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

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 $\varepsilon_{ct}$

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.
• Thanks Bob. When you say that $x_{ct}^2$ will have low variance you mean low variance in the country (cross-sectional) dimension, right? That still leaves room for variation along the time dimension, but my problem is that I believe this variation is already being captured by $\theta_t$ (time FE). In the end, it is a matter of multicollinearity between the dummy variables associated with $\theta_t$ and $x_{ct}^2$. I agree with you, though: neither $\theta_t$ nor $\theta_c$ should be removed because of the likely presence of unobserved confounders. Commented May 3, 2018 at 15:41