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So I am trying to fit a real data set with initial plot of acf beingenter image description here

But I don't recognize a time series with an ACF that looks similar to this.

Should I try differencing?

I attempted to remove seasonality on my original data and formed a residual series. For which the acf looks like

enter image description here

Plot of detrended data

PersonaA
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  • ACF's like this can often reflect a model that requires seasonal dummies as compared to seasonal autoregressive structure and/or/seasonal differencing. Only your data knows for sure !. Post your data and let us see what analyzing the data reveals. – IrishStat Nov 18 '17 at 21:06
  • A plausible model often includes time trends , level shifts , seasonal pulses , identified anomalies AND often an ARIMA component. – IrishStat Nov 18 '17 at 21:54
  • I can look at your data if you post it . There is no guarantee that ARIMA is not sufficient BUT it usually is not as it is presumtive purely auto-regresssive solution.. – IrishStat Nov 18 '17 at 22:19
  • pls post the actual data not just a picture. It is visually obvious that unusual activity has occurred and needs to be identified and incorporated into a plausible model.. – IrishStat Nov 18 '17 at 22:23
  • How ? It is a large data set – PersonaA Nov 18 '17 at 22:26
  • use dropbox or another web posting utility OR just add a csv file ..or if none of that works email it to me. – IrishStat Nov 18 '17 at 22:27
  • for whatever reason I can;t access your data so send it to me at dave@autobox.com – IrishStat Nov 18 '17 at 22:38
  • Did you receive? – PersonaA Nov 18 '17 at 23:04
  • yes 273 monthly values . please conform first point is 454.04 last point is 529.99 what date is the first point . It is nighttime here at FORECAST CENTRAL so I will pick up this thread and hopefully deliver a plausible model later. – IrishStat Nov 18 '17 at 23:12
  • The usual ARIMA approach with something like that could typically involve looking at seasonal differences (or sometimes even both seasonal and ordinary differences, though I'd tend to be cautious about differencing twice), but I agree with IrishStat that for practical problems you may want to consider a broader collection of possibilities than just ARIMA models. – Glen_b Nov 19 '17 at 02:02
  • Edit; now I see the original values in IrishStat's answer ... I wouldn't alter that suggestion of exploring more than just seasonal differencing in an ARIMA framework. – Glen_b Nov 19 '17 at 02:07

2 Answers2

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I assume these are monthly data. You have a pronounced effect with a 12-month-lag. This is yearly seasonality.

If you encode your time series as monthly with frequency=12, then auto.arima() or similar should give you a seasonal model.

This question is related: Understanding the blue dotted lines in an ACF from R

Stephan Kolassa
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  • It was given as monthly data but in a csv file . So, I stored as a TS object in R with frequency=12 yes. How about Holt Winters? – PersonaA Nov 18 '17 at 20:41
  • I tried differencing several times, but the resulting ACF plot still remains periodic. – PersonaA Nov 18 '17 at 20:51
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    Don't do simple differencing. Either do [seasonal differencing](http://otexts.org/fpp2/sec-seasonal-arima.html), best automatically via `auto.arima()`, or do seasonal exponential smoothing like [Holt-Winters](http://otexts.org/fpp2/sec-7-Taxonomy.html), best by using `ets()` in the `forecast` package. – Stephan Kolassa Nov 18 '17 at 21:05
  • I see. Thanks. Does it matter that I just want to fit an plausible ARMA model to explain the non stationary aspects, not an ARIMA? – PersonaA Nov 18 '17 at 21:26
  • I used seasonal methods to fit a model and formed the residual series. I added a picture in my post. Does that mean my fitted model is better to work with? – PersonaA Nov 18 '17 at 22:12
  • The new ACF looks much better, with some evidence for an AR(1) term. Your raw data as per the plot in IrishStat's answer shows what looks like a step change down, a few years on lower level, then a trend. Do you have any drivers of this behavior, such as market prices, that you could include in your model, e.g., by running regression with ARIMA errors? – Stephan Kolassa Nov 19 '17 at 08:56
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I took your 273 monthly values and introduced them to AUTOBOX. Here is the plot of the data enter image description here and the acf enter image description here . A plausible model was developed with the following acf of the model residuals enter image description here and a plot of the residuals enter image description here both suggesting a sufficient model that separated signal and noise.

The Actual/Fit/Forecast is here enter image description here with forecasts here enter image description here . The Cleansed/Actual graph presents the detection of deterministic structure enter image description here

The model building strategy is to iterate (not 1 step ) to a plausible/useful model. Since a very significant seasonal structure is present , AUTOBOX tested an ARIMA approach first BUT found that the dominant structure was causal/deterministic in form. Seasonal differencing /ar structure is one way to deal with seasonal data by using memory while a regression based model with seasonal dummies /level shifts/time trends is often superior as a starting model.

In this case AUTOBOX concluded through a comprehensive search that the best initial step was to incorporate seasonal dummies for the five months 1,2,6,7, and 8 suggesting that there is no needed seasonal deterministic response to months 3,4,5,9,10,11,12 . These seasonal factors are directly attributable above and beyond any autoregressive seasonal effect.

Additionally it found intercept changes at period 55 and 219 suggesting three regimes 1-54 ; 55-218 and 219-273. These three regimes are clearly evident from the descriptive graph. Additionally it was found necessary and sufficient to incorporate an AR(1) and an AR(12) component reflecting overall month-to-month dependence. One anomaly was detected at period 140.

enter image description herefinally the model form and here enter image description here and here enter image description here

Model formulation is done iteratively by examining the significance of included coefficients while examining the residuals for both deterministic and ARIMA structure.

In summary two level shifts are needed to make the series stationary. De-meaning is an alternate to differencing as both can render a non-statianary series to stationarity. Another often useful scheme to make a series stationary is to de-trend it using one or more trends via series like 0,0,0,0,0,0,1,2,3,3,4,5..... as predictors. Identifying which of these is the best remedy is handled by examining alternative strategies.

IrishStat
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