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I'm trying to show the following (exercise 3.11.4 from Durbin and Koopman (2012)):

Show that the state space model defined by $$ y_t=X_t\beta+Z_t\alpha_t+\epsilon_t\\ \alpha_{t+1}=W_t\beta+T_t\alpha_t+R_t\eta_t$$ can also be expressed as $$ y_t=X_t^*\beta+Z_t\alpha_t^*+\epsilon_t\\ \alpha_{t+1}^*=T_t\alpha_t^*+R_t\eta_t$$ for $X_t$ and $W_t$ are fixed matrices and all have appropriate dimensions.

I've been trying several things, but none seem fruitful. It seems to me as if the dimensions of $\alpha$ and $\alpha^*$ are the same, otherwise it would've been straightforward by just augmentation of the state vector. But I can't see how to do it as I should in this case. Suggestions are very welcome!

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  • $\begingroup$ 1. Add the self-study tag. 2. Should the last equation be $\alpha^\ast_{t+1}=\ldots$ instead of $\alpha_{t+1} = \ldots$? $\endgroup$ – Juho Kokkala Dec 16 '14 at 10:03
  • $\begingroup$ @JuhoKokkala Good catch, it should indeed. Tag added, too. $\endgroup$ – hejseb Dec 16 '14 at 10:24
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The way this is done, is to first establish the relationship between $\alpha_{t}$ and $\alpha_{t}^{\ast}$ and proceed from there. We take the initial state equations above and take

$$\alpha_{t}^{\ast} = \mathsf{T}_{t}^{-1}\mathsf{W}_{t}\beta + \alpha_{t},$$

we see that we can write

$$\alpha_{t + 1}^{\ast} = \mathsf{T}_{t}\alpha_{t}^{\ast} + \mathsf{R}_{t}\eta_{t}.$$

That's that one done. Now, for the first of the above equations we take

$$\mathsf{X}_{t}^{\ast} = \mathsf{X}_{t} - \mathsf{Z}_{t}\mathsf{T}_{t}^{-1}\mathsf{W}_{t},$$

and we can write

$$y_{t} = \mathsf{X}_{t}^{\ast}\beta + \mathsf{Z}_{t}\alpha_{t}^{\ast} + \epsilon_{t}.$$

You can convince yourself by substituting the expressions for $\alpha_{t}^{\ast}$ and $\mathsf{X}_{t}^{\ast}$ back into the equations and you will see that you get the initial ones back.

I hope this helps.

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