The Logarithmic Mean Divisia Index (LMDI) decomposition approach has many desirable properties. It has been recommended by Ang [2004a. Decomposition analysis for policymaking in energy: which is the preferred methods? Energy Policy 32, 1131–1139] for adoption in energy and energy-related environmental index decomposition analysis (IDA). A complication associated with this approach is the treatment of zero values in the data set. Several studies show that replacing the zero values with a small number gives converging results. In a recent paper, Wood and Lenzen [Zero-value problems of the logarithmic mean Divisia index decomposition method. Energy Policy, in press] argue that this strategy is not necessarily robust and recommend using the analytical limits proposed in Ang et al. [1998. Factorizing changes in energy and environmental indicators through decomposition. Energy 23, 489–495]. We compare these two strategies and extend earlier works by generalizing the analytical limits of LMDI.
In index decomposition analysis (IDA) applied to energy, the Logarithmic Mean Divisia Index (LMDI) decomposition approach which comprises the two decomposition methods LMDI I and LMDI II in both the additive and multiplicative forms has been espoused as the preferred decomposition approach (Ang, 2004a). However, due to the logarithmic terms in the LMDI formulae, complications arise when the data set contains zero values. Ang and Choi (1997)show that the zero values may be replaced by a small number δ and converging results are obtained when δ approaches zero. Thereafter, studies such as Choi and Ang (2001) and Choi and Ang (2002) also apply this strategy. They found that a value of δ between 10−10 to 10−20 generally gives satisfactory results. We shall refer to this strategy to tackle the zero-value problem as the “small value” (SV) strategy. As an alternative, Ang et al. (1998) present the limiting values for eight cases where zeros occur in the data set and use these analytical limits in an empirical study involving additive decomposition. An extension to multiplicative decomposition is reported in Ang et al. (2000). We shall refer to this alternative strategy as the “analytical limit” (AL) strategy. In a review of various IDA techniques, Ang (2004a)recommends the SV strategy as it has been found to be robust and easy to use.
In a recent study, Wood and Lenzen (in press) argue that the SV strategy is not necessarily robust because it would produce significant errors if applied in the decomposition of a data set containing a large number of zeros and/or small values. They recommend using the analytical limits given in Ang et al. (1998), i.e. the AL strategy. Wood and Lenzen (in press)also mention that for structural decomposition analysis (SDA) that involves large input–output tables, the AL strategy would drastically reduce computation times as compared to the SV strategy.
In this paper, we study the SV strategy in the context of IDA in greater detail, compare the two strategies, and extend the works of Ang et al. (1998) and Wood and Lenzen (in press)by generalizing the analytical limits for the LMDI approach. To be consistent with these earlier studies, we shall focus on the additive LMDI I method which has a simpler form in the LMDI family (Ang, 2004a). The general conclusions are applicable to the other LMDI methods except that the analytical limits are different.
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