Vector autoregression test When building a model with many regressors, Is there any statistical test to examine unidirectional versus mutual influence of variables?
 A: While your problem description is pretty brief, as far as I understand it you may be interested in Granger causality (Wikipedia). In a simple bivariate system $(x_t,y_t)$ you may ask whether $x_t$ Granger-causes $y_t$, denoted $x_t \xrightarrow{Granger} y_t$; or vice versa; or both. You may test that using $F$ test, for example. As an illustration, consider a VAR(1) model:
\begin{aligned}
x_t &= \varphi_{11} x_{t-1} + \varphi_{12} y_{t-1} + \varepsilon_{1,t}, \\
y_t &= \varphi_{21} x_{t-1} + \varphi_{22} y_{t-1} + \varepsilon_{2,t}.
\end{aligned}
You may test the following hypotheses:


*

*$H_0\colon \ x_t \not\xrightarrow{Granger} y_t$ by testing $H_0\colon \ \varphi_{21}=0$.

*$H_0\colon \ y_t \not\xrightarrow{Granger} x_t$ by testing $H_0\colon \ \varphi_{12}=0$.

*$H_0\colon \ x_t \not\xrightarrow{Granger} y_t \ \text{and} \ y_t \not\xrightarrow{Granger} x_t$ by testing $H_0\colon \ \varphi_{21}=\varphi_{12}=0$.


This way you may conclude over whether the relation is unidirectional (and then which way) or bidirectional. This also works in a multivariate system.
