= Invertible ARMA factorization of an AR(1)-plus-noise process
{title2=$(1-\phi B)Y=(1+\vartheta B)\varepsilon$}
Filtering the observations by $1-\phi B$ gives covariance $A=\sigma_z^2+(1+\phi^2)\sigma_w^2$ at zero, $C=-\phi\sigma_w^2$ at lag one and zero elsewhere. Set $\nu=(A+\sqrt{A^2-4C^2})/2$ and $\vartheta=C/\nu$. Then $\nu(1+\vartheta^2)=A$, $\nu\vartheta=C$ and $|\vartheta|<1$. Applying $(1+\vartheta B)^{-1}$ defines actual white-noise innovations with variance $\nu$, proving an at-most-(1,1) ARMA representation without assuming Gaussianity.
Back to article page
