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CHAPTER SEVEN

MULTI-DIMENSIONAL MEASURES OF POVERTY AND WELL-BEING BASED ON LATENT VARIABLE MODELS

Jaya Krishnakumar

7. 1 Introduction

Development is a multidimensional concept incorporating diverse social, economic,

cultural and political dimensions and economic growth, though necessary, is not

sufficient in itself to bring about development in this broad sense. According to

Nobel Prize Laureate Amartya Sen (e.g. Sen (1985, 1999)), the basic purpose of

development is to enlarge people's choices so that they can lead the life they want to.

In this approach, the choices are termed “capabilities” and the actual levels of

achievement attained in the various dimensions are called “functionings”. Thus

human development is the enhancement of the set of choices or capabilities of

individuals whereas functionings are a set of “beings” and “doings” which are the

results of a given choice. The concept of human development proposed by Mahbub

ul Haq, in the first Human Development Report in 1990 (see UNDP (1990)), largely

inspired from Sen's various works, represents a major step ahead in the

concretization of this extended meaning of development and in the effort to bring

people's lives to the centre of thinking and analysis. Since then, human development

and human deprivation have been the object of extensive theoretical and empirical

research. They have been studied from various angles: conceptual, methodological,

operational and policy-making. As it is not possible to directly observe and measure

human development in its broad sense or the lack of it, they are generally constructed

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as composite indices based on several variables (indicators).

The earliest among aggregate indices of welfare is the Physical Quality of Life Index

(PQLI) proposed by Morris (1979) which is a simple average of life expectancy at

age one, infant mortality and adult literacy. The first major operationalisation of

Sen's capability approach employed scaling and aggregation to construct the Human

Development Index (HDI). This index looks at the status of all the people in a

society on the basis of three dimensions: health and longevity, instruction and access

to knowledge and other dimensions for having a decent life (for which income is

taken as a proxy). Each one of the dimensions is determined by a single-dimensional

indicator. Health and longevity are measured by life expectancy at birth. Instruction

and access to knowledge are measured by a weighting average of adult literacy rate

(2/3) and school enrolment rate (1/3). Finally, the possibilities of having a decent life

are measured by real per capita GDP. The three dimensions are given equal weights

in the construction of the HDI. On the poverty side, we have the Human Poverty

Index which is a weighted average measuring deprivation in the three dimensions of

health (survival), education (illiteracy) and economic deprivation (itself a

combination of three elements – access to health, safe water and adequate

nourishment of children) for developing countries.

Over the recent years other indices have been proposed, derived from an underlying

theoretical model, that offer an explanation for the inclusion of the variables

composing the index as well as a better justification for the choice and values of the

weights in the construction of the index.

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A theoretical framework that is appealing in this context is a model which assumes

that the different dimensions of development or poverty are unobservable variables

observed through a set of indicators. Factor analysis, MIMIC (multiple indicators and

multiple causes) and structural equation models all fall into this line of reasoning.

Latent variable models are common in psychology and the reader can find an

excellent coverage of most of these models with applications in Bollen (1989),

Bartholomew and Knott (1999), Muthen (2002) and Skrondal and Rabe-Hesketh

(2004). Though principal components (PC) is not a latent variable model, we have

added it in our chapter for two reasons. First it is widely used in empirical

applications as an ‘aggregating’ technique and secondly the PC's can be shown to be

equivalent to the factor scores under certain conditions (see later). The principal

components method seeks linear combinations of the observed indicators in such a

way as to reproduce the original variance as closely as possible. But this method

lacks an underlying explanatory model which the factor analysis offers. In the factor

analysis model the observed values are postulated to be (linear) functions of a certain

number (fewer) of unobserved latent variables (called factors). Thus it provides a

theoretical framework for explaining the functionings by means of capabilities

represented by the latent factors. However this model does not explain the latent

variables (or the capabilities) themselves in that it does not say what causes these

capabilities to change. We believe it is as important to be able to say something

about the capabilities as it is to say how we can enhance them and promote human

development. It is not enough to be able to measure how much is achieved but it is

also essential to be able to say how things can be improved.

The MIMIC model (cf. Joreskog and Goldberger (1975)) represents a step further in

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the explanation of the phenomenon under investigation as it is not only believed that

the observed variables are manifestations of a latent concept but also that there are

other exogenous variables that “cause” and influence the latent factor(s). This

structure is highly relevant in our context as there are several institutional, political

and social arrangement factors which definitely influence human development and

need to be taken into account. One can even argue that not only do these factors

influence human development but they are also influenced by it. Let us take a simple

example. If access to education is facilitated leading to an increase in knowledge

capability of a given population, this may result in greater involvement in the

political sphere and bring about policy changes such as free education for all which

will in turn enhance knowledge and even ‘political’ capabilities. Adequate

institutional setups promote development but it is also true that development in turn

encourages favourable political and social arrangements by enforcing the

participatory element of progress. Thus there is a virtuous cycle by which human

development promotes its own “causal” factors. Unless this feedback mechanism is

taken into account we do not have a complete picture of the evolving nature of the

whole system. Therefore one has to go beyond one-way causal links towards

structural equation models (SEM) including several interdependent latent

endogenous variables and exogenous “causes”. Estimating the resulting econometric

model using real data, we will be able to verify our assumptions about the feedback

mechanism mentioned above. Furthermore the empirical model will also give us

estimates of capabilities (our latent variables) rather than functionings.

This chapter reviews the most important latent variable models which form the basis

of multidimensional indices of human development (or deprivation) starting from

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simpler ones such as factor analysis and going up to structural equation models. Only

those features of each model that are relevant for our context, namely the

construction of a welfare or poverty index, are presented in the review, directing the

reader to related references for further details. The final model will be one that

incorporates the simultaneous determination of the different (latent) dimensions of

well-being while accounting for the impossibility of their direct measurement. We

feel that it is the most suitable framework for our context.

At this point we would like to point out that we do not intend to cover non-statistical

approaches that have also been proposed in this context such as aggregation via

scaling (i.e. a projection of each variable onto a 0-1 range) and fuzzy sets theory. We

briefly discussed a few aggregate indices earlier in this section. Recently fuzzy

methodology has been gaining attention in the areas of inequality and well-being

analysis. In both cases, in fact, Sen's claim according to which “well-being and

inequality are broad and opaque concepts” (Sen, 1993) implicitly offers a

justification for drawing on mathematical tools that take into account ambiguity and

complexity when it comes to measuring individual well-being and deprivation from a

multidimensional perspective. Cerioli and Zani (1990), Qizilbash (1992), Cheli and

Lemmi (1995) and Chiappero Martinetti (2000) among others offer notable

contributions in the field of fuzzy measures.

The next section presents the principal components method and its relevance for

welfare measurement followed by the factor analysis model in Section 3. Section 4

discusses MIMIC models and SEM is examined in the following section. Some

major applications of SEM are reviewed in Section 6 and Section 7 ends the chapter

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with a few concluding remarks.

7.2 Principal Components Indices

The use of principal components (PC) or a combination of principal components is a

commonly used technique in the measurement of quality of life or well-being. One of

the earliest studies in this direction is Ram (1982) who first applies PC on the three

dimensions of PQLI mentioned above, namely life expectancy at age one, infant

mortality and adult literacy, and combines it with per capita GDP again using PC to

form a composite index. Slottje (1991) follows the same approach by selecting 20

attributes for 126 countries across the world, calculating a PC-based index and

comparing it with indices obtained using hedonic weighting procedures. This

method, which is essentially a data reduction technique, dates back to Hotelling

(1933) in the statistical literature with a wide range of applications in numerous

fields such as psychology, biology, anthropology and more recently in economics

and finance.

The basic idea behind this method is to determine orthogonal linear combinations of

a set of observed indicators chosen in such a way as to reproduce the original

variance as closely as possible. Here we introduce some notations that will be used

throughout the chapter. Let y denote a 1×k vector of observed variables (which we

already assume to be centered without loss of generality) and let Σ denote its

covariance matrix. Let us further denote by kθθ ,...,1 the k eigenvalues of Σ . For the

moment we assume Σ to be known (which will be replaced by its empirical version

in practice) and by kaa ,...,1 the corresponding eigenvectors. Then the principal

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components are given by:

kjyap jj 1,...,='=

or

yAp ′=

where ]...[= 1 kaaA is the matrix of eigenvectors of Σ . We have kIAAAA == ′′ and

AA ′ΘΣ = or ΘΣ′ =AA where kjdiag j 1,...=),(= θΘ with the jθ 's arranged in

descending order of magnitude. We also have AA ′ΘΣ −− 11 = . The variances of the

PC's are equal to the corresponding eigenvalues i.e. jpV jj ∀θ=)( .

One of the interpretations that is often made regarding the principal components is

that they are estimates of latent variables of which the observed values are indicators.

It should be remembered that this method is originally a purely descriptive technique

which tries to reproduce the observed variance or a large proportion of it using linear

combinations. The above interpretation is in fact the underlying assumption for the

factor analysis (FA) model to which we will turn in the next Section.

Before going to the FA model and the link between PC and FA, let us present the

indices derived from PC's. The two most commonly used are the first principal

component i.e the one corresponding to the greatest eigenvalue 1θ and a weighted

average of all the principal components jp 's, kj 1,...,= with the weights jw being

given by the proportion of the total variance explained by each PC.

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If we take the first principal component yap '= 11 as an aggregate index then we

have 11 =)( θpV . As for the weighted average its variance can be calculated as

follows. Let us write it as:

jj

k

j

pwH ∑1=

=ˆ

with

.=

1=j

k

j

jjw

θ

θ

∑

Denoting )(= jdiag θΘ and ]...[= 1 kwww′ and using jjpV θ=)( we have

wwHV Θ′=)ˆ(

where

.)(= 1 Θ′Θ′ − ιιιw

Thus

.)(

=)ˆ( 2

3

ιιιι

Θ′Θ′HV

In practice, Σ is unknown and hence has to be estimated and the eigenvalues and

eigenvectors of the estimator have to be used. These estimators are consistent (see

e.g. Anderson (1984)).

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Though these indices are often used in empirical studies, few (none to our

knowledge) give an estimation of their variance (or precision). Here we have a

convenient expression that can be easily implemented.

7.3 Factor Analysis Model

The FA model assumes that the observed variables (indicators) are all dependent on

one or more latent variables which are taken to be their common cause(s). In other

words the observed variables are different manifestations of one or more underlying

unobservable variables called factors. Thus the model is written as

ε+Λfy = (7.1)

where 1)( ×ky denotes the vector of observed variables, 1)( ×mf vector of latent

variables )<( km and Λ the )( mk × coefficient matrix. If there is only one latent

factor (for instance overall human development) then f is a scalar and Λ a 1)( ×k

vector.

Treating the latent factors as random, one assumes in general

ΨΦ =)(=)( εVandfV

with ΨΦ, positive definite. Let Σ denote the variance covariance matrix of the

observed vector y as before. Then

.= Ψ+Λ′ΛΦΣ

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This model uses the empirical estimators of Σ to find ΦΛ, and Ψ . It is usual to fix

I=Φ for identification purposes. For the same reason, it is also assumed that

ΛΨΛ′Γ −1= is diagonal. Maximum likelihood procedure is applied to the model to

estimate Λ and Ψ given Σ . Given ΨΛ, , one can derive minimum variance

estimators or predictors of f as follows:

yIf 11)(=ˆ −− ΨΛ′Γ+ (7.2)

This estimator minimises )ˆ( ffV − . It is also such that )|(=ˆ yfEf assuming joint

normal distribution for ),( fy .

Estimated in this way we do not have )|ˆ( fffE − equal to zero. If we add it as a

condition then we would obtain the following slightly different estimator (see e.g.

Bartholomew and Knott (1999)):

yyf 11111* )(==ˆ −−−−− ΨΛ′ΛΨΛ′ΨΛ′Γ (7.3)

which is the least squares estimator of f in model

Error! Reference source not found. given Λ,y .

It can be argued that 0=)|ˆ( fffE − may not be a pertinent condition when f is

not observed. In any case, the only difference between f̂ and *f̂ is that )( Γ+I in

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f̂ is replaced by Γ in *f̂ . Since Γ is diagonal this only means a rescaling of f̂ 's.

Let us now consider the special case I=Ψ . Then we get the following factor scores:

yIf Λ′ΛΛ′+ −1)(=~ (7.4)

and

yf Λ′ΛΛ′ −1* )(=~ (7.5)

for the ‘unbiased’ estimation.

Here we would like to point out two useful results on the link between PC and FA.

First, the estimators of the latent variables obtained above for I=Ψ can be shown to

be proportional to the (first m ) principal components say *p . Secondly, considering

the principal components to be potential estimators of the latent factors and deriving

an ‘unbiased’ variant say **p using the first m PC's in the sense 0=)|( ** ffpE − ,

we have *** ~= fp of (7.5) above. Both results are elaborated in Krishnakumar and

Nagar (2006).

The above results, in particular the identity between the ‘unbiased’ versions of PC's

and factor scores, provide the theoretical justification for the interpretation of

principal components as latent variable estimators.

7.4 MIMIC models

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This model initially proposed by Joreskog and Goldberger (1975) goes further in the

theoretical explanation by introducing “causes” of latent factors. According to this

model, the observed variables result from the latent factors and the latent factors

themselves are caused by other exogenous variables denoted here as x . Thus we

have a ‘measurement equation’ and a ‘causal’ relationship:

ςβελ+′+

xffy

==

(7.6)

In their model with f a scalar and x,,αβ vectors, the authors showed that the

estimator of f is given by

)()(1=ˆ 111 yxf −−− Ψ′+′Ω′− λαλλ

with Ψ+′ΩΨ λλσςε =,=)(,=)( 2 IVV .

The multivariate extension of this model is straightforward:

ςε

++Λ

Bxffy

==

(7.7)

with f a vector, B,Λ matrices of appropriate dimensions, and

IVV 2=)(,=)( σςε Ψ .

Then we have :

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).()(=ˆ 111 yBxIf −−− ΨΛ′+ΛΩΛ′−

Using the expression for the inverse of )(= Λ′Λ+ΨΩ , one gets (see

Krishnakumar and Nagar (2006))

yIBxIf 11111 )()(=ˆ −−−−− ΨΛ′ΛΨΛ′++ΛΨΛ′+ (7.8)

The above equation shows that the MIMIC latent factor estimator is a sum of two

terms: the first one is the “causes” term (function of x ) and the second one can be

called the “indicators” term. Note that the latter is nothing but the factor scores (7.2)

of the FA model. If there are no ‘causes’ then (7.8) reduces to the pure FA estimator

as one can expect.

Its variance is given by

)()()(=)ˆ( 111 ΛΨΛ′ΛΨΛ′++′ −−−IBxBVfV

7.5 Structural Equation Models

Recall that the main idea behind the latent variable approach is that the different

dimensions of development (or deprivation) cannot be directly measured but can be

represented by latent variables manifesting themselves through a set of achievements

(or the lack of it). At the same time these latent dimensions mutually influence one

another and hence it is important to explicitly specify these interactions in the form

of a structural model.

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Thus the most suitable extension to the above models is an interdependent system of

equations for the latent variables incorporating exogenous elements and a set of

measurement equations linking the unobserved variables to the observed indicators.

This is called the structural equation model (SEM), the most well-known in this

category being the LISREL model proposed by Joreskog (1973). This model

specifies a system of equations explaining the latent variables y* (which become the

endogenous variables of the model) by a set of exogenous (also latent) variables x*

and including mutual effects of the endogenous variables on one another. To this

system is added a set of equations to take account of the additional assumption that

these latent endogenous and exogenous elements are observed through some

indicators y and x respectively. This yields:

0=** uBxAy ++ (7.9)

ε+Λ *= yy (7.10)

ζ+ϒ *= xx (7.11)

with

ΞΨΣ =)(,=)(,=)( ζε VVuV

where (7.9) is the structural model and (7.10) and (7.11) constitute the measurement

equations. We assume that the observations are centered without loss of generality.

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One can represent the causal structure of the model as follows:

Although (7.11) does not pose any additional problem on the theoretical side, we will

remove it in the context of human development or well-being as the exogenous

variables (basically representing institutional and social structures) will generally be

observed. Though the literature in this area has seen several extensions of the above

model with ordinal/categorical variables and/or covariates (exogenous variables) in

measurement equations (cf. Muthen (1984, 2002), Joreskog (2002), Skrondal and

Rabe-Hesketh (2004)), we will continue with the above formulation for clarity of

exposition.

The parameters of (7.9) and (7.10) can be estimated by generalised method of

moments (GMM) by minimising the distance between the empirical variance

covariance matrix of the y 's and x 's and the theoretical expressions of the

covariance matrix given by (see e.g. Browne (1984)):

**1 ...,, myy

pyy , .. .,1

**1 ,..., kxx sxx ,...,1

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⎥⎦

⎤⎢⎣

⎡

Λ′ΛΨ+Λ′Σ+′Λ

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

−−−

)()()())(((

= 1'

11'1

xVAxVxVAABxBVA

xy

V

and taking into account any a priori constraints on the parameters. The distance is

optimally calculated in the metric (weight matrix) given by the inverse of the

asymptotic variance-covariance matrix of the vector of sample statistics. This

weighted least squares procedure is equivalent to a nonlinear GMM procedure on the

reduced form of the SEM.

An alternative procedure is the minimisation of the same distance between

theoretical and empirical variance matrices conditioning on x . This is often the case

as in general the mean and the variance of x are not restricted and are estimated by

their sample values.

Then one would minimise the distance between the sample variance-covariance of y

given x and )( 1'1 Ψ+Λ′ΣΛ −− AA under the same a priori constraints. Asymptotic

theory gives us the variance matrix of the resulting estimators and a ‘robust’ version

can be computed to account for non-i.i.d. behaviour using a heteroscedasticity-

consistent estimate of the same matrix.

One can also use (conditional) maximum likelihood (cf. e.g. Joreskog (1973),

Browne and Arminger (1995)) to estimate the parameters under (conditional)

normality of *y given x and correct its variance using the well-known ‘sandwich’

formula under non-normality (quasi- maximum likelihood, cf. White (1982),

Gouriéroux, Monfort and Trognon (1984)).

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Once the parameter estimates are obtained, the latent factors are estimated by their

posterior means given the sample, replacing the parameter values by their estimates.

This is called the Empirical Bayes estimator. For the above model (with observed x )

we get (see Krishnakumar and Nagar (2006)):

)()(=ˆ 1111111*iiii BxAyAAAABxAy −−′−−′−−− Λ−Ψ+Λ′ΣΛ′ΛΣ+

or

[ ] +ΛΨ+Λ′ΣΛ′ΛΣ− −−′−−′−−ii BxAAAAAIy 111111* )(=ˆ

iyAAAA 11111 )( −′−−′−− Ψ+Λ′ΣΛ′ΛΣ (7.12)

From the point of view of a substantive interpretation of the above expression (7.12),

it is important to notice that the factor scores are once again a combination of two

terms: one capturing the ‘causal’ influence and the other reflecting the ‘indicators’

relevance.

Its variance can be obtained as (see Krishnakumar and Nagar (2006))

111111111* )()(=)ˆ( −−−′−−′−−′−− ΣΛΨ+Λ′ΣΛ′ΛΣ+′ AAAAAAABxBVAyV i

An alternative method of obtaining factor scores is the maximum posterior likelihood

which leads to the same result as (7.12) for our SEM given by (7.9), (7.10) (see

Krishnakumar and Nagar (2006)).

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7.6 Some Empirical Applications

In this section we will briefly discuss some applications of SEM in the context of

multidimensional measures of well-being (or poverty). Such applications are still rare

but definitely growing.

An important contribution to this literature is Kuklys (2005) in which the author

applies the MIMIC model for measuring the unobserved functioning in health and

housing, each observed through a range of indicators. The model is estimated using

data from the British Household Panel Survey for 1991 and 2000. The factor

loadings are between 0.70 and 0.93 for health indicators whereas there is more

variation (between 0.29 and 0.72) for housing functionings. As far as the causes are

concerned, being a woman, being old and living in London all influence health

negatively in both the years under study. However the results are not stable over the

two years for housing. For instance income has no effect on housing conditions in

1991 but has a large positive significant effect in 2000. Similarly education and

being male are significant and positive in 2000 wheras it is not the case in 1991. Age

has a positive influence and living in London has a negative one in both years. On

deriving the latent variable scores, the author finds a lot of difference between

welfare viewed in the functioning space and that in the income space and hence

concludes that resource-based measures do not adequately capture the lack of

performance in the achievement space.

Another work (Di Tommaso (2003)) adopts the same MIMIC model to conceptualise

children's well being and applies it to Indian data. Four functionings are used to

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measure well-being: height for age, weight for age, enrolment into school and work

status of the child. The last two are found to be the most important for describing the

achievement in the well-being dimension. Literacy of the parents and the gender of

the child are the most influential ‘causes’ followed by income and wealth (assets and

durables).

Wagle (2005) uses a SEM for deriving multidimensional poverty measures using

household data from a survey conducted in Kathmandu, Nepal in 2002 and 2003.

Five major dimensions of well-being are considered: subjective economic well-

being, objective economic well-being, economic well-being, economic inclusion,

political inclusion and civic/cultural inclusion. Each of these dimensions is measured

by a series of indicators and they influence one another through a system of

simultaneous equations but there are no exogenous variables in the model. The

author finds that objective poverty is adequately measured by income and

consumption whereas the subjective notion of poverty is reflected in the

householders' view on the adequacy of income for food and non-food expenses.

Education, health and non-discrimination are seen to be the best indicators of

capability. The exclusion dimension of poverty is best captured by employment

status and access to financial resources in the case of economic exclusion, voting

frequency and involvement in political activities for political exclusion and

participation in social activities and networks for social exclusion. The author also

notes the strong correlations among the different dimensions of poverty.

Krishnakumar (2004, 2005) proposes a more general SEM with exogenous variables

in both the structural and measurement parts as an appropriate system for

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operationalising the capability approach. The first version proposes the theoretical

framework for deriving a multidimensional index of human development and the

second version completes the theory with an illustration using worldwide country-

level data. The data relate to a cross section of middle and low income countries

across the world for the year 2000 (or the year closest to it i.e. 1999 or 1998 for a few

variables). Even though we explored many international data sources theoretically

covering all countries, the number of countries with no missing values for any of the

selected variables was considerably reduced to 56. Based on the availability of data,

three fundamental capability dimensions namely knowledge (education), health and

political freedom, are considered. Income was not considered as a dimension in itself

due to its “instrumental” role in promoting human development rather than being a

component of it. All estimations are done by ‘robust’ maximum likelihood method

and implemented using the software MPLUS. In what follows we briefly present the

estimation results of this study as an example to point out the kind of understanding

that one can acquire from such an approach. Full results including data sources can

be found in the original paper.

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List of Variables

The latent endogenous variables :

y1*: Knowledge

y2*: Health

y3*: Political Freedom

The achievement indicators :

y1: Political Rights

y2: Civil Liberties

y3: Voice and Accountability

y4: Adult literacy rate (% age 15 and above)

y5: Combined primary, secondary and tertiary gross enrolment ratio (%)

y6: Life expectancy at birth (years)

y7: Infant mortality rate (per 1000 live births)

Possible exogenous variables (observed)

For the structural part

x1: Government Effectiveness

x2: Regulatory Quality

x3: Population using improved water sources (%)

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x4: Cellular mobile subscribers (per 1000 people)

x5: Public expenditure on health ({\%} of GDP)

x6: Total debt service (% of GDP)

x7: Density (persons per sq.km.)

x8: Political Stability

x9: Population Growth Rate (Annual %)

x10: Urban Population Growth Rate (Annual %)

x11: Youth Bulge (Pop. Aged 0-14 as a % of Total)

x12: Physicians (per 100,000 people)

x13: Press Freedom

x14: Democracy - Autocracy Index

x15: Total fertility rate (per woman)

x16: Foreign direct investment (PPP USD)

x17: Gross fixed capital formation (PPP USD)

x18: Trade (PPP USD)

For the measurement part

w1: Control of Corruption

w2: Rule of Law

w3: Population with access to essential drugs (%)

w4: Population using adequate sanitation facilities (%)

w5: Public expenditure on education (% of GDP)

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Table 7.1 below contains the estimation results of the measurement part of the

model. From these results one can evaluate the link between the observed indicators

and the latent dimensions. The coefficient of “knowledge” is positive and highly

significant for adult literacy rate and normalised for combined primary, secondary

and tertiary gross enrolment ratio. These coefficients are commonly known as factor

loadings. The situation is similar for life expectancy at birth and infant mortality rate

as indicators for “health” (the second one with a negative “health” loading) and the

four “political freedom” indicators.

We could only retain one of the two mortality indicators (infant and under-five) as

including both produced non-significant coefficients probably due to the high

correlation between the two. We therefore conclude that the selected indicators

reflect their latent dimension satisfactorily.

Table 7.1 Results of the Measurement Model

Dependent variable

y1 y2 y3 y4 y5 y6 y7

Explanatory variable

y1* - - - 1 (0)

0.71 (0.06)

- -

y2* - - - - - 1 (0)

-3.87 (0.34)

y3* 1 (0)

0.66 (0.04)

0.40 (0.02)

- - - -

w3 - - - - - 0.04 (0.03)

-0.10 (0.09)

w5 - - - 1.72 (0.82)

1.58 (0.83)

- -

R2 0.92 0.88

0.95 0.83 0.87 0.80 0.97

Figures inside parentheses are standard deviations.

From the above results one can also see the importance of certain supply side

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(exogenous) factors. The population with access to essential drugs has a significant

positive impact on life expectancy at birth whereas it has a negative though not

significant effect on the infant mortality rate. Public expenditure on education has a

positive and significant effect on adult literacy rate and combined primary, secondary

and tertiary gross enrolment ratio. None of the exogenous political factors turned out

to be significant in the measurement model. However some of them do have

significant coefficients in the structural model as we will see below.

Table 7.2 Results of the Structural Equation Model

Dependent variable y1*

y2* y3*

Explanatory variables

y1* - 0.01 (0.00)

y2* 1.37 (0.27)

- -

y3* - 0.28 (0.31)

w1 - 0.61 (0.18)

w4 - 0.07 (0.02)

x7 -0-03 (0.01)

x11 -64.39 (30.55)

x12 - 0.001 (0.01)

x13 - - 0.08 (0.01)

x14 0.58 (0.59)

-

x15 - -4.00 (0.48)

R2 0.82

0.80 0.89

Figures inside parentheses are standard deviations.

Let us now turn to the results of the structural model (Table 7.2). The results confirm

25

the interdependent nature of the three dimensions. The positive and significant

impact of health (y2*) on education (y1*) shows that better health is definitely an asset

for better performance in education, which is in turn an important factor in achieving

political rights as shown by the coefficient of y1* on y3*. Furthermore, greater

political freedom y3* leads to better health status (y2*) thus completing the loop. One

can therefore see that y3* indirectly affects y1* too because y3* affects y2* and y2*

affects y1* and hence all the three dimensions are interdependent.

The relevance of political, demographic, socio-economic and ‘environmental’ factors

in the determination of ‘capabilities’ is also confirmed by the results. The

democracy-autocracy index has an important positive effect on education (that is a

more democratic regime seems to favour higher achievement in education).

Population growth rate and population density have an important negative effect on

education. This can be explained by the increased pressure exerted by a higher

growth rate and density of population, on existing educational services and

government resources thereby affecting the overall achievement in this field. The

percentage of population using improved water sources and number of physicians per

100000 people have a positive and significant effect on health whereas fertility has a

negative effect as expected. Finally, press freedom and control of corruption have a

significant and positive effect on political freedom, the effects of regulatory quality,

government effectiveness and political stability not being significant. Lack of

corruption definitely implies more freedom and the stronger the ‘collective voice’ in

terms of freedom of the press, the better the political rights atmosphere. The

economic factors chosen were not significant for any of the three dimensions. This

does not mean that they are not important as such; they would have been if we had

26

explicitly included GDP in our model or if our model had a separate dimension

corresponding to material welfare. The R2 values in both Tables seem to indicate that

a relatively high percentage of the observed variance is explained by the equations of

the model, thus implying a reasonable fit.

One of the key messages of this paper is that a better social and political environment

not only implies a better conversion of capabilities into achievements but also

enhances the capabilities themselves. This emphasizes the powerful role that a State

can and should play in terms of providing better infrastructure and governance.

The paper goes on to construct ‘capability indices’ from the latent variable scores

and ranks countries according to them. These rankings are compared with those

obtained with the commonly used HDI or GDP per capita. Notable differences in the

rankings of countries are pointed out especially due to the inclusion of the political

dimension in the model.

Ballon and Krishnakumar (2005) present another application of the same model

using micro-level data on Bolivian households for analysing two basic capability

domains - knowledge (being able to be educated) and living conditions (being able to

be adequately sheltered).

These are the major studies involving comprehensive structural modelling (MIMIC,

SEM) in the context of multidimensional well-being/poverty measurement. There are

numerous other studies which apply the earlier procedures like principal components

(e.g. Ram (1982), Slottje (1991), Klasen (2000), Nagar and Basu (2001), Biswas and

Caliendo (2002), Rahman et al. (2003), Noorbaksh (2003), McGillivray (2005)) and

27

factor analysis (e.g. Massoumi and Nickelsburg (1988), Schokkaert and v.Ootegem

(1990), Balestrino and Sciclone (2000), Lelli (2001)) but it will be too lengthy to go

into their details here. The reader is invited to consult the original articles. We hasten

to add that we have consciously left out the large number of articles dealing with

SEM in other fields such as psychology (where it was first developed) and health, as

it is beyond the scope of this chapter.

7.7 Conclusions

This paper presents an overview of the various multidimensional measures of well-

being or poverty based on the latent variable approach. This approach is particularly

appealing as it postulates that human development or deprivation is unmeasurable as

such but can be observed through a set of indicators that are treated as outcomes of

the underlying situation. We begin with the PC method which is entirely descriptive

and represents a useful procedure for aggregation of several dimensions. Then we

have the FA model offering a theoretical relationship linking the observation and the

latent dimensions. MIMIC structures add exogenous ‘causes’ for the latent variables.

The SEM framework encompasses all these aspects and goes further in incorporating

interdependencies, exogenous influences and ‘conversion’ equations. Applications of

these methodologies in the field of welfare and poverty measurement are discussed

highlighting their main features and findings.

28

References

Anderson, T.W. (1984), An Introduction to Multivariate Statistical Analysis, John

Wiley and Sons, New York.

Balestrino, A. and N. Sciclone (2000), “Should we use Functionings instead of

Income to Measure Well-being? Theory, and Some Evidence from Italy”,

Mimeo, University of Pisa.

Ballon, P. and J. Krishnakumar (2005), “Estimating Basic capabilities: A Latent

Variable Approach Applied to Bolivian Data”, Paper presented at the 5th

International Conference on the Capability Approach: Knowledge and Public

Action, held in Paris, France.

Bartholomew, D.J. and M.Knott (1999), Latent Variable Models and Factor

Analysis, Edward Arnold, U.K.

Biswas B. and F. Caliendo (2002), “A Multivariate Analysis of the Human

Development Index”, Indian Economic Journal, Vol. 49, No. 4.

Bollen, K. A. (1989), Structural Equations with Latent Variables, John Wiley &

Sons, New York.

Browne, M.W. (1984), “Asymptotically Distribution-free Methods for the Analysis

of Covariance Structures” British Journal of Mathematical and Staistical

Psychology, 37, 62-83.

Browne, M.W. and G. Arminger (1995), “Specification and Estimation of Mean -

and Covariance- Structural Models”, in G. Arminger, C.C. Clogg & M.E.

Sobel (eds.) Handbook of Statistical Modelling for the Social and Behavioral

Sciences, 311-359, Plenum Press, Newbury Park.

Cerioli, A. and S. Zani (1990), “A Fuzzy Approach to the Measurement of Poverty”,

in Dagum and Zenga (eds.) Income and Wealth Distribution, Inequality and

29

Poverty, Springer Verlag Berlin.

Cheli, B. and A. Lemmi (1995), “A "Totally" Fuzzy and Relative Approach to the

Multidimensional Analysis of Poverty”, Economic Notes, vol. 24: 115-134.

Chiappero Martinetti, E. (2000), “A Multidimensional Assessment of Well-Being

based on Sen's Functionings Approach”, Rivista Internazionale di Scienze

Sociali, No 2: 207-239.

Gouriéroux, C., A. Monfort and A. Trognon (1984), “Pseudo-maximum Likelihood

Methods: Theory”, Econometrica, 52, 681-700.

Di Tommaso, M.L. (2006), “Measuring the Well-being of Children using a

Capability Approach: An Application to Indian Data”, Working Paper CHILD

no. 05/2006, University of Turin.

Hotelling, H. (1933), “Analysis of a Complex of Statistical variables into Principal

Components”, Journal of Educational Psychology, 24, 417-441.

Jöreskog, K. (1973), “A General Method for Estimatimg a Linear Structural

Equation system”, in A. S. Goldberger and O.D. Duncan (eds), Structural

Equation Models in the Social Sciences, Seminar Press, New York.

Jöreskog, K. (2002), “Structural Equation Modelling with Ordinal Variables using

LISREL”, http://www.ssicentral.com/lisrel/ordinal.htm

Jöreskog, K. and A. Goldberger (1975), “Estimation of a Model with Multiple

Indicators and Multiple Causes of a Single Latent Variable”, Journal of the

American Statistical Association, Vol. 70, No. 351.

Klasen, S. (2000), “Measuring Poverty and Deprivation in South Africa”, Review of

Income and Wealth, 46, 33-58.

Krishnakumar, J. (2004), “Going Beyond Functionings to Capabilities: An

Econometric Model to Explain and Estimate Capabilities”, Paper presented at

30

the 4th International Conference on the Capability Approach: Enhancing

Human Security, held in Pavia, Italy.

Krishnakumar, J. (2005), “Going Beyond Functionings to Capabilities: An

Econometric Model to Explain and Estimate Capabilities”, Revised version,

Paper presented at the Many Dimensions of Poverty Conference, organised by

the International Poverty Centre, UNDP, Brasilia.

Krishnakumar, J. and A.L. Nagar (2006), “On Statistical Properties of

Multidimensional Indicators based on Principal Components, Factor Analysis,

MIMIC and SEM”, Working paper.

Kuklys, W. (2005), Amartya Sen's Capability Approach: Theoretical Insights and

empirical Applications, Springer, Berlin.

Lelli, S. (2001), “Factor Analysis vs. Fuzzy Sets Theory : Assessing the influence of

different techniques on Sen's Functioning Approach”, Center for Economic

studies, K.U. Leuven.

Maasoumi, E. and G. Nickelsburg (1988), “Multidimensional Measures of Well-

being and an Analysis of Inequality in the Michigan Data”, Journal of

Business and Economic Statistics, 6(3), 327-334.

McGillivray, M. (2005), “Measuring Non-Economic Well-being Achievement”,

Review of Income and Wealth, 51,2, 337-364.

Morris, Morris D. (1979), Measuring the Condition of the World's Poor: The

Physical Quality of Life Index, Pergamon, New York.

Muthen, B. (1983), “Latent Variable Structural Equation modelling with Categorical

Data”, Journal of Econometrics, 22, 43-65.

Muthen, B. (1984), “A General Structural Equation Model with Dichotomous,

Ordered Categorical and Continuous Latent Indicators”, Psychometrika, 49,

31

115-132.

Muthen, B. (1989), “Latent Variable Modelling in Heterogenous Populations”,

Psychometrika, 54, 557-585.

Muthen, B. (2002), “Beyond SEM: General Latent Variable Modelling”,

Behaviormetrika, 29, 1, 81-117.

Nagar, A. L. and S. Basu (2001), “Weighting Socio-Economic Indicators of Human

Development (A Latent Variable Approach)”, National Institute of Public

Finance and Policy, New Delhi.

Noorbaksh, F. (2003), “Human Development and Regional Disparities in India”,

Discussion Paper, Helsinki: UNWIDER.

Qizilbash, M. (2002), “A Note on the Measurement of Poverty and Vulnerability in

the South African Context”, Journal of International Development14, 757-

772.

Rahman, T., R.C. Mittelhammer and P. Wandschneider (2003), “Measuring the

Quality of Life across Countries: A Sensitivity Analysis of Well-being

Indices”, WIDER International Conference on Inequality, Poverty and Human

Well-being, Helsinki, Finland.

Ram, R. (1982), “Composite Indices of Physical Quality of Life, Basic Needs

Fulfilment, and Income: A Principal Component Representation”, Journal of

Development Economics, 11, 227-247.

Schokkaert, E. and L. van Ootegem (1990), “Sen's Concept of the Living Standard

Applied to the Belgian Unemployed”, Recherches Economiques de Louvain,

56, 429-450.

Sen, A.K. (1985), Commodities and Capabilities, North-Holland, Amsterdam.

32

Sen A. K. (1993), Capability and Well-being, in Nussbaum M., Sen A. K. (eds), pp.

30-53.

Sen, A.K. (1999), Development as Freedom, Oxford : Oxford University Press.

Slottje, D. J. (1991), “Measuring the Quality of Life across Countries”, The Review

of Economics and Statistics, 73, 4, 684-693.

Skrondal, A. and S. Rabe-Hesketh (2004), Generalized Latent Variable Modeling:

Multilevel, Longitudinal, and Structural Equation Models, Chapman &

Hall/CRC, Boca Raton, U.S.A.

UNDP (annual, from 1990), Human Development Report, Oxford University Press,

U.K.

Wagle, U. (2005), “Multidimensional Poverty Measurement with Economic Well-

being, Capability and Social Inclusion: A Case from Kathmandu, Nepal”,

Journal of Human Development, 6, 3, 301-328.

White, H. (1982), “Maximum Likelihood Estimation of Misspecified Models”,

Econometrica, 50, 1-26.