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I'm learning the basic of panel data regression and there is something that i'm not sure about. There seems to be 2 different approachs to estimate FE models:
(1) Assuming that the slope is constant, and estimate the equation: $$Y_{it} = \alpha_i + \beta X_{it} + e_{it} $$ Which is equal to using the normal OLS to each sample $i$ individually.

(2) By applying the Fixed Effects Transformation to estimate: $$( Y_{it} - \overline{Y_i}) = \beta ( X_{it} - \overline{X_i}) + (\epsilon_{it} - \overline{\epsilon_i})$$

Is the result of the two estimation methods (significantly) different? What are the pros and cons of each method and how do you choose one or another? I've noticed in STATA the command xtreg,fe uses the latter method(2), then how is the intercept estimated by this command and is it equal the mean of $\alpha_i$ estimated by method(1)?

English is not my native language and I'm fairly new to this site so I'm well open for advices. Thank you.

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So the difference between pooled OLS and FE can be very large if time-invariant heterogeneity between groups relates to both your dependent and independent variable.

FE removes the time-invariant variation for each group, and hence controls for any OVB that doesn't change over time. Pooled OLS assumes that such OVB is uncorrelated with X and thus if that assumption fails, pooled OLS leads to inconsistent estimation.

The FE method is equivalent to including a dummy variable for each group. Thus each person has their own intercept. In Stata you can replicate the xtreg,fe command using reg by generating dummies for each person and including them in the regression.

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  • $\begingroup$ Thanks for your sharing. But then how is the _cons coefficent estimated in xtreg,fe if all the time-variant are eliminated? And what does this number mean? $\endgroup$
    – Hiếu Phẩy
    Sep 11, 2016 at 19:25
  • $\begingroup$ The constant is the average value of the fixed effects (that's why I recommended the dummy interpretation) :stata.com/support/faqs/statistics/… $\endgroup$
    – VCG
    Sep 11, 2016 at 19:57
  • $\begingroup$ Oh I've read that article. But in the demonstration of that article, the average value of the fixed effects estimated by the dummy method 4 + (-2.5 + 4.3333 + 10.3333)/3 is bigger than the number 7.545455 of xtreg,fe. So I wondered if there is any special computation method to the xtreg,fe command. $\endgroup$
    – Hiếu Phẩy
    Sep 11, 2016 at 20:18
  • $\begingroup$ EDIT: I think it should be 4 + (-2.5 + 4.3333 + 10.3333)/4 because there are 4 groups but the result is still slightly smaller than 7.545455 $\endgroup$
    – Hiếu Phẩy
    Sep 11, 2016 at 20:26
  • $\begingroup$ It's an unbalanced panel which makes computing it harder. I see your point and am calculating it myself. $\endgroup$
    – VCG
    Sep 11, 2016 at 20:26

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