Regularized estimation of linear functionals of precision matrices for high-dimensional time series

TL;DR

This paper introduces a regularized estimator for linear functionals of high-dimensional precision matrices in time series, providing explicit convergence rates that account for dependence and moment conditions, with applications in finance, neuroscience, and prediction.

Contribution

It develops a Dantzig-selector type estimator with explicit convergence rates applicable to various dependence and moment regimes, extending high-dimensional time series analysis.

Findings

Convergence rates depend on temporal dependence and moments.

Estimator performs well in portfolio optimization and fMRI classification.

Simulation studies confirm the theoretical results.

Abstract

This paper studies a Dantzig-selector type regularized estimator for linear functionals of high-dimensional linear processes. Explicit rates of convergence of the proposed estimator are obtained and they cover the broad regime from i.i.d. samples to long-range dependent time series and from sub-Gaussian innovations to those with mild polynomial moments. It is shown that the convergence rates depend on the degree of temporal dependence and the moment conditions of the underlying linear processes. The Dantzig-selector estimator is applied to the sparse Markowitz portfolio allocation and the optimal linear prediction for time series, in which the ratio consistency when compared with an oracle estimator is established. The effect of dependence and innovation moment conditions is further illustrated in the simulation study. Finally, the regularized estimator is applied to classify the cognitive states on a real fMRI dataset and to portfolio optimization on a financial dataset.