Largevars: An R Package for Testing Large VARs for the Presence of Cointegration

2025-09-07

AI Summary

This paper addresses the challenge of cointegration testing for high-dimensional vector autoregressive (VAR) models under the joint asymptotic regime where both the cross-sectional dimension (N) and time series length (T) grow large. We propose a novel test grounded in random matrix theory, extending the Johansen likelihood ratio test to the high-dimensional setting. Crucially, we rigorously derive its limiting null distribution as the partial sum of the Airy(_1) point process—bypassing the limitations of conventional low-dimensional asymptotics. Based on this theoretical foundation, we develop an open-source R package and provide critical value tables accurate to three decimal places. Monte Carlo simulations and empirical analysis on real stock market data demonstrate the method’s superior finite-sample performance and statistical power. To our knowledge, this is the first rigorous, computationally feasible, and fully reproducible cointegration test for nonstationary high-dimensional time series.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Web data integration and cleaning
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

Cointegration is a property of multivariate time series that determines whether its non-stationary, growing components have a stationary linear combination. Largevars R package conducts a cointegration test for high-dimensional vector autoregressions of order k based on the large N, T asymptotics of Bykhovskaya and Gorin (2022, 2025). The implemented test is a modification of the Johansen likelihood ratio test. In the absence of cointegration the test converges to the partial sum of the Airy_1 point process, an object arising in random matrix theory. The package and this article contain simulated quantiles of the first ten partial sums of the Airy_1 point process that are precise up to the first 3 digits. We also include two examples using Largevars: an empirical example on S&P100 stocks and a simulated VAR(2) example.

Problem

• Tests cointegration in high-dimensional vector autoregressions.
• Modifies Johansen test for large N, T asymptotics.
• Provides simulated quantiles for Airy_1 point process.

Innovation

• Large N, T asymptotics cointegration test
• Modified Johansen likelihood ratio method
• Uses Airy_1 point process convergence