# Interpreting regression coefficients and economic significance

I am having a difficult time understanding how to discern whether the regression coefficients I am getting are large or small relative to the data.

I have one regression that is cross-sectional. For example, a regression of log(tax revenue per capita + 1) on the share of the U.S. county that is black. The share of the population that is continuous, running from 0 to 1 (in theory, in practice, never that large). Suppose that the log(tax revenue per capita + 1) has a mean of 7.189 and a standard deviation on 0.469. Also, the mean of the share of the black population is 0.032 with a standard deviation of 0.048. Suppose I get a regression coefficient of -1.733 with a standard error of 0.708, such that it is clearly statistically significant at a conventional level. I understand the coefficient can be interpreted that a county with a 100% black population is expected, based on the regression, to have 173.3% lower tax revenues per capita. While that seems large, I want to compare that to the actual tax data to discern the economic significance. Do I compare it to the mean on the log(per capita tax + 1)? its standard deviation? its range? something else?

Could I perhaps say the follows? Suppose some counties have no blacks. Suppose that blacks in all other counties, on average, compose 11% of the population. Suppose that the average log(per capita tax rate + 1) in the counties with blacks is 8.29 whereas the same number in the group without blacks is 8.573. Can I say that the having a black population explains (1.733 * 0.11) / (8.57 - 8.29) = 0.19 / 0.28 = 67.8% of the difference in per capita tax revenue difference between the average county without any blacks relative to the average county with blacks?

The second situation I have is panel. Suppose I want to say the following. To what extent has the empirical change in the black population over time (that is, over the time periods) explain the change in per capita tax rate over time? Do I extract the mean of per capita taxes in the final period versus the first period, and compare that difference to the same difference in the share of the black population, multiplied by the regression coefficient?