I have a chicken-egg problem in time series:

How do we remove the deterministic trend in time series when you have a guess that the noise is non-normal? How do we check if the noise is non-normal without removing the deterministic trend?

For my time-series, when I plot the histogram of the raw values and fit a density a log-normal seems to fit very well. However, this is raw data, and it has a trend, and potential seasonality. If I remove the trend and seasonality, would the resulting density still remain a log-normal? How would I remove the trend and seasonality if I don't know the noise distribution? If my noise is indeed log-normal, then I don't think I can use an OLS regression, and have to use a GLM (which seems not to handle log-normal, somehow, but that's a separate question). Any thoughts on this, please?

I hope I have made my question clear, but please comment if you need more information.


1 Answer 1


If a series $\{ x_t\}_{t=1}^\infty$ has a linear deterministic trend, it can be written $x_t=y_t+k t$ . An estimator for $k$ is $k=\frac{X_T - X_t}{T-t}$. To remove the trend from data, you take $y_t=x_t-kt$.

  • $\begingroup$ You may need a further explanation to show that OP's concern about chicken and egg is not a problem here. $\endgroup$
    – Aksakal
    Feb 9, 2015 at 21:04
  • $\begingroup$ Thanks for your comments. In line with @Aksakal, even if you assume the trend is linear, you need an assumption on the noise to show that $k$ or an averaged or OLS version of $k$ is the right estimate. For instance, if your distribution looks lognormal, then we may prefer X_T/X_t as opposed to a subtraction (not claiming this is how it should be done, but just an example). $\endgroup$ Feb 10, 2015 at 20:53

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.