# What can be done in case with periodic acf?

So I am trying to fit a real data set with initial plot of acf being

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

• 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. Commented Nov 18, 2017 at 21:06
• A plausible model often includes time trends , level shifts , seasonal pulses , identified anomalies AND often an ARIMA component. Commented Nov 18, 2017 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.. Commented Nov 18, 2017 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.. Commented Nov 18, 2017 at 22:23
• How ? It is a large data set Commented Nov 18, 2017 at 22:26

I took your 273 monthly values and introduced them to AUTOBOX. Here is the plot of the data and the acf . A plausible model was developed with the following acf of the model residuals and a plot of the residuals both suggesting a sufficient model that separated signal and noise.

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

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.

finally the model form and here and 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.

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

• 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? Commented Nov 18, 2017 at 20:41
• I tried differencing several times, but the resulting ACF plot still remains periodic. Commented Nov 18, 2017 at 20:51
• Don't do simple differencing. Either do seasonal differencing, best automatically via auto.arima(), or do seasonal exponential smoothing like Holt-Winters, best by using ets() in the forecast package. Commented Nov 18, 2017 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? Commented Nov 18, 2017 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? Commented Nov 18, 2017 at 22:12