How do you know if you have a stochastic versus a deterministic trend (non-stationary). That is how do you test for which it is?

I know you can deal with non-stationarity by differencing. Do you deal with deterministic trends by specifying a cubic or quadratic form of time?

  • $\begingroup$ Chang, C.-W., Ushio, M., & Hseih, C. (2017). Empirical Dynamic Modeling for Beginners. Ecological Research, 32(6), 785–796. Empirical dynamic modeling approaches work for deterministic time series (as well as for stochastic). $\endgroup$
    – Alexis
    Commented Feb 5, 2022 at 16:26

1 Answer 1


No cubic, quartic , quadratic et al terms should ever be used as they fail to deliver reasonable forecasts .... see some salient quotes from opionated SE members.

How to extrapolate this simple trend line into the future for the purpose of forecasting in Matlab? and here Why is my high degree polynomial regression model suddenly unfit for the data?

Deterministic time trend series that are useful are described here Auto-regression versus linear regression of x(t)-with-t for modelling time series and here Time series trend

Differencing a series is often incorrect as de-meaning might be more appropriate in order to achieve stationarity of the model's error process. Often times time trend variables are more appropriate than differencing models with one or more drift parameters.

  • $\begingroup$ Hi: I don't know how to get access but you probably would find this paper useful. onlinelibrary.wiley.com/doi/pdf/10.1111/… Unfortunately, it's very difficult to get journal of time series analysis papers without coughing up some $ !!!!. $\endgroup$
    – mlofton
    Commented Apr 28, 2019 at 21:03
  • $\begingroup$ Thanks for the heads-up on this . I will try and reach out to my many university friends who have routine access. AUTOBOX solves the conundrum/puzzle/opportunity via search and evaluate strategies that culminate in the most efficient solution for each time series. If you want to chat about the approach feel free to contact me. $\endgroup$
    – IrishStat
    Commented Apr 28, 2019 at 21:11
  • $\begingroup$ thank you very much Irishstats. Your answered floored me because these are the approaches I had read as the norm for doing this in the literature over the years (in fact I read a Springer monograph today that brought this up). Especially in ARIMA. Does the answer change if you are not primarily interested in predicting future results, but trying to determine the impact of one variable on another over time in a regression? I have not heard of demeaning before, I will have to look that up. I don't have access to a statistical software other than SAS so autobox is out for me. $\endgroup$
    – user54285
    Commented Apr 28, 2019 at 23:00
  • $\begingroup$ As I read through the links provided the question I now have, is what is the best way to make a variable stationary (as noted I have no chance of getting software like autobox and am not familiar with R). This is the first time I have heard of demeaning, I will explore that. Given the vast literature on differencing, when is it appropriate to use that ( I am no longer sure :) ). $\endgroup$
    – user54285
    Commented Apr 28, 2019 at 23:28
  • $\begingroup$ demeaning is simply employing 2 or more means suggedted by data that has level shifts (which is different fro trends.) Level shifts are simply intercept changes like 1,1,1,1,2,2,2,2 OR 1,2,3,4,10,11,12,13 $\endgroup$
    – IrishStat
    Commented Apr 29, 2019 at 0:45

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