Simultaneous Inference for Nonlinear Time Series, a Sieve M-regression Approach

TL;DR

This paper develops a new framework for simultaneous inference in nonlinear time series using sieve M-regression, including uniform asymptotics, confidence regions, and a bootstrap method for practical implementation.

Contribution

It introduces a uniform Bahadur representation for sieve M-estimators with dependent data, enabling valid simultaneous inference over the entire predictor space.

Findings

Established a uniform Bahadur representation for dependent data.

Developed a convex Gaussian approximation for the estimator.

Designed a self-convolved bootstrap algorithm for practical inference.

Abstract

This paper studies simultaneous inference of conditional distributions in nonlinear time series from a sieve M-regression perspective. Existing literature on sieve M-regression has primarily focused on pointwise asymptotics, leaving the development of uncertainty quantification over the entire predictor space unexplored. We address this gap by establishing a uniform Bahadur representation for the sieve M-estimator, accommodating dependent data and a growing number of sieve basis functions. A novel high-dimensional empirical process theory is developed for temporally dependent data, and a specifically designed M-decomposition method is utilized to control high-dimensional complexities. Building on this representation, we develop a convex Gaussian approximation to characterize the asymptotic behavior of the estimator and construct valid simultaneous confidence regions (SCRs). To facilitate practical implementation, we introduce a self-convolved bootstrap algorithm that accurately approximates the distribution of the maximal deviation. Our inferential framework is supported by rigorous error bounds and validated through numerical simulations and real data applications.