A common factor can reverse an unreduced root classification

Both polynomials $1-2z$ have constant term one and the common factor $1-2z$ has root $z=1\mathrm{/}2$, so the unreduced pair is not coprime. Cancelling it leaves $(\varphi ,\theta )=(1,1)$, whose AR polynomial has no root.

With $X_t={ϵ}_t$, the reduced equation $X_t={ϵ}_t$ holds and has the causal representation ${\psi }_0=1$, ${\psi }_j=0$ for $j\ge 1$. By [F2] the reduced pair is causal. Multiplying both sides by the same finite filter $1-2B$ gives $(1-2B)X_t=(1-2B){ϵ}_t$, so the unreduced pair also displays the same process.

The unreduced $\varphi$ has a zero in $\mathrm{\mid }z\mathrm{\mid }\le 1$, but [F2] applies only after common-factor cancellation. Thus this raw root is not a valid classification of the process and explains the normalization clause.