<p>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.)</p>

CC BY-SA 3.0
2016-10-27T17:14:09.733Z
+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
CC BY-SA 3.0
2016-10-27T06:35:52.473Z
2016-10-27T06:35:52.473Z
chan1142
<p>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.</p>

CC BY-SA 3.0
2016-03-04T08:15:29.873Z
2016-03-04T08:15:29.873Z
Fuca26
<p>I asked exactly the same question on math.stackexchange:</p>

<p><a href="https://math.stackexchange.com/questions/1470490/fixed-effects-estimation">https://math.stackexchange.com/questions/1470490/fixed-effects-estimation</a></p>

<p>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.</p>

CC BY-SA 3.0
2016-03-04T16:55:04.707Z
2016-03-04T16:55:04.707Z
2017-04-13T12:19:37.213Z
ChinG
<p>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:</p>

<p>$y_{it} = constant + \beta x_{it} + v_i + u_{it}$</p>

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

<p>The second equation gives us an $\alpha^i$ for each individual and my question is whether $\beta =  \frac{1}{N} \sum_i^N \alpha^i$</p>

CC BY-SA 3.0
2016-03-04T15:34:19.773Z
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
CC BY-SA 3.0
2016-10-27T06:38:43.130Z
As for BKay's concern, I would suggest rephrasing the second equation to $y_{it} = constant_i + \alpha^i x_{it} + e_{it}$.
chan1142
<econometrics><regression><fixed-effects>
Intuition behind fixed effects estimator
user26750
# Intuition behind fixed effects estimator

(Asked by: username_0 on 2016-03-04)

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

**Tags:** econometrics, regression, fixed-effects

> **Comments:**
> **username_1**: 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.  
> **username_2**: As for username_1's concern, I would suggest rephrasing the second equation to $y_{it} = constant_i + \alpha^i x_{it} + e_{it}$.  

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## 3 answers

### Answer by username_5 on 2016-03-04. Score: 3 | Accepted answer

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.


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### Answer by username_2 on 2016-10-27. Score: 1

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 username_5'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.)

> **Comments:**
> **username_3**: +1. The form of $w_i$ also suggests that the FE estimator gives more weight to individuals/groups that have larger variation in $X$.  

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### Answer by username_4 on 2016-03-04. Score: 1

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.


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