Selection of covariates in Small Area Estimation with multilevel models

Mixed models can easily accommodate statistical relations between survey parameters and small‐area predictors, in order to provide indirect estimation of small domain‐level statistics. Moura and Holt (1999) introduced two‐level models, that integrates the use of regression‐synthetic estimators and area level covariates into a single model. A more complex feature of such a model (respect to the basic small area models) is that a two‐level mixed model allows for differences between slopes as well as the intercepts across small areas, and, in general, multilevel model (ML) involves the potential of the use of auxiliary information at both the unit and small area level. One of the advantages in using ML models over regression models is to recognize that small areas share common features. The small areas are not completely independent as could be assumed by using a separate linear regression model for each small area. With random coefficients we can model intercept and slope parameters in terms of area level covariates. There is a significant reduction of all the components of variance estimates after introducing explanatory area covariates Z, but in practice area‐level covariates are more difficult to find that unit‐level covariates (Rao, 2003). In the situation of no good area level explanatory variables, ML respect to standard unit level models can only avail of major flexibility in modelling, due to the presence of random slopes. If W is the complete set of variables in the available data, we currently partition W in the subset U, the unit level regression covariates, and A, the subset of remaining variables. In the situation in which there are no right area‐level predictors, we discuss on replacing the original candidate area covariates in A with some of their linear transformations, in order to reduce variance components in the model, and in consequence the mean squared error of the model‐based empirical estimator (Salvatore and Russo, 2008). We further analyze the reduction of components in the estimated mean squared error of the empirical predictor, related with the variations in model covariance parameters of random‐area effects and residual, based on these linear transformations. The simulation study consists on analyzing the mean squared error and the absolute relative error of the mean estimates by selecting samples of farms from the population of the Italian Agricultural Census, where the estimation problem is related to find estimates of the average working days at municipality level. We compare efficiency of several small area models, like traditional unit‐level and ML with and without covariates, and ML with explanatory area‐level covariates such as original and their transformed values. ML models variance components estimates are obtained by restricted maximum likelihood (REML) estimation method.