This section describes differences of factorisation, especially about detrending, between the original and our enhanced versions of X-13ARIMA-SEATS.

Factorisation of time series into trend, cycle and other factors consists important process of economic forecast. Turning point detections of the level, growth rate, and change of growth (acceleration) are included in this process. Our enhanced version of X-13ARIMA-SEATS has an additional spec \(\mathtt{ bundle\lbrace\ \rbrace}\) to save the related series into one file.

Detrending of \(\mathtt{x11}\) and \(\mathtt{ composite}\) specs.

Four arguments about detrend \(\mathtt{hpcycle, hplan, hprmls, hptarget}\) are added to \(\mathtt{x11}\) and \(\mathtt{ composite}\) specs as well as original \(\mathtt{seats}\) spec. An argument \(\mathtt{x11\lbrace hpcycle = yes \rbrace}\) factorises trend-cycle table \(\mathtt{d12}\) with its forecasts and backcasts into trend \(\mathtt{ltt}\) and cycle \(\mathtt{cyc}\).

Argument \(\mathtt{hptarget = sadjastci}\) of specs \(\mathtt{seats},\) \(\mathtt{ x11},\) and \(\mathtt{ composite}\) yields cycle-irregular series \(\mathtt{cyi}\), while results of original argument \(\mathtt{hptarget = sadj}\) and more precise \(\mathtt{hptarget = sadjastc}\) are recognised as cycle series.

Deriving trend and trend-cycle of \(\mathtt{ composite}\) series.

Assume \(Composite\ series\ =\ \sum\ element\ series\), and \(\sum\) (summation) means accumulation of element series to composite for simplicity. Also, let to calculate mean to factorise via regARIMA model or moving average.

Two ways exist; calculate for each element and accumulate (CA), or accumulate to composite series and calculate for composite (AC).

1. Calculate – accumulate (CA)
2. \(Composite\ original\ series\ =\ \sum\ element\ original\ series\)
   • and
3. \(Composite\ adjusted\ series\ =\ \sum\ element\ adjusted\ series\)

1. Accumulate – calculate (AC)
2. \(Composite\ original\ series\ =\ \sum\ element\ original\ series\)
   • and
3. \(Composite\ adjusted\ series\ =\ adjustment\ of\ composite\ original\ series\)

The original X-13-ARIMA-SEATS adopts CA from original series to final adjusted series. Final adjusted composite \(\mathtt{isa}\) is an accumulation of \(\mathtt{d11}\) or \(\mathtt{s11}\). Tables from trend-cycle and after are AC, calculated for composite series.

On the contrary, the author has been using CA for the series which have trend factors.

An argument \(\mathtt{trendlevel = [ accumulate | calculate ]}\) was added to spec \(\mathtt{composite}\). It controls the series which contains trend fator, because the original version calculates trend-cycle and trend from the composite time series, while the author has been applying as composite of component series’ trend-cycle and trend.