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I have an ARMA(2,1) model of the following form,

$$y_t=a_1y_{t-1}+a_2y_{t-2}+\epsilon_t+b_1\epsilon_{t-1}$$

Re-arranging and using lag operators:

$$(1-a_1L-a_2L^2)y_t=(1+b_1L)\epsilon_t$$

solving for $y_t$

$$y_t=\frac{(1+b_1L)}{1-(a_1L+a_2L^2)}\epsilon_t$$

Using the definition of an infinite geometric series

$$(1+b_1L)\sum^\infty_{j=0}(a_1L+a_2L^2)^j\epsilon_t$$

$$(1+b_1L)\sum^\infty_{j=0}(a_1\epsilon_{t-1}+a_2\epsilon_{t-2})^j$$

$$\sum^\infty_{j=0}(a_1\epsilon_{t-1}+a_2\epsilon_{t-2})^j+\sum^\infty_{j=0}(a_1b_1\epsilon_{t-1}+a_2b_1\epsilon_{t-2})^j$$

Using this solution to compute the variance:

$$Var(y_t)=Var(\sum^\infty_{j=0}(a_1\epsilon_{t-1}+a_2\epsilon_{t-2})^j+\sum^\infty_{j=0}(a_1b_1\epsilon_{t-1}+a_2b_1\epsilon_{t-2})^j)$$

$$=\sum^\infty_{j=0}Var(a_1^j\epsilon_{t-1}^j+a_2^j\epsilon_{t-2}^j)+\sum^\infty_{j=0}Var(a_1^jb_1^j\epsilon_{t-1}^j+a_2^jb_1^j\epsilon_{t-2}^j)$$

$$=\sum^\infty_{j=0}(a_1^{2j}+a_2^{2j})Var(\epsilon_{t-1}^j+\epsilon_{t-2}^j)+\sum^\infty_{j=0}(a_1^{2j}b_1^{2j}+a_2^{2j}b_1^{2j})Var(\epsilon_{t-2}^j+\epsilon_{t-3}^j)$$

I think I must have made a mistake along the way, I don't know what to do with the term $Var(\epsilon_{t-1}^j+\epsilon_{t-2}^j)$ any help would be much appreciated.

CorporateNationalism
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  • Use the rules for computing variance of linear combinations. See https://stats.stackexchange.com/questions/31177/does-the-variance-of-a-sum-equal-the-sum-of-the-variances/31181#31181 for details. You will need to use all the information about the $\epsilon_t^j$ that you haven't specified here (presumably, they are independent and have a constant variance whose value will appear in the final result). Does that solve your problem? – whuber Apr 12 '23 at 21:22
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    As you have asked many similar questions of variances/autocovariances of ARMA process, I highly recommend reading Section 3.3 of Time Series: Theory and Methods by Brockwell and Davis for a systematic treatment (in fact, three different treatments) to general ARMA($p$, $q$ ) processes. You can then treat all your asked questions as special cases of this textbook's derived results. – Zhanxiong Apr 12 '23 at 22:05

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