Análisis del cuestionario de Proyecto de Vida.

The empirical Bayes estimation of teacher effectiveness (TE) has become very popular within the VAM literature and other related topics, although the concept of predicting random effects have been discussed since early 1980s (Morris(1983);

Robinson(1991))

As the data are usually organised in clusters, the effects of these groups on a particular subject could be estimated by two alternatives approaches: (a) fixed effects, and (b) random effects. Whether to treat clusters as fixed effects or random effects depends on the question we want to answer with our model, then a suitable estimation methodology must be defined.

In those cases where the cluster or unobserved heterogeneity effects are not our main interest, and we want to control for potential bias when they are correlated to other explanatory variables, is recommendable to treat them as fixed effects. The common estimators in these cases are the Ordinary Least Square (OLS) with a dummy identification variable per group, and the Within-Group (WG) estimator which estimates mean deviations within groups in a panel data framework.

Alternatively, if we are interested in making inference on clusters effects and its dispersion is relevant for our analysis, we should treat the unobserved heterogeneity as random effects. Prediction of random effects can be made from different perspectives (e.g. Bayesian methods, empirical Bayesian, and frequentist prediction/estimation).

In pure Bayesian framework, all covariates are considered random, and ob- jective parameters (mean and variance) of prior distributions are required. Then, multiplying the prior densities with the maximum likelihood estimation we de- rive the posterior distribution. All inference regarding predicted random effects is made from the posterior distribution of the Bayesian estimation.

On the other hand, empirical Bayes estimation considers the prior distribu- tion as unknown, and uses the available data to estimate it. Therefore, empirical Bayes approach involve Bayesian concepts in a frequentist framework, where the estimation of random effects requires the recovery of the distribution of estimated parameters.

The linear regression models which combine fixed effects and multivariate random effects are known as: Bayesian hierarchical models, Mixed models, Linear random-intercept models, and Multilevel generalised models, among others. How- ever, all of them are Random Effects models that can be estimated by the same methodologies mentioned above.

Empirical Bayes estimation in a simplified framework

Setting a simple case for a linear mixed model to predict random teacher effects

τj.

yi,j =x0i,jβ+τj+εi,j (2.13)

xi,j corresponds to observable covariates with β fixed effects for student i, taught

by teacherj. Where i= 1...N and j = 1...J. In a matrix representation we have

y=xβ+ Γ + (2.14)

In the Bayesian approach the predictor forτj, will be given by the posterior

density function: f(τj|y,x;β) = f(y|τj,x;β)f(τj) R f(y|τj,x;β)f(τj) dτj (2.15)

Following Bayes Theorem and assuming all densities are proper probability densities functions, such us f(τj) > 0 and

R

f(τj)dτj = 1, we have that the pos-

terior distribution of τj conditional on the data is proportional to the conditional

likelihood times the prior density distribution:

f(τj|y,x;β)∝f(y|τj,x;β)f(τj) (2.16)

Given that the prior distribution of parameters is unknown, estimating teacher effects by traditional Bayes methods is unfeasible. However, using the empirical Bayes approach is possible when the model parameters are treated as known, and they are equal to those obtained with the maximum likelihood esti- mation (MLE). That meansE[ ˆβ_jM LE] =βj, then the posterior distribution can be

constructed or estimated.

Thus, the empirical Bayes prediction of teacher random effects will be the expected value of the empirical posterior distribution.

ˆ

τ_jEB =E[τj|y,x; ˆβjM LE] = Z

τjf(y|τj,x; ˆβjM LE) dτj (2.17)

Assuming a multivariate normal distribution structure for random effects Γ ∼ N(0, σ2

τIN), and error terms e ∼ N(0, σ2e)IJ. In addition to the classic

assumptions for random effects models such as; strict exogeneity E[e|x,Γ] = 0, and independence of unobserved heterogeneity with respect to other explanatory

variablesE[Γ|x] =E[Γ] = 0. We get that the conditional expectation of the vector responses is E[y|x,Γ] = x0β + Γ and its conditional variance is V ar[y|x,Γ] =

σ2

τIJ +σ2eIN.

Under this framework (linear model and joint normality assumptions) we obtain the same estimators of σ2

τ and σ2e using MLE. Therefore, the best linear

unbiased predictor (BLUP) will be the same as the mean of empirical Bayes pos- terior.

The empirical Bayes prediction is: ˆ τ_jEB=ψj(¯yj −x¯jβˆM LE) (2.18) whereψj = σ 2 τ σ2 τ+( σ2_u Nj)

is know as the shrinkage factor, which shrinks the MLE estimate towards the mean of prior distribution, in our case 0. The shrinkage factor is also considered as a “reliability” of the ˆτM LE

j measures obtained from the

total residuals of cluster j or ¯yj−x¯jβˆM LE.

The empirical Bayes predictions shrinks the maximum likelihood estimation ˆ

τM LE

j when the reliability decreases. That means if; (i) the number of observations

per teacher decreases, or (ii) the variance between teacher estimates decreases with respect to the total variance of the model. In the first case teachers with less observations will be adjusted toward 0 or prior mean. The second case there is less evidence of heterogeneity between clusters and the estimates adjustment will be made toward 0.

When the reliability increases, the empirical Bayes estimates rely more on the cluster information per group. That happens when the number of observation per teacher increases or the total variance of the model decreases with respect to the variance between group.

However, no matter what type of approach we use to estimate VAMs; whether fixed effects or random effects, it is important to address the main econo- metric issue to fulfil the exogeneity assumption of other covariates. In the absence of possible or valid instruments, researchers have used one of the two approaches mentioned above.

We next turn to possible causes of failures of the exogeneity assumption of included covariates in equations fromModel 2to4, induced solely by non-random assignments of students to schools/teachers and teachers to schools.