• Joint modeling of free and reduced-price percentages might improve the estimates. Because the two percentages are correlated (in both cases), joint modeling should improve efficiency.
  • More generally, cross-sectional and time-series models using several years of ACS data could be specified and estimated to improve efficiency. See, for example, Datta, Lahiri, and Lu (1999). This approach would be preferable to using the average of four 1-year estimates as a predictor variable.
  • While assuming that estimated eligibility percentages follow nor-mal distributions may be reasonable in some instances, it is not a good assumption for small samples (as for the school attendance areas in a small or medium-sized district) or for small percentages (such as reduced-price percentages) with skewed distributions or many estimates of 0. Better approaches include transformation of the percentage, assuming a discrete distribution, using a mixed distribution, or using a linking distribution defined in [0,1], such as the logistic or beta.
  • Variance estimation might be improved. For variances of direct estimates, the approach to GVF modeling should be compared to approaches in the literature. For estimating model variances, generalized maximum likelihood estimation methods have been developed that are consistent and strictly positive (in contrast to variance components methods). Another possibility is to use hierarchical Bayes or some simple approximations, such as the adjustment for density maximization method described in Morris and Tang (2011).
  • Exchangeability assumptions on regression coefficients and model variances could be relaxed by introducing heterogeneity using different regression coefficients and model variances for different groups based, for example, on administrative estimates of the percentage of students eligible for free or reduced-price meals, as well as the size of the resident population.

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