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An exact additive decomposition of the weighted arithmetic mean

Author

Listed:
  • Thomas von Brasch
  • Håkon Grini
  • Magnus Berglund Johnsen
  • Trond Christian Vigtel

    (Statistics Norway)

Abstract
The weighted arithmetic mean is used in a wide variety of applications. An infinite number of possible decompositions of the weighted mean are available, and it is therefore an open question which of the possible decompositions should be applied. In this paper, we derive a decomposition of the weighted mean that is rooted in functional analysis. 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 2020Q1 to 2021Q1 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 decomposition and the Bennet decomposition is substantial, and for some industries, the compositional effects are of opposite signs.

Suggested Citation

  • Thomas von Brasch & Håkon Grini & Magnus Berglund Johnsen & Trond Christian Vigtel, 2021. "An exact additive decomposition of the weighted arithmetic mean," Discussion Papers 944, Statistics Norway, Research Department.
    • Handle: RePEc:ssb:dispap:944
    as

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    File URL: https://www.ssb.no/en/forskning/discussion-papers/_attachment/438047?_ts=175daeaee98
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    References listed on IDEAS

    as
    1. Mick Silver, 2009. "An Index Number Formula Problem: The Aggregation of Broadly Comparable items," IMF Working Papers 2009/019, International Monetary Fund.
    2. Diewert W.Erwin & Lippe Peter von der, 2010. "Notes on Unit Value Index Bias," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 230(6), pages 690-708, December.
    3. Diewert, W. E., 1976. "Exact and superlative index numbers," Journal of Econometrics, Elsevier, vol. 4(2), pages 115-145, May.
    4. Mick Silver, 2010. "The Wrongs And Rights Of Unit Value Indices," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 56(s1), pages 206-223, June.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Weighted Arithmetic Mean; Decomposition; Index Theory;
    All these keywords.

    JEL classification:

    • J21 - Labor and Demographic Economics - - Demand and Supply of Labor - - - Labor Force and Employment, Size, and Structure
    • C43 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Index Numbers and Aggregation

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