
Causal Representation
Hi. I have this problem that I am a bit stuck on. Hope you can help me out...
$\displaystyle Let \ \{X_t\} \ denote \ the \ unique \ stationary \ solution \ of \ the \ the \ autoregressive \ equations$
$\displaystyle X_t = \phi X_{t1}+Z_t,\ t=0, \pm1, \ ...$
$\displaystyle where \{ Z_t \} \sim WN(0,1) \ and \ \phi>1.$
Define the new sequence :
$\displaystyle W_t = X_{t} \frac{1}{\phi}X_{t1},\ t=0, \pm1, \ ...$
$\displaystyle a) \ Show \ that \ \{X_t \} \ is \ causal \ in \ terms \ of \ \{W_t \} \ and \ compute \ its \ causal \ representation.$
This is what I did...
$\displaystyle W_t = X_{t} \frac{1}{\phi}X_{t1}$
$\displaystyle X_t = W_{t}+ \frac{1}{\phi}X_{t1}$
So it is now causal.
Compute its causal representation...
$\displaystyle X_t = W_{t}+ \frac{1}{\phi}X_{t1}$
$\displaystyle X_t = W_{t}+ \frac{1}{\phi}(W_{t1}+ \frac{1}{\phi}X_{t2})$
$\displaystyle X_t = W_{t}+ \frac{1}{\phi}W_{t1}+ \frac{1}{\phi^2}X_{t2}$
$\displaystyle X_t = W_{t}+ \frac{1}{\phi}W_{t1}+ \frac{1}{\phi^2}X_{t2}+...+\frac{1}{\phi^k}X_{tk}$
$\displaystyle X_t = (1+\frac{1}{\phi}B+ \frac{1}{\phi^2}B^2+...)W_t=\frac{1}{1 \phi^{1} B}W_t$
What am I doing wrong?
Thanks!