# Time series - Classic decomposition model

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. 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.

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

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)

• 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. – IrishStat Dec 10 '17 at 12:47
• 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 – Jackson Maike Jan 3 '18 at 23:03

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 . with statistics here and actual/forecast here . The original ACF is here ACTUAL/FIT AND FORECAST graph is revealing .. 