I am currently doing an online course on Time Series and this is a self-assessment question from the homework, I won't see the answer until I submit, so I would appreciate hints/leads.
I have made my attempt at the question, but I didn't think I went down the right path. This course is following Introduction To Time Series and Forecasting by Brockwell and Davis. I am using 3rd Edition.
here is the question:
Find the autocovariance function of the AR(3) process:
$(1 - 0.5B)(1 - 0.4B)(1 - 0.1B)X_{t} = Z_{t}, Z_{t} \sim WN{\lbrace0,\sigma^{2} \rbrace}$
I could think of 2 ways of doing this:
I could solve the autoregressive polynomial and prove the roots are greater than 1 in absolute value, which then prove the AR(3) is causal (in book 3.1.5); once this is done, I could then calculate the coefficients for $Z_{t}$, according to
$X_{t}=\sum_{j=0}^{\infty}\psi_{j}Z_{t-j}$ for all t, where
$\psi_{j} = \theta_{j} + \sum_{j=0}^{p}\phi_{k}\psi_{j-k}, j=0,1,.....$, $(1)$
I normally rewrite it as
$\psi_{j} = (\theta_{k}+\phi_{k})\phi^{j-1}, \psi^{0} = 1$
$k=1,2,....p$
$j=1,2,.....,$
in the context of the problem, we could think of it as ARMA process where moving average polynomial $\theta(z) \equiv 1$
To do this I wrote the l.h.s in its expanded form so the backward operators are removed and this gives me:
$X_{t}-X_{t-1}+0.29X_{t-2}-0.02X_{t-3}=Z_{t}$ $(2)$
and solve:
$ 1 - z + 0.29z^{2} - 0.02z^{3} = 0$
we could go ahead solve this cubic equation but I don't really think that is the way of solving this problem, it requires way too much calculation.
Another way I thought about was to use recursion where I rewrite $X_{t-1}$ and probably also $X_{t-2}$ in formula and plug them back into $(2)$, expecting the limit of the sum of all X_{t} terms to be zero and leaving us with only the $Z_{t}$ terms. this way we could also get all the coefficients of $Z_{t}$ and then solve it using $(1)$. But what I've got wasn't what I expected, I can't seem to get the limit of $x_{t}$ terms.
Could anyone see this from a different angle that I apparently missed out?