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Reinsdorf-Balk Transformation and Additive Decomposition of Indexes

Reinsdorf-Balk Transformation and Additive Decomposition of Indexes. I. Introduction II. Reinsdorf-Balk Transformation III. Additive Decompositions of Fisher Index IV. Additive Decomposition of Chain Indexes V. Numerical Illustrations (skip) VI. Concluding Remarks.

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Reinsdorf-Balk Transformation and Additive Decomposition of Indexes

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  1. Reinsdorf-Balk Transformationand Additive Decomposition of Indexes I. Introduction II. Reinsdorf-Balk Transformation III. Additive Decompositions of Fisher Index IV. Additive Decomposition of Chain Indexes V. Numerical Illustrations (skip) VI. Concluding Remarks Ki-Hong Choi(NPRI) and Hak K. Pyo(SNU)

  2. I. Introduction • Arithmetic mean index and decomposition of growth rate arithmetic mean index decomposition in %-change • Geometric mean index and decomposition of growth rate geometric mean index decomposition in log-change

  3. II. Reinsdorf-Balk Transformation • Literature Reinsdorf(1997, 2002), Balk(1999, 2004) Kohli(2007), “Truly remarkable” • Some need for improvements • Reinsdorf’s proof is hard to follow. • Though Balk’s proof is easy and clear, there is a better alternative.

  4. Balk’s Identity: from A index to G index Reinsdorf’s Identity: from G index to A index

  5. III. Additive Decompositions of Fisher Index • Conventional Decompositions • Decomposition in % changes van IJzeren(1952)=Dumagan(2002)=Ehemann et al(2002)

  6. Decomposition in log-changes Reinsdorf(1997), Reinsdorf et al(2002), Balk(2004)  Less satisfactory decomposition by Vartia(1976)

  7. New Decompositions Decomposition in % changes Reinsdorf et al.(2002), applying Reinsdorf identity to the geometric mean version of Fisher index.

  8. Reinsdorf et al(2002), “numerically identical”

  9. Decomposition in log-changes • Applying Balk’s identity to the van IJzeren form of Fisher index

  10. IV. Additive Decomposition of Chain Indexes • Difficulty with additive decomposition of cumulative (compound) growth rates of chain indexes • Cumulative growth rates is decomposable in log-changes • But decomposition in log-change is not good since it deviates from %-changes

  11. R-B transformation gives solution to this dilemma

  12. VI. Concluding remarks • R-B transformation is interesting since it makes arithmetic mean and geometric mean indexes interchangeable. • R-B transformation is useful since it can provide an additive decomposition of cumulative growth rates by chain indexes.  Thank you! 

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