Intuition behind fixed effects estimator I understand that the fixed effects estimator in a panel model (say, individuals, $i$ across years, $t$) can be understood either as a including a dummy for each $i$ or running OLS on the time demean-ed data. My question is whether the estimate from the FE model (that is, the within estimator) is equivalent to the average of the estimates from running OLS on each individual separately. Consider the following two approaches:

$y_{it} = constant + \beta x_{it} + v_i + u_{it}$

$y_t = constant + \alpha x_t + e_t$ for all $i \in (1, 2, ... N) $

The second equation gives us an $\alpha^i$ for each individual and my question is whether $\beta =  \frac{1}{N} \sum_i^N \alpha^i$
I asked exactly the same question on math.stackexchange:

https://math.stackexchange.com/questions/1470490/fixed-effects-estimation

In short, the answer is yes, it can be viewed as running separate OLS regressions- the weights, however, are not arbitrary. It is a weighted average of the separate OLS regressions.