The inverse of an arbitrary lag polynomial

Summary

• Given: a theta function for \(p=2\) , \(θ\left( L \right)=1-θ_1L-θ_2L^2\)
• Consider \(θ\left( x \right)=1-θ_1x-θ_2x^2\) . This function can be factored as

\[1-θ_1x-θ_2x^2=-θ_2\left( x-x_1 \right)\left( x-x_2 \right)\]

• where \(x_1\) and \(x_2\) are the roots of the equation \(1-θ_1x-θ_2x^2=0\) (may be complex).
• If \(x_1≠0\) and \(x_2≠0\) then we define \(ϕ_1=1/x_1\) and \(ϕ_1=1/x_2\) and

\[1-θ_1x-θ_2x^2=\left( 1-ϕ_1x \right)\left( 1-ϕ_2x \right)\]

• Result. \(θ\left( L \right)=1-θ_1L-θ_2L^2\) is invertible if and only if \(\left| ϕ_1 \right|<1\) and \(\left| ϕ_2 \right|<1\) . We have

\[{\left( 1-θ_1L-θ_2L^2 \right)}^{-1}={\left( \left( 1-ϕ_1L \right)\left( 1-ϕ_2L \right) \right)}^{-1}={\left( 1-ϕ_2L \right)}^{-1}{\left( 1-ϕ_1L \right)}^{-1}\]

• \(θ\left( L \right)\) is invertible if and only if \(\left| x_1 \right|>1\) and \(\left| x_2 \right|>1\) (root are outside the unit circle)
• This result generalizes to arbitrary \(p\) : \(θ\left( L \right)\) is invertible if and only if the roots of \(θ\left( x \right)=0\) are all outside the unit circle.
• A necessary condition for the stability of an AR(p) process is that \(θ\left( L \right)\) is invertible.