Intuition behind fixed effects estimator

Question

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 $\beta = \frac{1}{N} \sum_i^N \alpha^i$

Answer 1

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.

Answer 2

I believe the second approach you suggest is equivalent to using an OLS on the whole sample and with dummy variables for each person, country or whatever. Therefore, the answer to your question should be yes.

Answer 3

Let $\hat\alpha^i = \left[\sum_t (x_{it}-\bar{x}_i)^2 \right]^{-1} \sum_t (x_{it}-\bar{x}_i) (y_{it}-\bar{y}_i)$, estimator from the individual OLS regression. Let $\hat\beta$ be the FE estimator using the panel data. Then, math gives the identity $$ \hat\beta = \sum_{i=1}^N w_i \hat\alpha^i,\quad w_i = \frac{\sum_t (x_{it}-\bar{x}_i)^2}{\sum_{j=1}^N \sum_t (x_{jt}-\bar{x}_j)^2}. $$ This confirms ChinG's answer. (Note that the mean-group estimator is $N^{-1} \sum_{i=1}^N \hat\alpha^i$, which is different from the FE estimator.)