Hi. I have this problem that I am a bit stuck on. Hope you can help me out...

$\displaystyle Let \ \{X_t\} \ denote \ the \ unique \ stationary \ solution \ of \ the \ the \ autoregressive \ equations$

$\displaystyle X_t = \phi X_{t-1}+Z_t,\ t=0, \pm1, \ ...$

$\displaystyle where \{ Z_t \} \sim WN(0,1) \ and \ |\phi|>1.$

Define the new sequence :

$\displaystyle W_t = X_{t}- \frac{1}{\phi}X_{t-1},\ t=0, \pm1, \ ...$

$\displaystyle a) \ Show \ that \ \{X_t \} \ is \ causal \ in \ terms \ of \ \{W_t \} \ and \ compute \ its \ causal \ representation.$

This is what I did...

$\displaystyle W_t = X_{t}- \frac{1}{\phi}X_{t-1}$

$\displaystyle X_t = W_{t}+ \frac{1}{\phi}X_{t-1}$

So it is now causal.

Compute its causal representation...

$\displaystyle X_t = W_{t}+ \frac{1}{\phi}X_{t-1}$

$\displaystyle X_t = W_{t}+ \frac{1}{\phi}(W_{t-1}+ \frac{1}{\phi}X_{t-2})$

$\displaystyle X_t = W_{t}+ \frac{1}{\phi}W_{t-1}+ \frac{1}{\phi^2}X_{t-2}$

$\displaystyle X_t = W_{t}+ \frac{1}{\phi}W_{t-1}+ \frac{1}{\phi^2}X_{t-2}+...+\frac{1}{\phi^k}X_{t-k}$

$\displaystyle X_t = (1+\frac{1}{\phi}B+ \frac{1}{\phi^2}B^2+...)W_t=\frac{1}{1- \phi^{-1} B}W_t$

What am I doing wrong?

Thanks!