I’m currently working on a time series prediction model about calls incoming over one year with half hourly data. This time series has multiple and long seasonalities that I have identified. A daily one of 22, a weekly one of 152 (22 x 7) and a monthly one of 668.8 (22 x 30.4). I want also to include external regressors into the model as we have noticed that holidays and campaigns could have an effect on the total of calls.

Reading the literature, Dynamic Regression (A regression with ARMA errors) seems to be the perfect approach as it takes into account the complex seasonality (introduced by Fourier terms) and integrates the external regressors. But here, I’m a bit stuck on how to start..

My question is about stationarity. The literature emphasizes this as an important consideration that all the time series: Y and the different covariates are stationary.

I looked a my Y series and the ACF plot clearly shows that it is non stationary :

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ACF PLOT of the time series Y

I ran the KPSS test that suggests differencing with a lag of 12 but the ACF plot after differencing still shows a non-stationary time series. I tried a seasonal difference of 152 (22*7) but still the same conclusion.

Do you have any idea on how I can make this time series stationary?

For the covariates, most of them are dummies (the event has occurred or not). Should we apply the same methodology applied for the time series Y? Even if it is already stationary?

On which Y series, Fourier terms should be calculated : on the stationary one or the original time series?

Thank you for your help,

kr, Jérôme


1 Answer 1


As you mention, the time series you are analyzing is not stationary. Applying seasonal differences makes sense when considering stochastic seasonality (ARIMA approach). This is how it may take more than one difference to obtain a stationary time series (daily/weekly/monthly/yearly differences). The other alternative, as you comment, is to use Dynamic Regression using Fourier coefficients to encode seasonality in a deterministic manner. Dummy variables would be considered one more exogenous variable in that setup. Finally, the Fourier terms are not calculated "on a series", but rather, with respect to the period you want to encode with them (one day, one week, one year, etc.)


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