# 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