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Brockwell & Davis example 9.1.2. show the state-space representation for {$Y_t$}, an ARMA(1,1) process. They claim that the 2 equations below are the observation equation and state equation respectively for {$Y_t$}:

(1) ${Y}_t=[\theta $    $ 1] \begin{bmatrix} X_{t-1}\\X_t\end{bmatrix}$

(2) $\begin{bmatrix} X_{t}\\X_{t+1}\end{bmatrix}=\begin{bmatrix} 0 & 1\\0 & \phi \end{bmatrix}\begin{bmatrix} X_{t-1}\\X_t\end{bmatrix}+\begin{bmatrix} 0 \\Z_{t+1}\end{bmatrix}$

with $\mathbf X_1=\begin{bmatrix} \sum_{j=0}^{\infty}\phi^jZ_{-j} \\\sum_{j=0}^{\infty}\phi^jZ_{1-j}\end{bmatrix}$ and $Z_t$~$WN(0,\sigma^2)$

I have 2 questions here:

  1. How can I convince myself that the above is a valid state-space representation for the ARMA(1,1) process?
  2. How do I think about what the unobserved state vector $\mathbf X_t$=$\begin{bmatrix} X_{t-1}\\X_{t}\end{bmatrix}$ represents in this case?
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    $\begingroup$ Hint. For 1., use $Y_t = \theta X_{t-1} + X_t$, and apply $(1 - \phi B)$ to each side. $\endgroup$
    – Yves
    Nov 17, 2021 at 16:30
  • $\begingroup$ Note that the boldface notation $\mathbf{X}_t$ for the state vector in the book. It should be used for $\mathbf{X}_1$. The same is true for your question 2. $\endgroup$
    – Yves
    Nov 17, 2021 at 16:36
  • $\begingroup$ That was a very good hint; it did it for me; I was able to show that this is indeed the space-state representation. Very neat. Thank you. $\endgroup$ Nov 17, 2021 at 16:37
  • $\begingroup$ @Yves: why don't you put your hint as an answer (since it is the key point) and add an answer to part 2? $\endgroup$ Nov 17, 2021 at 16:39
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    $\begingroup$ You can use $\mathbf{X}_t$ to get $\mathbf{X}_t$. $\endgroup$
    – Yves
    Nov 17, 2021 at 16:55

1 Answer 1

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Provided that $|\phi| < 1$ we can as in the OP define $X_t := \sum_{j = 0}^\infty \phi^j Z_{t -j}$, so $(1-\phi B) \,X_t = Z_t$. Let us denote temporarily by $Y^\star_t$ the observation of the candidate State-Space (SS) representation. Since $Y_t^{\star} := \theta X_{t-1} + X_{t}$, by applying $(1 - \phi B)$ to each side of the last relation, we get the relation (1) with $Y^\star_t$ replacing $Y_t$. So, the same model holds for $Y^\star_t$ and $Y_t$. And the SS representation with $\mathbf{X}_t := [X_{t-1}, \, X_t]'$ gives the wanted model for $Y_t$.

The series $X_t$ applies a linear filter $1/(1 - \phi B)$ to the white noise $Z_t$. We can describe $X_t$ as a "coloured noise", yet I can not see a better interpretation to answer to question 2. To get an interpretable (equivalent) representation, we could go with the state vector $\mathbf{V}_t = [Y_{t},\, \hat{Y}_{t+1|t}]'$ and the observation equation $Y_t = [1,\, 0] \mathbf{V}_t$. This generalises to the $\text{ARMA}(p, \,q)$ case with $r := \max\{p,\,q +1\}$ by taking $\mathbf{V}_t := [Y_{t},\, \hat{Y}_{t+1|t}, \,\dots,\,Y_{t+r-1|t}]'$ with the obvious observation equation $Y_t = [1,\, 0, \, \dots,\,0]'\mathbf{V}_t$. This is sometimes called the Akaike SS representation.

Note also that in a SS representation, we need an initial covariance matrix, be it that $\mathbf{X}_0$ or that of $\mathbf{X}_1$, in both cases conditional on the empty observation set preceding the observation of $Y_1$. For the SS representation of the $\text{ARMA}(1,\,1)$ above, this covariance matrix is easily derived.

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  • $\begingroup$ This is terrific. Thank you again for all these insights. I can tell you are quite an expert in this field. $\endgroup$ Nov 18, 2021 at 12:03
  • $\begingroup$ Question 2 was sort of trick question as it is very difficult to conceptualize the states; but I like your "colored noise" description. This is a clue that spectral analysis of $X_t$ might give us some additional insights into its DGP. $\endgroup$ Nov 18, 2021 at 12:07

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