Assume we have the following state-space model: $$ z_{k} = \theta_{k} z_{k-1} + v_{k}\\ \theta_{k} = \phi \theta_{k-1} + w_{k}, $$ where $v_{k}$ and $w_{k}$ are independent and normal for all $k$. The space equation is the first one, i.e. the one with $z_{k}$. This state-space system is not linear, nevertheless, $cov(z_{k-1},v_{k}) = 0$.

Therefore, I rewrite the first equation as $$ z_{k} = H_{k} \theta_{k} + v_{k}, $$ where $H_{k} = z_{k-1}$ and I can use a standard linear Kalman filter. Basically, we end up with Kalman regression model, where the slope is the state variable. Is this correct?


Yes this is correct and you are allowed to do this. To learn more about this subject I suggest you read up on AR models with time varying coefficients.

I wrote a paper about robust estimation of such models, which also gives an introduction to the topic and an example that is very similar to the model you use: https://www.tandfonline.com/doi/abs/10.1080/03610918.2017.1422752 .

  • $\begingroup$ Dear @Ruben, thanks for the answer. Therefore, one can conclude that AR model with time varying coefficients can be solved with Kalman regression. Right? $\endgroup$
    – ABK
    Dec 2 '19 at 17:03
  • 1
    $\begingroup$ Yes, indeed you are right. $\endgroup$
    – Ruben
    Dec 3 '19 at 8:46

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