We study the likelihood ratio (LR) statistic for testing no cointegration in high-dimensional vector autoregressions. It has the form of a linear spectral statistic of a matrix C'ACB, where A is a sample covariance matrix of high-dimensional random walk, B is a sample covariance matrix of the random walk's innovations, and C is the sample cross-covariance between the random walk and its own innovations. We show that linear spectral statistics for C'ACB are asymptotically normal, and derive formulae for the corresponding asymptotic mean and variance. The formulae can be used to quickly obtain critical values of the LR test of no cointegration in high dimensions from the standard normal tables. This test substantially improves over the standard Bartlett-corrected LR tests based on complicated low-dimensional asymptotics.