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Using unbalanced panel-data (Unbalanced Panel: n=54, T=81-307, N=11138), I estimated both FE and FD models. My understanding that they are analogous for T>2 and if they produce different results, then it usually indicates a violation of the strict exogeneity assumption and therefore the FD is more consistent as it uses a weaker form of strict exogeneity than FE.

In R, using the package plm, the FD results show an estimate of the intercept (and in this case, it turns out to be significant). I am wondering what does the intercept mean and how to interpret its value?

Is it common/correct to force the intercept to zero? The results are a bit different when dropping the intercept but not by much.

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The FD intercept is the linear time trend coefficient in levels:

$$y_{i,t+1}-y_{i,t}=\alpha_i-\alpha_i + \beta\cdot (x_{i,t+1}-x_{i,t})+\gamma \cdot (t+1-t)+\varepsilon_{i,t+1}-\varepsilon_{i,t}$$

You can add time to your FE spec to check if this is the culprit.

Also some good ideas in this question on why $FD \ne FE$.

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