"Iterating" to show stationarity? - Help

[nacho-man]

I am not sure what is being done here, but I keep seeing statements like

Given $X_t - \phi X_{t-1} = Z_t$ $...(1)$

then
$$X_t = -\phi^{-1}Z_{t+1} + \phi^{-1}X_{t+1}$$
$$ = ... $$
$$= -\phi^{-1}Z_{t+1} - ... - -\phi^{-k-1}Z_{t+k+1}+\phi^{-1-k}X_{t+k+1}$$

What is going on here? What is the purpose of this?

If it helps, the concluding argument is that
$X_t = - \sum_{j=1}^{\infty} \phi^{-j}Z_{t+j}$ is the unique stationary solution of $(1)$

If I haven't provided enough information, let me know, and
I will also write out what the previous page says. I just can't figure out how they are equating those values in the first few steps, and what the relevance of it is. We start off with an AR(1) model, and soon they let it equal an AR(K) model... but how?

 

[Valkarie]

It looks like this statement is solving for the stationary solution of an autoregressive (AR) model. An AR model is a type of time series model that describes the relationship between the current value of a variable and its past values. In this case, the equation $(1)$ is describing an AR(1) model, with $X_t$ being the current value of the variable, $\phi$ being a coefficient, $X_{t-1}$ being the previous value, and $Z_t$ being a random disturbance term.The statement is then using the equation to solve for the unique stationary solution of the AR(1) model. The stationary solution is the value of $X_t$ that remains consistent over time. To solve for it, the statement is recursively substituting in the equation for $X_{t+1}$ and $Z_{t+1}$ until it reaches the limit $X_{t+k+1}$, which is the stationary solution. The statement is then rearranging the equation to get the desired result. In conclusion, the statement is solving for the unique stationary solution of an AR(1) model by recursively substituting in the equation for $X_{t+1}$ and $Z_{t+1}$.