This article explores the possibility of cointegration existing between processes integrated at different frequencies. Using the demodulator operator, we show that such cointegration can exist and explore its form using both complex- and real-valued representations. A straightforward approach to test for the presence of cointegration between processes integrated at different frequencies is proposed, with a Monte Carlo study and an application showing that the testing approach works well.

中文翻译：

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关于以不同频率集成的过程的协整

本文探讨了以不同频率集成的过程之间存在协整的可能性。使用解调器算子，我们证明了这种协整可以存在，并使用复数和实值表示来探索它的形式。提出了一种直接的方法来测试以不同频率集成的过程之间是否存在协整，蒙特卡罗研究和应用程序表明该测试方法运行良好。