= ARMA coefficient recursions
{c}
{title2=$\Phi(z)C(z)=\Theta(z),\quad\Theta(z)D(z)=\Phi(z)$}
For an <autoregressive moving-average model>, set $\Phi(z)=1-\sum_{k=1}^p\phi_kz^k$ and $\Theta(z)=1+\sum_{k=1}^q\theta_kz^k$. If both have no zeros on the closed unit disc, the causal and inverse filters have absolutely summable coefficients $C(z)=\Theta(z)/\Phi(z)=\sum_{j\geq0}c_jz^j$ and $D(z)=\Phi(z)/\Theta(z)=\sum_{j\geq0}d_jz^j$. Comparing <power series> coefficients gives $c_j=\theta_j+\sum_{k=1}^p\phi_kc_{j-k}$ and $d_j=a_j-\sum_{k=1}^q\theta_kd_{j-k}$, where $a_0=\theta_0=1$, $a_j=-\phi_j$ for $1\leq j\leq p$, and all out-of-range coefficients vanish. The <backshift operator> then gives the corresponding time-series filters.
Back to article page
