Panel with time fixed effects and a variable with high correlation across individuals

Question

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 $\varepsilon_{ct}$

Answer 1

The significance of a coefficient increases with the variance of the regressor. If $x^{2}_{ct}$ is highly correlated across countries and your data is country level, then you will have low variance. Unsurprisingly, this leads to an insignificant $\beta^2$, whether your have country FE or not. Taking out $\theta_c$ is unlikely to fix this problem.

Of course, if you can control for all relevant country characteristics, then you do not need the country FE. Nevertheless, it is very unlikely that you can.

Especially with cross-country data, regressions without country FE are suspect, because countries themselves can vary across so many dimensions that it is hard to interpret regression coefficients as being causal and it is near impossible to control for all relevant country factors. I would not recommend running a cross-country regression without country FE.