Solved – Time fixed effects

Based on the links in the comments the first test is the test between the within model and the within model with time dummies, i.e.

$$y_{it}=x_{it}beta+c_i+u_{it}$$

vs

$$y_{it}=x_{it}beta+c_i+u_{it} +lambda_{t},$$

where both of the models are estimated with within estimator, i.e. by time-demeaning and then OLS. The test used is Langrage multipliers test and it tests the hypothesis $H_0:lambda_t=0$, $t=1,…,T$.

The second test is between two-way within model and the pooled model, i.e.

$$y_{it}=x_{it}beta+c_i+u_{it} +lambda_{t}$$

vs

$$y_{it}=x_{it}beta+u_{it}$$

where now for the first model we use two-way within model (subtract time and cross-section means and add total mean and then OLS) and for the second the usual OLS. The test used is F-test and it tests the hypothesis $H_0:c_i=0,lambda_t=0$, $i=1,…,N$, $t=1,…,T$.

When written clearly it is evident that the null hypotheses are different, hence it is no surprise that the results are different too. Furthermore the second test assumes normality of the regression errors and is not valid under any asymptotics, since either for $Ntoinfty$ or $Tto infty$ the number of estimated coefficients is infinite. The first test would be valid under $Ntoinfty$, which is a norm for such type of panel data models.