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I want to find Yule-Walker equations for a causal ARMA(1,1)-process

$Y_t=\alpha Y_{t-1}+\epsilon_t+\theta\epsilon_{t-1}$.

My idea initially was to first multiply both sides by $Y_{t-1}$, so I would get

$\gamma_0=\alpha\gamma_1+\epsilon_t Y_t+\theta\epsilon_{t-1}Y_t$, where $\gamma_0$ is the variance and $\gamma_1$ is the first autocovariance.

But, I don't know how to continue from there.

Could someone help me out? Thanks in advance!

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    $\begingroup$ Yule walker is for AR models. This has an MA component $\endgroup$
    – Taylor
    May 6 '17 at 16:05
  • $\begingroup$ Go down the road of representing the process as an $AR(\infty)$? Does that get you anywhere? $\endgroup$ May 6 '17 at 17:32
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Yule Walker (for parameter estimation) is usually only used for AR models, but this method you're using is still a valid technique for finding the autocovariance function. I'm assuming that's what you're after.

Multiply both sides of an ARMA(p,q) model by $Y_{t-k} = \sum_{j=0}^{\infty}\psi_j \epsilon_{t-j-k}$ (this is where causality assumption comes in)

$$ Y_t Y_{t-k} - \phi_1 Y_{t-1}Y_{t-k} - \cdots - \phi_p Y_{t-p}Y_{t-k} = \left[\sum_{l=0}^q \theta_l \epsilon_{t-l}\right]\left[\sum_{j=0}^{\infty}\psi_j \epsilon_{t-j-k} \right]. \tag{1} $$

Then take expectations. Note that $\psi_j = 0$ for $j < 0$ and $\theta_j = 0$ for $j \not \in \{1,\ldots,q\}$. If $0 \le k < \text{max}(p,q+1)$ you get these $p$ (1 in your case) equations here $$ \gamma(k) - \phi_1 \gamma(k-1) - \cdots - \phi_p\gamma(k-p) = \sigma^2 \sum_{j=0}^{\infty}\theta_{k+j}\psi_j. \tag{2} $$ And if $k \ge \text{max}(p,q+1)$ you get $$ \gamma(k) - \phi_1 \gamma(k-1) - \cdots - \phi_p\gamma(k-p) = 0 \tag{3} $$ ($0$ on RHS because $l > q$).

You can solve your two equations by using results of homogeneous difference equations, or by just solving the system directly.

For your specific ARMA(1,1) model (I'll turn the $\phi$s back into $\alpha$s) you have

  1. $\gamma(0) - \alpha \gamma(1) = \sigma^2[1 + \theta_1 (\theta + \alpha)]$
  2. $\gamma(1) - \alpha \gamma(0) = \sigma^2[\theta]$
  3. $\gamma(k) - \alpha \gamma(k-1) = 0$ for $k \ge 2$

Plugging (2) into (1) gives us $$ \gamma(0) = \sigma^2\left[ 1 + \frac{(\theta+\alpha)^2 }{1 - \alpha^2 } \right] $$ (which is the same as before) and plugging this into (2) gives us $$ \gamma(1) = \sigma^2\left[ \theta + \alpha + \frac{(\theta+\alpha)^2\alpha }{1 - \alpha^2 } \right]. $$ If you want to get it at higher lags, just use (3) repeatedly.

When the model is a pure AR model, the RHS of (1) is simplified tremendously, and the resulting set of equations to solve is linear in the parameters $\phi_1,\ldots,\phi_p$.

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  • $\begingroup$ What are the $\psi$ terms that you introduce in the second paragraph? How are they related to the ARMA parameters? Is there no way to express the solution in a closed form, say, using matrix algebra? Thank you $\endgroup$
    – Confounded
    Nov 15 '20 at 14:27

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