# Is multivariate Granger-causality possible? Do I proceed as with univariate?

Basically, I have five time series. All are stationary. Now, let's call them $Y$, and $X_1$ to $X_4$.

Normally you'd do $$Y_{t} = \alpha + \beta_1 Y_{t-1} + \beta_2 X_{1,t-1} + \epsilon$$ and this could be redone for $X_2$ to $X_4$. But is it possible to just do $$Y_{t} = \alpha + \beta_0 Y_{t-1} + \beta_1 X_{1,t-1} + \beta_2 X_{2,t-1} + \beta_3 X_{3,t-1} + \beta_4 X_{4,t-1} + \epsilon \ ?$$ Will the $F$-test still be valid?

• Thanks, Richard. I didn't realise about the tick situation. I'm quite new to these forums. But yes, your answer was very helpful, thank you. Apologies it has taken me some time to come back. – eBopBob Jan 9 '18 at 19:09
• No problem, glad to help :) – Richard Hardy Jan 9 '18 at 20:00

Yes, you can examine multivariate Granger causality. You can examine causality

• from multiple series to one series,
• from one series to multiple series, and
• from multiple series to multiple series.

The idea of the test remains the same: restrict the lags of the series that supposedly causes the other and test whether the restriction holds in population. If you cannot reject the restriction, then you cannot reject the absence of Granger causality.

The $F$-test should be valid.

Read more in Lütkepohl "New Introduction to Multiple Time Series Analysis" Section 2.3.1 p. 42, starting with

The deﬁnition [of Granger causality] extends immediately to the case where $z_t$ and $x_t$ are $M$- and $N$-dimensional processes, respectively. ...