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The essential difference between fixed effects (FE) and random effects (RE) is whether $E(C_i|X_i)$ is a constant or not, where $C_i$ is the individual effects. The assumption of FE is more realistic but too weak. In contrast, RE has so strong assumption that it is not reasonable in practice. So we have the methods of Munlak and Chamblein between FE and RE, where the Chamblein has weaker assumptions than Munlak’s. According to Greene (Econometric Analysis), we know there are some approaches of estimation:

• Pooled Regression Model，which assuming that all the samples are drawn from the same population. Using OLS to estimate FE models will cause endogeneity because $E(C_i|X_i)\neq constant$. In RE models, we can use OLS. However, since $E(C_i|X_i)= constant$, RE models have autocorrelation so that we can estimate the models by OLS+robust covariance matrix.
• For the same reasons, the samples can be viewed as different clusters according to the different individuals. By transforming the models, we can use Group means+ robust covariance matrix、first difference or deviations (within, between, total) to estimate the models. All these methods are suitable to RE. Because of $E(C_i|X_i)\neq constant$, the Group means approach is not suitable to FE because of the existence of the identification problem.
• For FE, we can view it as LSDV model which actually putting the individual effects term into the models to avoid the endogeneity problem.  However, it will cost huge computation. Since we only have the interest on the parameters of the explanatory variables but not the individual effects, we generally use Partial Regression method to estimate LSDV. It is worth to notice that LSDV also works for RE.
• For RE, because of autocorrelation ,we can use GLS or WLS. However, these two approaches require the assumption of the covariance matrix. If covariance matrix is unknown, we can use FGLS and FWLS, where ‘F’ means feasible.