backshift operator notation

Question

• Original equation:

$$\begin{equation} z_t = \phi z_{t-1} + z_{t-1} - \phi z_{t-2} + \omega_t \end{equation}$$

• Rewrite the equation, re-arrange terms, and factorize them:

$$\begin{align} z_t - z_{t-1} &= \phi (z_{t-1} - z_{t-2}) + \omega_t \\ (z_t - z_{t-1}) - \phi (z_{t-1} - z_{t-2}) &= \omega_t \\ (1 - \phi B)(z_t - z_{t-1}) &= \omega_t \\ (1 - \phi B) \bigtriangledown z_t &= \omega_t \\ (1 - \phi B)(1 - B)z_t &= \omega_t \end{align}$$

BACKGROUND:

I'm trying to understand backshift operator notation in the context of this ARIMA example from machine learning time series https://github.com/etcrago/Tutorial-Arima-w-jeffrey-yau (lecture 2).

QUESTION:

In the 3rd line he somehow jumps to $(1-B)$, but my understanding is the $\phi$ part of the equation should be factored to $(B-B^2)$, not $(1-B)$. How does the author jump to that $(1-B)$ factorization ?

Answer 1

I misread this problem. The better way to look at it is that the author factored $(z_t−z_{t−1})$ out such that the equations do work when multiplied to $(1-\phi B)$.