A Graphical Investigation of the Size and Power of the Granger-Causality Tests in Integrated-Cointegrated VAR Systems

The size and power of the Wald, Edgeworth expansion corrected likelihood-ratio statistic (LRE), and bootstrap tests for Granger causality in integrated-cointegrated VAR systems are considered. By using Monte Carlo methods and simple graphical techniques, the p-value plots, and power-size plots, properties of the tests in two different forms, the standard and the Dolado and L ¨ utkepohl (1996) modified form, are studied. Regarding the size of the tests, we found that the Wald performs badly in small- and medium-size samples, but the corrected tests, and especially the LRE for large samples, exhibit good performances. The bootstrap test showed the best performance in almost all situations. When considering the power results, using the size-power curves on a correct-size-adjusted basis, we found that the bootstrap tests have adequate power.