# Does Normality of a Time Series imply Stationarity and Viceversa?

I have a theory question which never became completely clear to me. Reading Hamilton (1995) I understod that the stationarity requirement for time series data stands as the normality requirement for non time series (cross sectional data) before running an OLS because of the Central Limit Theorem. I have the following questions:

1. Does normality imply stationarity and viceversa (I don't think so, as I happen to notice) and if not why?
2. When a regression is run on a time series it is however checked that residuals are normal. Why not checking that they are stationary as well?
3. If the single time series of a regression are non stationary, but the residuals are, then we should run an ECM. What about the normality assumption?

Thank you!

• Normality is not a prerequisite but rather a result of the central limit theorem (CLT), i.e. the other way around from what you seem to be implying. CLT significantly reduces the importance of the normality assumption in many models. – Richard Hardy Dec 21 '20 at 11:14

1. You can see that normality does not imply stationarity by construction. A TS which starts as N(0,1) and transitions into N(0,2) is, on the whole, still normal as the sum of normals is normal. But clearly this violates the stationarity assumption of no heteroskedasticity. For the converse, again you can see by construction that this doesn't hold. Take N(0,1), transform it in any way (e.g. using the sinh-arcsinh transformation) to increase/decrease the kurtosis. Independently sample from this distribution, this will clearly be stationary but by construction isn't Normal.

EDIT: As Richard kindly pointed out, the first counter-example doesn't actually make sense. I think a counter-example that does work is sampling from a normal distribution and then ordering it. You then have a Normally distributed TS by construction but it is either monotonically increasing/decreasing which isn't stationary.

1. When running regressions you should check that you have stationary residuals. Autocorrelation and such problems means your model is not well calibrated to the problem you're trying to solve. If you're residuals suffer from heteroskedasticity you may have a regime change which your model isn't flexible enough to account for, etc.

I've not really done much work with cointegration so I'll let someone else answer that, hope that's somewhat helpful (hope it's correct as well!).

• as the sum of normals is normal: why should the sum of elements of a TS be relevant here? – Richard Hardy Dec 21 '20 at 9:46
• Hey Richard! I'm not suggesting the sum of the elements of the TS but the sum of the (independent) distributions those elements are sampled from. It's just to construct a counter-example, not suggesting anything that should be done in practice. – sports-modelling Dec 22 '20 at 12:57
• Could you explain this is more detail, step by step and ideally with some formulas? I might be misunderstanding what you are doing. – Richard Hardy Dec 22 '20 at 13:05
• Sorry yes you're right, just thought it through some more and the construction I've offered doesn't make sense, guess I meant the set of the combined samples but that then isn't Normal. Could instead construct it by sampling from a Normal distribution, then ordering the sample. This will be normal but clearly isn't stationary. – sports-modelling Dec 22 '20 at 13:28
• I see. I do you get your logic yet, but at least you got some of mine :) Consider editing the answer to reflect these considerations. – Richard Hardy Dec 22 '20 at 13:55