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I have a (probably simple) question on fixed effects estimation. I am trying to do baseline growth regressions of log GDP per capita against a number of covariates and, in line with the literature, I include both country and time dummies. The panel is unbalanced with 19 countries, 10-53 periods, resulting in 763 useful observations.

In R this is can be done with

plm(lgdp ~ lpop + lxr + lman + lman2 + lSB  + lSI, data = df, 
    model = "within", effect = "twoways")

or

lm(lgdp ~ lpop + lxr +lman + lman2 +lSB +lSI + factor(ISO) + 
          factor(year) - 1, data = df)

Function plm balance the R^2 for the inclusion of dummies, while lm clearly does not and result overfitted.

The problem is that I obtain (Adjusted R^2, AIC, BIC respectively):

  • 0.32, 1839.5, 1876.6 for the pooled ols
  • 0.67, 473.1, 593.7 with country fixed effects
  • 0.35, 1524, 1802.2 with time fixed effects
  • 0.065, 142.6, 504.3 with both country and time fixed effects. the R^2 is obtained from plm, while the AIC and BIC are computed on the lm.

The specification with two effects is clearly the best, but why does the adjusted R^2 fall so steeply?

I thought this could be due to multicollinearity, but a VIF analysis does not show particular problems concerning the country and time factors. Could it be due to the fact that 763/(6+19+54) = 9.8 observations per parameter on average?

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    $\begingroup$ What happens to $R^2$? Adjusted $R^2$ is $R^2$ corrected for the number of parameters. I presume you have a lot of additional parameters. $\endgroup$
    – Jeremy Miles
    Commented Jan 26 at 18:12
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    $\begingroup$ Do you mean that something fails, mathematically (the calculation doesn't tell you what it's supposed to tell you), or do you mean that you don't find the calculated value to be acceptable? $//$ I thought this could be due to multicollinearity What would that have to do with $R^2$, adjusted or not? $\endgroup$
    – Dave
    Commented Jan 26 at 18:13

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As @JeremyMiles 's comment implies, the answer to your last question is "yes". The adjustment is for additional terms in the model. You have a lot of terms!

I also wonder why you have year as a factor. That adds a lot of terms. You say this is "in line with the literature" but time is an interval level variable. If you are worried about nonlinearity, you could look at a spline.

Also, since you say you have 10 to 53 levels for period (which I assume is the same as "year" then you probably have some years that have only a few countries. That's another source of overfitting (and another reason not to treat year as a factor). And I think that affects the adjustment here as well, so the usual formula for adjusted $R^2$ is not right (I'm not sure about this last).

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