ARIMA modeling with non-stationary daily data, at what lag do I difference the series?

Question

If I have daily data set and it's a non-stationary series, then what is the lag I have to consider for the first difference 1 or 7?

Answer 1

This depends on the form of nonstationarity. If it is a because of a linear trend then you take a first difference. If it is a season of period 7 then it would be seven. However the method of detrending does depend on the type of trend. Knowing only that the series is nonstationary is not sufficient to answer the question.

Answer 2

If a series is non-stationary in the mean then there are two distinctly different ways to approach this problem. The actual data will tell you which is the best remedy. Let us for the moment assume that we have detected the presence of non-stationarity. In effect, we have a symptom and need to apply the appropriate remedy. We will consider two remedies 1) differencing the series and 2) detrending/demeaning. The first remedy is to create a surrogate series be it the differences of order k (1 or the frequency of the data). We will take enough, but not too many differences in order to approximate stationarity. The second approach has two options a) and b), both of which might be necessary. Option a is to adjust the original series for one or more deterministic trends (unspecified input series ) e.g.(1,2,3,4,....t ; 0,0,0,0,1,2,3,..t ; 0,0,0,0,0,0,0,0,1,2,...t) as needed. The second option b) is to adjust for a step/level shift in the mean by adjusting for unspecified input series e.g. 0,0,0,0,0,1,1,1,1,1,...1 ; By actually trying/evaluating these two alternative remedies one can directly deduce the "best remedy" for any particular data set. Box and Jenkins by ignoring the second of the two options and only focusing on the first (differencing) made a tactical mistake. Later researchers like I.Chang, Bill Bell, G. Tiao , D. Downing to name a few, introduced the second remedy as a possible alternative to differencing.