I'm looking for methods to derive an analytical formula for the variance of an ARMA(2,1) model.

Given $$(1-\phi_1B-\phi_2B^2)Z_t=(1-\theta_1B)a_t, $$

I've tried the usual "trick" $$\text{Var}[Z_t]=\text{E}[Z_t(\phi_1Z_{t-1}+\phi_2Z_{t-2} + a_t - \theta_1a_{t-1})], $$ which naturally gives me a relationship between the variance and the covariances $$\text{Var}[Z_t]=\gamma_0 \sim \gamma_1 + \gamma_2 + \text{const.}$$

I've tried solving this recursive relationship for $\gamma_0$ by generalizing the relationship to e.g. $$\gamma_k \sim \gamma_{k+1} + \gamma_{k+2} + \text{const.},$$ and writing out formulas for $k=\pm1$. But this introduces another unknown covariance $\gamma_3$.

I've also tried to expand $Z_t$ down to a known time value, say $t_0$. But this approach creates a messy formula, which I'm not able to evaluate for the variance.

Are there any other approaches that results in an analytical formula for the variance of an ARMA(2,1) model?

  • $\begingroup$ Is this self-study? If so, please add the tag and read its wiki. $\endgroup$ Commented Sep 22, 2015 at 10:50
  • $\begingroup$ Am I correct to assume that you are trying to find the unconditional variance of the process? $\endgroup$
    – Plissken
    Commented Sep 23, 2015 at 8:24
  • 4
    $\begingroup$ See this answer. If that's not the answer to your question, please give details on your question. $\endgroup$
    – javlacalle
    Commented Sep 26, 2015 at 18:14
  • $\begingroup$ See also this post. $\endgroup$
    – javlacalle
    Commented Sep 28, 2015 at 20:39
  • $\begingroup$ @javlacalle Yes, I was able to figure it out from the first answer you linked. Thanks. $\endgroup$
    – harisf
    Commented Oct 7, 2015 at 7:47


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