# Eigenvalues, stability, Granger causality for a VAR(1) model

I know I have to check the eigenvalues for the stability condition in general but I do not know how to process this exercise. How can I bring it in the right form? How do you check c) ?

Can anyone help with showing the single steps?

For c) I would say that c does not G-cause y becaue of the 0 in the matrix. But y G-causes c.

• Granger causality in the variable of some equation is present if any of the lags of the other variable(s - but here we only have one) are nonzero. Commented Jan 23, 2017 at 5:09
• Thank you. How would you attempt a); I would check for unit roots. Commented Jan 23, 2017 at 14:12
• @Christoph Hanck how can I brind the equation in world representation from? Commented Jan 23, 2017 at 21:37
• for a) you indeed may look at the appropriate eigenvalues Commented Jan 24, 2017 at 5:51
• How can I bring the given equation in the right format? Commented Jan 24, 2017 at 18:54

You need to check the roots of $$det(I-Az)=0,$$ where $A$ is your coefficient matrix on the lags.
Thus, $$det\left(I-\begin{pmatrix}\alpha&\beta\\0&\gamma\end{pmatrix}z\right)=det\begin{pmatrix}1-\alpha z&-\beta\\-0&1-\gamma z\end{pmatrix}=0$$ or $$(1-\alpha z)(1-\gamma z)=0$$ which has solutions $$z_1=1/\alpha\quad z_2=1/\gamma,$$ which need to be outside the unit circle. You can take it from here.
• Yeah, I have the same result. But then I also defined the parameters and said that $\alpha$ has to be between 0 and 1 (and between -1 and 0) as I need z bigger than 1 in absolute values to define a stable system. Commented Jan 28, 2017 at 11:20