Extreme canonical correlations and high-dimensional cointegration analysis

Abstract

We prove that the extreme squared sample canonical correlations between a random walk and its own innovations almost surely converge to the upper and lower boundaries of the support of the Wachter distribution when the sample size and the dimensionality go to innity proportionally. This result is used to derive previously unknown analytic expressions for the Bartlett-type correction coefficients for Johansens trace and maximum eigenvalue tests in a high-dimensional VAR(1). An analysis of cointegration among a large number of log exchange rates illustrates the usefulness of our theoretical results.