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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) ?

enter image description hereCan 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.

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    $\begingroup$ 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. $\endgroup$ Commented Jan 23, 2017 at 5:09
  • $\begingroup$ Thank you. How would you attempt a); I would check for unit roots. $\endgroup$ Commented Jan 23, 2017 at 14:12
  • $\begingroup$ @Christoph Hanck how can I brind the equation in world representation from? $\endgroup$ Commented Jan 23, 2017 at 21:37
  • $\begingroup$ for a) you indeed may look at the appropriate eigenvalues $\endgroup$ Commented Jan 24, 2017 at 5:51
  • $\begingroup$ How can I bring the given equation in the right format? $\endgroup$ Commented Jan 24, 2017 at 18:54

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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.

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  • $\begingroup$ For a) I would suggest that alpha and gamma need to be between 0 and 1 to have a stable process because then the roots are bigger than 1 in absolute value. $\endgroup$ Commented Jan 28, 2017 at 9:08
  • $\begingroup$ 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. $\endgroup$ Commented Jan 28, 2017 at 11:20
  • $\begingroup$ Right, that is what I'm saying. $\endgroup$ Commented Jan 28, 2017 at 11:24

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