My understanding of this article and this article is that:

  1. If a time series needs to be differenced to become stationary (i.e. is difference-stationary), then it has at least one unit root

  2. Trend-stationary time series do not necessarily require differencing to become stationary (and do not necessarily have a unit root) - they require only the removal of the underlying trend.

My question is as follows: while it is not necessary to difference a trend-stationary series to make it stationary, is it sufficient? I.e. will differencing (of some order) always suffice to make a trend-stationary series stationary? EDIT: is this anything to do with deterministic vs stochastic trends?

And a follow-up question - how is the 'trend' defined in 'trend-stationary'? In seasonal_decompose in statsmodels, the trend is just the moving average, but surely this isn't always an accurate reflection of the actual trend - e.g. what if it should be an exponentially-weighted average?


1 Answer 1


My understanding is that differencing a trend-stationary series leads to your model having extremely undesirable properties.

Take the trend-stationary model:

$$y_t = \alpha + \beta_t + u_t$$

And the model at $t-1$:

$$y_{t-1} = \alpha + \beta(t-1) + u_{t-1}$$

With differencing we get:

$$\Delta y_t = \beta + u_t - u_{t-1}$$

Therefore, by differencing, we have introduced a non-invertible MA error term on the RHS. The non-invertible MA cannot be expressed as an autocorrelated process, and the series $\Delta y_t$ would have some undesirable properties.

Above, I have assumed a linear deterministic trend.

  • 1
    $\begingroup$ EB3112 makes an excellent point. You want to de-trend a trend stationary process ( by trend stationarity they mean that there's a slope and it's due to time ) and difference a difference stationary process. The difference stationary process usually means that there's a random walk involved ( so a unit root ) but the formal definition may be something slightly different. Note that It's not always so easy to discern whether a series is trend or difference stationary. ( that's why nelson and plosser's, 1982 paper, got so much hype ). There's definitely literature on the net about doing that. $\endgroup$
    – mlofton
    Aug 26, 2022 at 20:45
  • $\begingroup$ That’s a really interesting answer, thanks. So can we roughly say that in a trend-stationary model we have a (linear or non linear) deterministic trend, whereas in a difference-stationary model we have a stochastic trend (I.e. a unit root)? And if we have both features, we should de-trend before differencing (so run our ARIMA model or whatever on the de-trended series? If so, then how should we find this deterministic trend - is just fitting a bunch of polynomials until we find the best one (eg best AIC) an alright approach? $\endgroup$
    – tobmo
    Aug 27, 2022 at 16:01

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