Random effects estimation

In statistics, random effects estimation is an estimation method used for the coefficients in multiple comparisons model in which the effects of different classes are random. In econometrics, random effects models are used in analysis of hierarchical or panel data when one assumes no fixed effects (i.e. no individual effects). The estimation can be done done via generalized least squares (GLS). If we assume random effects the error term in the model


 * $$y_{it}=x_{it}\beta+\alpha_{i}+u_{it},\,$$

where $$y_{it}$$ is the dependent variable, $$x_{it}$$ is the vector of regressors, $$\beta$$ is the vector of coefficients, $$\alpha_{i}=\alpha$$ are the random effects, and $$u_{it}$$ is the error term, then $$\alpha_{i}$$ should have a normal distribution with mean zero and a constant variance.

The coefficients can be estimated via


 * $$\widehat{\beta}=(X'\Omega^{-1} X)^{-1}(X'\Omega^{-1}Y),$$
 * $$\widehat{\Omega}^{-1}=\Iota \otimes \Sigma,$$

where X and Y are the matrix version of the regressor and independent variable, respectively, $$\Iota$$ is the identity matrix, $$\Sigma$$ is the variance of $$u_{it}$$ and $$\alpha$$, and $$\Omega$$ is the variance-covariance matrix.