Estimation of High-dimensional Nonlinear Vector Autoregressive Models

TL;DR

This paper introduces a flexible high-dimensional nonlinear VAR model using sparse additive methods, providing theoretical guarantees and demonstrating improved performance on gene expression data.

Contribution

It extends linear VAR models to nonlinear, non-Gaussian settings with convergence rates, model selection consistency, and sharp tail probability inequalities.

Findings

Theoretical convergence rates and model selection consistency established.

Sharp Bernstein-type inequalities derived for tail probabilities.

Numerical experiments show advantages of nonlinear VAR in gene expression analysis.

Abstract

High-dimensional vector autoregressive (VAR) models have numerous applications in fields such as econometrics, biology, climatology, among others. While prior research has mainly focused on linear VAR models, these approaches can be restrictive in practice. To address this, we introduce a high-dimensional non-parametric sparse additive model, providing a more flexible framework. Our method employs basis expansions to construct high-dimensional nonlinear VAR models. We derive convergence rates and model selection consistency for least squared estimators, considering dependence measures of the processes, error moment conditions, sparsity, and basis expansions. Our theory significantly extends prior linear VAR models by incorporating both non-Gaussianity and non-linearity. As a key contribution, we derive sharp Bernstein-type inequalities for tail probabilities in both non-sub-Gaussian linear and nonlinear VAR processes, which match the classical Bernstein inequality for independent random variables. Additionally, we present numerical experiments that support our theoretical findings and demonstrate the advantages of the nonlinear VAR model for a gene expression time series dataset.