Testing for Causation Between Two Variables in Higher-Dimensional VAR Models

Abstract

Different concepts of causality in vector autoregressive (VAR) models have been proposed in the literature. For instance, Granger (1969) defines a variable y 2 to be causal for y 1 if the former is helpful in predicting the latter. Formally, denoting by y 1,t (h|Ω t ) the optimal h-step predictor at origin t based on the set of all the relevant information in the universe Ω t , y 2 may be defined to be Granger-noncausal for y 1 if (1.1) $$ {y_{1,t}}\left( {h|{\Omega _t}} \right) = {y_{1,t}}\left( {h|{\Omega _t}\backslash \left\{ {{y_{2,s}} \le t} \right\}} \right),\quad h = 1,2,... $$ , where Ωt \ A denotes the set containing all elements of Ω t that are not in A. If y 1 and y 2 are generated by a bivariate VAR(p) process (1.2) $$ \left[ \begin{array}{l} {y_{1t}} \\ {y_{2t}} \\ \end{array} \right] = \sum\limits_{i = 1}^p {\left[ \begin{array}{l} {\alpha _{11,i}}\quad {\alpha _{12,i}} \\ {\alpha _{21,i}}\quad {\alpha _{22,i}} \\\end{array} \right]\left[ \begin{array}{l} {y_{1,t - i}} \\ {y_{2,t - i}} \\ \end{array} \right] + {u_t}} $$ and the information set is Ω t = {(y 1,s ,y 2,s )′|s ≤ t} then (1.1) is equivalent to (1.3) $$ {\alpha _{12,i}} = 0,\quad i = 1,2, \ldots ,p $$ . Under standard assumptions, these restrictions are easy to test.

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