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A decomposition model widely used in practice is

$$Z_t = \mu_t + N_t + a_t$$

where $\mu$ captures the deterministic part of the series and $N_t$ is a stationary process that can be adjusted by ARMA(p, q).

The series I'm studying shows no trend, only seasonality.

enter image description here

I tried to fit a dummy regression, however $N_t$ is non-stationary. Subsequently, I adjusted a sine-wave regression and the Anderson Darling test checked for $N_t$ stationarity.

enter image description here enter image description here

However when analyzing the fac and pafc of $N_t$ I have significant high order lags. Still pointing to the presence of something seasonal.

enter image description here enter image description here

Any idea how to get a gun model ARMA for $N_t$?

To read the series I used the following code:

x = scan()
0.94891
0.89786
0.83291
0.74823
0.69496
0.66354
0.58498
0.49182
0.45535
0.33249
0.31806
0.33588
0.94031
0.91268
0.88446
0.86162
0.81494
0.76389
0.68428
0.43835
0.33715
0.2918
0.29184
0.26639
0.91411
0.83015
0.79352
0.70579
0.67098
0.66247
0.65689
0.59713
0.60407
0.46496
0.38194
0.38118
0.98666
0.92302
0.80413
0.73738
0.62196
0.5825
0.54789
0.47124

z = ts(x[1:40], start = c(2014, 1, 1), freq = 12)
ts.plot(z)
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  • $\begingroup$ Your series may have deterministic effects such as seasonal pulses or level shifts or local time trends see stats.stackexchange.com/questions/316689/… . Post your data in a csv file and I will try and help further. $\endgroup$ – IrishStat Dec 10 '17 at 12:47
  • $\begingroup$ I updated the question by putting the values ​​in the series. I thought of adding a cyclic component in the model to see if later the resulting series $N_t$ follows a weapon model, but as I am not aware enough for this, I did not proceed with the idea. - @IrishStat $\endgroup$ – Jackson Maike Jan 3 '18 at 23:03
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I took your 68 values and introduced them to AUTOBOX my tool of choice. It automatically found after some iterations that there was 1 anomaly and a strong AUTO-PROJECTIVE seasonal model rather than any deterministic model .enter image description here with statistics here enter image description here and actual/forecast here enter image description here . The original ACF is here enter image description here

SUMMARY:

"A decomposition model widely used in practice" is innsufficient evidence to base statistical analysis on. Let the data speak and listen closely. In some (rare in my opinion except in very dated textbooks) your decomposition model may be applicable especially for highly aggregated national account series with 12 plus years BUT hardly ever for most circumstances as it assumes a very particular model with a ton of assumptions.

Sometimes deterministic model are appropriate .. sometimes autprojective (ARIMA) are appropriate and more often both components are needed. In this case the deterministic component was a pulse .

ACTUAL/FIT AND FORECAST graph is revealing ..enter image description here

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