Hello again! 
Hopefully for the last time in a while I have a general question. I would be very thankful if you can help me...as you always do.
Although I found multiple discussions about this problem here, I don't feel like there was a direct answer to this in previous topics. At least for me personally, there was not a clear answer.
I try to make it short: (I can post the actual test outputs if necessary, but I feel the question is not case specific)
Breusch Pagan Lagrange Multiplier test for random effects (xttest0): p-value > 10% -> Indicates Pooled OLS more appropriate than Random Effects model
Hausman Test: p-value < 5% -> Indicates Fixed Effects model more appropriate than Random Effects model
Joint F-test for fixed effects below the Fixed Effects regression table: p-value < 1% -> indicates Fixed Effects model is more appropriate than Pooled OLS
...so in this situation I am not sure if the Fixed effects model is the clear "winner", since I remember from a statistics course (not 100% sure if I remember it correctly!), that it was said: When the Breusch Pagan multiplier test couldn“t be rejected one has to run a pooled regression and ignore the panel data structure (So the BP LM test gives a statement for RE vs POLS and FE vs POLS)
Since I am not very comfortable with the proceeding to just ignore the panel structure, I searched online and found multiple contradictory statements:
E.g. Park, H.M. (2011) Practical Guides to Panel Data Modeling: A Step-by-Step Analysis Using Stata; p. 50, clearly states, that the given situation indicates the use of a Fixed effects model.
Nevertheless I also found multiple confirming statements.
Is there a clear proceeding for the choice between FE and POLS in this situation? Or is there at least a free choice between FE or Pooled OLS at this point?
Greetings and thank you in advance!

Hopefully for the last time in a while I have a general question. I would be very thankful if you can help me...as you always do.
Although I found multiple discussions about this problem here, I don't feel like there was a direct answer to this in previous topics. At least for me personally, there was not a clear answer.
I try to make it short: (I can post the actual test outputs if necessary, but I feel the question is not case specific)
Breusch Pagan Lagrange Multiplier test for random effects (xttest0): p-value > 10% -> Indicates Pooled OLS more appropriate than Random Effects model
Hausman Test: p-value < 5% -> Indicates Fixed Effects model more appropriate than Random Effects model
Joint F-test for fixed effects below the Fixed Effects regression table: p-value < 1% -> indicates Fixed Effects model is more appropriate than Pooled OLS
...so in this situation I am not sure if the Fixed effects model is the clear "winner", since I remember from a statistics course (not 100% sure if I remember it correctly!), that it was said: When the Breusch Pagan multiplier test couldn“t be rejected one has to run a pooled regression and ignore the panel data structure (So the BP LM test gives a statement for RE vs POLS and FE vs POLS)
Since I am not very comfortable with the proceeding to just ignore the panel structure, I searched online and found multiple contradictory statements:
E.g. Park, H.M. (2011) Practical Guides to Panel Data Modeling: A Step-by-Step Analysis Using Stata; p. 50, clearly states, that the given situation indicates the use of a Fixed effects model.
Nevertheless I also found multiple confirming statements.
Is there a clear proceeding for the choice between FE and POLS in this situation? Or is there at least a free choice between FE or Pooled OLS at this point?
Greetings and thank you in advance!

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