On cointegration for processes integrated at different frequencies

This paper explores the possibility of cointegration existing between processes integrated at di¤erent 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 di¤erent frequencies is proposed, with a Monte Carol study and an application showing that the testing approach works well.