Best prediction under a nested error model with log transformation

TL;DR

This paper develops analytical methods for predicting original-scale values in nested error models with log transformation, addressing bias and uncertainty assessment for small area estimation.

Contribution

It provides explicit formulas for best predictors and their mean squared errors under a nested error model with log transformation, including empirical versions and error estimators.

Findings

Derived analytical expressions for best predictors.

Established second order mean squared error approximations.

Illustrated methods with Mexican living conditions data.

Abstract

In regression models involving economic variables such as income, log transformation is typically taken to achieve approximate normality and stabilize the variance. However, often the interest is predicting individual values or means of the variable in the original scale. Back transformation of predicted values introduces a non-negligible bias. Moreover, assessing the uncertainty of the actual predictor is not straightforward. In this paper, a nested error model for the log transformation of the target variable is considered. Nested error models are widely used for estimation of means in subpopulations with small sample sizes (small areas), by linking all the areas through common parameters. These common parameters are estimated using the overall set of sample data, which leads to much more efficient small area estimators. Analytical expressions for the best predictors of individual values of the original variable and of small area means are obtained under the nested error model with log transformation of the target variable. Empirical best predictors are defined by estimating the unknown model parameters in the best predictors. Exact mean squared errors of the best predictors and second order approximations to the mean squared errors of the empirical best predictors are derived. Mean squared error estimators that are second order correct are also obtained. An example with Mexican data on living conditions illustrates the procedures.