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I am attempting to understand the intuition behind the local level model that incorporates a seasonal component. I am currently reading an introductory book regarding state space modelling.

For the sake of clarity, I highlighted the parts that I could not fully understand. As is easily noticed, I do not understand how the seasonal state equations are constructed and how they are computed.

For the seasonal component $\gamma_{i,t}$, the $i$ indices relates to the $i$th quarter. For this case, we consider quarterly data. While the indices $t$ relates to the time period. But what is meant with the time period? The number of time increments? The year?

Could somebody clarify the seasonal state equations of the local level model with seasonal component? I would appreciate it very much.

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EDIT

If I would follow the reasoning, I would arrive to following (trivial) identity:

$\gamma_{i,t}=\gamma_{i,t+1}$

However, the following identites adressed do not hold for the example I depicted below:

$\gamma_{2,t+1}=\gamma_{1,t}$

$\gamma_{3,t+1}=\gamma_{2,t}$

For example take $t=2$:

$\gamma_{2,3} \neq \gamma_{1,2}$

$\gamma_{3,3} \neq \gamma_{2,2}$

enter image description here

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  • $\begingroup$ Here $t$ seems to be indexing year given the descriptions in the text as $t = 1,...,n$ (just below figure 4.1). So you have quarter $i$ nested in year $t$. $\endgroup$ Dec 18, 2019 at 7:31
  • $\begingroup$ Thanks for the comment. I have posted an EDIT to clarify the issue with a simple example. Would you bother to take a look at it? $\endgroup$ Dec 18, 2019 at 11:10
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    $\begingroup$ Yeah I am a bit confused myself to be honest. The second highlighted part states the fourth equation -$\gamma_{2, t+1} = \gamma_{1, t}$ - implies that quarter $i$ of period $t+1$ equals quarter $i+1$ of period $t$. But this would mean $\gamma_{i, t+1} = \gamma_{i+1, t}$ which would mean $\gamma_{1, t+1} = \gamma_{2, t}$. I am clearly missing something here. Do you maybe need to substitute and do some re-arranging in Eq.4.1? $\endgroup$ Dec 18, 2019 at 12:48
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    $\begingroup$ That's exactly where I get stuck. I don't believe rearranging is helpful, as the identities should be taken as a 'fact of life'. Unfortunately, that 'fact of life' remains mysterious to me $\endgroup$ Dec 18, 2019 at 13:06

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