In decomposition analysis literature there has been a long debate about the residual term (see e.g. Muller 2007, p. 14ff). On the one hand, it is argued that each decomposition method represents an approximation to an integral path that is not known. Therefore, it is only obvious that each decomposition should show a residual term and not assign it arbitrarily to the variable effects. On the other hand, a residual term, which is not allocated to one specific variable, raises questions in explaining the results and can pose significant problems if the residual becomes big in comparison to changes due to variable changes. This leads to a trade-off between the arbitrariness in allocating residuals and a non-exact decomposition method with a residual term.
The Laspeyres index, as well as the Paasche index, have the advantage of clear interpretation of the decomposition components. The same holds true for the Edgeworth-Marshall decomposition as the arithmetic mean of both. The interaction terms hidden in the Divisia indices are explicitly accounted for in the Laspeyres index approach. The downside is the resulting residual that creates problems in result interpretation. Since the residual can be considerable when there are large changes in the underlying data, Ang and Liu (2007a, p. 1431) classified the Laspeyres index not to be a good choice.
The disadvantage with all Divisia indices is that it arbitrarily assigns interaction terms to the factors (Howarth et al. 1991, p. 137). Looking again at Equation (4.34), this problem becomes obvious. The expression can be rewritten into , thus is the sum of and . Since not only depends on variable i, but also on the change in other variables, the decomposition result for variable i can change even if solely the other variables (≠i) change. The same holds true for the logarithmic mean, for which there exists no reason from the integral and derivative approximations to use this weight. Consequently, the calculation of the Divisia indices are more difficult to understand because of their complex procedure to distribute the higher order terms. This applies especially to the Adaptive Weighting Divisia Index, which requires a relatively
complicated calculation and does not give a perfect decomposition. Within the Divisia indices, the Logarithmic Mean Divisia Index I and the Arithmetic Mean Divisia Index are the easiest of the perfect decomposition methods to understand.
The remaining indices, the Refined Laspeyres and the MRCI, are more complicated to assess. While the Refined Laspeyres index tries to logically distribute only those interaction terms (higher order terms) to a variable, in which the considered variable is involved, one cannot determine a logic in the MRCI concerning the elimination of the residual.
4.4.4 Rating
In the past, several researcher have attempted to rate decomposition methods and pick their preferred method. These ratings clearly depend on the preferences of the researcher, the problem to be solved, and on the criteria alongside the methods are compared.
Ang et al. (1994, p.88ff) comes to the conclusion (at a time where the LMDI and Refined Laspeyres methods had not yet been used in energy decomposition) that the Edgeworth-Marshall and the AMDI were the best decomposition methods. This selection is based on the robustness of the methods, i.e. giving stable results and not being subject to extreme results, and on the theoretical “superiority” of the AMDI compared to other methods. Ang et al. (1994) also generally prefer a small residual and ease of use in terms of computational complexity. The authors qualify this last aspect because computing no longer presents any significant limits, not taking into account the analyst‟s efforts.
Six years later, Ang et al. (2000, p.1165ff) come to another conclusion, namely that the LMDI I and the Refined Laspeyres are the most robust methods. In this study, the authors base their decision on some index tests, the importance of the residual and the complexity of the formula. Given its ease of calculation the LMDI I (proposed by Ang) is preferred over the Refined Laspeyres. In a paper on the preferred decomposition method for policymaking in energy, Ang (2004) proposes again the LMDI I as the
preferred decomposition method and recommended it for general use. This decision is based on the results of the factor reversal, time reversal, proportionality and consistency- in-aggregation tests, ease of use and ease of result interpretation. The consistency-in- aggregation test verifies that a single stage index can also be computed in two stages, i.e. by first computing the indices for subaggregates and from these the index for the aggregate indicator.
Ang (2004) reports that the LMDI I performed best in these tests, because the factor reversal test was judged to be the most important, a residual term is disapproved (complicating result interpretation) and the link between multiplicative and additive decomposition is easily established. Yet, from an unbiased point of view, there is no reason to prefer the factor reversal test over the others. From a mathematical point of view, rejecting a residual means accepting an arbitrariness in distributing the residual. Finally, the link between multiplicative and additive decomposition might be a beneficial feature, but is not essential when one concentrates only on additive decomposition, as in this thesis. Ang et al. (2009) give another reason in favour of the LMDI I, namely that this method distributes the residual term of each sub-category proportionately according the effects.
Diekmann et al. (1999, p. 100ff) base their decision on decomposition methods on the following criteria: size of the residual, theoretical soundness, complexity of calculation, comprehensibility of results and purpose of study. According to the authors, the purpose of the study can have an influence on the choice of the decomposition method depending on whether many sectors are considered and whether it is prospective or retrospective (to choose the index weights appropriately). Admitting that a silver bullet does not exist, Diekmann and his colleagues reach the conclusion that the complexity of calculation and the difficult comprehensibility of the AWDI outweigh the advantage of its theoretical soundness. Given its clarity and easier calculation, the authors recommend the Refined Laspeyres index to be generally used in decomposition analysis.
Muller (2007) questions the LMDI I method as the default best method because of reservations towards a zero residual and the consistency-in-aggregation. Nevertheless,
he recommends the LMDI I as the currently most reliable method based on its performance in comparison to other methods for a wide range of functional forms. Before coming to any conclusion, the statement of Diekmann et al. (1999) has to be reemphasised: there is clearly no superior decomposition method. If there were, the index number problem would no longer be one. Starting with the decomposition methods‟ theoretical soundness, Table 4.2 does not reveal one decomposition method as clearly the best. The preference of one decomposition method over another depends on the preference of certain tests or axioms. Based on the simplicity and comprehensibility of calculation, one should use either the Laspeyres, Paasche and Edgeworth-Marshall or one of the simple Divisia indices, the LMDI I or the AMDI. The Refined Laspeyres and the Adaptive Weighting Divisia Index are relatively complex to calculate. If a small residual is desired, one should use one of the perfect decomposition methods, such as the LMDI, the Refined Laspeyres or the MRCI.
Based on the previous discussion and on the requirements in this thesis, the LMDI I decomposition seems to be preferable over the other decomposition methods, because of its ease of use, its relatively easy comprehensibility and its zero residual. Even though this last property comes at the expense of an arbitrariness in assigning the residual term to a variable‟s effect.