Robust standard errors for panel data vs robust estimation for panel data It seems to me that rather than using non robust
estimation methods with  robust standard errors it would be better to use
robust estimation from the outset. I wonder what other people think.
 A: They're robust with respect to different things.
If you use robust regression to obtain an estimate of fixed effect in panel data, then you're computing an estimate that's resistant to outliers.
If you use robust standard errors for your OLS estimator, it's because you suspect that the assumption behind your error model is violated.  For example, in panel data, your errors may be autocorrelated and not iid, and your robust standard error offers protection against such phenomena.
But, robust standard errors don't guard against outliers, and robust regression doesn't necessarily account for autocorrelation.  Though I think there are techniques to provide robust standard errors for robust estimators, you would only use these when both conditions are true: your data have outliers and your error model assumption is violated.  
A: I don't know if I understand you correctly, but still I will give it a shot.
I think robust estimation from the outset would be better in most of the cases. 
Reason: 
If you estimation is not robust, outliers might severely affect your estimate. Still, in general, you will be far the true value. This may be also looked as the confidence interval not containing the true value.
In picture form: (This can happen, outliers cause the estimate to be very far from truth). If the estimate was found by robust methods, it would have been closer to the True Value.
----------||------------------------[------------------Not Robust Est---------------------]-
........True Value..................<--------CI found by Robust Method-------------->
