<p><span class="math-container">$$y_{it} =\beta_0 +\beta_1 x_{1it} + ...+\beta_k x_{k_it}+\alpha_i +u_{it} $$</span>
The random effects assumption is that <span class="math-container">$E[\alpha_i +u_{it}|X]=0$</span> where <span class="math-container">$X$</span> denotes all independent variables at all time periods.</p>
<p>If this assumption is true, then pooled OLS, RE, and FE are all consistent and it would not be surprising to get similar results from all 3.</p>
<p>EDIT: If the assumption fails and <span class="math-container">$E[\alpha_i +u_{it}|X]\ne 0$</span>, then FE and pooled OLS are biased. Thus, we would expect different coefficients from all 3 forms of estimation.</p>
<p>In this case, if one variable's coefficient does not differ across specifications, it is possibly because that variable is uncorrelated with all other independent variables (including dummy variables for individuals). In such a case, for that one variable, the coefficient would be similar across specifications.</p>
<p>Thanks to Jesper in the comments below who helped.</p>

CC BY-SA 4.0
2022-03-18T17:00:29.680Z
'+1' well spotted. However, the OP says 'I expected coefficients to differ quite significantly for the different techniques and this was true for all but one variable.' So I guess you would have to add something to your explanation like some subsets $X_1$ of variables and $X_2$ being independent but only one of them satisfying the condition $\mathbb E[\alpha_i + u_{it} \lvert X_j]$.
Jesper Hybel
CC BY-SA 4.0
2022-03-19T20:06:29.530Z
If any variable is correlated with the error, all coefficients are typically biased.
Michael Gmeiner
CC BY-SA 4.0
2022-03-19T22:35:33.863Z
True, but in the case where $X = (X_1,X_2)$ and $X_1$ is independent of $X_2$ and $X_1$ is correlated with error but $X_2$ is not the coefficients on $X_2$ are unbiased. In any case, you are explaining why coefficients do not change across estimation techniques, which is not what the OP is concerned with.
Jesper Hybel
CC BY-SA 4.0
2022-03-20T08:56:41.273Z
Well, the OP also contradicted himself/herself by first saying the one variable had different coefficients across techniques and then immediately saying the coefficients were "more or less identical". 

I really had no idea what the OP was asking or what the problem was, and tried to be as helpful as I could in response to what I think was a poorly worded question.
Michael Gmeiner
CC BY-SA 4.0
2022-03-20T10:33:53.717Z
I also think your answer is helpful hence the +1. However, I don't see any contradiction on behalf of the OP. I think the OP is quite clearly describing a case where the estimate for the coefficient of one variable is 'the same' or 'more or less identical' across specifications but the rest of the coefficients differ. The case you are explaining is the case where NO coefficients differ. I am merely trying to be helpful in pointing out how your explanation can be expanded to fit the situation the OP is describing. But nevermind ...
Jesper Hybel
CC BY-SA 4.0
2022-03-20T11:29:45.510Z
Ahhh... I understand now! I apologize if I was rude! Thank you for helping me interpret that sentence.
Michael Gmeiner
CC BY-SA 4.0
2022-03-20T11:32:27.640Z
I edited the response accordingly.
Michael Gmeiner
CC BY-SA 4.0
2022-03-20T11:58:03.020Z
No rudeness assumed, it's all good. My 'But nevermind ...' was also just triggered by the apparent inactivity of the OP.
Jesper Hybel
CC BY-SA 4.0
2022-03-18T09:19:56.547Z
2022-03-20T11:32:17.517Z
2022-03-20T11:32:17.517Z
Michael Gmeiner
<p>I have used pooled OLS, fixed effects and random effects to estimate a model and the results are as expected. I expected coefficients to differ quite significantly for the different techniques and this was true for all but one variable. This one variable has coefficients that are more or less identical in each technique and each has the same level of significance.</p>
<p>I was wondering why this could be? I know there is bias in these results and I wondered if anyone could tell me why this coefficient is consistently reported as the same.</p>
<p>Thanks in advance for any answers.</p>

<econometrics><applied-econometrics><paneldata><fixed-effects>
Pooled OLS, fixed effects and random effects yield very similar results
ejw_00
# Pooled OLS, fixed effects and random effects yield very similar results

(Asked by: username_0 on 2022-03-16)

I have used pooled OLS, fixed effects and random effects to estimate a model and the results are as expected. I expected coefficients to differ quite significantly for the different techniques and this was true for all but one variable. This one variable has coefficients that are more or less identical in each technique and each has the same level of significance.

I was wondering why this could be? I know there is bias in these results and I wondered if anyone could tell me why this coefficient is consistently reported as the same.

Thanks in advance for any answers.

**Tags:** econometrics, applied-econometrics, paneldata, fixed-effects

---

## 1 answers

### Answer by username_1 on 2022-03-18. Score: 3

$$y\_{it} =\beta\_0 +\beta\_1 x\_{1it} + ...+\beta\_k x\_{k\_it}+\alpha\_i +u\_{it} $$
The random effects assumption is that $E[\alpha\_i +u\_{it}|X]=0$ where $X$ denotes all independent variables at all time periods.

If this assumption is true, then pooled OLS, RE, and FE are all consistent and it would not be surprising to get similar results from all 3.

EDIT: If the assumption fails and $E[\alpha\_i +u\_{it}|X]\ne 0$, then FE and pooled OLS are biased. Thus, we would expect different coefficients from all 3 forms of estimation.

In this case, if one variable's coefficient does not differ across specifications, it is possibly because that variable is uncorrelated with all other independent variables (including dummy variables for individuals). In such a case, for that one variable, the coefficient would be similar across specifications.

Thanks to Jesper in the comments below who helped.

> **Comments:**
> **username_2**: '+1' well spotted. However, the OP says 'I expected coefficients to differ quite significantly for the different techniques and this was true for all but one variable.' So I guess you would have to add something to your explanation like some subsets $X_1$ of variables and $X_2$ being independent but only one of them satisfying the condition $\mathbb E[\alpha_i + u_{it} \lvert X_j]$.  
> **username_1**: If any variable is correlated with the error, all coefficients are typically biased.  
> **username_2**: True, but in the case where $X = (X_1,X_2)$ and $X_1$ is independent of $X_2$ and $X_1$ is correlated with error but $X_2$ is not the coefficients on $X_2$ are unbiased. In any case, you are explaining why coefficients do not change across estimation techniques, which is not what the OP is concerned with.  
> **username_1**: Well, the OP also contradicted himself/herself by first saying the one variable had different coefficients across techniques and then immediately saying the coefficients were "more or less identical".   I really had no idea what the OP was asking or what the problem was, and tried to be as helpful as I could in response to what I think was a poorly worded question.  
> **username_2**: I also think your answer is helpful hence the +1. However, I don't see any contradiction on behalf of the OP. I think the OP is quite clearly describing a case where the estimate for the coefficient of one variable is 'the same' or 'more or less identical' across specifications but the rest of the coefficients differ. The case you are explaining is the case where NO coefficients differ. I am merely trying to be helpful in pointing out how your explanation can be expanded to fit the situation the OP is describing. But nevermind ...  
> **username_1**: Ahhh... I understand now! I apologize if I was rude! Thank you for helping me interpret that sentence.  
> **username_1**: I edited the response accordingly.  
> **username_2**: No rudeness assumed, it's all good. My 'But nevermind ...' was also just triggered by the apparent inactivity of the OP.  

---

