# How to verify that a tentative state-space representation of an ARMA(1,1) process is valid

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?
• Hint. For 1., use $Y_t = \theta X_{t-1} + X_t$, and apply $(1 - \phi B)$ to each side.
– Yves
Nov 17, 2021 at 16:30
• 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.
– Yves
Nov 17, 2021 at 16:36
• 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. Nov 17, 2021 at 16:37
• @Yves: why don't you put your hint as an answer (since it is the key point) and add an answer to part 2? Nov 17, 2021 at 16:39
• You can use $\mathbf{X}_t$ to get $\mathbf{X}_t$.
– Yves
Nov 17, 2021 at 16:55

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.
• 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. Nov 18, 2021 at 12:07