# Convert ARMA(p,q) to MA$(\infty)$ and find ACF

$$Xt-0.4X_t+0.03X_{t-2}=Zt-0.4Z_{t-1}$$ 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 $$\psi_i$$.

• You need to express $\psi$ as a power series to read off $\psi_i$. Mar 31, 2023 at 23:49
• @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? Apr 1, 2023 at 0:36