June 2022 Cointegration in large VARs
Anna Bykhovskaya, Vadim Gorin
Author Affiliations +
Ann. Statist. 50(3): 1593-1617 (June 2022). DOI: 10.1214/21-AOS2164


The paper analyzes cointegration in vector autoregressive processes (VARs) for the cases when both the number of coordinates, N, and the number of time periods, T, are large and of the same order. We propose a way to examine a VAR of order 1 for the presence of cointegration based on a modification of the Johansen likelihood ratio test. The advantage of our procedure over the original Johansen test and its finite sample corrections is that our test does not suffer from overrejection. This is achieved through novel asymptotic theorems for eigenvalues of matrices in the test statistic in the regime of proportionally growing N and T. Our theoretical findings are supported by Monte Carlo simulations and an empirical illustration. Moreover, we find a surprising connection with multivariate analysis of variance (MANOVA) and explain why it emerges.

Funding Statement

V.G. acknowledges support from the NSF Grants DMS-1664619 and DMS-1949820.


The authors are grateful to Donald Andrews, Alexei Borodin, Giuseppe Cavaliere, Alice Guionnet, Bruce Hansen, Søren Johansen, Grigori Olshanski and anonymous referees for valuable suggestions.


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Anna Bykhovskaya. Vadim Gorin. "Cointegration in large VARs." Ann. Statist. 50 (3) 1593 - 1617, June 2022. https://doi.org/10.1214/21-AOS2164


Received: 1 January 2021; Revised: 1 October 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441133
zbMATH: 07547943
Digital Object Identifier: 10.1214/21-AOS2164

Primary: 60B20 , 62M10 , 62P20 , 91B84

Keywords: cointegration , High-dimensional VAR , Jacobi ensemble , Johansen test

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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