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.

翻译：本文研究了阶数为$k$的非平稳高维向量自回归模型VAR($k$)。允许添加趋势或季节性等确定性项。假设时间跨度$T$和变量维度$N$均较大且处于同一量级。在此框架下，推导了约翰森似然比（LR）检验、Pillai-Bartlett检验和Hotelling-Lawley协整检验的一阶渐近性质：检验统计量收敛于非随机积分。为进行更精细的分析，本文提出并分析了一种约翰森检验的修正形式。用于检验无协整关系的新检验统计量收敛于Airy$_1$点过程的部分和。支持性的蒙特卡洛模拟表明，在超出定理所述条件的许多情境中，该行为仍普遍存在。本文将该方法应用于对S$\&$P$100$股票及加密货币的实证分析。后者显著存在多重协整关系，而前者结果与无协整的原假设相一致。