We study the properties and linkages of some popular index decomposition analysis (IDA) methods in energy and carbon emission analyses. Specifically, we introduce a simple relationship between the arithmetic mean Divisia index (AMDI) method and the logarithmic mean Divisia index method I (LMDI I), and show that such a relationship can be extended to cover most IDA methods linked to the Divisia index. We also formalize the relationship between the Laspeyres index method and the Shapley value in the IDA context. Similarly, such a relationship can be extended to cover other IDA methods linked to the Laspeyres index through defining the characteristic function in the Shapley value. It is found that these properties and linkages apply to decomposition of changes conducted additively. Similar properties and linkages cannot be established in the multiplicative case. The implications of the findings on IDA studies are discussed.
The study by Ang (2004) classifies index decomposition analysis (IDA) methodology into additive and multiplicative. It further classifies those IDA methods often used by researchers and analysts into two groups, i.e. one is based on the concept of the Divisia index while the other based on that of the Laspeyres index or index numbers linked to the Laspeyres index. The study explains the properties and linkages of some of these methods, which include the arithmetic mean Divisia index (AMDI) method, logarithmic mean Divisia index method I (LMDI I) and method II (LMDI II), Laspeyres index method, Marshall–Edgeworth (M–E) method, Shapley/Sun (S/S) method and the conventional and modified Fisher ideal index methods.
As an extension to Ang (2004), this study introduces some new findings on the properties and linkages of IDA methods. In the case of methods linked to the Divisia index, Ang (2004)points out a simple relationship between the additive and multiplicative forms for both LMDI I and LMDI II. We further show that there exists a simple and meaningful relationship among most of the methods linked to the Divisia index, including that between AMDI and LMDI I.
In the case of methods linked to the Laspeyres index, Ang (2004) emphasizes the similarity between the method proposed by Sun (1998) and the Shapley value (Albrecht et al., 2002), and refers to them collectively as the S/S method.1 The S/S method is given in the additive form and, based on the classification in Ang (2004), its multiplicative counterpart is a modified Fisher ideal index method.2 In this study, we shall formalize the relationships between methods linked to the Laspeyres index and the Shapley value through defining the characteristic function in the Shapley value.
Our study is limited to IDA conducted additively, as the findings are applicable only to the additive form of decomposition, i.e. decomposing a change measured in terms of a difference. Similar relationships cannot be established when IDA is conducted multiplicatively, i.e. decomposing a change measured in terms of a ratio. With these findings, we are able to extend the findings in Ang (2004). This helps to improve our understanding of the properties of popular IDA methods and IDA methodology in general. The findings are also useful to analysts in method selection and decomposition result interpretation.
The next section gives an introduction to additive decomposition including the general formulae. It is followed by discussions on methods linked to the Divisia index in Section 3and methods linked to the Laspeyres index in Section 4. The conclusion is presented in Section 5.
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