There is little research directly linking model to potential outcomes for students and causal modeling. Bayesian methods have computational advantages and provide a principled method for making many desired inferences about teachers that account for uncertainty in all the model parameters. The model parameters, including the α (and ω) parameters, can be obtained via maximum likelihood or Bayesian methods. Traditional HLMs with students nested within classrooms are special cases with α (and ω) set to zero and correlation among ζ i 1 treated as zero. Cross-classified HLMs are a special case of model where again the α (and ω) are all set to one and ζ it = a i + b it, with ( a i, b i) assumed to be jointly normally distributed variables. For example, the layered model is a special case of model with no school effects (i.e., all the elements of all the γ st are set to zero), no explicit controls for students’ characteristics (i.e., all the elements of all the γ Xt are set to zero), and all the α are set to one so that there is no fadeout of teacher effects. Restrictions to this model recover most of the other commonly used statistical VAM. Extensions to more years of testing or multiple tests are direct. The α and ω parameters measure the fadeout of teacher and school effects and can be estimated from the data. Finally, ζ i = ( ζ i 1, ζ i 2, ζ i 3)′ are multivariate normal variables with mean zero and an unspecified covariance matrix Σ, and are independent of teacher, school, and other student effects. X it denotes vectors of student variables. The elements of γ st and γ Tt are normally distributed random variables representing school and teacher effects that are assumed to be independent across schools, teachers, and years. Where η t denotes overall means for the population each year, S it is a vector of indicators identifying each school’s share of the instruction for student i in school year t, and T it is an analogous vector for teachers.
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