Rewriting the characteristic polynomial using that the inverse roots are invertible

Question

I am looking at an I(1) process for an AR(3) model: $$x_t=\theta_1x_{t-1}+\theta_2x_{t-2}+\theta_3x_{t-3}+\varepsilon_t$$ I rewrite it using the lag-operator: $$(1-\theta_1L-\theta_2L^2-\theta_3L^3)x_t=\varepsilon_t$$ And the I factorise the characteristic polynomial: $$(1-\theta_1z-\theta_2z^2-\theta_3z^3)=(1-\phi_1z)(1-\phi_2z)(1-\phi_3z)$$ Where $\phi_1,\phi_2$ and $\phi_3$ are the inverse roots.

Because it is an I(1) process there is obviously only one unit root such that $\phi_1=1$ and it must hold that $|\phi_2|<1$ and $|\phi_3|<1$.

Where I am stuck is that many textbooks then say that because $|\phi_2|<1$ is invertible is can then rewrite it as: $$\frac{1}{1-\phi_2z}=1+\phi_2z+\phi_2^2z^2+\phi_2^3z^3+...$$

I hope someone can help me out with this :)

Cheers!

Answer 1

For any complex number $z$ for which $|\phi z|\lt 1,$ the right hand side is an absolutely convergent power series. Multiplying it by $1-\phi z$ gives

$$(\color{red}{1}-\color{blue}{\phi z})(1 + \phi z + (\phi z)^2 + \cdots) = \color{red}{ (1 + \phi z + (\phi z)^2 + \cdots) }- \color{blue}{(\phi z)(1 + \phi z + (\phi z)^2 + \cdots)}.$$

Because both terms (on either side of the "$-$") on the right are absolutely convergent series, we may subtract them term by term and we can do it in any order we please. So, compute the right hand side as

$$\color{red}{1}\ +\ \left(\color{red}{\phi z} - \color{blue}{(\phi z)(1)}\right)\ +\ \left(\color{red}{(\phi z)^2} - \color{blue}{(\phi z)(\phi z)}\right)\ +\ \cdots.$$

All the terms in parentheses after the initial $\color{red}{1}$ are of the form $\color{red}{(\phi z)^n} - \color{blue}{(\phi z)(\phi z)^{n-1}},$ which cancel, leaving $\color{red}{1}+0+0+\cdots = 1.$ Consequently, the right hand side is the multiplicative inverse of the left hand side in your equation: that's what is meant by the notation "$1/(1 - \phi z).$"

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BTW, lest these appeals to properties of power series seem overly fussy, notice that if we didn't pay attention to them we would wind up with some truly strange identities such as Euler's infamous

$$-1 = \frac{1}{1-2} = 1 + 2 + 4 + 8 + \cdots$$

when $\phi z = 2.$ (For that to be correct, you would need to be working in the ring of 2-adic integers ;-).)