^4)<\infty$, and let $μ_n$ denote the distribution of $\frac 1{\sqrt n}\sum_{i=1}^n [X_i^2-\mathbb E(X_0^2)]$. We derive a numerical approximation of $μ_n$ when $x_1,...,x_n$ are observed. This yields the derivation of confidence intervals for $μ= E(X_0^2)$ and we investigate the accuracy of these confidence intervals in comparison with standard ones based on normal approximation. Moreover, when the innovation process has heavy tail distribution, we improve the method using a new resampling method.

Remarks on stationary GARCH processes under heavy tail distributions

Abstract

Let be a GARCH process with , and let denote the distribution of . We derive a numerical approximation of when are observed. This yields the derivation of confidence intervals for and we investigate the accuracy of these confidence intervals in comparison with standard ones based on normal approximation. Moreover, when the innovation process has heavy tail distribution, we improve the method using a new resampling method.
Paper Structure (10 sections, 7 theorems, 42 equations, 2 tables)

This paper contains 10 sections, 7 theorems, 42 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Local and global approximations
  3. Association
  4. Resampling
  5. Stable integrals
  6. Resampling of GARCH processes
  7. Inference on the variance of GARCH(1,1) processes
  8. Estimation using Theorem \ref{['theo:2.4']}
  9. Estimation using Theorem \ref{['theo:4.2']}
  10. Numerical simulations

Key Result

Theorem 2.1

BHH Let $(X_n)_{n\in \mathbb Z}$ be a stationary GARCH($1,1$) process, $X_0\in L_2$ and $\xi_0\in L_4$. Then there exists $\tau^2\ge 0$ such that the statistics converge weakly to a normal distribution with expectation 0 and variance $\tau^2$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Definition 3.1
  • Remark 3.2
  • proof : Proof of Theorems \ref{['theo:2.1']}--\ref{['theo:2.4']}
  • Proposition 3.3
  • proof
  • ...and 4 more