0
$\begingroup$

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?

$\endgroup$
0

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
2
  • 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.