ESTIMATION FOR A NONSTATIONARY SEMI-STRONG GARCH(1,1) MODEL WITH HEAVY-TAILED ERRORS

研究问题: 当 GARCH(1,1) 不存在平稳解且误差可能重尾、具有时间依赖时，LAD 与高斯 QMLE 的极限性质有何差异，应如何开展推断？
方法与思路: 作者分析半强模型的平稳解条件及非平稳情形下 LAD 和 QMLE 的极限，并在误差独立同分布时提出 percentile-t 子抽样 bootstrap。
材料陈述的贡献: 摘要称，在所述矩条件下 LAD 局部极小值具有正态极限，而重尾条件下高斯 QMLE 可具有稳定律极限；数值结果支持相关理论。
阅读关注点: 应核对半强误差的依赖与矩条件、LAD 的收敛速度、稳定律指数的对应关系，以及子抽样推断为何额外要求误差独立同分布。
阅读边界: 所给 Crossref 摘要的收敛速度公式显示不完整，稳定律指数关系也未展开；不能将正态性推广到任意重尾误差或推断版本变化。

GPT-6.1 Sol · 高 · 2026-10-03

原文摘要

This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate $\root \of n $ to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate $\root \of n $ to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.

摘要来源：Crossref

笔记与分组

分类依据与纠错

ESTIMATION FOR A NONSTATIONARY SEMI-STRONG GARCH(1,1) MODEL WITH HEAVY-TAILED ERRORS

代码、数据与补充材料

重尾误差下非平稳半强 GARCH(1,1) 模型的估计

原文摘要

同方向的基础文献

同方向的基础文献