Decomposition analysis: when to use which method?

Figures & data

TABLE 1. Summary for the case of two factors.

Number of ones	Combinations	Number of duplicates	Weight
0	{0}	1	$1/2$$1/2$
1	{1}	1	$1/2$$1/2$

TABLE 2. Fulfilment of relevant properties of ideal indices and indicators.

	Fisher	Montgomery-Vartia	Sato-Vartia	Bennet	Montgomery	ASV
Monotonicity	Yes	No	No	Yes	No	No
Proportionality	Yes	No	Yes	N.A.	N.A.	N.A.
Consistency-in-aggregation	No	Yes	No	Yes	Yes	No

TABLE 3. Decomposition of change in value into price and quantity effect of ideal methods.

Name	Weight	Price effect	Quantity effect
Fisher index	N.A.	${\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^1{q}_i^0}{\sum _{i=1}^N{p}_i^0{q}_i^0}}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^1{q}_i^1}{\sum _{i=1}^N{p}_i^0{q}_i^1}}\right)}^{{\displaystyle \frac{1}{2}}}$${\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^1{q}_i^0}{\sum _{i=1}^N{p}_i^0{q}_i^0}}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^1{q}_i^1}{\sum _{i=1}^N{p}_i^0{q}_i^1}}\right)}^{{\displaystyle \frac{1}{2}}}$	${\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^0{q}_i^1}{\sum _{i=1}^N{p}_i^0{q}_i^0}}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^1{q}_i^1}{\sum _{i=1}^N{p}_i^1{q}_i^0}}\right)}^{{\displaystyle \frac{1}{2}}}$${\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^0{q}_i^1}{\sum _{i=1}^N{p}_i^0{q}_i^0}}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{\sum _{i=1}^N{p}_i^1{q}_i^1}{\sum _{i=1}^N{p}_i^1{q}_i^0}}\right)}^{{\displaystyle \frac{1}{2}}}$
Bennet indicator	N.A.	$\sum _{i=1}^N{\displaystyle \frac{\left(q_i^0+q_i^1\right)}{2}}\left(p_i^1-p_i^0\right)$$\sum _{i=1}^N{\displaystyle \frac{\left(q_i^0+q_i^1\right)}{2}}\left(p_i^1-p_i^0\right)$	$\sum _{i=1}^N{\displaystyle \frac{\left(p_i^0+p_i^1\right)}{2}}\left(q_i^1-q_i^0\right)$$\sum _{i=1}^N{\displaystyle \frac{\left(p_i^0+p_i^1\right)}{2}}\left(q_i^1-q_i^0\right)$
Montgomery indicator	$w_i^M=L\left(v_i^1,v_i^0\right)$$w_i^M=L\left(v_i^1,v_i^0\right)$	$\sum _{i=1}^N{w}_i^M\text{ln}\left({\displaystyle \frac{p_i^1}{p_i^0}}\right)$$\sum _{i=1}^N{w}_i^M\text{ln}\left({\displaystyle \frac{p_i^1}{p_i^0}}\right)$	$\sum _{i=1}^N{w}_i^M\text{ln}\left({\displaystyle \frac{q_i^1}{q_i^0}}\right)$$\sum _{i=1}^N{w}_i^M\text{ln}\left({\displaystyle \frac{q_i^1}{q_i^0}}\right)$
Montgomery-Vartia index	$w_i^{MV}={\displaystyle \frac{w_i^M}{L\left(V^1,V^0\right)}}$$w_i^{MV}={\displaystyle \frac{w_i^M}{L\left(V^1,V^0\right)}}$	${\displaystyle \prod _{i=1}^N{\left(\frac{p_i^1}{p_i^0}\right)}^{w_i^{MV}}}$${\displaystyle \prod _{i=1}^N{\left({\displaystyle \frac{p_i^1}{p_i^0}}\right)}^{w_i^{MV}}}$	${\displaystyle \prod _{i=1}^N{\left(\frac{q_i^1}{q_i^0}\right)}^{w_i^{MV}}}$${\displaystyle \prod _{i=1}^N{\left({\displaystyle \frac{q_i^1}{q_i^0}}\right)}^{w_i^{MV}}}$
Sato-Vartia index	$w_i^{SV}={\displaystyle \frac{L\left(s_i^1,s_i^0\right)}{\sum _{i=1}^{N}L\left(s_i^1,s_i^0\right)}}$$w_i^{SV}={\displaystyle \frac{L\left(s_i^1,s_i^0\right)}{\sum _{i=1}^{N}L\left(s_i^1,s_i^0\right)}}$	${\displaystyle \prod _{i=1}^N{\left(\frac{p_i^1}{p_i^0}\right)}^{w_i^{SV}}}$${\displaystyle \prod _{i=1}^N{\left({\displaystyle \frac{p_i^1}{p_i^0}}\right)}^{w_i^{SV}}}$	${\displaystyle \prod _{i=1}^N{\left(\frac{q_i^1}{q_i^0}\right)}^{w_i^{SV}}}$${\displaystyle \prod _{i=1}^N{\left({\displaystyle \frac{q_i^1}{q_i^0}}\right)}^{w_i^{SV}}}$
Additive Sato-Vartia indicator	$w_i^{ASV}=L\left(V^1,V^0\right)w_i^{SV}$$w_i^{ASV}=L\left(V^1,V^0\right)w_i^{SV}$	$\sum _{i=1}^N{w}_i^{ASV}\text{ln}\left({\displaystyle \frac{p_i^1}{p_i^0}}\right)$$\sum _{i=1}^N{w}_i^{ASV}\text{ln}\left({\displaystyle \frac{p_i^1}{p_i^0}}\right)$	$\sum _{i=1}^N{w}_i^{ASV}\text{ln}\left({\displaystyle \frac{q_i^1}{q_i^0}}\right)$$\sum _{i=1}^N{w}_i^{ASV}\text{ln}\left({\displaystyle \frac{q_i^1}{q_i^0}}\right)$

TABLE 4. Summary of names of methods.

Ratio change		Difference change
Index	Multiplicative decomposition	Indicator	Additive decomposition
Fisher	Dietzenbacher- Hoen-Los or Generalized Fisher	Bennet	Dietzenbacher- Los or Sun-Shapley
Montgomery-Vartia	LMDI- I	Montgomery	LMDI- I
Sato-Vartia	LMDI- II	Additive Sato-Vartia	LMDI- II

TABLE 5. Name of the method, the weight, the effect of a factor and the decomposition.

Name	Weight	Effect of factor	Decomposition
Montgomery (LMDI-I additive)	$w^M=L\left(v^1,v^0\right)$$w^M=L\left(v^1,v^0\right)$	$D_{x_f}^M=\sum _{\left\{I\right\}}{w}^M\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$$D_{x_f}^M=\sum _{\left\{I\right\}}{w}^M\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$	$\sum _{f=1}^n{D}_{x_f}^M$$\sum _{f=1}^n{D}_{x_f}^M$
Montgomery- Vartia (LMDI-I multiplicative)	$w^{MV}={\displaystyle \frac{w^M}{L\left(V^1,V^0\right)}}$$w^{MV}={\displaystyle \frac{w^M}{L\left(V^1,V^0\right)}}$	$\text{ln}\left(R_{x_f}^{MV}\right)=\sum _{\left\{I\right\}}{w}^{MV}\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$$\text{ln}\left(R_{x_f}^{MV}\right)=\sum _{\left\{I\right\}}{w}^{MV}\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$	$\prod _{f=1}^n{R}_{x_f}^{MV}$$\prod _{f=1}^n{R}_{x_f}^{MV}$
Sato-Vartia (LMDI-II multiplicative)	$w^{SV}={\displaystyle \frac{L\left(s^1,s^0\right)}{\sum _{\left\{I\right\}}L\left(s^1,s^0\right)}}$$w^{SV}={\displaystyle \frac{L\left(s^1,s^0\right)}{\sum _{\left\{I\right\}}L\left(s^1,s^0\right)}}$	$\text{ln}\left(R_{x_f}^{SV}\right)=\sum _{\left\{I\right\}}{w}^{SV}\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$$\text{ln}\left(R_{x_f}^{SV}\right)=\sum _{\left\{I\right\}}{w}^{SV}\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$	$\prod _{f=1}^n{R}_{x_f}^{SV}$$\prod _{f=1}^n{R}_{x_f}^{SV}$
Additive Sato-Vartia (LMDI-II additive)	$w^{ASV}=L\left(V^1,V^0\right)w^{SV}$$w^{ASV}=L\left(V^1,V^0\right)w^{SV}$	$D_{x_f}^{ASV}=\sum _{\left\{I\right\}}{w}^{ASV}\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$$D_{x_f}^{ASV}=\sum _{\left\{I\right\}}{w}^{ASV}\text{ln}\left({\displaystyle \frac{x_f^1}{x_f^0}}\right)$	$\sum _{f=1}^n{D}_{x_f}^{ASV}$$\sum _{f=1}^n{D}_{x_f}^{ASV}$

TABLE 6. Summary for the case of five factors.

Appendix A Equation	Number of ones	Combinations				Number of duplicates	Weight
(A.1)	0	{0,0,0,0}				24	1/5
(A.2)	1	{1,0,0,0}	{0,1,0,0}	{0,0,1,0}	{0,0,0,1}	6	1/20
(A.3)	2	{1,1,0,0}	{1,0,1,0}	{1,0,0,1}		4	1/30
(A.4)	2	{0,0,1,1}	{0,1,0,1}	{0,1,1,0}		4	1/30
(A.5)	3	{0,1,1,1}	{1,0,1,1}	{1,1,0,1}	{1,1,1,0}	6	1/20
(A.6)	4	{1,1,1,1}				24	1/5

TABLE 7. Formulas for the methods based on the logarithmic mean.

Factor	Montgomery	Montgomery-Vartia	Sato-Vartia	Additive Sato-Vartia
$E_i$$E_i$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^M\text{ln}\left(\frac{e_i^1}{e_i^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^M\text{ln}\left(\frac{e_i^1}{e_i^0}\right)}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{e_i^1}{e_i^0}\right]}^{w_{ijk}^{MV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[\frac{e_i^1}{e_i^0}\right]}^{w_{ijk}^{MV}}}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{e_i^1}{e_i^0}\right]}^{w_{ijk}^{SV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[\frac{e_i^1}{e_i^0}\right]}^{w_{ijk}^{SV}}}}$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left(\frac{e_i^1}{e_i^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left(\frac{e_i^1}{e_i^0}\right)}}$
$D_i$$D_i$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^M\text{ln}\left(\frac{d_{ij}^1}{d_{ij}^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^M\text{ln}\left({\displaystyle \frac{d_{ij}^1}{d_{ij}^0}}\right)}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{d_{ij}^1}{d_{ij}^0}\right]}^{w_{ijk}^{MV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{d_{ij}^1}{d_{ij}^0}}\right]}^{w_{ijk}^{MV}}}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{d_{ij}^1}{d_{ij}^0}\right]}^{w_{ijk}^{SV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{d_{ij}^1}{d_{ij}^0}}\right]}^{w_{ijk}^{SV}}}}$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left(\frac{d_{ij}^1}{d_{ij}^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left({\displaystyle \frac{d_{ij}^1}{d_{ij}^0}}\right)}}$
$B_i$$B_i$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^M\text{ln}\left(\frac{b_{jk}^1}{b_{jk}^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^M\text{ln}\left({\displaystyle \frac{b_{jk}^1}{b_{jk}^0}}\right)}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{b_{jk}^1}{b_{jk}^0}\right]}^{w_{ijk}^{MV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{b_{jk}^1}{b_{jk}^0}}\right]}^{w_{ijk}^{MV}}}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{b_{jk}^1}{b_{jk}^0}\right]}^{w_{ijk}^{SV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{b_{jk}^1}{b_{jk}^0}}\right]}^{w_{ijk}^{SV}}}}$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left(\frac{b_{jk}^1}{b_{jk}^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left({\displaystyle \frac{b_{jk}^1}{b_{jk}^0}}\right)}}$
$U_i$$U_i$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^M\text{ln}\left(\frac{u_k^1}{u_k^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^M\text{ln}\left({\displaystyle \frac{u_k^1}{u_k^0}}\right)}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{u_k^1}{u_k^0}\right]}^{w_{ijk}^{MV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{u_k^1}{u_k^0}}\right]}^{w_{ijk}^{MV}}}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{u_k^1}{u_k^0}\right]}^{w_{ijk}^{SV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{u_k^1}{u_k^0}}\right]}^{w_{ijk}^{SV}}}}$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left(\frac{y^1}{y^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left({\displaystyle \frac{y^1}{y^0}}\right)}}$
$Y_i$$Y_i$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^M\text{ln}\left(\frac{y^1}{y^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^M\text{ln}\left({\displaystyle \frac{y^1}{y^0}}\right)}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{y^1}{y^0}\right]}^{w_{ijk}^{MV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{y^1}{y^0}}\right]}^{w_{ijk}^{MV}}}}$	${\displaystyle \prod _{j=1}^N\prod _{k=1}^m{\left[\frac{y^1}{y^0}\right]}^{w_{ijk}^{SV}}}$${\displaystyle \prod _{j=1}^N{\displaystyle \prod _{k=1}^m{\left[{\displaystyle \frac{y^1}{y^0}}\right]}^{w_{ijk}^{SV}}}}$	${\displaystyle \sum _{j=1}^N\sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left(\frac{y^1}{y^0}\right)}$${\displaystyle \sum _{j=1}^N{\displaystyle \sum _{k=1}^m{w}_{ijk}^{ASV}\text{ln}\left({\displaystyle \frac{y^1}{y^0}}\right)}}$

TABLE 8. Carbon dioxide emissions (million kg), ratio and difference change, and total final demand (million €) for the five largest emitters and for the total economy.

#	Sector	CO2 2004	CO2 2005	RCO2	DCO2	Final 2004	Final 2005
25	Electricity and gas supply	56,538	55,076	0.974	−1,462	7,689	7,657
13	Chemicals; man-made fibres	15,149	15,215	1.004	66	19,978	20,605
12	Petroleum products, cokes, etc.	12,941	12,826	0.991	−115	13,217	13,097
36	Air transport	12,425	12,940	1.041	515	5,910	6,332
34	Land transport	8,821	8,478	0.961	−343	8,121	8,195
	Five largest Emitters	105,874	104,534		−1,340	54,915	55,866
	Total economy	171,419	168,599	0.984	−2,820	580,936	592,633
	% five largest in total economy	67.9 (33.0)	68.2 (32.7)		52.2 (51.8)	10.4 (1.3)	10.4 (1.3)

TABLE 9. Results of the multiplicative decompositions.

Sector	Method	RE	RD	RB	RU	RY
25	Fisher	0.9093	1.0670	0.9839	1.0010	1.0201
	Sato-Vartia	0.9093	1.0655	0.9853	1.0003	1.0201
	Montgomery-Vartia	0.9102	1.0646	0.9854	1.0003	1.0119
13	Fisher	0.9850	0.9899	1.0002	1.0088	1.0201
	Sato-Vartia	0.9857	0.9899	1.0005	1.0084	1.0201
	Montgomery-Vartia	0.9857	0.9899	1.0005	1.0084	1.0201
12	Fisher	0.9960	0.9962	0.9732	1.0073	1.0201
	Sato-Vartia	0.9949	0.9962	0.9734	1.0071	1.0201
	Montgomery-Vartia	0.9949	0.9962	0.9734	1.0071	1.0201
36	Fisher	0.9734	1.0070	1.0354	1.0065	1.0201
	Sato-Vartia	0.9734	1.0064	1.0356	1.0063	1.0201
	Montgomery-Vartia	0.9734	1.0064	1.0356	1.0063	1.0201
34	Fisher	0.9449	0.9920	0.9980	1.0024	1.0201
	Sato-Vartia	0.9449	0.9920	0.9974	1.0024	1.0201
	Montgomery-Vartia	0.9499	0.9920	0.9974	1.0024	1.0201

TABLE 10. Results of the three additive decompositions.

Sector	Method	DE	DD	DB	DU	DY
25	Bennet	−5,314	3,662	−912	17	1,115
	Montgomery	−5,253	3,492	−821	18	1,102
	Additive S-V	−5,304	3,540	−829	18	1,112
13	Bennet	−219	−154	3	132	303
	Montgomery	−218	−153	8	128	303
	Additive S-V	−219	−153	8	128	303
12	Bennet	−66	−49	−351	94	257
	Montgomery	−66	−49	−347	91	257
	Additive S-V	−66	−49	−348	91	257
36	Bennet	−342	81	441	82	253
	Montgomery	−341	81	443	80	253
	Additive S-V	−342	81	443	80	253
34	Bennet	−445	−69	−22	20	172
	Montgomery	−445	−70	−22	21	172
	Additive S-V	−445	−70	−22	21	172

TABLE 11. Summary of methods and of their relevant properties.

Name	Change-in-sign robust	Proportionality*	Consistent- in- aggregation	Complete at disaggregate level	Simplicity of implementation
Fisher	Yes	Yes	No	Yes	Low
Montgomery- Vartia	No	No	Yes	Yes	High
Sato-Vartia	No	Yes	No	Yes	High
Bennet	Yes	N.A.	Yes	Yes	Low
Montgomery	No	N.A.	Yes	Yes	High
Additive Sato-Vartia	No	N.A.	No	No	High

*There is no analogue of the axiom of proportionality for indicators.