A Two-Stage Bennet Decomposition of the Change in the Weighted Arithmetic Mean

The weighted arithmetic mean is used in a wide variety of applications. An infinite number of possible decompositions of the change in the weighted mean are available, and it is therefore an open question which of the possible decompositions should be applied. In this article, we derive a decomposition of the change in the weighted mean based on a two-stage Bennet decomposition. Our proposed decomposition is easy to employ and interpret, and we show that it satisfies the difference counterpart to the index number time reversal test. We illustrate the framework by decomposing aggregate earnings growth from 2020Q4 to 2021Q4 in Norway and compare it with some of the main decompositions proposed in the literature. We find that the wedge between the identified compositional effects from the proposed two-stage Bennet decomposition and the one-stage Bennet decomposition is substantial, and for some industries, the compositional effects have opposite signs.