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I know that Pooled OLS is not efficient if there is existence of the individual-specific error component (one that doesn't vary over time) because the usual standard errors are incorrect and the tests (t-,F-,Wald) based on them.

The presence of individual-specific error components will bias the standard errors downward.

My question is that is it always the case? In which cases the bias can go the other way?

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It's not always the case. It depends on the correlation between the observed explanatory variables and the individual fixed effect, and the correlation between the fixed effect and the outcome. For example, if you regress $$y_{it} = \beta S_{it} + \delta A_i + \epsilon_{it}$$ where $y$ are individual wages for person $i$ at time $t$, $S_{it}$ is an individual's education (suppose this is time varying for the argument) and $A_i$ is individual ability.

Ability is fixed over time but it is unobserved, so your estimated coefficient on education is $$\widehat{\beta} = \beta + \delta \frac{Cov(S_{it},A_i)}{Var(S_{it})}$$

and since higher ability workers have higher wages $\delta>0$. Also people with higher ability often acquire more education $Cov(S_{it},A_i)>0$. In which case your estimated coefficient is upward biased.

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