A restricted eigenvalue condition for unit-root non-stationary data

TL;DR

This paper introduces a restricted eigenvalue condition tailored for unit-root non-stationary data, proving its validity under Gaussian innovations and demonstrating the consistency of the lasso estimator in high-dimensional cointegrated time series.

Contribution

It develops a new restricted eigenvalue condition for non-stationary data and applies it to establish lasso estimator consistency in high-dimensional cointegration models.

Findings

Validity of the restricted eigenvalue condition under Gaussian innovations

Consistency of lasso estimator in ultra high-dimensional cointegrated data

Method relies on matrix concentration inequalities

Abstract

In this paper, we develop a restricted eigenvalue condition for unit-root non-stationary data and derive its validity under the assumption of independent Gaussian innovations that may be contemporaneously correlated. The method of proof relies on matrix concentration inequalities and offers sufficient flexibility to enable extensions of our results to alternative time series settings. As an application of this result, we show the consistency of the lasso estimator on ultra high-dimensional cointegrated data in which the number of integrated regressors may grow exponentially in relation to the sample size.