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Robust standard error for a time series mean.

Andreas Klein 
Hello.

I have a non-stationary time series of returns and I would like to calculate the standard error for the mean of that series.

When I use the White-estimator or assume constant variance I got the same results. When I use the Newey-West-estimator, which also cares about autocorrelation, the standard errors increase a lot.

1. What is the right estimator?
2. Is the Newey West estimator strongly affected by the non-stationarity?


Does anyone have any literature source or a link to a paper or something like that? 


I hope someone can help me.


Thank you in advance.


Sincerely,
Andy.