# Outlier in differentiated series

I'm studying methods for time series analysis, using gretl. I have this time series. I used TRAMO and X12-ARIMA to detect probable outlier, but I found nothing. So I used difference-log of first order to make the serie stationary, and I had this: It seems that there is something in the end of 2008. Infact TRAMO found a temporary change on November 2008

 106 TC    (11 2008)


First question: Is it possible that the series before being differentiated had no outlier and after yes? I continued the analysis, linearizing the serie, obtaining So I can begin to study the ACF and PACF TRAMO suggested me to use ARIMA(1,0,0)(0,1,1), but I found that a simple AR(1) give the same result. These are the ACF-PACF of ARIMA(1,0,0)(0,1,1) residuals: and these of AR(1) residuals Comparing AIC and BIC, theory suggests to choose the AR(1). This is the output for the ARIMA

    Modello 3: ARIMA, usando le osservazioni 2001:03-2017:09 (T = 199)
Stimato usando il metodo BHHH (MV condizionale)
Variabile dipendente: (1-Ls) ld_Finla_xl

coefficiente   errore std.      z        p-value
------------------------------------------------------------
const       7,67509e-05   0,000137004     0,5602   0,5753
phi_1       0,307572      0,0671470       4,581    4,64e-06  ***
Theta_1    −0,765016      0,0511360     −14,96     1,33e-050 ***

Media var. dipendente  0,000170   SQM var. dipendente    0,008292
Media innovazioni      0,000046   SQM innovazioni        0,006505
Log-verosimiglianza    719,6341   Criterio di Akaike    −1431,268
Criterio di Schwarz   −1418,095   Hannan-Quinn          −1425,937
Note: SQM = scarto quadratico medio; E.S. = errore standard

Reale   Immaginario   Modulo  Frequenza
-----------------------------------------------------------
AR
Radice  1           3,2513     0,0000     3,2513     0,0000
MA (stagionale)
Radice  1           1,3072     0,0000     1,3072     0,0000
-----------------------------------------------------------


this for the AR model

    Modello 1: ARMA, usando le osservazioni 2000:03-2017:09 (T = 211)
Stimato usando i minimi quadrati (MV condizionale)
Variabile dipendente: ld_Finla_xl

coefficiente   errore std.     z     p-value
--------------------------------------------------------
const      0,000623196    0,000402829   1,547   0,1219
phi_1      0,312608       0,0644261     4,852   1,22e-06 ***

Media var. dipendente  0,000935   SQM var. dipendente    0,006078
Media innovazioni      0,000000   SQM innovazioni        0,005776
Log-verosimiglianza    789,1005   Criterio di Akaike    −1574,201
Criterio di Schwarz   −1567,497   Hannan-Quinn          −1571,491
Note: SQM = scarto quadratico medio; E.S. = errore standard

Reale   Immaginario   Modulo  Frequenza
-----------------------------------------------------------
AR
Radice  1           3,1989     0,0000     3,1989     0,0000
-----------------------------------------------------------


Second question: TRAMO suggests to use seasonale difference, but from the ACF/PACF it seems that it's not necessary. I know that seasonal difference is requested when there is no stationary caused by the seasonal component. Is it true?

There are three kinds of additive outliers viz, Pulses, Level Shifts and Seasonal Pulses. Tramo Seats does not treat Seasonal Pulses correctly or perhaps not all due to ineffective implementation as it is non-trivial to implement. A Seasonal Pulse is an old-fashioned (but very useful) seasonal dummy which doesn't necessarily start at the beginning of the data. For example if we have 213 historical values but starting at 2004/12 (period 60) there is a consistently higher observation by .306 reflecting Christmas. This would lead to a dummy indicator of the form 59 0's followed by a 1 at the next period and each successive December. Additionally there is detected seasonal dummy required for February starting at period 86 reflecting a trading-day effect as February has fewer business days. The identified equation is here . IMO the fact that only two months have a hint of seasonality in no way suggests seasonal differencing.

The model statistics are here using AUTOBOX a piece of siftware that I have helped develop and here

The plot of the residuals is here with companion acf

The cleansed vs actual plot is revealing and the actual/fit/forecast here with possible anomalies in the future.

1) Is it possible that the series before being differentiated had no outlier and after yes? Yes as the outlier may be hidden due to predictive structure that has yet to be identified much like being unable to "see" if your eyeglasses(filter/model) have a crack in them.

2) TRAMO suggests to use seasonale difference, but from the ACF/PACF it seems that it's not necessary. No there is no need for seasonal differencing just better analytics from Tramo-Seats identifying the need for Seasonal Pulses and their starting point. Unwarranted seasonal differencing INJECTS structure into the residuals (Slutsky effect ? e.g differencing a white noise series creates a new series with greater variance) rather than eliminating structure .

For an extended discussion of the flaws in TS see https://www.researchgate.net/publication/264993978_About_model-based_time_series_procedures_some_remarks_to_TRAMOSEATS_and_CENSUS_X-12-ARIMA

and less so here