On Measuring Social Exclusion: A New Approach with an Application to FYR Macedonia

Abstract

This chapter proposes an original methodology to measure social exclusion and applies it to the UNDP/UNICEF 2010 survey dataset on social exclusion in FYR Macedonia. Following a brief presentation of some general features of the economic and social situation in FYR Macedonia, this chapter explains first that the multivariate technique called correspondence analysis allows one to aggregate in one factor, separately for each domain of social exclusion, the variables that are available to characterize this domain. Then an approach commonly used in productivity analysis and called stochastic production frontier is presented. This technique allowed us to derive a latent vector assumed to represent the level of overall social exclusion of each of the individuals in the survey. In the last section of this chapter, an attempt is made to find out what the determinants of social exclusion are by regressing the latent variable assumed to describe social exclusion on a certain number of explanatory variables such as the level of education, the marital status, and the age of the individual.

Appendices

Appendix 1. List of Variables Used

• 1. Various domains of social exclusion which have been distinguished and to which correspondence analysis was applied (list of variables used in each domain and number of the question in the original UNDP questionnaire):

  First domain: Employment

  First variable:

  • During the last month, have you worked for payment (in cash or in kind) or for any other income at least for 1 day? (question 4)

  • If you do not work, what is the main reason why you did not look for work:

    • Have a temporary/occasional work

    • Got tired of searching, thought no work available

    • Unable to work due to long-term illness or disability (question 8)

  Second variable:

  • During the last month, have you worked for payment (in cash or in kind) or for any other income at least for 1 day? (question 4)

  • If “no job,” are you registered with the employment services in any capacity?

    • Yes, as a person looking for work

    • Yes, in another capacity

    • No (question 5)

  • If “no job,” have you made any efforts within the past 4 weeks to find work or established a business or enterprise? Yes or no. (question 7)

  Second domain: Assets

  • First variable: Do you have a bank account (including a credit card and saving deposit) registered under your own name? (question 33)

  • Second variable: Does your household own land? (question 37)

  • Third variable: Does your accommodation in which you live have running water? (question 43)

  • Fourth variable: Does your accommodation in which you live have flushing indoor toilet? (question 43)

  • Fifth variable: Does your accommodation in which you live have central heating or a local heating system? (question 43)

  • Sixth variable: Does your accommodation in which you live have electricity supply? (question 43)

  • Seventh variable: Does your accommodation in which you live have sewage system? (question 43)

  • Eighth variable: Does your accommodation in which you live have central gas supply? (question 43)

  Third Domain: Living Standards

  • 1. Eleven following variables: There are some things that many people cannot afford. For each of the following items on the card, can I just check how often your household could afford it in the past 12 months? The possible answers are never, seldom, sometimes, often. (question 46)

    • Buying food for three meals a day

    • Paying regularly the bills

    • Keeping your home adequately warm

    • Buying new clothes and shoes that you or your household needed

    • Buying medication that you or your household needed

    • Regular dental checks for every child in your household

    • Buying school materials/books for every child in your household

    • Having friends or family for a drink or meal at least once a month

    • Paying for a week’s annual holiday away from home/abroad

    • Traveling to family celebrations/for family events

    • Buying books, cinema or theater tickets

  • 2. Twenty following variables: Could you tell me whether your household has it, your household does not have it because you cannot afford it, or your household does not have it because you do not need it? (question 47)

    • Television

    • Computer

    • Internet

    • Cellular phone

    • Satellite/cable TV

    • Car (not motorcycle)

    • Washing machine

    • Freezer/refrigerator

    • Landline telephone

    • Radio receiver

    • Gas oven

    • Electric oven

    • Generator

    • Electric iron

    • Outdoor metal stove

      • Electric sewing/knitting machine

      • Electric room heater

      • Kerosene lamp

    • Microwave oven

    • Bed for each household member

    • Living room furniture

    • Vacuum cleaner

  Fourth domain: Subjective well-being

  • 1. Thirteen following variables: There are many situations that could negatively affect you or your household. Please tell me how worried you are about each of the following situations, assessing each type from 1 to 5 (“1” not worried at all and “5” very worried) (question 62)

  • Lack of sufficient incomes

  • Hunger

  • Denied access to health-care practitioners

    • Lack of housing (eviction)

    • Poor sanitation-related diseases

    • Street crime

  • Denied opportunity to practice your religion

  • Organized crime (e.g., racketeering you business)

  • Local religious conflicts (conflicts between different religious groups)

  • Local interethnic conflicts (conflicts between different ethnic groups)

    • Corruption of officials

    • Pollution

  • Global warming

  Fifth domain: Social capital

  • First variable: In your country, parents and children usually help each other. How is it in your family? Parents rather support the children or everyone takes care of themselves or children rather support the parents (question 87)

  • Second variable: Can you tell me about your close friends? These are people whom you trust, can talk to about private matters, or call on for help. With respect to close friendship, would you say that you do have close friends or you do not have any close friends, only acquaintances? (question 91)

  • Third variable: How often (never, seldom, a few times each month, a few times a week, almost every day) do you spend your free time with

    • Family/relatives

    • Neighbors

    • Your friends (question 93)

  • Fourth variable: Generally speaking, do you think most people can be trusted? (question 95)

  • Fifth variable: Have you attended any cultural event (theater, museums, concert, etc.) in the last 3 months? (question 98)

• 2. List of explanatory variables used as explanatory variables in regression analysis where the dependent variable is the individual degree of social exclusion obtained after applying the stochastic production frontier technique

  Variable	Question
  Level of education	Question 77
  Ethnic group	Question 104
  Type of settlement (village, small town,regional/economic center, capital)	Question 128
  Age	General questions about household
  Square of age	General questions about household
  Region of residence	General questions about household
  Marital status (married or not)	General questions about household
  Gender	General questions about household
  Size of accommodation	General questions about household
  Property is barrack or slum	General questions about household

Appendix 2. On Correspondence Analysis

Correspondence analysis (CA) was originally introduced by Benzécri and Benzécri (1980). It is strongly related to principal components analysis (PCA), but while PCA assumes that the variables are quantitative, CA has been designed to deal with categorical variables. More precisely, CA offers a multidimensional representation of the association between the row and column categories of a two-way contingency table. In short, the goal of CA is to find scores for both the row and column categories on a small number of dimensions (axes) that will account for the greatest proportion of the χ ² measuring the association between the row and column categories. There is thus a clear parallelism between CA and PCA, the main difference being that PCA⁵ accounts for the maximum variance. A clear presentation of CA is given in Asselin and Vu Tuan Anh (2008) and Chap. 7 in Kakwani and Silber (2008).

Let us first recall what the main features of PCA. It is in fact a data reduction technique that consists of building a sequence of orthogonal and normalized linear ­combinations of the K primary indicators that will exhaust the variability of the set primary indicators. These orthogonal linear combinations are evidently latent variables and usually called “components.” In PCA, the first component has the greatest variance, and all subsequent components have decreasing variances.

Let N be the size of the population, K the number of indicators ${I}_{k}$. The first component ${F}^{1}$may be expressed for observation i as

$$ {F}_{i}^{1}={\displaystyle \sum _{k=1}^{K}{w}_{k}^{1}{I}_{i}^{*k}}$$

where ${I}^{*k}$refers to the standardized primary indicator ${I}^{k}$. Note that $[{w}_{k}^{1}]$is the (first) factor score coefficient for indicator k. It turns out that the scores $[{w}_{k}^{1}]$are in fact the multiple regression coefficients between the component ${F}^{1}$and the standardized primary indicators ${I}^{*k}$. It is very important to understand that PCA has some limitations, of which the most important is probably the fact that PCA has been developed for quantitative variables.

It is therefore better not to use PCA when some of the variables are of a qualitative nature. Multiple correspondence analysis (MCA) is in fact the data reduction technique that should be used in the presence of categorical variables.

Let us therefore assume now that the K primary indicators are categorical ordinal and that the indicator ${I}^{k}$has ${J}^{k}$categories. Note that if some of the variables of interest are quantitative, it is always possible to transform them into a finite number of categories. To each primary indicator ${I}^{k}$, we therefore associate the set of ${J}^{k}$binary variables that can only take the value 0 or 1.

Let us now call $X(N,J)$the matrix corresponding to the N observations on the Kindicators which are now decomposed into ${J}^{k}$variables. Note that $J={\displaystyle \sum _{k=1}^{K}{J}^{k}}$ represents now the total number of categories. Call ${N}_{j}$the absolute frequency of category j. Clearly ${N}_{j}$is equal to the sum of column j of the matrix X. Let ${N}_{\mathrm{..}}$refer to the sum of all the $(N\text{by}K)$elements of the matrix X. Let also ${f}_{j}$be the relative frequency $({N}_{j}/{N}_{\mathrm{..}})$, ${f}^{i}$be the sum of the ith line of matrix X, ${f}_{ij}$be the value of cell (i, j), and ${f}_{j}^{i}$be equal to the ratio $({f}_{ij}/{f}^{i})$. Finally, call $\left\{{f}_{j}^{i}\right\}$the set of all ${f}_{j}^{i}$’s for a given observation i (j  =  1 to J). This set will be called the profile of observation i.

As stressed previously, CA is a PCA process applied to the matrix X but with the $[{c}^{2}]$-metric on row/column profiles, instead of the usual Euclidean metric. This $[{c}^{2}]$-metric is in fact a special case of the Mahalanobis distance developed in the 1930s. This metric defines the distance ${d}^{2}({f}_{j}^{i},{f}_{j}^{{i}^{\prime }})$between two profiles i and i′ as

$$ {d}^{2}({f}_{j}^{i},{f}_{j}^{{i}^{\prime }})={\displaystyle \sum _{j=1}^{J}(1/{f}_{j})(}{f}_{j}^{i}-{f}_{j}^{{i}^{\prime }}{)}^{2}$$

Note that the only difference with the Euclidean metric lies in the term $(1/{f}_{j})$. This term indicates that categories which have a low frequency will receive a higher weight in the computation of distance. As a consequence, CA will be overweighting the smaller categories within each primary indicator. It can be shown that

$$ {w}_{j}^{1,k}=\frac{1}{({N}_{j}^{k}/N)}Cov({F}^{1*},{I}_{j}^{k})$$

where ${w}_{j}^{1,k}$is the score of category ${j}_{k}$on the first (non-normalized) factorial axis, ${I}_{j}^{k}$is a binary variable taking the value 1 when the population unit belongs to the category ${j}_{k}$, ${F}^{1*}$is the normalized score on the first axis, and ${N}_{j}^{k}$is the frequency of the category ${j}_{k}$of indicator k.

It is also very interesting to note that CA offers a unique duality property since it can be shown that

$$ {F}_{1}^{i}=\frac{{\displaystyle \sum _{k=1}^{K}{\displaystyle \sum _{j=1}^{{J}_{k}}\frac{{w}_{j}^{1,k}}{{\lambda }_{1}}{I}_{i,j}^{k}}}}{K}$$

where K is the number of categorical indicators, ${J}_{k}$is the number of categories for indicator k, ${w}_{j}^{1,k}$is the score of category ${j}_{k}$on the first (non-normalized) factorial axis, ${I}_{i,j}^{k}$is a binary variable taking the value 1 when unit i belongs to category ${j}_{k}$, and ${F}_{1}^{i}$is the (non-normalized) score of observation i on the first factorial axis.⁶

Reciprocally it can be shown that

$$ {w}_{j}^{1,k}=\frac{{\displaystyle \sum _{i=1}^{N}\frac{{F}_{1}^{i}}{{\lambda }_{1}}}}{{N}_{j}^{k}}$$

This duality relationship implies thus that the score of a population unit on the first factor is equal to the average of the standardized factorial weights of the K categories to which it belongs. Conversely the weight of a given category is equal to the average of the standardized scores of the population units belonging to the corresponding category.

Appendix 3. On Frontier Efficiency Measurement

Duality and the Concept of Input Distance Function in Production Theory

Let ${x}_{i}=({x}_{1i},\dots,{x}_{ji},\dots,{x}_{ki})$denote the vector of levels of social exclusion in the various $k$domains of social exclusion for individual $i$and let ${y}_{i}$denote the overall level of social exclusion for individual $i$. An individual’s performance, as far as social exclusion is concerned, may hence be represented by the pair $({x}_{i},{y}_{i})$, i  =  1, … I.

A theoretical social exclusion index $\text{SE}$can then be estimated using a Malmquist input quantity index:

$$ SE(y,{x}^{s},{x}^{t})={D}_{input}(y,{x}^{s})/{D}_{input}(y,{x}^{t})$$

where x ^s and x ^t are two different “social exclusion input” vectors and ${D}_{input}$is an input distance function. The idea behind the Malmquist index is to provide a reference set against which to judge the relative magnitudes of the two vectors of “social exclusion inputs.” That reference set is the isoquant $L(y)$and the radially farther ${x}_{i}$is from $L(y)$, the higher the overall level of social exclusion of individual $i$is, for ${x}_{i}$must be shrunk more to move back onto the reference set $L(y).$

There is, however, a difficulty because the Malmquist index depends generally on $y$. One could use an approximation of this index such as the Tornqvist index, but such an index requires price vectors as well as behavioral assumptions.⁷ Since we do not have prices for the “social exclusion inputs,” we have to adopt an alternative strategy. The idea is to get rid of $y$by treating all individuals equally and assume that each individual has the same overall level of social exclusion: one unit for each “social exclusion input.” Let e represent such a vector of “social exclusion inputs”—a $k$-dimensional vector of ones. Thus, the reference set becomes L(e) and bounds the vectors of “social exclusion inputs” from below. Individuals with “social exclusion vectors” onto L(e) share in fact the lowest level of “overall social exclusion,” with an index value of unity, whereas individuals with large vectors of “social exclusion inputs” will then have higher overall level of social exclusions, with index values above unity.

To estimate the distance function, let $\lambda =(1/{x}_{k})$and define a $(k-1)$-dimensional vector $z$as $z=\left\{{z}_{j}\right\}=({x}_{j}/{x}_{k})$with $j=1,\dots,k-1$. Then ${D}_{input}(z,e)=(1/{x}_{k}){D}_{input}(x,e)$, and since ${D}_{\text{input}}(x,e)\le 1$, we have

$$ (1/{x}_{k})\le {D}_{input}(z,e)$$

This implies that we may also write

$$ (1/{x}_{k})={D}_{input}(z,e)\mathrm{exp}(e),\text{{1em}}e\le 0$$

By assuming that ${D}_{input}(z,e)$has a translog functional form, we have

$$ \mathrm{ln}(1/{x}_{k})={a}_{0}+{\displaystyle \sum _{j=1}^{k-1}{a}_{j}\mathrm{ln}{z}_{j}}+(1/2){\displaystyle \sum _{j=1}^{k-1}{\displaystyle \sum _{h=1}^{k-1}{a}_{jh}}}\mathrm{ln}{z}_{j}\mathrm{ln}{z}_{h}+\varepsilon$$

Estimates of the coefficients ${a}_{j}$and ${a}_{jh}$may be obtained using corrected ordinary least squares (COLS) or maximum-likelihood methods (see below) while the input distance function ${D}_{input}({z}_{i},e)$for each individual i is provided by the transformation

$$ {D}_{input}({z}_{i},e)=\mathrm{exp}\{\mathrm{max}({e}_{i})-{e}_{i}\}.$$

This distance will, by definition, be greater than or equal to one (since its logarithm will be positive) and will hence indicate by how much an individual “social exclusion input vector” must be scaled back in order to reach the “social exclusion inputs” frontier. This procedure guarantees therefore that all “social exclusion input vectors” lie on or above the resource frontier $L(e)$. The overall level of social exclusion for individual i will then be obtained by dividing ${D}_{input}({z}_{i},e)$by the minimum observed distance value—which by definition equals 1.

Estimation Procedures: The Stochastic Production Frontier Approach

Let us take as a simple illustration the case of a Cobb–Douglas production function. Let $\mathrm{ln}{y}_{i}$be the logarithm of the output of firm $i(i=1\text{to}N)$and ${x}_{i}$a $(k+1)$row vector, whose first element is equal to one and the others are the logarithms of the $k$inputs used by the firm. We may then write that

$$\mathrm{ln}({y}_{i})={x}_{i}\beta-{u}_{i}\text{}\,i=1\text{to}N$$

where $b$is a $(k+1)$column vector of parameters to be estimated and ${u}_{i}$a nonnegative random variable, representing the technical inefficiency in production of firm $i$.

The ratio of the observed output of firm $i$to its potential output will then give a measure of its technical efficiency ${\text{TE}}_{i}$so that

$$[{\text{TE}}_{i}={y}_{i}/\mathrm{exp}({x}_{i}\beta)=\mathrm{exp}({x}_{i}\beta-{u}_{i})/\mathrm{exp}({x}_{i}\beta)=\mathrm{exp}(-{u}_{i})]$$

One of the methods allowing the estimation of this output-oriented Farrell measure of technical efficiency TE _i (see Farrell 1957) is to use an algorithm proposed by Richmond (1974) which has become known as corrected ordinary least squares (COLS). This method starts by using ordinary least squares to derive the (unbiased) estimators of the slope parameters. Then in a second stage, the (negatively biased) OLS estimator of the intercept parameter ${b}_{0}$is adjusted up by the value of the greatest negative residual so that the new residuals have all become nonnegative. Naturally the mean of the observations does not lie any more on the estimated function: the latter has become in fact an upward bound to the observations.

One of the main criticisms of the COLS method is that it ignores the possible influence of measurement errors and other sources of noise. All the deviations from the frontier have been assumed to be a consequence of technical inefficiency. Aigner et al. (1977) and Meeusen and van den Broeck (1977) independently suggested an alternative approach called the stochastic production frontier method in which an additional random error ${v}_{i}$is added to the nonnegative random variable ${u}_{i}$. We therefore write

$$ \mathrm{ln}({y}_{i})={x}_{i}\beta+{v}_{i}-{u}_{i}$$

The random error ${v}_{i}$is supposed to take into account factors such as the weather and the luck, and it is assumed that the ${v}_{i}$’s are i.i.d. normal random variables with mean zero and constant variance σ ²_v . These ${v}_{i}$’s are also assumed to be independent of the ${u}_{i}$’s, the latter being taken generally to be i.i.d. exponential or half-normal random variables. For more details on this maximum-likelihood estimation procedure, see Battese and Corra (1977) and Coelli et al. (1998), as well as programs such as FRONTIER (Coelli 1992) or LIMDEP (Green 1992). The same methods (COLS and maximum likelihood) may naturally be also applied when estimating distance functions.