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

• Shouldn't there be a $x_{i,t}$ somewhere in the second equation? How does the second equation give you an $\alpha^i$? It doesn't seem to contain a value of $\alpha^i$ at all.
– BKay
Mar 4, 2016 at 15:34
• As for BKay's concern, I would suggest rephrasing the second equation to $y_{it} = constant_i + \alpha^i x_{it} + e_{it}$. Oct 27, 2016 at 6:38

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.

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.

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.)

• +1. The form of $w_i$ also suggests that the FE estimator gives more weight to individuals/groups that have larger variation in $X$.
– Paul
Oct 27, 2016 at 17:14