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There are different ways to formulate and implement ARIMA. The one discussed in the Chapter 8 is the conventional way, and the model in that case can be estimate directly, assuming that its initialisation happens some time before the Big Bang: the conventional ARIMA assumes that there is no starting point of the model, we just observe a specific piece of data from a population without any beginning or end. Obviously this assumption is idealistic and does not necessarily agree with reality (imagine the series of infinitely lasting sales of Siemens S45 mobile phones. Do you even remember such thing?).

But besides the conventional formulation, there are also state space forms of ARIMA, the most relevant to our topic being the one implemented in SSOE form (Chapter 11 of Hyndman et al., 2008). adapted this state space model for supply chain forecasting, developing an order selection mechanism, sidestepping the hypothesis testing and focusing on information criteria. However, the main issue with that approach is that the resulting ARIMA model works very slow on the data with large frequencies (because the model was formulated based on Chapter 11 of ). Luckily, there is an alternative SSOE state space formulation, introduced in Chapter 5.1. This model is already implemented in msarima() function of smooth package and was also used as the basis for the ADAM ARIMA.

In this chapter, we discuss the state space ADAM ARIMA for both pure additive and pure multiplicative cases, the conditional moments from the model and parameter space, then move to the distributional assumptions of the model (including the conditional distributions) and finish the chapter with the discussion of implications of ETS+ARIMA model. The latter has not been discussed in the literature and might make model unidentifiable, so an analyst using the combination should be cautious.

### References

• Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D., 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.
• Svetunkov, I., Boylan, J.E., 2020b. State-space ARIMA for supply-chain forecasting. International Journal of Production Research. 58, 818–827. https://doi.org/10.1080/00207543.2019.1600764