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!

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.