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?


1 Answer 1


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, 2019 at 17:03
  • 1
    $\begingroup$ Yes, indeed you are right. $\endgroup$
    – Ruben
    Dec 3, 2019 at 8:46

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