How to deal with a single Yearly spike with ARIMA?

I have a time series which shows an yearly spike around summer but otherwise is predictable by an AR(1) model. The tests on the data also show that the time series shows stationarity and is non-seasonal. How do I model the spike? (More details about the time series here: What does the following ACF curve mean ? (Picture attached))

• The claim that the data show stationarity appears false. If it were true, the annual traces would be more nearly superimposed. – Nick Cox Jul 7 '15 at 8:51
• @NickCox : Thats what i had also though but the ADF tests show otherwise Dickey-Fuller = -489.5881, Lag order = 0, p-value = 0.01 alternative hypothesis: stationary  – UD1989 Jul 7 '15 at 9:01
• These tests don't serve you well when you let them overrule simple facts: compare 2006 and 2015. If stationarity is correct then the real series won't stand out from the yearly series shuffled randomly. – Nick Cox Jul 7 '15 at 9:05
• Perhaps the spike could be associated with a particular event? Month 3 or 4 could be Easter, but you say it is in the summer, so perhaps something else. If you knew the event and its date, you could use a dummy variable to account for the spike. – Richard Hardy Aug 4 '15 at 14:10

There are many possible approaches, I list below a couple of the ways that you could go about this:

i) You could try and apply a RegARIMA model. You're probably going to have to have 2 or 3 covariates, as it appears this 'spike' doesn't occur on the same month each year.

Better yet, if you had some other variable that was correlated with the occurrence of the spike, you would use this in your regARIMA model. Indeed, looking at your data, I wouldn't be surprised if the occurrence of the spike was correlated with Easter. It just so happens this is exactly what you observe in many time series, as Easter occurs in March or April. And, in your data, the spike appears in the third or fourth month.

For more information on this approach, you'd be best served to look how it is implemented in X12ARIMA by the US Census Bureau (or any other NSO).

or

ii) you could give up the idea that the data is not seasonal. If you are getting a spike at the same time each year, that would seem to indicate seasonality... have you done a test for (moving) seasonality? If the data is seasonal, you of course would apply a SARIMA model.

When you have a series that is consistently related month to month (1 year ago) this is routinely handled by a seasonal arima component. When you have just a few months that are related in this manner a much more efficient procedure (ignored by most of the free and commercial arima software approaches) is to use/add seasonal dummies for those effected months. Empirical identification of these special months is done via Intervention Detection or very often by eye via an examination of the residual plots from the deficient arima model . The acf is not useful as the seasonal effect only exist for the few special months. I would be glad to demonstrate this for you if you post 1 of your time series.

• Thats a good suggestion. I'll try that out, Thanks! – UD1989 Jul 7 '15 at 18:34