Moving-average polynomial ($\Theta (z)=1+{\sum }_{j=1}^q{b}_j{z}^j$)

The moving-average polynomial records the coefficients of the finite noise filter. Evaluating it at the backshift operator gives $\Theta (B)w_t=w_t+{\sum }_{j=1}^q{b}_j{w}_{t-j}$. In a reduced ARMA representation, invertibility is equivalent to having no roots in the closed unit disc.

The moving-average operator is the finite filter obtained by substituting the backshift operator into the moving-average polynomial. It maps a white noise sequence to its weighted finite sum of present and past values.