Testing no cointegration in large VARs

We study the asymptotic behavior of Johansen's (1988, 1991) likelihood ratio test for no cointegration when the number of observations and the dimensionality of the vector autoregression diverge to infinity simultaneously and proportionally. We find that the empirical distribution of the squared canonical correlations that the test is based on converges to the so-called Wachter distribution. This finding provides a theoretical explanation for the observed tendency of the test to find "spurious cointegration" in the data. 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, as a quick check of the null hypothesis of no cointegration in high-dimensional VARs.