Alternative asymptotics for cointegration tests in large VARs

TL;DR

This paper investigates the asymptotic distribution of cointegration test statistics in high-dimensional VARs, revealing convergence to the Wachter distribution and explaining spurious findings in large samples.

Contribution

It introduces a new asymptotic framework for cointegration tests in large VARs, showing convergence to the Wachter distribution and proposing a graphical assessment tool.

Findings

Distribution converges to Wachter distribution

Explains spurious cointegration in high dimensions

Proposes a scree-plot-like graphical method

Abstract

Johansen's (1988, 1991) likelihood ratio test for cointegration rank of a Gaussian VAR depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data. We study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the VAR diverge to infinity simultaneously and proportionally. We find that the distribution almost surely weakly converges to the so-called Wachter distribution. This finding provides a theoretical explanation for the observed tendency of Johansen's test to find "spurious cointegration". It also sheds light on the workings and limitations of the Bartlett correction approach to the over-rejection problem. We propose a simple graphical device, similar to the scree plot, for a preliminary assessment of cointegration in high-dimensional VARs.