Adaptive Huber regression on Markov-dependent data

Abstract

High-dimensional linear regression has been intensively studied in the community of statistics in the last two decades. For the convenience of theoretical analyses, classical methods usually assume independent observations and sub-Gaussian-tailed errors. However, neither of them hold in many real highdimensional time-series data. Recently [Sun, Zhou, Fan, 2019, J. Amer. Stat. Assoc., in press] proposed Adaptive Huber Regression (AHR) to address the issue of heavy-tailed errors. They discover that the robustification parameter of the Huber loss should adapt to the sample size, the dimensionality, and the moments of the heavy-tailed errors. We progress in a vertical direction and justify AHR on dependent observations. Specifically, we consider an important dependence structure -Markov dependence. Our results show that the Markov dependence impacts on the adaption of the robustification parameter and the estimation of regression coefficients in the way that the sample size should be discounted by a factor depending on the spectral gap of the underlying Markov chain.

Factor and Idiosyncratic VAR-Ito Volatility Models for Heavy-Tailed High-Frequency Financial Data

“…For (III), we have Consider (4.11). Similar to the proofs of Proposition 1 in Fan et al (2019), we can show…”

Volatility Models for Stylized Facts of High-Frequency Financial Data

“…To handle this heavy-tail problem, Sun et al (2020) introduced the adaptive Huber regression for independent observations, which can obtain the optimal convergence rate with only the finite b-th moment for b ∈ (1, 2]. Fan et al (2019) extended the adaptive Huber regression to the dependent observations, such as Markov dependence. On the other hand, Mikosch et al (2006) investigated the stable limits of the usual Gaussian quasi-likelihood maximum estimator (QMLE) with the heavy-tailed observations.…”

Volatility models for stylized facts of high‐frequency financial data

“…Fan et al . (2022) extended the adaptive Huber regression to the dependent observations, such as Markov dependence. On the other hand, Mikosch and Straumann (2006) investigated the stable limits of the usual Gaussian quasi‐likelihood maximum estimator (QMLE) with the heavy‐tailed observations.…”

“…To handle this heavy-tail problem, Sun et al (2020) introduced the adaptive Huber regression for independent observations, which can obtain the optimal convergence rate with only the finite bth moment for b ∈ (1, 2]. Fan et al (2022) extended the adaptive Huber regression to the dependent observations, such as Markov dependence. On the other hand, Mikosch and Straumann (2006) investigated the stable limits of the usual Gaussian quasi-likelihood maximum estimator (QMLE) with the heavy-tailed observations.…”