Does Normality of a Time Series imply Stationarity and Viceversa?

Question

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!

Answer 1

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.

2. 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!).