Triangle for Parameters of AR(2) Stationary Processes

We’ve seen yesterday conditions on\phi_1,\phi_2) so that the canonical process,, satisfying\phi_1%20X_{t-1}+\phi_2%20X_{t-2}+\varepsilon_t

The condition is rather simple, since\phi_1,\phi_2) should be a triangular region. But the proof is a bit more tricky…

Recall that we want to parametrize the region\{(\phi_1%20,\phi_2)\in\mahtbb{R}^2:%201-\phi_1z-\phi_1z^2\neq%200,\forall%20z\in\mathbb{C},\vert\vert%20z\vert\vert%20\leq%201\}

Since we have a true process, then\phi_2\neq%200. Our polynomial is here\Phi(z)=1-\phi_1z-\phi_1z^2=\left(1-\frac{z}{\lambda_1}\right)\left(1-\frac{z}{\lambda_2}\right)

where\lambda_i‘s are the roots – in\mathbb{C} – of\Phi(\cdot). Consider now some kind of dual version of that polynomial,\tilde\Phi(z)=\left(1-{z}{\lambda_1}\right)\left(1-{z}{\lambda_2}\right)=1+\frac{\phi_1}{\phi_2}z+\frac{1}{\phi_2}z^2

Having the roots of\Phi(\cdot) outside the unit circle is the same as having the roots of\tilde\Phi(\cdot) inside the unit circle. Obserse that we can write\tilde\Phi(z)=\frac{1}{\phi_2}(\underbrace{z^2-\phi_1%20z-\phi_2}_{\bar{\Phi}(z)})

Roots of\bar{\Phi}(\cdot)} are then\xi%20=%20\frac{1}{2}\left(\phi_1\pm\sqrt{\phi_1^2+4\phi_2}\right)

From this point, we should discuss a little bit, depending on the value of\Delta=\phi_1^2+4\phi_2.

  • if\Delta=\phi_1^2+4\phi_2=0

Then there is one root, and only one. So we need to have\vert\phi_1\vert%20%3C2 or equivalently\phi_2%3E-1.

  • if\Delta=\phi_1^2+4\phi_2%3E0

Then we got roots in\mathbb{R}, and\frac{1}{2}\left(\phi_1\pm\sqrt{\phi_1^2+4\phi_2}\right)%3C%201

means, equivalently, that\phi_2%3E-1%20\%20;%20\%20\phi_2-\phi_1%3C1%20\%20;%20\%20\phi_2+\phi_1%3C1

  • if\Delta=\phi_1^2+4\phi_2%3C0

Then we have two (conjugate) roots in\mathbb{C}, and the square of norm of those roots is\vert\vert%C2%A0\xi\vert\vert^2=-\phi_2. Thus,\phi_2%3E-1.

We get what was mention in the course: the canonical has a stationary solution if, and only if\left\{\begin{array}{l}%20\phi_2-\phi_1%3C1%20\\\phi_2+\phi_1%3C1\\%20\vert\phi_2\vert%3C1\end{array}\right.

which is a triangular region, see

2 thoughts on “Triangle for Parameters of AR(2) Stationary Processes”

  1. Can you explain more of how you arrived at “some kind of dual version of that polynomial”?

    Also, should the signs be – instead of + in that dual version? If not, I’m not sure how factoring out \phi_{2} flips the signs here.

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