Why do random effect models require the effects to be uncorrelated with the input variables, while fixed effect models allow correlation?

From Wikipedia

There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a random effects model) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If the random effects assumption holds, the random effects model is more efficient than the fixed effects model. However, if this assumption does not hold (i.e., if the Durbin–Watson test fails), the random effects model is not consistent.

I was wondering why random effect models require the random effects to be uncorrelated with the input variables, while fixed effect models allow the effects to be correlated with the input variable?

Thanks!

When you include any variable in a regression, its coefficient is estimated holding all other variables in the model fixed. If the variable is correlated with another variable that isn't included in your model, its coefficient cannot be estimated holding that omitted variable constant. This leads to omitted variables bias.

The fixed effects approach adds variables to the model representing the individuals or groups of interest. As a result, the other coefficients in the model can be calculated holding the individual or group fixed. This is known as the within (individual or group) estimator.

The random effects approach does not add variables to the model representing the individuals or groups. Instead, it models the correlations structure of the error terms. Essentially, the random effect is seen as an unestimated parallel shift in the regression line and this same shift applies to all observations for a particular individual or group. This leads all these within individual or group observations to be correlated. Random effects models this correlation.

The random effect model basically omits the fixed effect and overcomes the omission by modeling the error structure. This is fine so long as the omitted fixed effect isn't correlated with any included variables. As discussed above, such omitted variables lead to biased coefficient estimates.

The benefit to excluding the fixed effects, as the random effects procedure does, is that variables that do not vary within an individual or group's observations cannot be included with fixed effects due to multicolinearity; random effects are the only way to estimate the coefficients for such variables.

• What do you think of modeling the estimated fixed effect as function of the time-invariant variables? – Dimitriy V. Masterov Jan 17 '14 at 16:03
• @Tim Might you consider accepting this answer and mark this thread as answered? – Charlotte R Sep 20 '16 at 15:17
• No, random effects models definitely include variables representing the individuals or groups. They're just subject to shrinkage. See for example this answer: stats.stackexchange.com/a/111896/11646 – Paul Feb 27 '18 at 16:29

From what I know, the random effects is a sort of extension of an OLS model, in which the constant is included in the vector of regressors, and the error is composed by both an unobserved effect (time invariant), and an observed error (time variant).

I don't know very well how to answer your question, but I would simply say that RE models require the error to be uncorrelated with the independent variables because, if they are correlated, it means that you are in case in which FE estimates are more appropriate. You can test which of them interprets better your dataset by performing a Hausman test once you have run the regression with both specifications.

This is from Econometric Analysis of Cross Section and Panel Data, by Wooldridge: (...) 