  The paper studies nonstationary high-dimensional vector autoregressions of
order $k$, VAR($k$). Additional deterministic terms such as trend or
seasonality are allowed. The number of time periods, $T$, and the number of
coordinates, $N$, are assumed to be large and of the same order. Under this
regime the first-order asymptotics of the Johansen likelihood ratio (LR),
Pillai-Bartlett, and Hotelling-Lawley tests for cointegration are derived: the
test statistics converge to nonrandom integrals. For more refined analysis, the
paper proposes and analyzes a modification of the Johansen test. The new test
for the absence of cointegration converges to the partial sum of the Airy$_1$
point process. Supporting Monte Carlo simulations indicate that the same
behavior persists universally in many situations beyond those considered in our
theorems.
  The paper presents empirical implementations of the approach for the analysis
of S$\&$P$100$ stocks and of cryptocurrencies. The latter example has a strong
presence of multiple cointegrating relationships, while the results for the
former are consistent with the null of no cointegration.
