An extension to Sun's decomposition methodology: The Path Based approach

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Energy Economics 30 (2008) 1020–1036
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An extension to Sun's decomposition methodology:The Path Based approach

Esteban Fernández ⁎, Paula Fernández

Department of Applied Economics, Faculty of Economics, University of Oviedo, Spain

Received 1 June 2006; received in revised form 5 January 2007; accepted 5 January 2007Available online 6 March 2007

Abstract

This paper proposes the use of the Path Based approach as a general and flexible method to carry out exactdecompositions when some additional information for intermediate periods is available. This additionalinformation is used in a Maximum Entropy estimation procedure to arrive at parameter estimates thatdetermine the temporal paths followed by the decomposition factors. An empirical illustration, decomposingchanges in greenhouse gas emissions in the EU 15 from 1990 to 2002 by both the Path Based and Sun'speriodwise method, is carried out. The outcome shows that the Path Based approach leads to more accurateresults than Sun's periodwise method when both are compared with a complete time series decomposition.© 2007 Elsevier B.V. All rights reserved.

JEL classification: C43; C13; Q53

Keywords: Change decomposition; Path Based approach; Sun's methodology; Greenhouse gas emissions; EU 15

1. Introduction

Since the 1980s a number of papers have dealt with the decomposition of the change in anaggregate environmental or energy indicator between any two benchmark years to study theunderlying mechanisms of change at the macro level. Authors such as Jenne and Cattell (1983),Reitler et al. (1987), Boyd et al. (1988), Liu et al. (1992), Ang and Lee (1994), Sun andAng (2000) orAlbrecht et al. (2002) have dealt with methodological aspects and have introduced a framework forformulating and applying decomposition methods. The development of the decomposition methodsis basically the application of economic indices to decompose changes in an aggregate variable.

⁎ Corresponding author. Tel.: +34 985105056; fax: +34 985105050.E-mail address: [email protected] (E. Fernández).

0140-9883/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.eneco.2007.01.004

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1021E. Fernández, P. Fernández / Energy Economics 30 (2008) 1020–1036

Thus, the methodological problems are analogous to the well known index number problemstated by Fisher (1922) and developed by Diewert (1980).

The basic decomposition models include those formulated using the Laspeyres, Paasche andMarshall Edgeworth indices. The common problem shared by these basic models is that they donot lead to a perfect decomposition. The residual influences the model accuracy and, conse-quently, the results of decomposition can be notably different. An exact decomposition modelwould remove the problems caused by the presence of a residual term.

Extended and refined models developed by Ang and Choi (1997) or Sun (1998) lead to aperfect decomposition where the residual is not longer an interaction that is still an incognita tothe researcher. Ang (1995) proposes a logarithmic mean weight function while Sun (1998)assumes the principle of “jointly created and equally distributed” to the residual.

Albrecht et al. (2002) present a Kaya Identity based technique that leads to perfect andsymmetric decomposition. Ang (2004) compares different decomposition methods and makessome recommendations when applying one of them. Other recent development concerningexhaustive decompositions can be found in Ang and Zhang (2000), Chung and Rhee (2001), Anget al. (2003, 2004) or Boyd and Roop (2004). Fernández (2004) proposes the Path Based method(PB) as an alternative technique where some additional information is available for some of thevariables in intermediate periods. This information is used in a Maximum Entropy (ME)estimation procedure to arrive at parameter estimates that determine the temporal paths followedby the factors. In this paper we apply this approach to decomposition problems that deal withenvironmental or energy topics. In Section 2, we briefly recall the “non uniqueness” problem bymeans of a simple decomposition analysis with two determinants and also refer to the solutionpreviously proposed by Sun (1998). Section 3 shows the basis of the so-called Path Based (PB)decomposition methodology, which offers a much broader class of solutions than that introducedin Section 2. In Section 4, the principles of ME estimation are highlighted, and we show how MEestimation techniques can be used to estimate the parameters of interest with some additionalinformation. Section 5 discusses several situations with different availability of additional infor-mation. In Section 6, we present an empirical illustration of the approach. We will study changesin greenhouse gas emissions in the EU 15 between 1990 and 2002 with a decomposition modelthat considers as determinants the intensity of greenhouse gas emissions, the national weight ofevery country in the European GDP and the whole GDP of the EU 15. Moreover, we alsocompare the outcomes obtained by the different approaches with an annual average decom-position, in order to check which of them yields the closest solution. Finally, Section 7 presentsthe main conclusions of the paper.

2. The non uniqueness problem in decomposition analyses

Assume that the value of an endogenous variable v is given as the product of a set of nexogenous variables or determinants x1,x2,…,xn.

1:

v ¼ x1x2 N xn ð1ÞWe will assume that each determinant could change without a necessarily accompanying

change in the values of one or more of the other determinants and that the difference in v to bestudied relates to a difference over time.

1 The endogenous and exogenous variables can be represented by scalars, vectors and/or matrices.

Fig. 1. Polar and straight line paths.

1022 E. Fernández, P. Fernández / Energy Economics 30 (2008) 1020–1036

Focusing on an additive decomposition:

Dv ¼ v1−v0 ¼ x11x12 N x

1n−x

01x

02 N x

0n ð2Þ

and considering the case in which n=2:

Dv ¼ v1−v0 ¼ x1y1−x0y0 ð3Þtwo possible decomposition forms are:

Dv ¼ Dxy0 þ x0Dyþ DxDy ð4Þ

Dv ¼ Dxy1 þ x1Dy−DxDy ð5ÞEq. (4) uses a Laspeyres weighting structure whereas (5) corresponds to a Paasche decom-

position. Both expressions are categorized as “approximate” or “non exhaustive” decompositionsbecause the sum of the effects of the factors does not necessarily equal the whole change in theendogenous variable v. The last term ΔxΔy is often labelled the “interaction effect” and can betaken as the residual term in the decomposition required to obtaining the whole variation Δv.Sometimes these approximate forms may be preferred to exhaustive forms, for example if a clearinterpretation can be given to the interaction term. If the numbers of determinants increases, nonexhaustive decompositions will contain several interaction terms, for which straightforwardinterpretations are not always to hand. For this reason “exact” or “exhaustive” decompositionmethodologies are usually preferred to these non exhaustive ones.

The specification of a particular temporal path for the determinants implies a particulardecomposition form to split up the interaction term.

In Fig. 1, the average of the two extreme alternative paths (PP1 and PP2) would imply an equaldivision of the interaction rectangle and this is the result attained by Sun's (1998) method.2 Sunhimself refers to his solution as the implication of a “jointly created and equally distributed”principle (Sun, 1998, p. 88). In a general case as the one depicted in expression (1) in which v is

2 A similar solution was proposed in the Input Output field for Structural Decomposition Analysis by Dietzenbacherand Los (1998).


the product of n determinants, the expression of the effect of changes in a factor xi if we followSun's approach will be:

Dxi Effect ¼Z t¼1

t¼0j

n

jpi

xjdxidt

dt ¼ ji−1

jbi

x0j

" #Dxi j

n

jN1

x0j

" #þ ð6aÞ

þXnjpi

12j

i−1

kbi

x0kDxijj−1

ibkbj

x0kDxjjn

kNj

x0k

" #þ ð6bÞ

þXnjpi

Xnlpj;i

13j

i−1

kbi

x0kDxijj−1

ibkbj

x0kDxjjl−1

jbkbl

x0kDxljn

kNl

x0k

" #þ ð6cÞ

þ 1n j

n

j¼1

Dxj

" #ð6dÞ

This methodology fulfils thatPn

i¼1Dxi Effect ¼ Dv, i.e. it is an exhaustive decompositionmethodology that removes any residual term by splitting it up among all the xi determinants.

In the next sections, we will study the main features of a more general decomposition method.

3. The Path Based approach

Building on the earlier work by Vogt (1978) and connecting with a more recent work byHarrison et al. (2000), we propose an alternative exhaustive decomposition method that we willcall the Path Based (PB) approach. It that can be taken as a generalization of the approachdescribed in the last section and its main advantage is that allows us to use additional informationwhen available in order to reduce arbitrariness.

The alternative setup starts from the assumption that the determinants xi change continuouslyover time, between time 0 and time 1. Hence, we can write:

vðtÞ ¼ x1ðtÞx2ðtÞ N xnðtÞ ð7Þand, assuming differentiability of each xi(t), an infinitesimal change in v can be expressed as:

dv ¼ AvAx1

dx1dt

dt þ N þ AvAxn

dxndt

dt ð8Þ

Finally, the total change in v can be expressed as the sum of all the infinitesimal changesbetween time 0 and time 1:

Dv ¼Z t¼1

t¼0

dvdt

dt ¼Z t¼1

t¼0

Xni¼1

AvAxi

dxidt

dt ð9Þ

The effects of the determinants xi can now be written as:

Dxi Effect ¼Z t¼1

t¼0

AvAxi

dxidt

dt ¼Z t¼1

t¼0j

n

jpixjdxidt

dt ð10Þ


Eq. (10) shows that the derivatives of the determinants xi to time t play an importantrole in the size of the effects attributed to changes in these determinants. Consequently,the specification of the temporal path that each factor follows between initial and final periods,xi(t) = fi(t), can have a big impact on the measurement of their effects that together add up to thevariation in v. Harrison et al. (2000) proposed the solution be reached by assuming straight linepaths of the variables xi:

xiðtÞ ¼ x0i þ ðx1i − x0i Þt ¼ x0i þ Dxit ð11ÞActually, this approach yields the same solution as Sun's ‘equal shares’ method (1998).

However, empirical values of the variables x and y at some intermediate point might show that thestraight line assumption is very unlikely to be tenable. In this work, we suggest a method to takesuch information explicitly into account in attributing parts of the interaction effects to the effectsof the respective determinants. The proposed methodological innovation relies on relaxing thestrict assumption of a straight line, by considering more flexible forms for the functions fi(t).Specifically, we choose to consider a specific class of monotonic functions:

xiðtÞ ¼ x0i þ Dxithi ; 8hi N 0 ð12Þ

The class of paths considered contains all possible monotonic paths from x0 to x1 that do nothave inflexion points (Fig. 2). Although this is a limitation for sure, it still can be considered as amore flexible approach than a straight line. Obviously, the temporal path of xi will be a straightline if θi equals 1. If this holds for all i(i=1,…,n), the solution obtained by the PB methodologyintroduced here will be identical to Sun's solution (1998).

The basic idea is that the specific path implied by the parameter values θi determine the sharesof the interaction effect that is attributed to the distinct determinants. The paths PP1(θx/θy→0) andPP2(θx/θy→∞), and the straight line path (θx/θy=1) are included as special cases of this generalclass. P1 and P2 are intermediate cases.

For the most general case in which a change in v is decomposed into the effects of ndeterminants xi, the expression for the respective contributions for any possible set of n time paths

Fig. 2. Generalized monotonic temporal paths.


was already given in Eq. (12). Substituting the more specific temporal paths assumed in Eq. (12)into Eq. (10), we can write:

Dxi Effect ¼Z t¼1

t¼0j

n

jpi

xjdxidt

dt ¼ ji−1

jbi

x0j

" #Dxi j

n

jNi

x0j

" #þ ð13aÞ

þXnjpi

hihi þ hjj

i−1

kbi

x0kDxijj−1

ibkbj

x0kDxjjn

kNj

x0k

" #þ ð13bÞ

þXnjpi

Xnl pj;i

hihi þ hj þ hlj

i−1

kbi

x0kDxijj−1

ibkbj

x0kDxjjl−1

jbkbl

x0kDxljn

kNl

x0k

" #þ ð13cÞ

þ hiPnj¼1

hjj

n

j¼1

Dxj

" #ð13dÞ

The importance of the values of the θi parameters for the measurement of the determinant'scontributions is clear. The higher the value of θi in comparison to the remaining θj, the greater theportions of the interaction effects attributed to xi and, thus, the greater its contribution to the wholechange in variable v. In the next sections we will turn to methods to infer on plausible values forθi, which allow us to apply Eqs. (13a) (13b) (13c) (13d) to interesting empirical problems.

4. Generalized maximum entropy econometrics

If there is no information about the evolution of the determinants over time, the most plausibleassumption would be that the temporal path parameters are equal to each other (θ1=θ2=…=θn). Ifinformation about the evolution of all the determinants is available for intermediate periods, onemay carry out a dynamic time series decomposition. Nevertheless, the most of the times, onlypartial information for some of the determinants in the intermediate periods is available. In suchcases, this paper proposes a methodology that allows the use of this partial information and leadsto more accurate results than the case of no additional information at all.

We propose Generalized Maximum Entropy (GME) to estimate the temporal path parametersθI.

3 We want to estimate the parameters of a linear model in which a variable y depends on Hexplanatory variables xh:

y ¼ Xqþ e ð14Þwhere y is a (T×1) vector of observations, X is the (T×H) matrix of explanatory variables for eachobservation, θ is the (H×1) vector of parameters to be estimated, and e is a (T×1) vector ofrandom disturbances. We estimate this model using the maximum entropy estimator proposed byGolan et al. (1996).4

3 See Kapur and Kesavan (1993) or Golan et al. (1996) for a detailed analysis of properties of the estimators obtainedby means of these techniques.4 There are several applied papers that have used this estimator. See, for example, Paris and Howitt (1998), Fraser

(2000), Golan et al. (2001) and Gardebroek and Oude Lansink (2004).


The starting point is the specification of each parameter θh as a discrete random variable thatcan take (M≥2) different values which are grouped in the so-called support vector: bh=(bh1,bh2,…,bhM). The support vector is chosen using priors on the value of the parameter. For example, ifthe priors determine a range of possible values for the parameter the support vector could containsM values in this interval. Each value of the support vector has a probability of being the ‘real’parameter. Hence, the support vector M has an associated probability vector ph′=(ph1,ph2,…,phM)such that

PMm¼1 phm ¼ 1; 8h.

Finally, each parameter in the model is written as a linear combination of the elements of thesupport vector (chosen) weighted by their probability (unknown):

hh ¼ bh ph ¼XMm¼1

bhm phm ð15Þ

The vector of H unknown parameters in the model can be written as:

q ¼h1h2vhH

2664

3775 ¼ Bp ¼

b1 0 N 00 b2 N 0v v ⋱ v0 0 N bH

2664

3775

p1p2vpH

2664

3775 ð16Þ

A similar procedure is used to specify the vector of random disturbance of the model (e). Foreach element et of the vector, we assume the existence of a support vector v=(v1,v2,…,vJ)

5 withprobabilities wt′=(wt1,wt2,…,wtJ) where J≥2. The vector of random disturbances e can be writtenas:

e ¼e1e2veT

2664

3775 ¼ Vw ¼

v 0 N 00 v N 0v v ⋱ v0 0 N v

2664

3775

w1

w2

vwT

2664

3775 ð17Þ

And the value of the random disturbance, for an observation t is:

et ¼ vwt ¼XJj¼1

vjwtj ð18Þ

Finally, the model in Eq. (16) can be written as:

y ¼ XBpþ Vw ð19ÞIn this specification, the problem of estimating the unknown vector of parameters θ is

transformed into the estimation of M×H probabilities of the values of the support vectors of theparameters and J×T probabilities of the values of the support vector of the error. Using theestimated probabilities, an estimate of each parameter can be recovered as:

hh ¼XMm¼1

phmbhm; 8h ¼ 1; N ;H ð20Þ

5 The common practice is to choose a set of values centered about zero.


ME estimates the probabilities in (20) by maximizing an entropy function. The entropyfunction is a measure of uncertainty or ignorance about the outcomes of an event represented by arandom variable. The entropy function defined by Shannon (1948) is:

EFðpÞ ¼ −XMm¼1

pm ln pm ð21Þ

where EF is the value of the entropy function and p=[ p1,p2,…,pM] are the probabilities of Mpossible outcomes x1,x2,…,xM of a discrete random variable x, such that

PMm¼1 pm ¼ 1.

The interpretation of the entropy function as a measure of ignorance is quite intuitive. Thefunction EF( p) goes to zero when the probability of one of the possible outcomes of the randomvariable goes to one. In other words, the function goes to zero as uncertainty vanishes. This is theminimum value of the entropy function since it can not take negative values. In the otherhand, the entropy function achieves its unrestricted maximum for the uniform distribution:pm ¼ 1

M ; 8m ¼ 1; N ;M� �

. In empirical analysis, we are somewhere between these two polarcases (no uncertainty and complete uncertainty). In this case, the focal point is the level ofignorance that can be claimed when there are some observations of the outcome of a randomvariable. The existence of data consisting of several observations of the outcomes of the randomvariable reduces the uncertainty and, therefore, the entropy. In fact, it is possible to rejectprobability distributions that could not have generated these data. For example, two differentrealizations of a random variable mean that the sample could not come from a probabilitydistribution that attributes a probability of one to a realization and zero to the others. Themaximum level of ignorance compatible with a given dataset is measured by the maximumentropy conditional on the constraints on probability defined by the dataset. The maximumentropy principle consists of choosing as estimates the probabilities associated with the maximumentropy conditioned on the dataset.

The estimation of the model in (19) requires the estimation of the probabilities of the elementsof the support vectors. The probabilities can be calculated using the following optimizationprogram:

Maxp;w

EFðp;wÞ ¼ −XHh¼1

XMm¼1

phm ln phm−XTt¼1

XJj¼1

wtj ln wtj ð22aÞ

Subject to:

XHh¼1

XMm¼1

bhm phm xht þXJj¼1

vjwtj ¼ yt; 8t ¼ 1; N ; T ð22bÞ

XMm¼1

phm ¼ 1; 8h ¼ 1; N ;H ð22cÞ

XJj¼1

wtj ¼ 1; 8t ¼ 1; N ; T ð22dÞ

Eq. (22a) is the entropy function of Shannon adapted to the estimation of M×H+ J×Tprobabilities. Eq. (22b) contains sample information in terms of the model in (19) ensuring

Fig. 3. A dynamic decomposition.


that the estimated probabilities are compatible with the T available observations. Eqs. (22c)and (22d) ensure that probabilities add up to one. The solution of the optimizationprogram in Eqs. (22a) (22b) (22c) (22d) gives the estimates of the probabilities of theelements of the support vectors. The parameters θ of the model can be recovered usingexpression (20).6

5. The use of additional information

If there is some available additional information for intermediate periods between t=0 andt=1, this can help to reduce the non uniqueness problem present in the decompositionanalysis. Suppose that for the simplest case where v(t) = x(t)y(t), we know the values of x(t′)and y(t′), being 0b t′b1. In such a case, we can incorporate the additional information andconsider a two stages decomposition like v1− v0 = (v1− vt′)+ (vt′− v0). Fig. 3 illustrates thissituation:

The sum of the interaction terms for the two stages of the decomposition (two grey shadedareas) is smaller than the interaction term ΔxΔy of Figs. 1 and 2. If the number of stages isincreased, i.e. if the number of intermediate periods with observations for factors x and yincreases, the size of the interaction terms decreases and this also reduces the seriousness of thenon uniqueness problem.7 But such a solution requires a large amount of available data, which isnot always available. In fact, the unavailability of regular information for energy or environmentalvariables, especially in regional or small scale analyses, makes this “dynamic” approach oftenunfeasible. This paper suggests the possibility of including intermediate observations for some,thought not all of the factors.

To illustrate this idea simply, let us continue with the simplest case with two factors x and y.Suppose that for an intermediate period t′; we have collected the value of x(t′), although y(t′) isunknown. Obviously, this situation disables a two stages decomposition like the one represented

6 The large sample properties of the ME estimators are analyzed in Golan et al. (1996; pp. 96–123 and 131–133). MEestimators are shown to be consistent and asymptotically normal. These authors analyze also the small sample propertiesusing Monte Carlo simulation.7 Note that in the hypothetical case when an infinite number of intermediate points, the interaction terms would vanish

and there would be a unique decomposition form.


in Fig. 3, but the additional information x(t′) can be used to obtain a decomposition different fromthe average of equations (n) and (n), which would be the most appropriate solution if no additionalinformation was available. Applying the PB approach to this context, we have the followingtemporal paths for factor x, which is not exactly the same as before:

xðtÞ ¼ x0 þ Dxthx eet ð23ÞExpression (23) includes a stochastic component εt that allows x(t) to diverge from the

deterministic path (14). Taking logarithms, we have:

x⁎ðtÞ ¼ hxt⁎þ et ð24Þ

where x⁎ tð Þ ¼ ln xðtÞ−x0x0

� �and t⁎ ¼ ln tð Þ. Since Eq. (24) is a linear model with one parameter to

be estimated, it is possible to apply the GME technique analyzed in the previous section, and (24)can be written as:

x⁎ðtÞ ¼XMm¼1

bm pmt⁎þXJj¼1

vjwtj ð25Þ

which will be taken as a constraint in the following maximization problem:

Maxp;w

EFðp;wÞ ¼ −XMm−1

pm ln ðpmÞ−XTt¼1

XJj¼1

wtj ln ðwtjÞ ð26aÞ

subject to:

x⁎ðtÞ ¼XMm¼1

bmpmt⁎þXJj¼1

vjwtj; 8t ¼ 1; N ; T ð26bÞ

XMm¼1

pm ¼ 1 ð26cÞ

XJj¼1

wtj ¼ 1; 8t ¼ 1; N ; T ð26dÞ

Solving this problem yields an estimate for parameter θx. For factor y for which there is noadditional information, the estimates θy should equal 1 to resemble the linear path, just becausethe central value b⁎ should be set to 1. Upon having obtained estimates for θx and θy, it isimmediately possible to obtain the estimated respective contributions of changes in the deter-minants just by substituting their values in the following equations:

Dx Effect ¼ Dxy0 þ hxhx þ hy

DxDy ð27aÞ

Dy Effect ¼ x0Dy þ hyhx þ hy

DxDy ð27bÞ


which are the reduced versions for the two factor case of Eqs. (13a) (13b) (13c) (13d). Theuse of the framework outlined above can cause nontrivial problems if observations forintermediate periods of x(t) are rather unlikely to be generated by a time path belonging tothe class of paths defined by Eq. (12). We deal with such observations by fitting themost appropriate monotonic paths. Appendix A explains in detail how we handle thesesituations.

6. Illustration: decomposition of changes in greenhouse gas emissions in the EU 15

In this section the technique previously developed will be applied to study the contributions ofthree factors to the changes in the greenhouse gas emissions in the European Union with 15countries8 (EU 15) over the period from 1990 to 2002. It should be stressed already from thebeginning that the aim of this section is not so much to provide a detailed description of thedynamics of greenhouse gas emissions in the EU 15, but rather to provide an illustration of themethodology proposed in the present paper. The required data were taken from the Eurostat: NewCronos data base.

6.1. The decomposition model and the estimation procedure

The point of departure is a well known equation that expresses the quantity of greenhousegas emissions in a country i(Ei) as the product of three factors, i.e. greenhouse gas emissions perunit of national GDP (intensity factor Ii), the weight of the GDP of the country over the GDP ofthe whole EU 15 (structural factor Si) and, finally, the GDP of the whole EU 15 (productionfactor Y ):

Ei ¼ IiSiY ð28Þ

where Ii ¼ EiYi; Si ¼ YiP15

i¼1 Yi¼ Yi

Y and Yi denotes the GDP of a country i. The objective is to

decompose the total change ΔEi into the following three components:

E2002i −E1990

i ¼ DEi ¼ DIi Effectþ DSi Effect þ DY Effect ð29Þ

We assume the following temporal paths for the elements of the factors:

IiðtÞ ¼ I1990i þ DIithIi ; i ¼ 1; N ; 15 ð30Þ

SiðtÞ ¼ S1990i þ DSithSi ; i ¼ 1; N ; 15 ð31Þ

Y ðtÞ ¼ Y 1990 þ DYthY ð32Þ

8 These countries are: Belgium (B), Denmark (DK), Germany (D), Greece (EL), Spain (E), France (F), Ireland (IRL),Italy (I), Luxembourg (L), The Netherlands (NL), Austria (A), Portugal (P), Finland (FIN), Sweden (S) and UnitedKingdom (UK).


According to Eqs. (13a) (13b) (13c) (13d), the contributions of changes in elements of Ii, Siand Y to the changes in Ei can be written as:

DIi Effect ¼ DIiS1990i Y 1990 þ hIi

hIi þ hSiDIiDSiY

1990 þ hIihIi þ hY

DIiS1990i DY

þ hIihIi þ hSi þ hY

DIiDSiDY ð33aÞ

DSi Effect ¼ I1990i DSiY1990 þ hSi

hIi þ hSiDIiDSiY

1990 þ hSihSi þ hY

I1990i DSiDY

þ hSihIi þ hSi þ hY

DIiDSiDY ð33bÞ

DY Effect ¼ I1990i S1990i DY þ hYhIi þ hY

DIiS1990i DY þ hY

hSi þ hYI1990i DSiDY

þ hYhIi þ hSi þ hY

DIiDSiDY ð33cÞ

As argued before, assuming that parameters θ are the same for all the factors (i.e. θIi=θSi=θY,∀i) would yield Sun's solution (1998). This would be a natural reasoning if no informationwere available for the years between 1990 and 2002.

In order to illustrate the techniques outlined in the previous sections, we will estimate some ofthe θ parameters by supposing one situation with some additional information. Specifically, wewill assume a scenario where the additional information is the national GDPs of the countries ofthe EU 15 in three intermediate years: 1992, 1996 and 1999. Note that this information is relatedto only two of the factors of the decomposition model, namely Si and Y, and it will be possible toestimate the parameters θSi and θY by means of GME econometrics.

Firstly, we had to decide on the values to be assigned to the a priori distributions contained inthe support vector b and the possible realizations for the random term in vectors v. The followingvector was used for all parameters throughout the empirical analyses below:9

b ¼ ½−8:0; −5:0; −2:0; 1:0; 4:0; 7:0; 10:0� VNote that to the central value of b, namely b⁎, is assigned a value 1 since the solution of the

constrained maximization without additional information yields estimates equal to this value b⁎.This means that, with a lack of information, there is no reasons to assume either a convex (i. e.,θb1) or concave (i. e., θN1) path. As a consequence, for factor Ii for which we suppose that thereis no additional information, the estimates θ1i should equal 1. For the supporting vector v we hadtaken the most popular 3 σ rule proposed in Golan et al. (1996), consisting in taking as limits ofthis vector three times the standard deviation of the dependent variable.10

9 Golan et al. (1996, p. 138) asserted that the estimation results are generally not very sensitive to the choice of aparticular set of vectors if the true parameters are bounded between the established limits. Nevertheless, in Appendix A asimple numerical analysis is made to test the sensitivity of the results to the choice of the supporting vectors.10 It is important to note that if we had assumed a situation with less information the methodology could be applied aswell: for example, supposing that the additional information was only the overall GDP for the EU, in such a case it wouldimply the estimation of only one parameter (θY).

Table 1Additional information a and estimates of the parameters

Country (1) (2) (3) (4) (5) (6)

Yi(1993) Yi(1996) Yi(1999) θIi θSi θY

B 18,4466.3 212,421.5 235,683 1.00 1020 1.27DK 118,541.2 144,155.2 162,430.1 1.00 0.684 1.27D 1,711,383.9 1,921,660.5 2,012,000 1.00 1020 1.27EL 79,771.3 97,972.9 117,849.5 1.00 0.907 1.27E 425,936 480,535.4 565,419 1.00 1020 1.27F 1,089,369.4 1,224,606.3 1,355,102 1.00 3.618 1.27IRL 42,569.9 57,468.8 89,457 1.00 2.186 1.27I 849,036.8 971,065 1,107,994.2 1.00 0.373 1.27L 11,804.7 14,297.3 18,739.1 1.00 0.819 1.27NL 276,821.7 324,479.1 374,070 1.00 0.681 1.27A 161,880.6 186,282.8 200,025.3 1.00 0.298 1.27P 73,635.4 88,309.8 108,029.9 1.00 0.678 1.27FIN 73,770.7 100,623.6 119,985 1.00 0.298 1.27S 169,274.6 213,177.1 235,767.8 1.00 0.414 1.27UK 822,693.5 937,100.4 1,371,052.3 1.00 4.384 1.27

a Values of columns (1) to (3) measured in millions of Euro.


Let us define hSi ¼ b VpSi ¼PM

m¼1 bm pSim andhY ¼ b VpY ¼ PMm¼1 bm pYm where pSi and pY are

unknown probability distributions. We rewrite the temporal paths for factors Si and Y including astochastic element in the same fashion as we commented in the previous section for Eq. (25).The estimates obtained here will be used to obtain the respective contributions for the effect ofchanges in the three factors considered.

6.2. Contributions of changes in the determinants

In order to apply the methodology proposed in this paper we need to consider some additionalinformation of the factors involved in the decomposition model. As commented in the previoussubsection, we will include as additional information the national GDP values for some intermediateperiods. Columns (1) to (3) of Table 1 report the values of the national GDP of the countries of theEU 15 in three intermediate years: 1992, 1996 and 1999, these data being obtained from theEurostat: New Cronos data base. These observations lead to the estimates of the parameters θIi,θSiand θY of columns (4) to (6) that characterize the temporal paths (Eqs. (32) (33a) (33b) (33c) (34)).Note that the additional information of columns (1) to (3) is related to factors Si and Y, and wesuppose that there is no information in intermediate years for factor Ii.

Consequently, the estimates of θIi equal 1 for each country, as we reasoned before.11

Table 2 reports the periodwise decomposition of changes in greenhouse gas emissions (column(1)) through the PB approach (columns (2) to (4)). Note that the decomposition is exhaustivebecause the sum of the effects equals the whole variation in the greenhouse gas emissionsreported in column (1). These effects are obtained by including the estimates of the parametersfrom Table 1 in the decomposition forms (Eqs. (33a) (33b) (33c)).

11 See Appendix A for a more detailed explanation of how these estimates are obtained depending on the type ofintermediate observations of Yi.

Table 2Decomposition results a of the PB approach

Country (1) (2) (3) (4) (5) (6) (7)

ΔEib ΔIi Eff ΔSi Eff ΔY Eff ρI ρS ρY

B 4257.4 −73,851.4 −1269.8 79,378.6 97.39 97.56 97.53DK −260.0 −38,619.2 1423.5 36,935.7 97.13 96.70 97.13D −232,606.4 −801,346.0 −61,256.9 629,996.5 100.04 87.30 98.65EL 30,597.2 −62,390.3 28,149.3 64,838.2 97.53 96.70 99.05E 115,176.8 −71,489.3 10,022.8 176,643.3 95.62 113.46 97.53F −10,844.8 −279,649.3 −35,316.8 304,121.3 98.66 98.39 98.59IRL 15,455.1 −59,718.4 43,777.8 31,395.8 85.24 89.64 85.60I 44,854.9 −148,688.4 −77,683.4 271,226.6 93.70 91.51 94.03L −1920.4 −14,831.4 5816.0 7095.0 98.98 98.91 98.77NL 2381.1 −141,422.4 27,013.9 116,789.6 98.05 97.38 98.25A 6873.1 −36,693.6 266.7 43,300.0 97.03 94.14 97.51P 23707.9 −35,334.7 20,603.4 38,439.1 99.60 96.14 101.82FIN 5192.9 −14,625.7 −19,259.3 39,077.9 90.38 88.50 90.59S −2538.9 −23,322.5 −15,112.6 35,896.2 92.71 92.92 92.32UK −107,781.3 −613,140.7 147053.1 358,306.2 90.00 85.53 89.23Total −107,455.4 −2,415,123.4 74,227.9 2,233,440.0 95.74 87.08 95.86a Values of columns (1) to (4) measured in 1000 tons of CO2 equivalent.b Columns (2) to (4) do not always add up to the numbers in column (1) due to rounding.


Table 2 also compares the results of the proposed decomposition method with those obtainedby Sun's periodwise methodology, resulting from a situation where θIi=θSi=θY, i.e. a situationwhere no additional information of the factors is considered. This comparison is important for thefollowing reason: if there were no important differences between these two techniques, thesupplementary computational work that the application of the PB approach requires it would benot worthwhile. Ratios ρ of columns (5) to (7) make this comparison in percentage.12

The values for the EU 15 as a whole in the bottom row are obtained by simply adding thenational results. It can be seen how there has been a great reduction in the greenhouse gasemissions in the EU 15 between 1990 and 2002 (column (1)), mainly given by a reduction inFrance, Germany and the United Kingdom. Clearly, the shift to less contaminant technologieswould have led to lower greenhouse gas emissions in all the countries if nothing else had changed(the ΔIi effect is negative for all of them). In contrast, positive results for the ΔY Effect suggestthat the GDP in the whole EU 15 has changed in such a way that greenhouse gas emissions wouldhave increased, if the structural and intensity effect had equalled zero. Finally, the contributions ofthe ΔSi Effect are generally positive, although countries like Belgium, Germany, France, Italy,Finland and Sweden are the exceptions. For these nations, changes in the weight of their nationalGDPs over the whole production of the EU 15 have reduced their greenhouse gas emissions,whereas the opposite outcome has happened for the rest of the countries.

Besides the results of the decomposition, columns (5) to (7) are also interesting, because theypresent ratios that compare the effects obtained by the PB approach with those obtained by Sun'smethodology.13 Avalue of the ratio equal to 100 implies that the same result would be obtained by

12 Specifically, these ρ ratios have been computed as 100 PB effectSun′s method effect

.13 The reader could think that is not possible that all the ratios are smaller than 100, since positive and negativedeviations have to compensate each other if we add up the three effects. However, note that negative values for some ofthe effects prevent this.

Table 3Comparison of the results with a Sun's time series decomposition a

ΔIi Eff ΔSi Eff ΔY Eff eI eS eY Total

Sun's time series −2,391,228 95,476.8 2,188,296 – – – –Sun's periodwise −2,522,667.2 85,242.6 2,329,969.2 131,439.0 10,234.2 269.2 14,1942.4PB approach −2,415,123.4 74,227.9 2,233,440.0 23,895.1 21,248.9 228.4 45,372.4a Values measured in 1000 tons of CO2 equivalent.


both approaches, in other words, the information included does not lead to results remarkablydifferent from a “non informative situation”.

As one can see from these columns, the deviations are sometimes considerable. Although ingeneral terms the results obtained by the PB approach are quite close to Sun's decompositionresults for ΔY and ΔIi Effects, for the ΔSi Effect there is a more remarkable difference around13% between both methodologies. This is a rather logical result, since the additionalinformation included is about this last factor. The results for some specific countries show alarger variability: the case of Ireland is a good example, because if we apply the PB approach tothis country we will obtain results for the ΔIi, ΔSi and ΔY Effects around 15%, 11% and 15%smaller, respectively, than the results obtained by Sun's decomposition model. Finland or theUnited Kingdom are other examples of great variations for all the effects.

6.3. Comparison with a Sun's time series decomposition

Outcomes of Table 2 show that the results of using the PB approach were remarkably differentfrom Sun's methodology for some specific cases. This result suggests that the technique proposedin the paper could be considered as an alternative decomposition methodology to more traditionalprocedures. Consequently, a question arises: which one (the PB approach or the Sun's decom-position method) obtains the most “accurate” outcomes? This subsection is devoted to testing ifthe results of PB decomposition (that allows the use of limited additional information) aresomehow more “consistent” than those obtained by using Sun's method.

Section 5 of the present paper pointed out how a dynamic approach reduces the gravity of the nonuniqueness problem (see Fig. 3). The disadvantage of dynamic decompositions is that a largeamount of data is usually required. For instance, a time series decomposition with annual data from1990 to 2002 of the three factors could have been computed, since Eurostat provides thisinformation. Note that for this time series decomposition is necessary to get observations for all thefactors involved in the decomposition equation. Given these data, the Sun's method has beenapplied in each one of the 11 stages (obtaining a Sun's time series decomposition). These outcomeshave been taken as the benchmark to compare the results of the PB and Sun's methodology (boththem periodwise), since we will consider as the “most accurate” decomposition methodology theone that obtains outcomes most similar to this base scenario. Table 3 reports the overall resultsobtained by the Sun's time series decomposition, the Sun's periodwise decomposition and the PBapproach and makes a comparison among them obtaining the differences:14

Note that, considering the sum of the three effects, the results obtained by the PB technique arethe closest to the Sun's time series decomposition. The conclusion would be that, using only alimited amount of additional information, one would obtain more “accurate” results than applyingthe solution proposed by Sun that considers only initial and final data. Of course, this result

14 The detailed results of the squared differences for each country and type of effect are shown in Appendix B.


applies only to the example taken as illustration, so more empirical exercises have to be made tocompare the performance of the PB technique compared with other decomposition metho-dologies. This could be of special interest if the decomposition analysis is made at regional orsmaller scale level, where the lack of availability of information for all the factors often disablesthe application of a dynamic time series decomposition.

7. Conclusions

A well known problem of decomposition analysis is that the results often strongly dependon the specific decomposition formula chosen, whereas numerous formulae are equivalentfrom a theoretical point of view. A previous solution to this non uniqueness problemwas proposed by Sun (1998). This paper suggests a decomposition methodology usingGeneralized Maximum Entropy (GME) econometrics to select the decomposition formula thatprovides an optimal “fit” to additional empirical information. The point of departure is thePath Based (PB) method, showing that this technique can be seen as a generalization of Sun'smethodology. Since the solutions of this method depend on unknown parameters thatcharacterize the paths of the determinants, these parameters can be estimated even if theavailable data is very limited. If some limited additional information for the factors isavailable for intermediate periods, a non linear GME program can be solved to estimate themand obtain a unique decomposition.

We applied the methodology to quantify the contributions of three determinants to the changesin the greenhouse gas emissions in the EU 15 from 1990 to 2002, i.e. greenhouse gas emissionsper unit of national GDP, the weight of the national GDP over the EU 15 GDP and the GDP of thewhole EU 15. As additional information, data of the national GDP of the countries of the EU 15 in1992, 1996 and 1999 were included. The results show that, firstly, the use of additional infor-mation in the PB approach can well yield results that differ substantially from the Sun'speriodwise decomposition. Secondly, when a Sun's time series decomposition is carried out to beused as a reference in order to compare both PB and Sun's approaches, the PB method yieldscloser results to time series decomposition than those obtained by Sun's periodwise methodology.This result leads us to think that the PB method provides an interesting alternative to previousdecomposition approaches in the literature.

Acknowledgement

The authors are grateful to one anonymous referee whose comments have improved the finalquality of the paper.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, atdoi:10.1016/j.eneco.2007.01.004.

References

Albrecht, J., Francois, D., Schoors, K., 2002. A Shapley decomposition of carbon emissions without residuals. EnergyPolicy 30 (9), 727–736.

Ang, B.W., 1995. Decomposition methodology in industrial energy demand analysis. Energy 20 (11),1081–1095.


Ang, B.W., 2004. Decomposition analysis for policymaking in energy: which is the preferred method? Energy Policy 32(9), 1131–1139.

Ang, B.W., Lee, S.Y., 1994. Decomposition of industrial energy consumption: some methodological and applicationissues. Energy Economics 16 (2), 83–92.

Ang, B.W., Choi, K.H., 1997. Decomposition of aggregate energy and gas emission intensities for industry: a refinedDivisia Index Method. Energy Journal 18 (3), 59–74.

Ang, B.W., Zhang, F.Q., 2000. A survey of index decomposition analysis in energy and environmental studies. Energy 25(12), 1149–1176.

Ang, B.W., Liu, F.L., Chew, E.P., 2003. Perfect decomposition techniques in energy and environmental analysis. EnergyPolicy 31 (14), 1561–1566.

Ang, B.W., Liu, F.L., Chung, H.S., 2004. A Generalized Fisher Index approach to energy decomposition analysis. EnergyEconomics 26 (5), 757–763.

Boyd, G.D, Roop, J.M., 2004. A note on the Fisher Ideal Index decomposition for structural change in energy intensity.Energy Journal 25 (1), 87–101.

Boyd, G., Hanson, D.A., Sterner, T., 1988. Decomposition of changes in energy intensity: a comparison of the DivisiaIndex and other methods. Energy Economics 10, 309–312.

Chung, H.S., Rhee, H.C., 2001. A residual-free decomposition of the sources of carbon dioxide emissions: a case of theKorean industries. Energy 26 (1), 15–30.

Dietzenbacher, E., Los, B., 1998. Structural decomposition techniques: sense and sensitivity. Economic Systems Research10, 307–323.

Diewert, W.E., 1980. Recent developments in the economic theory of index numbers: capital and the theory of pro-ductivity. American Economic Review 70 (2), 260–267.

Eurostat: New Cronos data base, Statistical Office of the European Communities Regions, Luxembourg.Fernández, E. The Use of Entropy Econometrics in Decomposing Structural Change. Unpublished PhD thesis, University

of Oviedo (Spain); 2004.Fisher, I., 1922. The Making of Index Numbers, 3rd ed. Houghton Mifflin, Boston.Fraser, I., 2000. An application of maximum entropy estimation: the demand for meat in the United Kingdom. Applied

Economics 32 (1), 45–59.Gardebroek, C., Oude Lansink, A., 2004. Farm-specific Adjustment costs in Dutch pig farming. Journal of Agricultural

Economics 55 (1), 3–24.Golan, A., Judge, G., Miller, D., 1996. Maximum Entropy Econometrics: Robust Estimation with Limited Data. John

Wiley, Chichester, UK.Golan, A., Perloff, J.M., Shen, E.Z., 2001. Estimating a demand system with nonnegativity constraints: Mexican meat

demand. Review of Economics and Statistics 83 (3), 541–550.Harrison, W.J., Horridge, J.M., Pearson, K.R., 2000. Decomposing simulation results with respect to exogenous shocks.

Computational Economics 15, 227–249.Jenne, C., Cattell, R., 1983. Structural change and energy efficiency in industry. Energy Economics 5 (2),

114–123.Kapur, J.N., Kesavan, H.K., 1993. Entropy Optimization Principles with Applications. Academic Press,

New York.Liu, X.Q., Ang, B.W., Ong, H.L., 1992. The application of the Divisia Index to the decomposition of changes in industrial

energy consumption. Energy Journal 13 (4), 161–177.Paris, Q., Howitt, R.E., 1998. An analysis of ill-posed production problems using maximum-entropy. American Journal of

Agricultural Economics 80, 124–138.Reitler, W., Rudolph, M., Schaefer, M., 1987. Analysis of the factors influencing energy consumption in industry: a revised

method. Energy Economics 9, 145–148.Shannon, J., 1948. A mathematical theory of communication. Bell System Technical Bulletin Journal 27,

379–423.Sun, J.W., 1998. Changes in energy consumption and energy intensity: a complete decomposition model. Energy Eco-

nomics 20, 85–100.Sun, J.W., Ang, B.W., 2000. Some properties of an exact decomposition model. Energy 25, 1177–1188.Vogt, A., 1978. Divisia Indices on different paths. In: Eichhorn, W., et al. (Ed.), Theory and Application of Economic

Indices. Physica-Verlag, Wurzburg.

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An extension to Sun's decomposition methodology: The Path Based approach