Pooled OLS, fixed effects and random effects yield very similar results

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.

$$y_{it} =\beta_0 +\beta_1 x_{1it} + ...+\beta_k x_{k_it}+\alpha_i +u_{it}$$ 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.

• '+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 $\mathbb E[\alpha_i + u_{it} \lvert X_j]$. Mar 18, 2022 at 17:00
• If any variable is correlated with the error, all coefficients are typically biased. Mar 19, 2022 at 20:06
• 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. Mar 19, 2022 at 22:35
• 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. Mar 20, 2022 at 8:56
• 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 ... Mar 20, 2022 at 10:33