Statistical inference of high-dimensional vector autoregressive time series with non-i.i.d. innovations

TL;DR

This paper develops a statistical inference framework for high-dimensional vector autoregressive models with dependent, possibly non-stationary innovations, using a post-selection estimator and bootstrap methods to handle complex covariance structures.

Contribution

It introduces a novel inference approach for high-dimensional VAR models with dependent innovations, relaxing the i.i.d. assumption and providing practical bootstrap algorithms.

Findings

Validates the proposed methods through simulations.

Demonstrates effectiveness on real-life data.

Shows the approach outperforms traditional methods assuming independence.

Abstract

Independent or i.i.d. innovations is an essential assumption in the literature for analyzing a vector time series. However, this assumption is either too restrictive for a real-life time series to satisfy or is hard to verify through a hypothesis test. This paper performs statistical inference on a sparse high-dimensional vector autoregressive time series, allowing its white noise innovations to be dependent, even non-stationary. To achieve this goal, it adopts a post-selection estimator to fit the vector autoregressive model and derives the asymptotic distribution of the post-selection estimator. The innovations in the autoregressive time series are not assumed to be independent, thus making the covariance matrices of the autoregressive coefficient estimators complex and difficult to estimate. Our work develops a bootstrap algorithm to facilitate practitioners in performing statistical inference without having to engage in sophisticated calculations. Simulations and real-life data experiments reveal the validity of the proposed methods and theoretical results. Real-life data is rarely considered to exactly satisfy an autoregressive model with independent or i.i.d. innovations, so our work should better reflect the reality compared to the literature that assumes i.i.d. innovations.