The concept of Granger causality is increasingly being applied for the\ncharacterization of directional interactions in different applications. A\nmultivariate framework for estimating Granger causality is essential in order\nto account for all the available information from multivariate time series.\nHowever, the inclusion of non-informative or non-significant variables creates\nestimation problems related to the 'curse of dimensionality'. To deal with this\nissue, direct causality measures using variable selection and dimension\nreduction techniques have been introduced. In this comparative work, the\nperformance of an ensemble of bivariate and multivariate causality measures in\nthe time domain is assessed, focusing on dimension reduction causality\nmeasures. In particular, different types of high-dimensional coupled discrete\nsystems are used (involving up to 100 variables) and the robustness of the\ncausality measures to time series length and different noise types is examined.\nThe results of the simulation study highlight the superiority of the dimension\nreduction measures, especially for high-dimensional systems.\n