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A note on input-output linkage measures
Umed Temurshoev
Department of Economics and BusinessUniversity of Groningen, PO Box 8009700 AV Groningen, The Netherlands
This project is funded by the European Commission, Research Directorate Generalas part of the 7th Framework Programme, Theme 8: Socio-Economic Sciences and
Humanities. Grant Agreement no: 225 281
1st WIOD Conference, Vienna, Austria, June 25-28, 2010
(RuG) Linkages RuG 1 / 22
Outline
1 Introduction
2 IO linkages: traditional approach
3 Generalized IO linkages
4 Empirical exercises
5 Conclusion
(RuG) Linkages RuG 2 / 22
Introduction
Input-output (IO) linkages
Applications of IO linkages (Miller and Blair, 2009, Chapter 12)
Comparative analyses of production structures of economiesStructural change, impact analysisKey sector identification (output, income, greenhouse gases, etc.)
We consider six widely used IO linkage measures
gross output approachgeneralized IO setting (less used, but more relevant)
(RuG) Linkages RuG 3 / 22
Introduction
Input-output (IO) linkages
Applications of IO linkages (Miller and Blair, 2009, Chapter 12)
Comparative analyses of production structures of economiesStructural change, impact analysisKey sector identification (output, income, greenhouse gases, etc.)
We consider six widely used IO linkage measures
gross output approachgeneralized IO setting (less used, but more relevant)
(RuG) Linkages RuG 3 / 22
Introduction
Purpose of this note
How are the IO linkages interrelated?
Derive the closed-form expressions for three hypothetical extractionlinkages.
Propose a net forward linkage as a ‘counterpart’ of Oosterhaven andStelder’s (2002) net backward linkage measure.
Empirical evidence on the link between these linkages.
(RuG) Linkages RuG 4 / 22
Introduction
Purpose of this note
How are the IO linkages interrelated?
Derive the closed-form expressions for three hypothetical extractionlinkages.
Propose a net forward linkage as a ‘counterpart’ of Oosterhaven andStelder’s (2002) net backward linkage measure.
Empirical evidence on the link between these linkages.
(RuG) Linkages RuG 4 / 22
Introduction
Purpose of this note
How are the IO linkages interrelated?
Derive the closed-form expressions for three hypothetical extractionlinkages.
Propose a net forward linkage as a ‘counterpart’ of Oosterhaven andStelder’s (2002) net backward linkage measure.
Empirical evidence on the link between these linkages.
(RuG) Linkages RuG 4 / 22
Introduction
Purpose of this note
How are the IO linkages interrelated?
Derive the closed-form expressions for three hypothetical extractionlinkages.
Propose a net forward linkage as a ‘counterpart’ of Oosterhaven andStelder’s (2002) net backward linkage measure.
Empirical evidence on the link between these linkages.
(RuG) Linkages RuG 4 / 22
IO linkages: traditional approach
Leontief model
x = Ax + y, where A = Zx̂−1 is the input matrix. Thus,
x = Ly, (1)
where L = (I− A)−1 is the Leontief inverse (Leontief 1936, 1941).
The total backward linkage of sector i is
bi = moi =
n∑k=1
lki . (2)
(RuG) Linkages RuG 5 / 22
IO linkages: traditional approach
Leontief model
x = Ax + y, where A = Zx̂−1 is the input matrix. Thus,
x = Ly, (1)
where L = (I− A)−1 is the Leontief inverse (Leontief 1936, 1941).
The total backward linkage of sector i is
bi = moi =
n∑k=1
lki . (2)
(RuG) Linkages RuG 5 / 22
IO linkages: traditional approach
Ghosh model
x′B + v′ = x′, where B = x̂−1Z is the output matrix. Thus,
x′ = v′G, (3)
where G = (I− B)−1 is the Ghosh inverse (Ghosh 1958).
The total forward linkage of sector i is
fi =n∑
k=1
gik . (4)
(RuG) Linkages RuG 6 / 22
IO linkages: traditional approach
Ghosh model
x′B + v′ = x′, where B = x̂−1Z is the output matrix. Thus,
x′ = v′G, (3)
where G = (I− B)−1 is the Ghosh inverse (Ghosh 1958).
The total forward linkage of sector i is
fi =n∑
k=1
gik . (4)
(RuG) Linkages RuG 6 / 22
IO linkages: traditional approach
Hypothetical extraction method
To find the (relative stimulative) importance of sector i , (Strassert 1968,Schultz 1977),
delete the i-th row and column of the input matrix A,
using Leontief model compute the reduced outputs (with yi = 0),
find the difference between total output before and after theextraction, i.e,total linkage of sector i = ı′x− ı′x−i (ı - summation vector).
(RuG) Linkages RuG 7 / 22
IO linkages: traditional approach
Hypothetical extraction method
To find the (relative stimulative) importance of sector i , (Strassert 1968,Schultz 1977),
delete the i-th row and column of the input matrix A,
using Leontief model compute the reduced outputs (with yi = 0),
find the difference between total output before and after theextraction, i.e,total linkage of sector i = ı′x− ı′x−i (ı - summation vector).
(RuG) Linkages RuG 7 / 22
IO linkages: traditional approach
Hypothetical extraction method
To find the (relative stimulative) importance of sector i , (Strassert 1968,Schultz 1977),
delete the i-th row and column of the input matrix A,
using Leontief model compute the reduced outputs (with yi = 0),
find the difference between total output before and after theextraction, i.e,total linkage of sector i = ı′x− ı′x−i (ı - summation vector).
(RuG) Linkages RuG 7 / 22
IO linkages: traditional approach
Non-complete HEM
Distinguish between backward and forward HE linkages(Dietzenbacher and van der Linden 1997)
DenoteA−i
c - nullify i-th column of AB−i
r - nullify i-th row of B
Then, x−ic = (I− A−i
c )−1y and (x−ir )′ = v′(I− B−i
r )−1
Hypothetical extraction backward and forward linkages of i are
bhi = ı′x− ı′x−i
c and f hi = ı′x− ı′x−i
r (5)
(RuG) Linkages RuG 8 / 22
IO linkages: traditional approach
Non-complete HEM
Distinguish between backward and forward HE linkages(Dietzenbacher and van der Linden 1997)
DenoteA−i
c - nullify i-th column of AB−i
r - nullify i-th row of B
Then, x−ic = (I− A−i
c )−1y and (x−ir )′ = v′(I− B−i
r )−1
Hypothetical extraction backward and forward linkages of i are
bhi = ı′x− ı′x−i
c and f hi = ı′x− ı′x−i
r (5)
(RuG) Linkages RuG 8 / 22
IO linkages: traditional approach
Non-complete HEM
Distinguish between backward and forward HE linkages(Dietzenbacher and van der Linden 1997)
DenoteA−i
c - nullify i-th column of AB−i
r - nullify i-th row of B
Then, x−ic = (I− A−i
c )−1y and (x−ir )′ = v′(I− B−i
r )−1
Hypothetical extraction backward and forward linkages of i are
bhi = ı′x− ı′x−i
c and f hi = ı′x− ı′x−i
r (5)
(RuG) Linkages RuG 8 / 22
IO linkages: traditional approach
Non-complete HEM
Distinguish between backward and forward HE linkages(Dietzenbacher and van der Linden 1997)
DenoteA−i
c - nullify i-th column of AB−i
r - nullify i-th row of B
Then, x−ic = (I− A−i
c )−1y and (x−ir )′ = v′(I− B−i
r )−1
Hypothetical extraction backward and forward linkages of i are
bhi = ı′x− ı′x−i
c and f hi = ı′x− ı′x−i
r (5)
(RuG) Linkages RuG 8 / 22
IO linkages: traditional approach
Net backward linkage
Oosterhaven and Stelder’s (2002) key sector indicator
Net backward linkage of sector i
bni =
biyi
xi, (6)
i.e., output generated in all sectors due to final demand of sector ioutput generated in sector i due to final demands of all sectors .
factorback
(RuG) Linkages RuG 9 / 22
IO linkages: traditional approach
Output worth measure
What is the closed form of the complete HEM linkage, ı′x− ı′x−i?
It is given by the output worth of i (Temurshoev 2010)
ωoi =
bixi
lii(7)
Besides multiplier bi , the size xi and input self-dependency lii areconsidered
Three step HEM procedure is redundant
hembf
(RuG) Linkages RuG 10 / 22
IO linkages: traditional approach
Output worth measure
What is the closed form of the complete HEM linkage, ı′x− ı′x−i?
It is given by the output worth of i (Temurshoev 2010)
ωoi =
bixi
lii(7)
Besides multiplier bi , the size xi and input self-dependency lii areconsidered
Three step HEM procedure is redundant
hembf
(RuG) Linkages RuG 10 / 22
IO linkages: traditional approach
Output worth measure
What is the closed form of the complete HEM linkage, ı′x− ı′x−i?
It is given by the output worth of i (Temurshoev 2010)
ωoi =
bixi
lii(7)
Besides multiplier bi , the size xi and input self-dependency lii areconsidered
Three step HEM procedure is redundant
hembf
(RuG) Linkages RuG 10 / 22
IO linkages: traditional approach
Output worth measure
What is the closed form of the complete HEM linkage, ı′x− ı′x−i?
It is given by the output worth of i (Temurshoev 2010)
ωoi =
bixi
lii(7)
Besides multiplier bi , the size xi and input self-dependency lii areconsidered
Three step HEM procedure is redundant
hembf
(RuG) Linkages RuG 10 / 22
IO linkages: traditional approach
Non-complete HEM linkages
What about bhi = ı′x− ı′x−i
c and f hi = ı′x− ı′x−i
r ?
Result 1
The closed-form expressions of the backward and forward linkages resultingfrom the hypothetical extraction of sector i are given, respectively, by
bhi =
(bi − 1)xi
liiand f h
i =(fi − 1)xi
lii. (8)
No extraction needed (e.g., PyIO). Note that ωoi = bh
i + xilii
.worth factorhem
(RuG) Linkages RuG 11 / 22
IO linkages: traditional approach
Connections between the IO linkages:
Result 2
The following identities between the linkage measures always hold:
bhi =
ωoi (bi − 1)
bi=
ωoi (bi − 1)
bni
si and f hi =
ωoi (fi − 1)
bi=
ωoi (fi − 1)
bni
si ,
(9)where si = yi/xi is the share of net output in gross output of sector i .
res4
(RuG) Linkages RuG 12 / 22
Generalized IO linkages
Generalized IO linkages
Let π be the vector of direct factor coefficients
The total factor backward linkage of sector i is (b′π = π′L)
bπi = mπ
i =n∑
k=1
πk lki . (10)
The total factor forward linkage of sector i is (fπ = Gπ)
f πi =
n∑k=1
gikπk . (11)
Note: total factor requirements to satisfy y = total factor usageaccompanying v, i.e.,
b′πy = π′x = x′π = v′fπ.
(RuG) Linkages RuG 13 / 22
Generalized IO linkages
Generalized IO linkages
Let π be the vector of direct factor coefficients
The total factor backward linkage of sector i is (b′π = π′L)
bπi = mπ
i =n∑
k=1
πk lki . (10)
The total factor forward linkage of sector i is (fπ = Gπ)
f πi =
n∑k=1
gikπk . (11)
Note: total factor requirements to satisfy y = total factor usageaccompanying v, i.e.,
b′πy = π′x = x′π = v′fπ.
(RuG) Linkages RuG 13 / 22
Generalized IO linkages
Generalized IO linkages
Let π be the vector of direct factor coefficients
The total factor backward linkage of sector i is (b′π = π′L)
bπi = mπ
i =n∑
k=1
πk lki . (10)
The total factor forward linkage of sector i is (fπ = Gπ)
f πi =
n∑k=1
gikπk . (11)
Note: total factor requirements to satisfy y = total factor usageaccompanying v, i.e.,
b′πy = π′x = x′π = v′fπ.
(RuG) Linkages RuG 13 / 22
Generalized IO linkages
Generalized IO linkages
Let π be the vector of direct factor coefficients
The total factor backward linkage of sector i is (b′π = π′L)
bπi = mπ
i =n∑
k=1
πk lki . (10)
The total factor forward linkage of sector i is (fπ = Gπ)
f πi =
n∑k=1
gikπk . (11)
Note: total factor requirements to satisfy y = total factor usageaccompanying v, i.e.,
b′πy = π′x = x′π = v′fπ.
(RuG) Linkages RuG 13 / 22
Generalized IO linkages
Oosterhaven and Stelder’s factor net backward linkage of i is
bπ,ni =
bπi yi
πixi, (12)
i.e., factor generated in all sectors due to final demand of sector ifactor generated in sector i due to final demands of all sectors .
What is the closed-form of the ‘generalized’ HEM indicator,π′x− π′x−i?
It is given by the factor worth of i (Temurshoev 2010)
ωπi =
bπi xi
lii. (13)
netback netforw
(RuG) Linkages RuG 14 / 22
Generalized IO linkages
Oosterhaven and Stelder’s factor net backward linkage of i is
bπ,ni =
bπi yi
πixi, (12)
i.e., factor generated in all sectors due to final demand of sector ifactor generated in sector i due to final demands of all sectors .
What is the closed-form of the ‘generalized’ HEM indicator,π′x− π′x−i?
It is given by the factor worth of i (Temurshoev 2010)
ωπi =
bπi xi
lii. (13)
netback netforw
(RuG) Linkages RuG 14 / 22
Generalized IO linkages
Oosterhaven and Stelder’s factor net backward linkage of i is
bπ,ni =
bπi yi
πixi, (12)
i.e., factor generated in all sectors due to final demand of sector ifactor generated in sector i due to final demands of all sectors .
What is the closed-form of the ‘generalized’ HEM indicator,π′x− π′x−i?
It is given by the factor worth of i (Temurshoev 2010)
ωπi =
bπi xi
lii. (13)
netback netforw
(RuG) Linkages RuG 14 / 22
Generalized IO linkages
What about the ‘generalized’ non-complete HEM linkages,
bπ,hi = π′x− π′x−i
c and f π,hi = π′x− π′x−i
r ? (14)
Result 3
The closed-form expressions of the hypothetical extraction factor backwardand forward linkages of sector i are given, respectively, by
bπ,hi =
(bπi − πi )xi
liiand f π,h
i =(f π
i − πi )xi
lii. (15)
Thus, ωπi = bπ,h
i + πixilii
.outputhem
(RuG) Linkages RuG 15 / 22
Generalized IO linkages
Interconnections between the linkage measures bπi , f π
i , bπ,ni , ωπ
i , bπ,hi and
f π,hi :
Result 4
The following identities between the linkage measures always hold:
bπ,hi =
ωπi (bπ
i − πi )
bπi
=ωπ
i
bπ,ni
bπi − πi
πisi ,
f π,hi =
ωπi (f π
i − πi )
bπi
=ωπ
i
bπ,ni
f πi − πi
πisi ,
(16)
where si = yi/xi is the share of net output in gross output of sector i .
res2
(RuG) Linkages RuG 16 / 22
Generalized IO linkages
Forward counterpart of bπ,ni
Forward ‘counterpart’ of Oosterhaven and Stelder’s net backward linkage:
The factor net forward linkage of sector i is
f π,ni =
vi fπi
πixi, (17)
i.e., factor usage by all sectors associated with value-added of sector ifactor used by sector i accompanying value-added of all sectors .
Hence, with πk = 1 for all k, net forward linkage of sector i is
f ni =
vi fixi
.
netback
(RuG) Linkages RuG 17 / 22
Generalized IO linkages
Forward counterpart of bπ,ni
Forward ‘counterpart’ of Oosterhaven and Stelder’s net backward linkage:
The factor net forward linkage of sector i is
f π,ni =
vi fπi
πixi, (17)
i.e., factor usage by all sectors associated with value-added of sector ifactor used by sector i accompanying value-added of all sectors .
Hence, with πk = 1 for all k, net forward linkage of sector i is
f ni =
vi fixi
.
netback
(RuG) Linkages RuG 17 / 22
Generalized IO linkages
Interconnections between the linkage measures involving f π,ni :
Result 5
The following identities between the linkage measures always hold:
bπ,hi =
ωπi
f π,ni
f πi
bπi
bπi − πi
πis̃i and f π,h
i =ωπ
i
f π,ni
f πi
bπi
f πi − πi
πis̃i , (18)
where s̃i = vi/xi is the share of sector i ’s value-added in its gross output.
(RuG) Linkages RuG 18 / 22
Empirical exercises
Empirical exercises
How similar (or different) are the outcomes of the linkage indicators?
The OECD IO Database, 2006 edition: 48 sectors (commodities), twotime periods
Six countries: Austria, Greece, India, Indonesia, Japan, and the USA
Compute output and income linkages
Compute Spearman’s rank correlation coefficients
(RuG) Linkages RuG 19 / 22
Empirical exercisesB BH BN F FH FN W B BH BN F FH FN W
B 0.31 0.23 0.53 0.32 -0.01 0.18 0.34 -0.09 0.64 0.35 0.23 0.21BH 0.41 0.68 0.03 0.76 0.10 0.98 0.41 0.55 0.05 0.68 0.05 0.97BN 0.31 0.70 -0.38 0.21 -0.24 0.69 -0.06 0.36 -0.59 -0.05 -0.50 0.61F 0.52 0.03 -0.38 0.46 0.72 -0.01 0.62 0.18 -0.57 0.55 0.82 -0.03FH 0.39 0.77 0.25 0.40 0.52 0.73 0.36 0.72 -0.14 0.61 0.55 0.64FN 0.00 -0.02 -0.36 0.77 0.39 0.16 0.13 0.09 -0.40 0.77 0.57 0.06W 0.28 0.97 0.70 -0.03 0.76 0.04 0.26 0.97 0.43 0.11 0.71 0.14
B 0.57 0.56 0.30 0.32 -0.16 0.34 0.10 0.16 0.00 -0.23 -0.58 -0.09BH 0.66 0.43 0.27 0.76 0.06 0.91 -0.13 0.45 -0.45 0.60 -0.44 0.97BN 0.43 0.42 -0.35 -0.09 -0.29 0.40 -0.17 0.37 -0.93 -0.31 -0.79 0.42F 0.44 0.31 -0.32 0.68 0.73 0.16 0.32 -0.41 -0.94 0.27 0.72 -0.46FH 0.43 0.80 -0.03 0.67 0.54 0.71 -0.32 0.73 -0.13 0.07 0.30 0.65FN -0.14 0.05 -0.21 0.63 0.46 0.19 -0.49 -0.19 -0.60 0.52 0.38 -0.32W 0.42 0.89 0.39 0.22 0.77 0.20 -0.35 0.95 0.37 -0.45 0.77 -0.05
B 0.02 0.11 0.14 -0.12 -0.63 -0.17 0.46 0.38 0.58 0.41 0.24 0.40BH 0.11 0.36 -0.29 0.66 -0.29 0.96 0.04 0.66 0.27 0.79 0.35 0.99BN 0.14 0.41 -0.90 -0.24 -0.69 0.40 -0.06 0.19 -0.16 0.22 -0.04 0.65F 0.07 -0.35 -0.89 0.34 0.58 -0.38 0.37 -0.08 -0.90 0.63 0.84 0.27FH -0.13 0.52 -0.33 0.41 0.32 0.61 0.12 0.66 -0.45 0.52 0.70 0.81FN -0.58 -0.38 -0.79 0.73 0.42 -0.21 -0.37 -0.20 -0.84 0.68 0.40 0.39W -0.10 0.96 0.44 -0.42 0.52 -0.30 -0.06 0.99 0.18 -0.11 0.66 -0.14
B 0.31 0.34 0.35 0.38 0.32 0.28 0.47 0.39 0.20 0.38 0.07 0.63BH 0.29 0.64 0.02 0.79 0.02 0.96 0.45 0.57 -0.03 0.70 -0.05 0.95BN 0.23 0.66 -0.40 0.21 -0.40 0.64 0.51 0.38 -0.58 0.01 -0.57 0.59F 0.34 -0.03 -0.42 0.44 1.00 0.05 0.16 0.11 -0.56 0.49 0.98 -0.02FH 0.31 0.79 0.26 0.38 0.44 0.74 0.34 0.74 -0.12 0.59 0.47 0.63FN 0.30 -0.03 -0.42 1.00 0.38 0.05 0.10 0.12 -0.57 0.99 0.60 -0.08W 0.24 0.94 0.65 0.01 0.76 0.02 0.59 0.96 0.43 0.12 0.72 0.11
B 0.28 0.06 0.65 0.39 0.51 0.56 0.11 0.01 0.20 0.18 0.09 0.31BH 0.18 0.39 0.12 0.75 0.12 0.84 0.33 0.40 -0.44 0.64 -0.45 0.94BN 0.02 0.38 -0.38 -0.11 -0.38 0.28 0.12 0.33 -0.91 -0.27 -0.93 0.30F 0.69 0.08 -0.36 0.60 0.97 0.32 0.11 -0.27 -0.87 0.29 0.98 -0.28FH 0.34 0.79 -0.07 0.54 0.61 0.70 0.42 0.77 -0.11 0.26 0.28 0.70FN 0.50 0.06 -0.31 0.93 0.52 0.27 0.03 -0.27 -0.87 0.97 0.24 -0.32W 0.53 0.80 0.24 0.34 0.74 0.24 0.60 0.92 0.30 -0.14 0.80 -0.17
B 0.06 -0.03 0.27 0.08 0.10 0.35 0.56 0.34 0.54 0.49 0.48 0.60BH 0.23 0.34 -0.33 0.70 -0.29 0.91 0.17 0.63 0.33 0.81 0.29 0.98BN 0.38 0.42 -0.82 -0.18 -0.77 0.33 0.05 0.18 -0.14 0.24 -0.13 0.60F -0.39 -0.40 -0.89 0.26 0.94 -0.18 -0.02 -0.17 -0.95 0.69 0.99 0.38FH -0.07 0.48 -0.34 0.44 0.30 0.66 0.09 0.67 -0.41 0.45 0.67 0.83FN -0.39 -0.40 -0.89 1.00 0.44 -0.18 -0.11 -0.20 -0.94 0.98 0.43 0.34W 0.25 0.92 0.35 -0.29 0.51 -0.29 0.24 0.97 0.14 -0.15 0.66 -0.18
Japan 1995 and 2000 USA 1995 and 2000
Austria 1995 and 2000 Greece 1995 and2000
India 1993-94 and 1998-99 Indonesia 1995 and 2000
Japan 1995 and 2000 USA 1995 and 2000
Rank correlations of gross output IO linkages
Rank correlations of income IO linkagesAustria 1995 and 2000 Greece 1995 and2000
India 1993-94 and 1998-99 Indonesia 1995 and 2000
(RuG) Linkages RuG 20 / 22
Conclusion
Conclusions
1 We discuss how IO linkages are interrelated
2 Provide closed-form expressions for three HEM linkage measures
3 Propose a factor net forward linkage
4 Some are highly correlated: (1) ωπi , bπ
i , and f πi ; (2) ωπ
i and bπ,ni ; (3)
f πi and f π,n
i ; (4) bπ,ni , on average, is negatively correlated with the
class of forward linkages
5 Each linkage is intended to address different goal (stemming fromtheir economic interpretations), thus as expected in general there isno consistent relations between these indicators.
(RuG) Linkages RuG 21 / 22