# How to include seasonality in the data ARMAX model that has multiple periodicities?

I am doing regression with ARIMA errors. The residuals are as shown in the figure below. Clearly, the scatter plot shows that this time series has some sort of periodicity or seasonality, but its very hard to identify the seasonality.

I also plotted the ACF and PACF of the residuals clearly it shows that model is non-stationary as the ACF is very slowly decaying. To estimate periodicity I plotted the periodogram of the time series.

As it can be seen in the periodogram that its very hard to estimate its seasonality, clearly there are more than one significant frequencies. How do I go about from here?

I also estimated the ACF and PACF after first differencing, looks like its being over differenced? How do I go about solving this problem?

• What is your tool of choice,?if it's $R$ then it's not possible. – forecaster Jul 21 '15 at 15:42
• @forecaster I am using astsa package in R. What other tool can be used to solve it ? – Spandy Jul 21 '15 at 15:44
• To my knowledge there is no ARIMA model for multiple periodicity in $R$. You might want to use SAS which is one of the few software that I know has ability to model any periodicity as an example (0,1,1)(0,1,1)7,(0,1,1)365 for daily data with yearly and weekly seasonality can be easily done in SAS – forecaster Jul 21 '15 at 15:48
• What is the main problem here? Is it detecting an unknown number of periods? Or is it modeling data that has multiple known periods (e.g. hourly data that has a period of 24 hours as well as a period of 1 year)? The body does not quite match the main title. Anyway, you may check Rob J. Hyndman's blog, search for "seasonality" or similar. See also tbats function in "forecast" package in R for modeling data with two periods. I am also more optimistic about capabilities of R than forecaster is, at least when it is not yet completely clear what you are up to. – Richard Hardy Aug 3 '15 at 17:22
• @RichardHardy My question is, after finding the periodicity of the time series using periodogram, what should be my next step ? – Spandy Sep 10 '15 at 16:31