I have panel data of countries and I am estimating a Fixed Effects model similar to
$$ y_{ct} = \underbrace{\theta_t}_{\text{Time FE}} + \underbrace{\theta_c}_{\text{Country FE}} + \beta^1x^1_{ct} + \beta^2x^2_{ct} + \varepsilon_{ct} $$
For every $t$, the variable $x^2_{ct}$ is highly correlated across countries. That is, $x^2_{ct}$ shows high correlation in the cross-sectional dimension. This means that almost all of its variation can be captured by $\theta_t$. Thus, when I fit this model, I get that $\beta^2$ is not significant at all.
However, theory suggest that $x^2_{ct}$ should have a strong causal impact on $y_{ct}$. So the question is: assuming I have controlled for all confounders with $(\theta_t, \theta_c, x^1_{ct})$, is it possible to estimate $\beta^2$ using this setup?
P.S.: for the estimation, I assume two-way clustering in $\varepsilon_{ct}$