Decomposition methods

After having discussed the origin of index decomposition in conjunction with the index number problem, the following discussion gives an overview of the additive decomposition methods used in the energy/environment context and their derivation. Early decomposition studies in the energy/environment context predominantly did not mention the method used, but it can be assumed that they used Laspeyres decomposition method, which was the most common at that time. In the late 1980s, Reitler et al. (1987) and Boyd et al. (1988) were the first to discuss methodological questions in the context of energy decomposition.

Laspeyres

The Laspeyres index decomposition, where all variables except for the explaining one are held constant in the base period, is rooted in the Laspeyres price index. Since it is one of the simplest decomposition forms, it was used from the very beginning of decomposition analysis (Hankinson and Rhys 1983).

Starting from Equation (4.5), the absolute change in the aggregate indicator due to variable xi in the Laspeyres form is mathematically described in the following way

(4.21)

Paasche

In contrast to the Laspeyres decomposition, the Paasche decomposition (named according to the Paasche price index) holds all variables except for the explaining variable constant at the observed period, resulting in the following formula

(4.22) This decomposition method has been relatively rarely applied, one example is Thomas et al. (1982).

Edgeworth-Marshall

The Edgeworth-Marshall decomposition combines, like its corresponding price index, the Laspeyres and Paasche index by taking the arithmetic average of the values in the base and observed period.

_(4.23)

This decomposition form was already applied in the 1980s by Reitler et al. (1987). Refined Laspeyres

The Refined Laspeyres decomposition has been known under different names, which is due to the fact that it can be formulated in different ways. Sun (1998) was the first to suggest this method in the context of decomposition analysis. He started from the

common Laspeyres index and had a closer look at the disregarded terms of higher order or so-called interactions terms. He coined the phrase “jointly created, equally distributed” to distribute the change arising from the interaction terms to the involved variables. An example in the three variable case looks like

(4.24)

where the interaction terms are proportionally attributed to variable 1.

Another formulation dates back to a game theoretic approach of Shapley (1953). He gave a formula to evaluate the real power of a voter with transferable utility in a coalition voting game.

Dietzenbacher et al. (1998) used the same approach within structural decomposition analysis. They started from a combination of the Laspeyres and Paasche index, i.e. holding some of the variables in the base period and some in the observed period. Following this technique, there exist n! possible permutations and accordingly exactly the same number of decomposition methodologies. Taking the average of all possible permutations yields a complete decomposition.

Albrecht et al. (2002) suggested this method the first time within index decomposition analysis, referring to Shapley (1953). As some combinations appear more than once in the permutation, these are weighted according to the Shapley value (Shapley 1953, p. 311) , where n is the total number of variables and s the number of variables held at the observed period T plus the studied variable. In the three variable case this results in

(4.25)

It can be shown that this formulation is equivalent to the refined Laspeyres formulation (Equation (4.24)). For a more detailed discussion on the equality of the Shapley and Refined Laspeyres decomposition, see Ang et al. (2003) and Lenzen (2006).

In the n-factor case, the refined Laspeyres decomposition takes the following form (4.26)

The above formula demonstrates that the equation becomes more complicated as the number of factors increases. This can be a significant disadvantage in the context of CO2 emission analyses with several explanatory variables.

The multiplicative equivalent to the refined Laspeyres is the generalised Fisher index (Ang et al. 2004).

Mean Rate of Change Index

In response to possible distortions in the Refined Laspeyres decomposition method, Chung and Rhee (2001) proposed the Mean Rate of Change Index (MRCI), which uses a more complicated form compared to the previous decomposition methods. The Refined Laspeyres, which is equal to the Edgeworth-Marshall decomposition in the two-factor case, was criticised for its uniform allocation of the residual term without taking into account the relationship between the original effects (Casler 2001, p. 146). In the presence of a comparably large residual, this can lead to a considerable change even with only a small scale effect. Therefore, de Bruyn (2000, p. 171) called for a method that is based on the condition that the relative increase due to the allocation of the residual remains the same for all effects. He introduced the concept of relative growth rates for the two-factor case to achieve this goal. Chung and Rhee (2001) extended this concept to a situation with multiple factors.

The decomposition makes use of the weight term Mi,r(*), which involves the rates of

change of all the relevant variables Ai,r(*)

(4.27)

(4.28) and (4.29)

In words, the equation for the change due to one variable is described by the total change of the aggregate indicator times the relative change (the authors call it mean rate of change) of the analysed variable

divided by the sum of the

relative change of all variables . In this way, the weighting method makes sure that the decomposition does not leave a residual and is therefore perfect. However, in particular the sum of the relative change of all variables is prone to distortions in the case of summands having mixed positive and negative signs. It can result in a situation where the sign in the decomposition term is opposite to the one in the relative change of the variable. In the extreme, this can lead to the sum becoming zero and therefore rendering the MRCI undefined. A general expression for the MRCI, which is not only restricted to mid-point weights, is given by Lenzen (2006, p. 193).

Arithmetic Mean Divisia Index

All of the following Divisia decomposition methods have their origin in the Divisia price index (Divisia 1925). The big difference between the previous decomposition forms and the Divisia index is that the Divisia decomposition methods are usually based on a logarithmic change in contrast to a change on a percentage basis. The general Divisia decomposition can be derived by differentiating Equation (4.5), which yields the total differential (4.30)

Integrating one factor i, Equation (4.30) on both sides gives

_(4.31)

(4.32) Incorporating (4.33)

An approximation to this path-dependent integral, proposed by Hulton (1973), is the arithmetic mean, which results in

_(4.34)

Boyd et al. (1988) suggested the Arithmetic Mean Divisia Index the first time within the scope of decomposition analysis. It multiplies the logarithmic change in variable i with the mid-point weight of the aggregate indicator.

The multiplicative equivalent to the additive version of the Arithmetic Mean Divisia Index was first introduced in Boyd et al. (1987).

Adaptive Weighting Divisia Index

A further and more flexible way of specifying the weighting is presented in the Adaptive Weighting Divisia Index (AWDI) (Liu et al. 1992a), which is based on the generalised first mean value theorem. According to this theorem (see e.g. Spiegel 1963, p. 82), if f(x) and g(x) are continuous in [a,b], and g(x) does not change sign in the interval, then there is a point in (a,b) such that

_(4.35)

According to this theorem, Equation (4.33) can be solved for i=1,2,n

(4.36) _(4.37) (4.38)

α,β,ω determines the weighting, where 0 would be a Laspeyres weighting, 0.5 a

Edgeworth-Marshall weighting and 1 a Paasche weighting.

If one applies the mean value theorem to Equation (4.31), the result is as follows for

i=1,2,n _{(4.39) (4.40) (4.41)}

The underlying condition in equations (4.36-4.41) is 0 ≤ α,β,...,ω ≤ 1. Analogue to the first three equations, the three equations above are called Parametric Divisia Method 2 (Liu et al. 1992a). The difference to method 1 is that it regards absolute changes and not logarithmic changes.

As Equations (4.36-4.38) and (4.39-4.41) are two different analytical expressions, but are mathematically equivalent, one can specify the parameters α,β,...,ω. For the parameter α we have by identifying the αr-term in Equation (4.36) and the term in

Equation (4.39) (4.42) Transformation yields: (4.43)

The parameter values, such as in this example , can then be entered in Equation (4.36) or (4.39) to solve the equation. The parameters give an indication of how to weight the basis and avoid any arbitrary guess work. Instead of choosing a value, as is done in the Arithmetic Mean Divisia Index or the Laspeyres index, the parameters can be analytically estimated based on the two deterministic approximations. In this way,

the AMDI provides a basis for choosing the parameter values. Consequently, this method is superior to other decomposition methods on theoretical grounds.

Logarithmic Mean Divisia Index I

Instead of using the arithmetic mean for an approximation of the Divisia integral, Ang et al. (1998) proposed to use the logarithmic mean. This mean was independently described by Sato (1976, p.224) and Törnqvist et al. (1985, p.44). According to Törnqvist et al. (1985), this logarithmic mean was already presented by himself in a report for the Bank of Finland in 1935. The first to publish an approximation of the Divisia index based on the logarithmic mean (without using the term logarithmic mean) was Montgomery (1937, p.51f).

The logarithmic mean is defined as follows

(4.44)

where both x and y are positive numbers and x ≠y. Two further special cases are defined: L(x,x)=x and L(0,0)=0. For nonnegative numbers the logarithmic mean lies between the arithmetic and the geometric mean (Vartia 1976, p. 122).

If one uses the logarithmic mean from (4.44), then equation (4.33) becomes

_(4.45)

which gives a perfect decomposition.

Like the Arithmetic Mean Divisia Index, the Logarithmic Mean Divisia Index I (LMDI I) uses the logarithmic change of the variable. The weighting consists of the logarithmic mean instead of the arithmetic mean, which gives higher values than the logarithmic mean.

The equivalent multiplicative decomposition of the LMDI I is given in Ang et al. (2001).

Logarithmic Mean Divisia Index II

Another Logarithmic Mean Divisia Index, suggested by Ang et al. (2003) is the Logarithmic Mean Divisia Index II (LMDI II), which is different to the LMDI I concerning its weighting.

Within the context of price index numbers Vartia (1976) proposed two different logarithmic mean index formulas for a multiplicative index. The Vartia Index I is used in the LMDI I and Vartia Index II in the LMDI II. Assuming

, the weights in the Vartia Index I are

(4.46)

while in the Vartia Index II they are defined as

(4.47)

Using Vartia Index II equation (4.33) is transformed into

(4.48)

In Equation (4.48),

describes the logarithmic change of the variable,

the logarithmic mean of the aggregate indicator and

a normalised weight function. The normalised weight function is used, because on its own does not add up to unity (Sato 1976, p. 224). The Vartia II weighting is more complicated and is in contrast to the Vartia I weighting not consistent in aggregation (Ang et al. 2003).

The equivalent multiplicative decomposition of the LMDI II is given in Ang et al. (1997).