Causality root criterion for an autoregressive model
= Causality root criterion for an autoregressive model
{title2=$\Phi(z)\ne0\quad(|z|\leq1)$}
A <causal time series> satisfying $\Phi(B)X_t=\varepsilon_t$ with nondegenerate <white noise> has a square-summable present-and-past filter. The identity $\Phi(z)\Psi(z)=1$ excludes roots inside the unit disk; a pole of $1/\Phi$ on its boundary also prevents square summability. Thus all roots of $\Phi$ lie strictly outside the <unit circle>.