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I'm currently trying to make a forecast of the use of prepaid payment

instruments using ARIMA modelling in Stata. I have a time series data set, containing monthly oberservations from April 2011 to October 2016 (67 observations).

This is the data set I am using (copied from csv format. If you have an other suggester how to better upload my data, give it to me):

month,Vol_PPIs,lnVol_PPIs
Apr 2011,3.08,1.12493
May 2011,3.21,1.166271
Jun 2011,2.59,.9516579
Jul 2011,2.98,1.091923
Aug 2011,.68,-.3856625
Sep 2011,2.38,.8671005
Oct 2011,2.22,.7975072
Nov 2011,.58,-.5447271
Dec 2011,3.08,1.12493
Jan 2012,3.19,1.160021
Feb 2012,2.99,1.095273
Mar 2012,3.62,1.286474
Apr 2012,3.6,1.280934
May 2012,3.71,1.311032
Jun 2012,3.260761,1.181961
Jul 2012,3.744257,1.320223
Aug 2012,4.070717,1.403819
Sep 2012,5.12,1.633154
Oct 2012,5.700896,1.740623
Nov 2012,6.184041,1.821972
Dec 2012,7.18,1.971299
Jan 2013,6.682868,1.899547
Feb 2013,7.329987,1.991974
Mar 2013,10.353717,2.337346
Apr 2013,8.680184,2.161043
May 2013,9.19,2.218116
Jun 2013,9.037625,2.201396
Jul 2013,10.09,2.311545
Aug 2013,10.892093,2.388037
Sep 2013,10.804673,2.379979
Oct 2013,10.359762,2.337929
Nov 2013,9.64968,2.266925
Dec 2013,10.784192,2.378081
Jan 2014,13.460249,2.599741
Feb 2014,14.11,2.646884
Mar 2014,16.571989,2.807714
Apr 2014,15.725985,2.755314
May 2014,16.248582,2.788006
Jun 2014,16.584616,2.808475
Jul 2014,18.709978,2.929057
Aug 2014,20.003466,2.995906
Sep 2014,22.842242,3.128612
Oct 2014,31.08,3.436564
Nov 2014,26.379108,3.272572
Dec 2014,28.98,3.366606
Jan 2015,33.492338,3.511317
Feb 2015,30.312295,3.411553
Mar 2015,54.104713,3.990921
Apr 2015,74.361793,4.308942
May 2015,46.914643,3.84833
Jun 2015,55.426342,4.015055
Jul 2015,64.533643,4.167187
Aug 2015,52.948234,3.969315
Sep 2015,58.00953,4.060607
Oct 2015,61.823247,4.124279
Nov 2015,62.659297,4.137712
Dec 2015,68.670216,4.229316
Jan 2016,65.248441,4.178202
Feb 2016,65.369233,4.180052
Mar 2016,72.052751,4.277399
Apr 2016,69.299994,4.238445
May 2016,70.946566,4.261927
Jun 2016,76.980277,4.343549
Jul 2016,77.851066,4.354797
Aug 2016,96.282689,4.567288
Sep 2016,97.074452,4.575478
Oct 2016,126.90475,4.843437

In order to make the data stationary, I took first differences of the data. Plotting the data it looks quite stationary, but the variance is increasing over time. Plot of first differnce of data

Nevertheless, the Augmented-Dickey Fuller test did reject the null hypothesis of a unit root, so I concluded that the differenciated data is roughly stationary.

In order to find the appropriate ARIMA specification I follow the general procedure plotting the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the differenciated data. Unfortunately, my PACF shows a very uncommon pattern: The partial autocorrelation are increasing extremly from lag 17 onwards.

Partial Autocorrelation Function of differenctiated data

I was wondering if the increasing partial autocorrelations are maybe due to the heteroskedasticity of my data. Taking the logarithm, the Problem of heteroskedasticity seems to be avoided (despite two outliers in the beginning).

Plot of log of first difference data

Nontheless, the PACF has its Peak with the last lag.

Partial Autocorrelation of log of differentiated data

So my question is if anyone can explain to me what is the reason for the PACF to behave like this? And how can I counteract the increasing partial autocorrelations?

I appreciate any hint you can give me. I did a lot of Research but have never seen a PACF like mine anywhere.

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The DF tests have a SEVERE drawback: they generally assume you have already removed other things that are making the data non-stationary. You may have to include variance stabilizing weights as suggested here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html . This diagnostic check was implemented into AUTOBOX some 10 years ago. Other developers have seemingly ignored this very important aspect of time series modelling.

Alternatively it may be necessary to simply transform the data via Box-Cox as I answered here http://stats.stackexchange.com/questions/18844/when-and-why-to-take-the-log-of-a-distribution-of-numbers.

Often times (not your case) simple Intervention Detection remedies the apparent need for a Box-Cpx transformation as was presented here http://autobox.com/cms/index.php/blog/entry/u-didnt-need-logs

Please post your actual data and I will take this a step further.

EDITED After receipt of data .

Time series data is nearly never trivial or simple thus simple brute-force methods rarely work. The data presented hereenter image description here aptly called THEGREAT suggests that some complexity is present. Note that ARIMA modelling even with empirically identified deterministic structure is non-causal . The identified structure can be a hint as to the nature of omitted causals.

I introduced the data to an automatic package that I have helped to develop (AUTOBOX) and it delivered the following useful equation . enter image description here a few anomalies , 3 seasonal pulses , an AR(3) structure based upon a significant deterministic change in error variance at or around period 43 of 67.

The Actual, Fit and Forecast is here enter image description here with cleansed data and actual presented here. enter image description here

In standard fashion the plot of the residuals is here enter image description here with ACF here enter image description here enter image description here . Both suggesting sufficiency .

The plot of the forecasts is here enter image description here

Hope this helps ....

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  • $\begingroup$ Thanks a lot! I uploaded my data set in the edited version of my question and I am trying now to realize your suggestions concerning the variance stabilizing weights. $\endgroup$ – Theodora Apr 11 '18 at 10:07

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