ARMA(1,1) causal and inverse coefficients ($c_j=(\varphi +\theta ){\varphi }^{j-1},d_j=-(\varphi +\theta )(-\theta {)}^{j-1}$)

For $|\varphi |<1$ and $|\theta |<1$, the autoregressive moving-average model $(1-\varphi B)X_t=(1+\theta B){ϵ}_t$ has $c_0=d_0=1$ and the displayed coefficients for $j\ge 1$. These follow by multiplying the two geometric power series. If $\varphi +\theta =0$, all later coefficients vanish and the filtered process is white noise. Thus root restrictions on an uncancelled nonminimal representation are sufficient but need not be necessary for the process after common factor cancellation in an ARMA model.