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As my question illustrates, we know the state space model (SSM) is more general compared with ARIMA, and every ARIMA model could be transformed into a state space form, and some SSMs could be transformed into ARIMA as well, like the local linear trend model could be viewed as an approximate of ARIMA(0,2,2). This question has talked about the difference of SSMs and ARIMA before.

Generally speaking, I am wondering which kind of SSMs can not be transformed into an ARIMA model or you may give me some examples. I came up with one example, the pseudo-additive structural time series model (gaussian noises), but not very sure whether it is correct. $$ y_t=T_t(S_t+I_t-1) $$ $$ T_{t+1} = T_t + \eta_t $$ $$ S_t= -\sum_{i=1}^{s-1} S_{t-i} + \omega_t $$

More strictly, are there any linear SSMs that don't have corresponding ARIMA forms?

Thanks :)

LincolnG96
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  • This [question](https://stats.stackexchange.com/questions/178937/find-state-space-model-to-compare-with-box-jenkins-arima-model/178950) has talked about the case of how to transform a SSM to ARIMA. But I want some counter-example here... Anyone knows? – LincolnG96 Jun 03 '20 at 02:55

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