FE Regression Interpretation

[Sally Silk]

Dear Community,

I am currently running a fixed-effect regression with time and country-fixed effects. I had already posted about this regression some time ago, but I have a few more questions and would appreciate any help! Just for some background my outcome variable of interest is GDP growth, with my main covariates of interest being economic policy uncertainty and democracy/polyarchy (ranges from 0.000 to 1.000).

Here is my output

xtreg gdpg epu polyarchy polyarchy2 polyepu eys mys trade gcf mort infl leab i.year, fe

Fixed-effects (within) regression Number of obs = 504
Group variable: country Number of groups = 24

R-squared: Obs per group:
Within = 0.6513 min = 21
Between = 0.0901 avg = 21.0
Overall = 0.2108 max = 21

F(31,449) = 27.05
corr(u_i, Xb) = -0.8331 Prob > F = 0.0000

------------------------------------------------------------------------------
gdpg | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
epu | -.026194 .0059042 -4.44 0.000 -.0377972 -.0145907
polyarchy | 47.10087 13.77964 3.42 0.001 20.02027 74.18147
polyarchy2 | -36.38016 9.615551 -3.78 0.000 -55.27723 -17.48309
polyepu | .0303808 .0073758 4.12 0.000 .0158855 .0448761
eys | -.2563044 .164398 -1.56 0.120 -.5793896 .0667807
mys | -1.684913 .3283187 -5.13 0.000 -2.330145 -1.039681
trade | .0418149 .0071237 5.87 0.000 .0278151 .0558148
gcf | .1956006 .0315612 6.20 0.000 .1335746 .2576266
mort | -.0979374 .0458212 -2.14 0.033 -.187988 -.0078869
infl | -.2217516 .0573018 -3.87 0.000 -.3343646 -.1091385
leab | .0571427 .1547104 0.37 0.712 -.2469037 .3611891
|
year |
2001 | -2.1258 .5398174 -3.94 0.000 -3.186682 -1.064917
2002 | -1.777253 .5485515 -3.24 0.001 -2.855301 -.6992063
2003 | -1.374254 .5614857 -2.45 0.015 -2.47772 -.270788
2004 | -.0329219 .5844018 -0.06 0.955 -1.181424 1.11558
2005 | -.8107316 .6042395 -1.34 0.180 -1.99822 .376757
2006 | -.230025 .6345811 -0.36 0.717 -1.477143 1.017093
2007 | -.4702987 .6545978 -0.72 0.473 -1.756755 .8161572
2008 | -2.868747 .6801661 -4.22 0.000 -4.205451 -1.532043
2009 | -6.338044 .7005902 -9.05 0.000 -7.714887 -4.961201
2010 | -.3762288 .7260767 -0.52 0.605 -1.803159 1.050702
2011 | -1.41363 .7656697 -1.85 0.066 -2.918371 .0911115
2012 | -2.461866 .786467 -3.13 0.002 -4.007479 -.9162523
2013 | -2.027235 .8212383 -2.47 0.014 -3.641183 -.4132871
2014 | -1.025382 .8505386 -1.21 0.229 -2.696912 .6461492
2015 | -.4443696 .8643188 -0.51 0.607 -2.142982 1.254243
2016 | -1.216199 .8902149 -1.37 0.173 -2.965704 .5333063
2017 | -.3554842 .912558 -0.39 0.697 -2.148899 1.437931
2018 | -.6058677 .9358398 -0.65 0.518 -2.445038 1.233302
2019 | -1.353752 .9688951 -1.40 0.163 -3.257884 .5503803
2020 | -7.227344 .9229527 -7.83 0.000 -9.041187 -5.4135
|
_cons | 2.170436 13.9184 0.16 0.876 -25.18286 29.52373
-------------+----------------------------------------------------------------
sigma_u | 4.3451978
sigma_e | 1.8559754
rho | .84570729 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(23, 449) = 6.23 Prob > F = 0.0000

My questions are as follows:
1. I have included the square of democracy, as well as the regular values of democracy.
The non-squared values of democracy have a positive coefficient and hence my interpretation is as follows: a one-unit increase in the level of democracy leads to an increase in GDP growth by 47.10 percentage units.
Whereas, the square of democracy is negative, suggesting that there is a turning point where greater democracy does not benefit GDP growth.

When I say a one-unit increase in democracy leads to an increase in GDP growth by 47.10, does that mean going from 0.000 to 1.000 will lead to an increase by 47.10 percentage units in GDP growth?

Is this correct reasoning about the meaning of the negative coefficient on the squared term of democracy? And if so, is there a way for me to calculate the value at which democracy reaches is maximum effect on GDP growth?

Thank you very much!

Sally Silk

[JJ Kovach]

1. From your results, a one unit increase in polyarchy is associated with a 47 unit increase in gdpg. Since polyarchy is between 0-1, it probably makes more sense to think of a 0.1 unit increase in polyarchy is associated with a 4.7 unit increase in gdpg. Or perhaps something even smaller depending on the average within-country variation of polyarchy in your sample.
2. Take the derivative of your regression equation and set to 0 to find the maximum value. In this case, when polyarchy is 47/36=1.3, which is outside the range of the variable. So, the association is effectively increasing at a decreasing rate from 0 to 1.

As an fyi, with multiple fixed effects, -reghdfe is often preferred as it will run faster and drops singletons which -xtreg preserves that could bias the results.

[Jeff Wooldridge]

The turning point is 47/(2*36), or about 0.65. So it may be within the range of polyarchy.

You can't ignore the quadratic when obtaining the partial effect. Use c.polyarchy and c.polyarchy#c.polarchy and then use the margins command at various values of polyarchy.

[JJ Kovach]

Yep, I forgot to bring down the 2 during the derivative.

[Sally Silk]

Thank you for all the clarifications and suggestions. I really appreciate them!

Sally