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I have a very basic question on including constant in Arima models. I'll illustrate this by an example. I have the following ACF and PACF of a weekly time series that is differenced at lag 1 (trend) and lag 52 (seasonality). The series becomes stationary after differencing. In looking at the ACF and PACF I see that the appropriate model is an MA(1) model or a simple exponential smoothing (SES), so I end up with (0,1,1)X(0,1,0)x52.

  • When I include a constant I get a linear trend aka. Arima with drift or SES with drift.
  • When I do not include a constant, I get a flat forecast which is reasonable for the series at hand.

How can I objectively determine, if we need to include a constant term or not ?

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  • $\begingroup$ Logically speaking, the existence of a drift in a first-differences specification, means that there is a "core" increase in this difference as time passes. In other words the series, when in levels, grows "exponentially", it should look like a convex function (with fluctuations of course). $\endgroup$ – Alecos Papadopoulos Jul 27 '15 at 15:52
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    $\begingroup$ @AlecosPapadopoulos, a drift in first-differences implies a linear trend in levels, not exponential growth. By drift I mean a constant term in the model for the first differences. Or am I wrong? Or is this an issue of terminology? ||| forecaster, are you willing to use subject-matter knowledge? The existence of a trend can be intuitive or counter-intuitive from a subject matter perspective. If subject-matter consideration are out of question, what about comparing the AIC or BIC values of a model with drift versus a model without drift? Would that make sense? $\endgroup$ – Richard Hardy Aug 3 '15 at 17:03
  • $\begingroup$ @RichardHardy Thank you for your feedback. From these two source1 and source 2 it is clear that for d= 0 always include constant, for d=2 do not include constant. I ended up with no constant model because (d+D) =2. The fuzzy area is when I have d=1. as suggested I'll use either subject matter or IC criteria to determine if I need constant or not. $\endgroup$ – forecaster Aug 3 '15 at 18:05

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