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

Abstract

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.1 Unfortunately, the name of said researcher as well as the conference he presented at slipped the authors' memories; as if choosing the adjective 'young' was not enough for proving that one grew older. 2 There exists a multitude of sub-variants of each model class, for example, hybrid versions of MIDAS LASSO (Siliverstovs, 2017), Markov-switching MF models (Foroni et al., 2015a) and MF-VARs with time-varying parameters and stochastic volatility Hauzenberger, 2018 or Cimadomo andD'Agostino, 2016), among many others. GC IN POSSIBLY (CO)INTEGRATED MF-VARS 915Particularly the latter model class, MF-VAR models, has received a lot of attention recently, predominantly in two related fields of application: forecasting (Schorfheide and Song, 2015; Götz and Hauzenberger, 2018, among others) and Granger causality (GC) testing 2018;Götz et al. 2016). Both topics are of immense interest to practitioners at, for example, central banks, who routinely forecast key variables like the gross domestic product (GDP) using a variety of, usually higher-frequent, indicators, or investigate causal patterns between the time series they monitor. We will focus on GC testing, generally introduced in Granger (1969), whereby both concepts are obviously related when defining GC as predictability of one series conditional on past observations of the other (Dufour and Renault, 1998). Under this definition, GC testing in fact boils down to testing a set of zero restrictions on the parameters of the respective system. The three articles mentioned above cover different aspects of such GC testing within a MF-VAR: Ghysels et al. (2016) discuss the general theory of associated hypothesis tests in detail, which -while asymptotically valid -suffer from size distortions and a loss of power in case the number of HF observations is large relative the low-frequency (LF) period. Götz et al. (2016) and Ghysels et al. (2018) then introduce various wa...