1

A SEM approach for composite indicators building

Michel Tenenhaus & Carlo LauroMichel Tenenhaus & Carlo Lauro

2

Economic inequality and political instability Data from Russett (1964), in GIFI

Economic inequalityAgricultural inequality

GINI : Inequality of land distributions

FARM : % farmers that own half of the land (> 50)

RENT : % farmers that rent all their land

Industrial developmentGNPR : Gross national product per

capita ($ 1955)

LABO : % of labor force employed in agriculture

Political instabilityINST : Instability of executive

(45-61)

ECKS : Nb of violent internal war incidents (46-61)

DEATH : Nb of people killed as a result of civic group violence (50-62)

D-STAB : Stable democracy

D-UNST : Unstable democracy

DICT : Dictatorship

3

Economic inequality and political instability (Data from Russett, 1964)

47 countries

4

GINI

FARM

RENT

GNPR

LABO

Agricultural inequality (X1)

Industrialdevelopment (X2)

ECKS

DEATH

INST

Politicalinstability (X3)

1

2

3

++

+

+

-

+

+

+

+

-

Economic inequality and political instability

5

Building composite indicators

1. Separately for each block (without taking into account the other blocks).

2. For each block, taking into account all the other blocks (multi-block data analysis).

3. For each block, taking into account the causal model (Structural Equation Modelling).

6

1

Politicalinstability

deathe31

eckse21

inste11

1. Using SEM for factor analysis

Measurement model

1

2

3

1

2

3

1

2

3

0 0

0 0

0 0

xΘ

1

2

3

xΛ

7

ULS algorithm

S = Observed covariance matrix for MV

2'

2

ˆ ˆ ˆ( )1GFI

x x εS Λ Λ Θ

S

Goodness-of-fit Index (Jöreskog & Sorbum):

2'

,

Minimize ( ) x ε

x x εΛ Θ

S Λ Λ Θ = PCA when 0εΘ

8

First result

1.00

Politicalinstability

death

.82

e31

ecks

-1.43

e21.551

.40

GFI=1.000

inst

.94

e1.21

1

This solution is not admissible because 2 2

ˆ ( ) 1.43Var e

9

A solution

The variance of residual e2 is fixed to a small value

2 2

ˆ ( ) .05Var e

1

Politicalinstability

deathe31

ecks

.05

e21

inste11

10

Result 2

The variance of residual e2 is fixed to a small value: 2 2

ˆ ( ) .05Var e

1.00

Politicalinstability

death

.61

e31

ecks

.05

e2.981

.60

GFI=.994

inst

.90

e1.27

1

11

Bootstrap ResultsRegression Weights:

Composite indicator

*Political instability i iX

12

Principal component analysis with SEM

The variance of the residuals are fixed to 0 : ˆ ( ) 0i iVar e

1

Politicalinstability

death

0

e31

ecks

0

e21

inst

0

e11

13

Result 3

The variance of the residuals are fixed to 0 : ˆ ( ) 0i iVar e

1.00

Politicalinstability

death

.00

e31

ecks

.00

e2.901

.81

GFI=.758

inst

.00

e1.49

1

14

Bootstrap Results

Regression Weights:

Conclusion

Coefficient of INST is not significant.

Parameter Estimate Lower Upper P Ecks <--- Political instability .900 .614 1.034 .010 Death <--- Political instability .813 .490 1.028 .010 Inst <--- Political instability .487 -.013 .996 .112

15

Result 4

The variance of the residuals are fixed to 0 : ˆ ( ) 0i iVar e

1.00

Politicalinstability

death

.00

e2

ecks

.00

e1

GFI=.950

1

1

.89

.89

16

Bootstrap ResultsRegression Weights:

Composite indicator

Political instability Ecks + Death

Parameter Estimate Lower Upper P Ecks <--- Political_instability .892 .665 1.045 .010 Death <--- Political_instability .892 .665 1.059 .010

17

2. Using SEM for multi-block data analysis

1

Agriculturalinequality

gini

e1

1

farm

e2

1

rent

e3

1

1

Industrialdeveloment

gnpr

e4

1

labo

e5

1

1

Politicalinstability

death

e8

1

ecks

e7

1

inst

e6

1

18

Result 5

1.00

Agriculturalinequality

gini

.21

e1

.87

1

farm

-.12

e2

1.05

1

rent

.79

e3

.44

1

1.00

Industrialdeveloment

gnpr

.12

e4

.93

1

labo

.24

e5

-.86

1

1.00

Politicalinstability

death

.44

e8

1

ecks

.28

e7

.84

1

-.39

.74

GFI=.868

inst

.28

.90

e6

1

.40 -.71

This solution is notadmissible because

2 2

ˆ ( ) .12Var e

19

Result 6

Var(e2) is fixedto a small value

2 2

ˆ ( ) .05Var e

1.00

Agriculturalinequality

gini

.10

e1

.94

1

farm

.05

e2

.99

1

rent

.82

e3

.40

1

1.00

Industrialdeveloment

gnpr

.10

e4

.94

1

labo

.25

e5

-.85

1

1.00

Politicalinstability

death

.34

e8

1

ecks

.40

e7

.76

1

-.31

.80

GFI=.978

inst

.27

.91

e6

1

.50 -.73

20

Result 7

MacDonald (1996)proposal

All Var(e) are fixedto 0:

ˆ ( ) 0i iVar e

1.00

Agriculturalinequality

gini

.00

e1

.95

1

farm

.00

e2

.98

1

rent

.00

e3

.57

1

1.00

Industrialdeveloment

gnpr

.00

e4

.96

1

labo

.00

e5

-.93

1

1.00

Politicalinstability

death

.00

e8

1

ecks

.00

e7

.87

1

-.26

.88

GFI=.888

inst

.39

.00

e6

1

.41 -.60

21

Bootstrap ResultsRegression Weights:

Parameter Estimate Lower Upper P Gini <--- Agricultural inequality .947 .800 1.052 .010 Farm <--- Agricultural inequality .979 .813 1.089 .010 Rent <--- Agricultural inequality .566 .053 .909 .051 Gnpr <--- Industrial develoment .957 .807 1.076 .010 Labo <--- Industrial develoment -.928 -1.052 -.739 .010 Ecks <--- Political instability .875 .624 1.038 .010 Death <--- Political instability .883 .642 1.044 .010 Inst <--- Political instability .395 -.019 .788 .115

Conclusion

Coefficient of INST is not significant.

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Result 8MacDonald (1996)Proposals:

(1) All Var(e) are fixed to 0:

ˆ ( ) 0i iVar e

1.00

Agriculturalinequality

gini

.00

e1

.95

1

farm

.00

e2

.98

1

rent

.00

e3

.57

1

1.00

Industrialdeveloment

gnpr

.00

e4

.96

1

labo

.00

e5

-.92

1

1.00

Politicalinstability

death

.00

e8

1

ecks

.00

e7

.87

1

-.27

.92

GFI=.933

.42 -.60

(2) Composite indicator:

Political instability

.87*Ecks +.92*Death

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3. Causal model estimation using SEM-ULS

Agriculturalinequality

gini

e1

1

1

farm

e2

1

rent

e3

1

Industrialdevelopment

gnpr

e4

1

1

labo

e5

1

Politicalinstability

death

e8

1

ecks

e7

1

d11

1

inst

e6

1

24

Result 9

This solution is notadmissible because

2 2

ˆ ( ) .09Var e

.83

Agriculturalinequality

gini

.15

e1

1.00

1

farm

-.09

e2

1.14

1

rent

.82

e3

.43

1

.88

Industrialdevelopment

gnpr

.10

e4

1.00

1

labo

.25

e5

-.91

1

Politicalinstability

death

.34

e8

1

ecks

.40

e7

.95

1

-.26

.25

d11

1.00

GFI=.979

-.55.26

inst

.34

.91

e6

1

25

Result 10

Var(e2) is fixedto a small value

2 2

ˆ ( ) .05Var e

.88

Agriculturalinequality

gini

.10

e1

1.00

1

farm

.05

e2

1.05

1

rent

.82

e3

.42

1

.88

Industrialdevelopment

gnpr

.10

e4

1.00

1

labo

.25

e5

-.91

1

Politicalinstability

death

.34

e8

1

ecks

.40

e7

.95

1

-.27

.25

d11

1.00

GFI=.978

-.54.25

inst

.34

.91

e6

1

26

Bootstrap ResultsRegression Weights:

Conclusion

Coefficient of INST is not significant.

Parameter Estimate Lower Upper P Political instability <--- Industrial development -.544 -.792 -.371 .010 Political instability <--- Agricultural inequality .254 .057 .451 .031 Gini <--- Agricultural inequality 1.000 1.000 1.000 ... Farm <--- Agricultural inequality 1.053 .937 1.190 .010 Rent <--- Agricultural inequality .423 .047 .735 .051 Gnpr <--- Industrial development 1.000 1.000 1.000 ... Labo <--- Industrial development -.911 -1.215 -.685 .010 Ecks <--- Political instability .952 .594 1.405 .010 Death <--- Political instability 1.000 1.000 1.000 ... Inst <--- Political instability .337 -.029 .662 .130

27

Result 11

Var(e2) is fixedto a small value

2 2

ˆ ( ) .05Var e

Compositeindicator:

Political instability

.89*Ecks + 1.00*Death

.87

Agriculturalinequality

gini

.10

e1

1.00

1

farm

.05

e2

1.06

1

rent

.82

e3

.42

1

.90

Industrialdevelopment

gnpr

.07

e4

1.00

1

labo

.27

e5

-.88

1

Politicalinstability

death

.29

e8

1

ecks

.43

e7

.89

1

-.28

.28

d11

1.00

GFI=.982

-.55.26

28

Bootstrap ResultsRegression Weights:

Conclusion

All coefficients are significant.

Parameter Estimate Lower Upper P Political_instability <--- Industrial_development -.545 -.807 -.369 .010 Political_instability <--- Agricultural_inequality .265 .063 .449 .015 Gini <--- Agricultural_inequality 1.000 1.000 1.000 ... Farm <--- Agricultural_inequality 1.055 .939 1.194 .010 Rent <--- Agricultural_inequality .423 .044 .717 .051 Gnpr <--- Industrial_development 1.000 1.000 1.000 ... Labo <--- Industrial_development -.882 -1.189 -.680 .010 Ecks <--- Political_instability .892 .554 1.312 .010 Death <--- Political_instability 1.000 1.000 1.000 ...

29

Conclusion

• Agricultural inequality and Industrial development are drivers of political instability

• Russet hypotheses are validated:

• Other composite indicators:

2

Political instability .26*Agricultural inequality - .55*Industrial development

(p = .015) (p = .01)

R .596

Agricultural inequality 1.00*Gini + 1.06*Farm + .42*Rent

Industrial development 1.00*GNPR - .88*Labo

30

31

Conclusion

• Agricultural inequality above the average

• Industrial development below the average

DICTATORSHIP