Find (1-p)/(1-q) where p and q are polynomials (AR-representation)

Question

In the book Analysis of Financial Time Series by Tsay, there is a certain representation of an AR-model there is a little algebraic identity I cannot see why it is true, highlighted in blue in the picture below.

Answer 1

You can check $$ (1-\phi_1B)(1-\theta_1B)\sum_{i=0}^{\infty}(\phi_1B)^i = 1-\theta_1B $$ to see that the equality is indeed true.

To get an intuitive feeling where it comes from you can use the identity for a geometric series. $$ \frac{1-\theta_1B}{1-\phi_1B} = (1-\theta_1B)\sum_{i=0}^{\infty}(\phi_1B)^i. $$

However, if you want to formally derive this you need to calculate the Laurent series representation of this function on an annulus around the unit circle (note that you then assume there are no poles on the unit circle). You can check this text bij Van Der Vaart, specifically the beginning of chapter 8.