# Time series forecasting via decomposition and component-wise modelling

Consider a time series $t_{k} = d_{k} + s_{k}$, where $d_{k}$ is a deterministic series (trend or periodic component, for example) and $s_{k}$ - a stochastic process, for example, ARMA(p,q)-GARCH(P,Q).

1. Is it correct to fit ARMA-GARCH part after $d_{k}$ vanishing?

2. Assuming we don't know anything about the type of $d_{k}$, what is the best way to vanish the trend? Is wavelet thresholding a good technique for this?

3. Is it correct to predict $k+1, k+2, \dots$ values of $t_{k}$ by extrapolating $d_{k}$, predicting $s_{k}$ and summing up these values?

• What do you mean by "trend vanishing"? Do you mean you model the trend and then remove the fitted trend (then I would avoid the word "vanish")? Or does the trend somehow get smaller over time and towards the end of the sample $d_k\approx 0$ and thus $t_k\approx s_k$ (which could be called "vanishing over time")? – Richard Hardy Oct 17 '16 at 18:06

## 1 Answer

1. If you mean decomposing the series into a trend and an ARMA-GARCH part and then modelling them separately (the focus being on the ARMA-GARCH part in this part of the question), then this is commonly done and discussed in time series textbooks, e.g. Hyndman & Athanasopoulos "Forecasting: Principles and Practice" Chapter 6.6 "Forecasting with decomposition" or Anderson & Semmelroth "Statistics for Big Data For Dummies" (arrived at via a website on "Time Series Analysis: Forecasting with Decomposition Methods").
2. Not sure, I am not familiar with wavelet thresholding.
3. What is "correct" can be debated, but the strategy of (1) decomposing the series, (2) predicting the components, and (3) adding the forecasting of the components to form the forecast of the original series can be useful and is widespread.
• thank you. 1) Yes, I meant this. 2) Why? What're other techniques of this? 3) I meant that this method is useful and doesn't give incorrect prediction. What're other methods? And could you give me books names for 1 and 3 puncts? Thank you much. – Dmitriy Oct 17 '16 at 19:06
• Regarding 3., I do not have a canonic reference, but you will see in applied studies (journal publications) that this is what is typically done. Often it is not even mentioned because it is perceived as self evident. Regarding 1., I am trying to find one. Often the textbooks get lost in details and do not bring the pieces back together, which is exactly what is needed here. I might have been overly optimistic saying the texbooks mention that. But there are really so many time series textbooks that it would take a while to go over them... – Richard Hardy Oct 17 '16 at 19:12
• so, the best way to look articles by this tag:"Time series forecasting via decomposition and component-wise modelling", isn't it? – Dmitriy Oct 17 '16 at 19:22
• This way you would probably find some methodological contributions but would miss lots of applications that do this without even mentioning. Try googling "forecast decompose" (without quotes), you will find quite a few examples. – Richard Hardy Oct 17 '16 at 19:35
• thatnk you very-very much!!! If you will be in Moscow - say me, I'll buy you beer.)))) – Dmitriy Oct 17 '16 at 19:42