Estimating Structured Vector Autoregressive Model

TL;DR

This paper develops bounds for estimating high-dimensional structured VAR models with dependent data, showing that the estimation error matches that of independent cases across various norms, supported by theoretical analysis and experiments.

Contribution

It provides the first non-asymptotic error bounds for structured VAR models with dependent samples, applicable to any norm and validated through experiments.

Findings

Estimation error is comparable to independent sample cases.

Bounds hold for any norm used to capture structure.

Experimental validation on synthetic and real data supports theoretical results.

Abstract

While considerable advances have been made in estimating high-dimensional structured models from independent data using Lasso-type models, limited progress has been made for settings when the samples are dependent. We consider estimating structured VAR (vector auto-regressive models), where the structure can be captured by any suitable norm, e.g., Lasso, group Lasso, order weighted Lasso, sparse group Lasso, etc. In VAR setting with correlated noise, although there is strong dependence over time and covariates, we establish bounds on the non-asymptotic estimation error of structured VAR parameters. Surprisingly, the estimation error is of the same order as that of the corresponding Lasso-type estimator with independent samples, and the analysis holds for any norm. Our analysis relies on results in generic chaining, sub-exponential martingales, and spectral representation of VAR models. Experimental results on synthetic data with a variety of structures as well as real aviation data are presented, validating theoretical results.