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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.

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    $\begingroup$ I think clarifying what kind of standard error we are talking about here would be important. $\endgroup$
    – Papayapap
    Jun 11 at 13:15
  • $\begingroup$ I mean, standard error of variables when we run panel data regression. Thank yoiu so much $\endgroup$
    – Louise
    Jun 12 at 5:40
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    $\begingroup$ 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 $\endgroup$
    – ChinG
    Jun 12 at 12:27
  • $\begingroup$ @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? $\endgroup$
    – Louise
    Jun 14 at 0:48
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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:

$$SE_{gdp}=\frac{\sigma_{gdp}}{\sqrt{T}} << \frac{\sigma}{\sqrt{T}} = SE_{i}$$

That's my argument for why (generally) the standard errors at the national level should not be larger than at the firm level. I'm 99% sure you can cook up examples that flip this result with the right covariance structure. Nevertheless, it does not have to be the case that standard errors of country level variables are larger than firm level ones, and in the simplest case, the opposite is true.

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