Granger causality testing in mixed-frequency Vars with possibly (co)integrated processes

We analyze Granger causality testing in mixed-frequency VARs with possibly (co)integrated time series. It is well known that conducting inference on a set of parameters is dependent on knowing the correct (co)integration order of the processes involved. Corresponding tests are, however, known to often suffer from size distortions and/or a loss of power. Our approach, which boils down to the mixed-frequency analogue of the one by Toda and Yamamoto (1995) or Dolado and Lutkepohl (1996), works for variables that are stationary, integrated of an arbitrary order, or cointegrated. As it only requires an estimation of a mixed-frequency VAR in levels with appropriately adjusted lag length, after which Granger causality tests can be conducted using simple standard Wald test, it is of great practical appeal. We show that the presence of non-stationary and trivially cointegrated highfrequency regressors (Goetz et al., 2013) leads to standard distributions when testing for causality on a parameter subset, without any need to augment the VAR order. Monte Carlo simulations and two applications involving the oil price and consumer prices as well as GDP and industrial production in Germany illustrate our approach.