# Why standard errors in country-level variables are higher than that in firm-level variables?

From this dicussion, the commentor said

Lastly, firm fixed effects may absorb more variation and likely reduced the size of their standard errors.

In practice, I also mainly see that the standard error for country-level mainly higher than that of firm-level variables. I am wondering if there is any mathematical or intuitive way to explain this phenomenon.

• I think clarifying what kind of standard error we are talking about here would be important. Jun 11 at 13:15
• I mean, standard error of variables when we run panel data regression. Thank yoiu so much Jun 12 at 5:40
• Well- these are very different models. When you include country fixed effects, you are including a country specific dummy term, which can account for a lot of variation at the country level. Consequently, the residual variance reduces, as you end up with a much better fit. Variance of the point estimates depends directly on residual variance, thus being lessened as well Jun 12 at 12:27
• @ChinG thank you so much, so, do you mean that the country-level variable being active first and the absorb a part of variation already, and then therefore, the variation in firm-level variables will be smaller as a result? Jun 14 at 0:48

Consider a global panel model where there are many firms M >> N, the number of countries. If we are expansive about what we mean by firms to include small businesses, sole-proprietorships, and freelancers, then all domestic economic aggregates are going to be the sum of the firm level aggregates (employment is the sum of domestic firm employment, national output is the sum of domestic firm employment, and so on). Aggregate variability of these statistics is going to be less than the firm level variability. In THE GRANULAR ORIGINS OF AGGREGATE FLUCTUATIONS BY XAVIER GABAIX, he shows that if firms are identical and have the output variance $$\sigma^2$$ (variance as percent deviations from the mean), then the GDP standard deviation (of percentage deviations from the mean) is $$\sigma_{gdp}=\frac{\sigma}{\sqrt{n}}$$
This going to make GDP much less variable than the average firm and approximately zero. An insight of his wonderful paper is that variability in firm size matters a great deal, and in practice: $$\sigma_{gdp}=\frac{\sigma}{\log{n}}$$ This is much, much larger, but still $$\sigma_{gdp} << \sigma$$.
Of course, you asked about the standard errors and not the standard deviations. Recall that the basic calculation of the standard error of the mean calculated from independent and identically distributed random variables is: $$SE = \frac{\sigma}{\sqrt{T}}$$, where T is the number of observations. While things get much more complicated as we allow for heteroskedasticity, serial correlation, clustering, and the like, the basic idea remains that you need more observations to shrink the standard errors. Since we established that $$\sigma_{gdp} << \sigma$$, if we observe firms and nations the same number of times (T) and the observations are IID:
$$SE_{gdp}=\frac{\sigma_{gdp}}{\sqrt{T}} << \frac{\sigma}{\sqrt{T}} = SE_{i}$$