Granger Causality Testing in Mixed-Frequency VARs with Possibly (Co)Integrated Processes

We analyze Granger causality (GC) testing in mixed-frequency vector autoregressions (MF-VARs) with possibly integrated or cointegrated 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 works for MF variables that are stationary, integrated of an arbitrary order, or cointegrated. As it only requires the estimation of a MF-VAR in levels with appropriately adjusted lag length, after which GC tests can be conducted using simple standard Wald tests, it is of great practical appeal. In addition, we show that the presence of non-stationary and trivially cointegrated high-frequency regressors leads to standard distributions when testing for causality on a subset of parameters, sometimes even 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.