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

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)

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)

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.

Introduction

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)

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)

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∑

gik . (4)

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∑

gik . (4)

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).

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

Non-complete HEM

DenoteA−i

c )−1y and (x−ir )′ = v′(I− B−i

r )−1

Non-complete HEM

DenoteA−i

c )−1y and (x−ir )′ = v′(I− B−i

r )−1

Non-complete HEM

DenoteA−i

c )−1y and (x−ir )′ = v′(I− B−i

r )−1

Net backward linkage

Oosterhaven and Stelder’s (2002) key sector indicator

Net backward linkage of sector i

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

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 =

lii(7)

Besides multiplier bi , the size xi and input self-dependency lii areconsidered

Three step HEM procedure is redundant

ωoi =

lii(7)

ωoi =

lii(7)

ωoi =

lii(7)

Non-complete HEM linkages

What about bhi = ı′x− ı′x−i

Result 1

The closed-form expressions of the backward and forward linkages resultingfrom the hypothetical extraction of sector i are given, respectively, by

(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

Connections between the IO linkages:

Result 2

The following identities between the linkage measures always hold:

ωoi (bi − 1)

si and f hi =

ωoi (fi − 1)

(9)where si = yi/xi is the share of net output in gross output of sector i .

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∑

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.

bπi = mπ

i =n∑

πk lki . (10)

f πi =

n∑k=1

gikπk . (11)

bπi = mπ

i =n∑

πk lki . (10)

f πi =

n∑k=1

gikπk . (11)

bπi = mπ

i =n∑

πk lki . (10)

f πi =

n∑k=1

gikπk . (11)

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

bπ,ni =

bπi yi

πixi, (12)

ωπi =

bπi xi

lii. (13)

netback netforw

bπ,ni =

bπi yi

πixi, (12)

ωπi =

bπi xi

lii. (13)

netback netforw

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

Interconnections between the linkage measures bπi , f π

i , bπ,ni , ωπ

i , bπ,hi and

f π,hi :

Result 4

bπ,hi =

ωπi (bπ

i − πi )

bπ,ni

bπi − πi

πisi ,

f π,hi =

ωπi (f π

i − πi )

bπ,ni

f πi − πi

πisi ,

where si = yi/xi is the share of net output in gross output of sector i .

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

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

Interconnections between the linkage measures involving f π,ni :

Result 5

bπ,hi =

f π,ni

bπi − πi

πis̃i and f π,h

i =ωπ

f π,ni

f πi − πi

πis̃i , (18)

where s̃i = vi/xi is the share of sector i ’s value-added in its gross output.

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

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

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.

Conclusion

Thanks for your attention!

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