Interpreting correlation between fixed effect and explanatory variable I'm dealing with panel data and I chose to do a regression with cross-sectional fixed effects. Some of my explanatory variables (such as $debt_{it}$) do not show very much variation over the time dimension. In fact, the most interesting variation in $debt_{it}$ (but also in some other variables), is probably the cross-sectional variation (i.e. one country having a higher debt over the entire period under investigation) rather than the variation of debt over time within each country.

Therefore, I believe that demeaning the data by using cross-section fixed effects may elimate a lot of interesting information. The $H_{0}$ of redundant fixed effects is strongly rejected, and the Hausman test strongly rejects the consistency of the random effects estimator. Doing a pooled estimation, rather than F.E. estimation, more than doubles the estimated coefficient to debt.

In an attempt to 'retrieve' the valuable cross-sectional information, I plotted each country's fixed effect against each country's average level of debt over the time period. There was a significant correlation $(p&lt;0,01)$ between the averaged debt and the fixed effect. Intuitively, it seems to me that I can conclude that (even though the coefficient to $debt_{it}$ is possibly insignificant, depending on the specification, I'm still working on this) a higher debt does result in higher interest rate spreads (which is the dependent variable).

These are my questions: 


Is my intuition correct? 
If it is not, which I assume to be the case since I have never come across a similar reasoning, where exactly does my reasoning fail? Can you provide some intuition or an accessible reference dealing with the issue? 
Are there any estimators that are better equipped to use cross-sectional variation in the data than the within-estimator?
You have done two different things.

Your fixed-effects model captures the within-group over-time functional relationship between $debt_{it}$ and $y_{it}$ (that is, how much average difference in $y_{it}$ is there between two periods with a 1-unit difference in $debt_{it}$ within a country). In your data, there is limited within-group variability in $debt_{it}$, which probably lead to a large standard error and a resulting insignificance.

Your reasoning about the correlation between the fixed effects and the explanatory variable is about the cross-sectional functional relationship. You found that countries with higher $debt_{it}$ have higher $y_{it}$ values (as opposed to periods with higher $debt_{it}$ having higher $y_{it}$ within a country). This is not what you originally intended to investigate when you set up your fixed-effects model.

Your Hausman test says that the within-group functional relationship and the cross-sectional functional relationship are different.

That said, yes, it is true that "a higher debt does result in higher interest rate spreads", but it is true only in the specific sense that "countries with higher debts show higher interest rate spreads". You should be careful when saying "result in".

Overcoming the issue of limited over-time variation and having significant results is hard. There is no magic. Abandon the fixed effects model, and try to control for many time-varying and time-invariant regressors, enough for you to argue that you controlled for most country-specific factors. (You can use RE or POLS estimation.) People might still criticize that you didn't control for enough factors, but you will need to defend yourself somehow. You could also move over to dynamic models or others, but that's a different story.