Random effects model

In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. In econometrics, random effects models are used in the analysis of hierarchical or panel data when one assumes no fixed effects (it allows for individual effects). The random effects model is a special case of the fixed effects model. In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. In econometrics, random effects models are used in the analysis of hierarchical or panel data when one assumes no fixed effects (it allows for individual effects). The random effects model is a special case of the fixed effects model. Contrast this to the biostatistics definitions, as biostatisticians use 'fixed' and 'random' effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables). Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from the data through differencing, for example by taking a first difference which will remove any time invariant components of the model. Two common assumptions are made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption 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, the random effects model is not consistent. Suppose m large elementary schools are chosen randomly from among thousands in a large country. Suppose also that n pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let Yij be the score of the jth pupil at the ith school. A simple way to model the relationships of these quantities is where μ is the average test score for the entire population. In this model Ui is the school-specific random effect: it measures the difference between the average score at school i and the average score in the entire country. The term Wij is the individual-specific random effect, i.e., it's the deviation of the j-th pupil’s score from the average for the i-th school.