Impact measurement and dimension reduction of auxiliary variables in calibration estimator using the Shapley decomposition

In multipurpose surveys several interest variables and a very large number of auxiliary variables are collected. Auxiliary variables are usually considered in calibration for improving estimates. But, very often, some of them are included for the sole purpose of increasing consistency. Consistency is an important point for National Statistical Institutes especially as a means for promoting credibility in published statistics. As a direct result, the number of auxiliary variables considered in calibration continue to grow over time. In literature, several methods show how to manage many auxiliary variables in order to prevent some unpleasant consequences on the accuracy of estimates. They consist mainly in variable selection or dimension reduction and they are very useful for deriving calibrated estimates more accurately. However, looking at them, it is not easy to infer how much the contribution of each auxiliary variable is, especially when there are plenty of them. The Shapley decomposition applied in the calibration context could be a useful tool to better understand the net effects of auxiliary variables, and, in addition, it provides further information for supporting researchers in choosing the best calibration system. It provides a direct measure of the change with respect to Horvitz–Thompson estimates and to related sampling variances due to the introduction of each auxiliary variable in the calibration. The method has been applied to real data of the Italian Labour Force Survey that makes an extensive use of auxiliary variables in calibration.