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In this paper we derive permanent‐transitory decompositions of non‐stationary multiple times series generated by (r)nite order Gaussian VAR(p) models with both cointegration and serial correlation common features. We extend existing analyses to the two classes of reduced rank structures discussed in Hecq, Palm and Urbain (1998). Using the corresponding state space representation of cointegrated VAR models in vector error correction form we show how decomposition can be obtained even in the case where the number of common feature and cointegration vectors are not equal to the number of variables. As empirical analysis of US business fluctuations shows the practical relevance of the approach we propose.

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Reproduction permitted only if source is stated.ISBN 978-3-95729-217-9 (Printversion) Non-technical summary Research QuestionWhen dealing with time series sampled at various frequencies it has become common practice to directly incorporate high-frequency information into the econom(etr)ic model at hand. These specifications were first restricted to the single regression case; with the development of the (stacked) mixed-frequency vector autoregressive (MF-VAR) system (Ghysels, 2015) it is now possible to treat all series similarly and investigate causal effects between them. However, if the difference in frequencies between the series involved is large (as, e.g., in a month/working day scenario), estimation accuracy of the system coefficients is exacerbated, implying the detection of causal effects to be potentially inaccurate. To overcome this issue various parameter reduction techniques are introduced and analyzed. These methods are then evaluated in terms of their ability to detect causality patterns between the series under consideration in the resulting restricted model. ContributionTwo parameter reduction techniques are discussed in detail: three reduced rank regression (RRR) model variants and a Bayesian MF-VAR. Using a Monte Carlo experiment both approaches are compared in terms of their Granger causality testing behavior with an unrestricted VAR, a (time-aggregated) low-frequency VAR and the max-test (Ghysels, Motegi, and Hill, 2015a). To further enhance their finite sample properties we develop and evaluate (whenever possible) two bootstrap variants of these tests. Finally, the methods are applied to U.S. data by investigating channels of causality between the monthly growth rate of the industrial production index (IPI) and daily bipower variation (BV) of the S&P500 index. ResultsWe find that, depending on the direction of causality under consideration, a different set of tests results in the best Granger non-causality testing behavior. For the direction from the high-to the low-frequency series, standard testing within the Bayesian MF-VAR, the max-test and the restricted bootstrap version of the Wald test in two RRR model versions performs best. For the reverse direction, the unrestricted bootstrap variants of the Bonferroni-corrected Wald tests within the unrestricted VAR and the RRR models dominate. As far as our application is concerned, Granger causality from BV to IPIgrowth is clearly supported by the data; evidence for causality in the reverse direction, however, only comes from a subset of tests. Maastricht UniversityAbstract We analyze Granger causality testing in a mixed-frequency VAR, where the difference in sampling frequencies of the variables is large. Given a realistic sample size, the number of high-frequency observations per low-frequency period leads to parameter proliferation problems in case we attempt to estimate the model unrestrictedly. We propose several tests based on reduced rank restrictions, and implement bootstrap versions to account for the uncertainty when estimating fac...