While translating an MA(1) process $y_t=\epsilon_t+\phi\epsilon_{t-1} $with $\epsilon_{t}$~WN$(0,\sigma^2) $ to a space state model can I use the zero matrix as the transition matrix like this:

$$\begin{bmatrix} \epsilon_t\\ \epsilon_{t-1}\end{bmatrix} = \begin{bmatrix}0&0\\0&0\end{bmatrix} \begin{bmatrix} \epsilon_{t-1}\\ \epsilon_{t-2}\end{bmatrix} + \begin{bmatrix} \epsilon_{t}\\ \epsilon_{t-1}\end{bmatrix} \\ y_t= \begin{bmatrix}1&\phi\end{bmatrix} \begin{bmatrix} \epsilon_t\\ \epsilon_{t-1}\end{bmatrix} $$


1 Answer 1


Expanded from a comment:

You can have a zero matrix but you cannot have that the state errors are correlated across time, so this is not a state space model by the usual definition.

Your model looks something like this:

$$X_t = 0 \cdot X_{t-1} +\eta_t$$ $$y_t = \begin{bmatrix}1& \phi\end{bmatrix} X_t$$

I've written the state $X_t = \begin{bmatrix}\epsilon_t\\\epsilon_{t-1}\end{bmatrix}$ and the state error $\eta_t = \begin{bmatrix}\epsilon_t\\\epsilon_{t-1}\end{bmatrix}$.

In a state-space model, the $\eta_t$ are assumed uncorrelated across time. However, you have that:

$$\text{cov}(\eta_t, \eta_{t-1}) = \begin{bmatrix}0 & 0 \\ \sigma^2 & 0 \end{bmatrix}$$

The non-zero term is $\text{cov}(\epsilon_{t-1},\epsilon_{t-1}) = \sigma^2$.

One valid way of writing an MA(1) model as a state-space model (there are others) is this:

$$\begin{bmatrix} \epsilon_t\\ \epsilon_{t-1}\end{bmatrix} = \begin{bmatrix}0&0\\1&0\end{bmatrix} \begin{bmatrix} \epsilon_{t-1}\\ \epsilon_{t-2}\end{bmatrix} + \begin{bmatrix} \epsilon_{t}\\ 0\end{bmatrix} \\ y_t= \begin{bmatrix}1&\phi\end{bmatrix} \begin{bmatrix} \epsilon_t\\ \epsilon_{t-1}\end{bmatrix} $$

That way, the state error is independent across time, and the transition matrix is just used to mechanically "keep track" of the previous error and make it available in the state in the following period.


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