 Econometric texts and panel data analysis typically focus on two different approaches for modeling the unobserved effect. One is the random effects model where the unobserved effect or the effect that we use for dealing with unobserved originating is put into the random part of the model. Then we have the fixed effects approach where the unobserved effect is modeled as in the fixed part of the model. But there is also a third approach called the correlated random effects approach that in a way unifies these two approaches. So typically we have the question of which technique of these two we model and we apply and the random effects and the fixed effects approach both have strengths and weaknesses. The fixed effects approach has the weakness that it cannot include variables that are constant within clusters. So it only models within effects but you can't model between effects or contextual effects using these approach because it eliminates all between cluster variation. As another disadvantage it is inefficient when the random effects assumption holds. So the idea here was that when the random effects assumption holds then the between effect and the within effect are the same and we get more precise estimate of the within effect by using the between effect to help in estimation. In fixed effects approach you eliminate any between effects and therefore you lose efficiency compared to the GLS random effects approach. As an advantage these fixed effects approach is consistent even when the random effects assumption does not hold. Let's go on to the random effects model. So the GLS random effects approach can include variables as an advantage variables that vary only between clusters. So you can model between effects and you can model contextual effects. This is efficient compared to the GLS fixed effects when the random effects assumption holds but it's inconsistent and biased if the random effects assumption doesn't hold. In practice we quite often use the Hausmann test to compare these two modeling approaches and then pick the random effects approach if the Hausmann test is not rejected. If the Hausmann test rejects and all hypothesis then we pick the GLS fixed effects approach. What if we want to model the effects of level 2 variables but we have good reasons to believe that the random effect assumption doesn't hold. So what do we do because if we just choose between these two we have the GLS random effects which is inconsistent under this scenario and we have the GLS fixed effects approach which can't model variables that are constant within clusters or level 2 variables. So what do we do? Fortunately there is the third approach called correlated random effects model. And the idea of the correlated random effects model is that we take this random effects model so we model the unobstructed originating in the random part and we add cluster means of all level 1 predictors to the model. And that allows us to have level 2 predictors which of course equal their cluster means as predictors in the model. So what's the advantage of this model and what kind of effects the model produces. Let's do another comparison. This is from our article here and in the random effects model we have the fixed part containing the recursion coefficients and the random part that contains the variation that the model doesn't explain including the unobstructed term. Then we have the fixed effects model where the unobstructed term is modeled as a specific fixed effect for which we estimate a value for each case and then we have this normal error term in a recursion analysis. In the correlated random effects model we have fixed part and random part as well. So we retain the random part here from the random effects model but we add the cluster means to the fixed part and depending on whether we cluster mean center the original variable this cluster means gives either the contextual effect or the between effect. So why does this not require the unobstructed term to be uncorrelated with the predictors. The reason is that if we add this X bar or mean of X here as predictor the cluster mean then that including that predictor in the model will account for any level 3 effect that the unobstructed term would have. So if X is correlated with U then adding X bars of control will make U and X conditionally independent from one another. So how that can be proven but I will go through the proof here. The point here is that this CRE correlated random effects model will always give you the within effect for the X and it will give you the same estimate and the fixed effects model and this X bar cluster mean will give you the contextual effect or the between effect. We provide an example in our article in this table for and we can see here that we have these two fixed effects approaches. We have OLS regression with dummies and we have GLS fixed effects approach where we eliminate the other fixed effects by cluster means entering the data within the GLS procedure. So these are fixed effects model we only estimate the within effect and no other effects. Then we have a variable Z that is only varies between a level 2 units so X varies within levels Z varies between levels and then we have X bar the mean cluster mean of X. We have cluster mean centered X and grand mean centered X. Then we can compare that regardless of how we apply the cluster means if we can use it in OLS regression we can use it in GLS random effects model. We can use it in maximum likelihood estimated random effects model we can use it in maximum likelihood estimated random effects model with cluster mean centered predictors. We always get the same within effect so we get the same 0.51 effect of all these approaches. So which one of these approaches should we then use? Well typically if an estimation technique produces you more information then it's more useful than a technique that produces you less information. So here these fixed effects approaches eliminate all the contextual effects and all the between effects. These techniques estimate the within effect or the between effect depending on whether we mean center the X predictor. So this is a between effect here and all these are within effects they are the same value regardless of how we estimate the model. And we can also use the Z level 2 predictor. So this is the CRE correlated random effects model produced is the same within effect as the fixed effects approach. But it allows you to also model level 2 predictors here Z which the fixed effects approaches don't allow you to do and it provides you information about the contextual effect here or the between effect if that happens to be of interest. These standard errors also should be the same except that these are cluster robot standard errors which differ because they are because of how they are calculated. They depend on the residuals and the residuals in these different models are not the same. But these standard errors will nevertheless be consistent so in large samples they should generally converge. So these techniques can produce the same results but CRE produces more results and can be applied in scenarios for if it can. So that seems a preferable alternative at least to me. So let's take a summary of these approaches or the correlated random effects approach. So the advantages of correlated random effects is that it doesn't do the random effects assumption that the unobserved term is uncorrelated with all the predictors. Another advantage is that it estimates the within effect consistently the same way as the fixed effects estimation approach or modeling approach does. And it can accommodate level 2 variables so whereas fixed effects models can't have any variables that don't vary within levels then these correlated random effects can. And it provides an estimate of the contextual effect or the between effect. The contextual effect is probably in more commonly more useful than the between effect but there's one disadvantage in correlated random effects model. And the disadvantage is that it is less efficient than random effects models if the random effects assumption holds. The correlated random effects model is sometimes also referred to as the Mandelach procedure, the hybrid approach or the within between model. So this is the terminology for this modeling approach is not as standardized as for the GLS fixed effects and GLS random effects model. But you can encounter this model under all these different terms in the literature.