Can causal relationships be inferred from a random effects panel model? Can a causal relationship between two variables be inferred from a random effects panel model?
I estimated the following one-way random effects (RE) panel model:
$\mathrm{Gini}_{it} = \alpha + \beta \mathrm{intraEUtrade}_{it} + \delta \Theta_{it} + \mu_i + v_{it}$, and found that the coefficient for intra-EU trade ($\beta$) is negative and significant. $\Theta$ is a vector of relevant control variables, which coefficients ($\delta$) are significant as well.
Can I conclude from these results that intra-EU trade has a causal impact in Gini, or is it impossible to conclude this from the results of a RE model? (The tests for the right model specification were conducted and showed that RE is the right model specification here)

I would say, yes, causal relationships can be inferred from random effects panel models under certain circumstances, especially random assignment of a treatment. Unfortunately, I would say no, you cannot infer causality for your model (from what we know). That does not mean the finding is meaningless, though.
I'm assuming you mean the Hausman test, regarding the specification. Personally, I'm always a little suspicious of random-effects models when I expect to see unit-level heterogeneity. This is a step in the right direction, though, because this means that your model is less likely to be mistaking within-unit variation for between-unit variation. There are still several sources of endogeneity that could affect the findings, though.
Is it possible that reverse causality is occurring? Could an increase (decrease) in inequality possibly cause a decrease (increase) in trade? Maybe, as the worst-off gain income, they buy more goods. Perhaps there is a strong theoretical argument against this. That could build your case.
Could there be a confounding factor that effects both trade and inequality? This is not my actual theory but shows what I mean: perhaps liberalization of markets and borders occurs simultaneously. Now, the poorest residents of an unequal country can leave to earn more money, elsewhere. This means the country is more equal. The liberalization of tariffs also causes more trade. Your model would report this negative relationship, even though the trade, itself, has no causal relationship with inequality.
Finally, without time (two-way) fixed-effects, there is a chance that all of this is just an artifact of general trends among countries. Trade in all countries on average increased over the panel and/or inequality in all countries on average decreased over the panel.
