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$Xt-0.4X_t+0.03X_{t-2}=Zt-0.4Z_{t-1}$ This process is causal and invertible.

For the $MA(\infty)$ representation, I wrote it as $X_t=\frac{1-0.4B}{1-0.4B+0.03B^2}Z_t$, where $\psi(B)=\frac{1-0.4B}{1-0.4B+0.03B^2}$.

So, $X_t = \psi(B)Z_t=\sum _{i=0}^{\infty }\:\psi _i\:Z_{t-i}$

Now, I want to find the acf, this is where I get stuck.

For MA($\infty$) model, have that $\gamma_X(h)=\sigma^2\sum _{i=0}^{\infty }\:\psi _i\psi_{i+|h|}$

I'm not quite sure how to compute the acvf or the acf here, because I'm not sure how exactly I get the values for each $\psi_i$.

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  • $\begingroup$ You need to express $\psi$ as a power series to read off $\psi_i$. $\endgroup$
    – Zhanxiong
    Mar 31, 2023 at 23:49
  • $\begingroup$ @Zhanxiong I'm trying to follow the method from the Brockwell and Davis book: Intro to Time series and Forecasting, chapter 3 (3.1,3.2): home.iitj.ac.in/~parmod/document/… I don't think they used a power series to solve it out, but would it be easier to do it that way? $\endgroup$
    – eddie
    Apr 1, 2023 at 0:36

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