# 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 $(p<0,01)$ 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?
• By specification, the parameter estimates in a FE model captures time series variation after controlling for time invariant unobserved heterogeneity. Results like yours basically say that the explanatory variable have no explanatory power. This should be viewed as a failure of the model. FE need not be exogenous and cannot form a basis of causal inference. Otherwise lots of, if not most, panel regressions with no significant results can be salvaged like this. – Michael Apr 22 '16 at 8:51
• @Michael, thank you for your reaction. I agree that when my expl variables come with insignificant coeffs, I should conclude that they have no explanatory power. Nevertheless, in this case, I thought that such insignificant results might just result from the specificities of my expl variables (large cross-sectional variation, small over-time variation). E.g. assume that the relationship exists, and suppose that, due to little time-variation the coefficient in the FE-model is insignificant. Would the relationship not show up in the FEs (regardless of whether or not inference is possible)? – Wecon Apr 22 '16 at 13:52

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.