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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?

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Income can't be 173% lower - that would indicate substantial negative income, and that's not possible. But that comes from extrapolating the data and is not necessarily a problem.

What your regression results mean is that, for each additional "point" of "proportion Black" the DV declines by 1.73. But here one "point" is the whole range. So, for each 1/100th of a point (that is, one percent) increase in proportion Black, the DV declines by 0.0173.

However, the DV is logged which makes this result not have intuitive appeal. The best way, I think, to develop that appeal is to plot the results: With just one IV this is simple. The x-axis could be proportion Black (ranging from 0 to the maximum for any county) and the y-axis the predicted value of income.

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  • $\begingroup$ I showed you how to interpret the coefficient. You can't interpret it for 100% Black because that is extrapolating way beyond the data, which often leads to problems like this. (Aas you noted there are no all Black counties). That has nothing to do with the DV being on a log scale. I showed you what it means. The economic significance is up to you to interpret. $\endgroup$ – Peter Flom♦ Oct 28 '12 at 17:42
  • $\begingroup$ In addition, though, there may be (probably are) nonlinearities in the relationship. You could plot the raw data along with the regression line and a loess line to see if these nonlinearities are severe. $\endgroup$ – Peter Flom♦ Oct 28 '12 at 17:43
  • $\begingroup$ Thank you for your feedback. My question is about the economic significance, with regards to comparing the coefficient to the dependent variable data and the independent variable of interest data. Could I take the two group and compare the mean differences like I wrote above? Does the 67.8% interpretation I came up with make sense? $\endgroup$ – user1690130 Oct 28 '12 at 17:52
  • $\begingroup$ I don't think you want to take groups. You can look at the predicted value for the DV at any level of the IV. But only look at levels of the IV that actually exist. If you want a number, you could compare the predicted value of the DV at the 25%tile and 75%tile of the IV; but, really, I think a graph is the best way to interpret. $\endgroup$ – Peter Flom♦ Oct 28 '12 at 18:11
  • $\begingroup$ Well, suppose the underlying question of interest is what is the impact of having any blacks in the community relative to having zero blacks? I want to say something like the follows: The regression implies that having blacks explains x% of the average difference in per capita taxes between counties that have no blacks and those that do. Would the mean comparison that I described above that yields 67.8% indicate that? $\endgroup$ – user1690130 Oct 28 '12 at 18:22

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