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

$$ {y_{1,t}}\left( {h|{\Omega _t}} \right) = {y_{1,t}}\left( {h|{\Omega _t}\backslash \left\{ {{y_{2,s}} \le t} \right\}} \right),\quad h = 1,2,... $$

((1.1))

, where Ω_t \ 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

$$ \left[ \begin{array}{l} {y_{1t}} \\ {y_{2t}} \\ \end{array} \right] = \sum\limits_{i = 1}^p {\left[ \begin{array}{l} {\alpha _{11,i}}\quad {\alpha _{12,i}} \\ {\alpha _{21,i}}\quad {\alpha _{22,i}} \\\end{array} \right]\left[ \begin{array}{l} {y_{1,t - i}} \\ {y_{2,t - i}} \\ \end{array} \right] + {u_t}} $$

((1.2))

and the information set is Ω _t = {(y _1,s ,y _2,s )′|s ≤ t} then (1.1) is equivalent to

$$ {\alpha _{12,i}} = 0,\quad i = 1,2, \ldots ,p $$

((1.3))

. Under standard assumptions, these restrictions are easy to test.

Parts of this research have been presented in seminars at Berkeley,San Diego, USC/UCLA and Stanford and at a workshop on “Applied Econometrics” in Munich. The author has greatly benefitted from comments of the participants and especially the discussant Walter Krämer. The paper was also presented at the World Congress of the Econometric Society 1990 in Barcelona. Financial support was provided in part by the Deutsche Forschungsgemeinschaft.