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I'm new to this type of modeling, and I'm trying to make sense of the seasonal parameters of the SARIMA model:

  1. In what type of scenarios there will be a difference between the non-seasonal p,q parameters and the seasonal P,Q. Most of the examples I found had the same value for both.

  2. I understand that for the non-seasonal p,q parameters, it is instructive to look at the ACF and PACF of the stationary data (after differencing) to see the lags. However, for the seasonal P,Q parameters, I'm not sure what's the intuition behind figuring their order, and what are the suggested steps one should take in order to find good values for them

  3. What is the difference between d and D? Is there an example that can clarify when they are inherently different?

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Some thoughts for your questions:

  1. The choice of $(p, d, q)$ is depending on the long term behavior of time-series and $(P, D, Q)$ is depending on its seasonal behavior (if any). Most of the examples around consider small integers for $p, P, q, Q$ and $d, D$ , because they generate rather simple models. Thus, it is just a matter of coincidence than $(p, d, q)$ is not significantly different from $(P, D, Q)$.

  2. Seasonality means that every $m$ steps analogous values are observed. The ACF diagram may be helpful to decide what is the value of $m$ since there will be seasonal significant highs at every $m$ steps. Further evidence for $m$ comes from PACF diagram where, in case of $m$-seasonality, those highs are in significantly decreasing order. So, lets break the original series $\{Y_t\}_{t \geq 0}$ into the $m$ parts $\{Y_{i + km}\}_{k \geq 0}$ , $i = 0, 1, ..., m-1$. For each one subseries a model of the form $(P, 0, 0)$ or $(0, 0, Q)$ or even $(P, 0, Q)$ may be well adapted. In that step, the goal is to find a single model that is well adapted for all subseries, that is the original time series. In practice, various possible models are proposed and methods as the AIC criterion are applied to help finding the best.

  3. In case that a seasonal part exist, the difference part $D$ may be utilized to heal a non stationary time series. If, for a particular choice of $(P, D, Q)$ the seasonal part is described well (as the residuals show) and there is still a significant trend, then the small $d$ may be utilized to heal the remaining trend and creates a stationary series. One more time, a selection criterion should be applied in deciding what is the best choice among various models that seems appropriate for a given time series.

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