I have a question regarding the test of time fixed effects. In the scientific literature there are two ways to test for time fixed effects.

The first possibility is to test for time fixed effects by running a pFtest on the basis of fixed time effects and fixed effects.

-> pFtest(fixed.time, fixed)

The alternative is to run a pFtest on the fixed effects model and the pooling model.

->pFtest(fixed.time, pooling)

The outcomes are totally different as one rejects the null whereas the other does not reject the null. However, I have no idea which test I have to use. My data has N=15 and T=10.

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#### Best Answer

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.

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