Negative-value problems of the logarithmic mean Divisia index decomposition approach

An issue that has not been fully resolved in the Logarithmic Mean Divisia Index (LMDI) decomposition approach is how to deal with negative values in the data set. We provide an analytical solution to this problem and illustrate with an example. With the issue resolved, the LMDI approach can now be generally applied to any decomposition situation.

The Logarithmic Mean Divisia Index (LMDI) approach has been recommended for adoption in index decomposition analysis (IDA) by Ang, 2004, Ang, 2005. Lately, there have been a growing number of studies using this approach to analyse changes in an aggregate, including energy consumption or intensity, greenhouse gas emissions or intensity, electricity generation efficiency, water consumption, manufacturing process quality, and inventory turnover.1 The LMDI approach uses models that have logarithmic terms. Complications arise when the data set contains zero or negative values. The occurrence of zero or negative values is rare in energy IDA studies but it is likely in gas emissions IDA studies and in structural decomposition analysis (SDA). The zero-value problem has been resolved and the proposed procedures can be found in Ang and Liu (in press) and Wood and Lenzen (in press).

The negative-value problem of LMDI was first raised by Chung and Rhee (2001) where the CO2 emissions decomposition problem they study has negative values in the data set. Ang et al. (2004) and Lenzen (2006) point out that the LMDI approach is not negative-value robust. Ang (2004) mentions the need to opt for an alternative decomposition approach when the data set has negative values. In a recent paper, Rhee and Chung (in press)reiterate this limitation of the LMDI approach.

The first attempt to deal with the negative-value problem of LMDI is reported in Wu et al. (in press). In their study on the dynamics of energy-related CO2 emissions in China, negative values occur for variables such as exports and stock changes. They tackle the problem in two different situations which will be pointed out in the next section. In this paper, we extend and refine their ideas and integrate them with the procedure for handling zero values in Ang and Liu (in press), and then present the guidelines for dealing with negative and/or zero values in a unified framework. An illustrative example will be presented. In line with Ang and Liu (in press), we shall focus on the additive LMDI I method which has a simpler functional form in the LMDI family of decomposition methods (Ang, 2004). The analysis, however, can be similarly extended to other LMDI methods.

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