Testing Causality between Two Vectors in Multivariate Autoregressive Moving Average Models

Abstract

In the analysis of economic time series, a question often raised is whether a vector of variables causes another one in the sense of Granger. Most of the literature on this topic is concerned with bivariate relationships or uses finite-order autoregressive specifications. The purpose of this article is to develop a causality analysis in the sense of Granger for general vector autoregressive moving average (ARMA) models. We give a definition of Granger noncausality between vectors, which is a natural and simple extension of the notion of Granger noncausality between two variables. In our context, this definition is shown to be equivalent to a more complex definition proposed by Tjostheim. For the class of linear invertible processes, we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis. This result is then specialized to the class of stationary invertible ARMA processes. Further, relatively simple necessary and sufficient conditions are obtained for two important cases: (1) the case where the two vectors reduce to two variables inside a larger vector including other variables; and (2) the case where the two vectors embody all the variables considered. Test procedures for these necessary and sufficient conditions are discussed. Among other things, it is noted that the necessary and sufficient conditions for noncausality may involve singularities at which standard asymptotic regularity conditions do not hold. To deal with such situations, we propose a sequential approach that leads to bounds tests. Finally, the tests suggested are applied to Canadian money and income data. The tests are based on bivariate and trivariate models of changes in nominal income and two money stocks (M1 and M2). In contrast with the evidence based on bivariate models, we find from the trivariate model that money causes income unidirectionally.

Key Words:

• Bounds test
• Causality test
• Granger causality
• Invertible linear process
• Sequential procedure