Q:

Why is the coefficient in this Washington Post fixed effects regression output considered significant?

I'm trying to understand the multiple regression fixed effects model the Washington Post used for a story.
See the outputs here:

What is confusing me is the first predictor: midwhitedeathrate.
The Post says in their story that the median white death rate is the most predictive variable for Donald Trump support.
But I'm having a hard time with this. The coefficient is basically zero, with most of the model explained by the employment population ratio.
Am I missing something? My understanding is the absolute value of the coefficient should be used to interpret the importance of a predictor. Maybe the t score should be used, as that normalizes the SE?
This is the author's Twitter feed where he has posted similar regression output tables.

A:

It's important to note what the author actually says in the article:

Even after controlling for these other factors, the middle-aged white
  death rate in a county was still a significant predictor of the share
  of votes that went to Trump.

In other words, the article does not say that "the median white death rate is the most predictive variable." 
The confusion lies in the fact that the author introduces the piece and the regression with his finding of a pairwise correlation between Trump votes and the median white death rate. After controlling for the other factors in the model, the median white death rate is still significant -- that's all he's saying.
Another important point of confusion concerns the interpretation of the coefficients. The author does not state that he used standardized predictors. Given that, the coefficients are in no way a measure of variable importance in the regression since they are expressed as the change in Y given a unit change in X. In other words, a small coefficient for death rate is appropriate for its unit of measure.
There are many approaches to unpacking relative variable importance in a regression. Ulrike Groemping has written papers and an R module (RELAIMPO) that reviews this literature in considerable detail...
https://www.jstatsoft.org/article/view/v017i01
A readily developed heuristic for relative variable importance that avoids the computationally intensive methods Groemping recommends is simply to take the absolute value of the parameter's t-statistic. Since the t-value is derived by dividing the coefficient by its standard error, it is a useful proxy for importance. And having tested it against Groemping's more rigorous metric, I can testify that they are strongly associated. 
Using the absolute values of the t-statistic, the most important predictor in relative terms is bachelorspercent. Median white death rate comes in fourth.

