The Global Multidimensional Poverty Index Report: The Most Detailed Picture to Date of the World’s Poorest People presents the global MPI 2018, a newly revised index based on a short but powerful list of 10 deprivations. During the launch of the Global MPI 2018, Achim Steiner further highlights that if development is about being more precise about directing the limited resources governments have and the indicators of the global MPI can aid that process. The revised global MPI is the joint work of OPHI and UNDP. The overarching aim of the revised MPI is to better align the global MPI with the SDGs (Alkire and Jahan 2018).
Chapter 1 provides a global overview of findings from the global MPI 2018. Chapter 2 focuses on India, presenting a case study on MPI from 2005/06 to 2015/16, with analyses of trends by age, state, caste, and religion, and a direct mapping of poverty at the district level in 2015/16. Turning first to the youngest on our planet, Chapter 3 assesses child poverty across all countries. Multidimensional poverty varies both within and across major geographic regions like Latin America or East Asia and the Pacific, and Chapter 4 presents some notable highlights. Going within countries, Chapter 5 scrutinises poverty levels and composition across rural and urban areas. Finally, Chapter 6 zooms in to investigate circumstances within and across countries according to subnational regions.
Download the Report here.
An article following the launch of Afghanistan Multidimensional Poverty Index 2016–2017: Report and Analysis on 31 March 2019. Available here.
An introduction to Afghanistan Multidimensional Poverty Index 2016–2017: Report and Analysis, co-authored by OPHI, which was published in March 2019. The article is available here.
A piece introducing the Afghanistan Multidimensional Poverty Index 2016–2017: Report and Analysis which was co-authored by OPHI and published in March 2019. The article is available here.
A short overview of the Afghanistan Multidimensional Poverty Index 2016–2017: Report and Analysis, following its launch in March 2019. The article is available here.
A piece introducing the Afghanistan Multidimensional Poverty Index 2016–2017: Report and Analysis which was co-authored by OPHI and published in March 2019. Read the full article here.
An article in Afghanistan Times, following the launch of Afghanistan Multidimensional Poverty Index 2016–2017 which was published in March 2019 and was co-authored by OPHI. Read the article here.
10 Some Regression models for AF measures
From a policy perspective, in addition to measuring poverty we must perform some vital analyses regarding the transmission mechanisms between policies and poverty measures. Issues we may wish to explore with a regression model include the determinants of poverty at the household level in the form of poverty profiles or the elasticity of poverty to economic growth, while controlling for other determinants. We may also be interested in understanding how macro variables such as average income, public expenditure, decentralization, information technology, and so on relate to multidimensional poverty levels or changes across groups or regions—and across time. Through regression analysis, we can partially study these transmission mechanisms by looking at the determinants of multidimensional poverty. In a regression model, we can account for the effect or the ‘size’ of determinants of multidimensional poverty, which would not be possible with a purely descriptive analysis.
Such analyses are routinely performed for income poverty using what we will term ‘micro’ or ‘macro’ regressions. As is explained below, the term ‘micro’ refers to analyses in which the unit of analysis is a person or household; the term ‘macro’ refers to analyses in which the unit of analysis is a subgroup, such as a district, a state, a province, or a country. This section provides the reader with a general modelling framework for analysing the determinants of Alkire–Foster poverty measures, at both micro and macro levels of analyses.
In general in micro regressions, the focal variable to be modelled may be a binary variable denoting a person’s status as poor (or non-poor) or a variable denoting the deprivation score assigned to the poor. In macro regressions, the focal variable to model is a subgroup poverty measure like the poverty headcount ratio or any other Foster–Greer–Thorbecke (FGT) poverty measure. As with regressions that model the monetary headcount ratio or the poverty gap, macro regressions with 
-dependent variables must respect their nature as cardinally meaningful values ranging from zero to one. In these cases, a classic linear regression is not the appropriate model. The common assumptions of the classic linear regression fall short because the range of the dependent variable is bounded and may not be continuous or follow a normal distribution that is often assumed in linear regression models.
Generalized linear models (GLMs), by contrast, are preferred as the data analytic technique because they account for the bounded and discrete nature of the AF-type dependent variables. GLMs extend classic linear regression to a family of regression models where the dependent variable may be normally distributed or may follow a distribution within the exponential family—such as the gamma distribution, bernoulli distribution, or binomial distribution. GLMs encompass models for quantitative and qualitative dependent variables, such as linear regression models, logit and probit models, and models for fractional data. Hence they offer a general framework for our analysis of functional relationships.[1]
This section presents the GLM as an overall framework to study micro and macro determinants of multidimensional poverty. Within this framework we are able to account for the bounded nature of the Adjusted Headcount Ratio and the incidence 
while modeling their determinants. We are also able to model these determinants for the probability of being multidimensionally poor.
This Chapter is structured as follows. We begin by differentiating micro and macro regression analyses. For this purpose, we review the measure of the AF class, its consistent partial indices, and the type of variables they represent in a regression framework. We then present the general structure and possible applications of the GLMs to AF measures. We begin with an exposition of linear regression models and how these extend to models for binary dependent variables—logit and probit—and fractional[2] data. We assume readers have some background in applied statistics and key elements of estimation and inference. Our exposition deals with cross-sectional data but could be easily extended to panel data.[3] Before we begin we should point out that the notation used in this chapter is self-contained. Some notation may duplicate that used in other sections or chapters for different purposes. When the notation is linked to discussions in other sections or chapters, it will be specified accordingly.
The AF measures can be used to analyse poverty determinants[4] for a household or person (henceforth we use the term ‘household’) and for a population subgroup. We could study determinants of household or subgroup poverty in a ‘micro’ and a ‘macro’ context. In what follows, the term ‘micro’ refers to regressions where the unit of analysis is the person or household. The term ‘macro’ refers to regressions where the unit of analysis is some spatial or social aggregate, such as a district, state, province, ethnic group, or country. Micro regressions are useful for describing the distinctive features of multidimensional poverty profiles across households (in a given country) or to understand the determinants of poverty. Macro regressions, on the other hand, are useful for studying the determinants of poverty at the province, district, state, or country levels. Both types of regressions use specific components of the AF measures. In the case of micro regressions, the focal variable is the (household) censored deprivation score. From the exposition of Chapter 5, we know that if the deprivation score of a household 
is equal to or greater than the multidimensional poverty cutoff (
), the household is identified as multidimensionally poor. This poverty status of a household is represented by a binary variable (indicator function) that takes the value of one if the household is identified as multidimensionally poor and zero otherwise.
A natural question that arises is how to analyse the ‘causes’ (in the sense of determinants) that underlie the (multidimensional) poverty status of a household. An intuitive way would be to model the probability of a household becoming multidimensionally poor or falling into multidimensional poverty. A crucial point should be noted here, which may be more particular to multidimensional notions of poverty than their monetary counterparts: when modelling the probability of a household being in monetary poverty, various health- and education-related variables, which are not embedded in the monetary poverty measures, are used as exogenous variables.[5] In a multidimensional case, these exogenous variables may be used directly to construct the poverty measure and so the probability models at the household level are subject to a potential endogeneity issue. For example, if among the explanatory variables we include an asset variable like car ownership, and if that indicator was also included among the ‘assets’ indicator that appears in the multidimensional poverty measure, there will be an endogeneity issue in the model. A typical approach to deal with endogeneity is to use an instrumental variable, but often it is very difficult to find a valid instrument.[6] An alternative approach would be to restrain the set of explanatory variables of the household regression model to non-indicator measurement variables[7]—like certain demographic variables—or additional socioeconomic characteristics of the household. From such a perspective one would be interested in examining household poverty profiles. Sample research questions would be: are female-headed households more likely to be multidimensionally poor? Are larger households more prone to be multidimensionally poor? How does the probability of being multidimensionally poor vary by household size and composition, caste, or ethnicity?
In the case of cross-sectional macro regressions, the focal variables are the 
measures at the province, district, state, or country levels, or some other population sub-group or aggregate which leads to a proper sample size.[8] If the focus is on the Adjusted Headcount Ratio , the focal variables in a macro regression could comprise or could use the intensity 
and incidence of multidimensional poverty. However, from Chapter 5 we know that and are partial indices that do not enjoy the same properties as the measure. In this Chapter we do not further consider regression models for . Although is also a partial index, which violates dimensional monotonicity, we do discuss its analysis, given the prominence of existing studies using the unidimensional poverty headcount ratio.
As already noted, and are bounded between zero and one. In statistical terms, and are fractional (proportion) variables that lie in the unit interval. Their restricted range of variation limits the use of the linear regression model because these models assume continuous variables comprised between 
and +
. A natural model to be considered is one that reflects the fractional nature of any of these two indices (see section 10.4).
Our exposition of GLMs draws on Nelder and Wedderburn (1972), McCullagh and Nelder (1989) and Firth (1991). We treat GLMs in an applied manner covering the basic structure of the models, estimation, and model fitting. We do not provide a detailed exposition of the method itself. Readers interested in a complete statistical treatment of GLMs can refer to McCullagh and Nelder (1989) or to Dobson (2001). The former presents an excellent and comprehensive statistical overview of GLMs, but assumes an advanced statistics background on the part of the reader. The latter presents a briefer and more synthetic exposition of GLMs at a moderate level of statistical complexity.
Generalized linear models are an extension of classic linear models. The linear regression model has found widespread application in the social sciences mainly due to its simple linear formulation, easy interpretation, and estimation. In monetary poverty analysis, linear regression analysis has been used to study the determinants of household consumption expenditures or to model the growth elasticity of per capita income or income poverty aggregates like the headcount ratio or the poverty gap index.[9] Linear regressions are also used to model changes in (i) the income share of the poorest quintile (Dollar and Kray 2004); (ii) adjusted GDP incomes (Foster and Szekely 2008); (iii) the poverty rate (Ravallion 2001); and (iv) the growth rates of real per capita GDP (Barro 2003).
We begin with a brief review of the classic linear regression model and its notation and build on this to present the more generic case of GLMs. The classic linear regression model (LRM) assumes that the endogenous or dependent variable (
) (hitherto referred to as ‘endogenous’) is a linear function of a set of 
exogenous[10] variables (
). The LRM assumes that the endogenous variable is continuous and distributed with constant variance. In addition the LRM may also assume that the endogenous variable is normally distributed. However this assumption is not needed for estimating the model but only to obtain the exact distribution of the parameters in the model. In the case of large samples one may not need to assume normality in a LRM as inference on parameters is based on asymptotic theory (c.f Amemiya, 1985). These assumptions may be inappropriate if the endogenous variable is discrete (binary or categorical)—or continuous but non-normal.[11] GLMs overcome these limitations. They extend classic linear regression to a family of models with non-normal endogenous variables. In what follows, random variables are denoted in uppercase and observations in lowercase; vectors are represented with lowercase bold and matrices with uppercase bold.
Consider a sample of 
observations of a scalar dependent variable () and a set of K exogenous variables (
). This data is specified as 
, where 
is a 
column vector. Each observation 
is assumed to be a realization of a random variable 
independently distributed with mean 
. The classic regression model with additive errors for the 
observation can be written as
|
|
(10.1) |
where 
denotes the conditional expectation[12] of the random variable 
given 
, and 
is a disturbance or random error. From equation (10.1) we see that the dependent variable is decomposed into two components: a systematic or deterministic component given the exogenous variables and an error component. The deterministic component is the conditional expectation , while the error component, attributed to random variation, is .
Equation (10.1) is a general representation of regression analysis. It attempts to explain the variation in the dependent variable through the conditional expectation without imposing any functional form on it. If we specify a linear functional form of the conditional expectation 
we obtain the classic linear regression model. Then, the systematic part of the model may be written
|
|
(10.2) |
where 
is the value of the 
exogenous variable for observation 
. To show the relation between a linear regression model and a generalized linear model it will become convenient to denote the right-hand side of equation (10.1) by 
, referred to as the predictor in the generalized linear model. Thus we can write
|
|
(10.3) |
and the systematic part can be expressed as
|
|
(10.4) |
Equations (10.1) to(10.4) lead to the familiar linear regression model:
|
|
(10.5) |
where 
, 
,…, 
are parameters whose values are unknown and need to be estimated from the data.[13] Note that in the linear regression model of equation (10.6), the conditional expectation is equal to the linear predictor:
|
|
(10.6) |
The LRM additionally assumes that the errors 
are independent, with zero mean, constant variance 
and follow a Gaussian or normal distribution.[14] Often the assumptions on 
are conditional on the exogenous variables, as these are possibly stochastic or random. Then, the errors have zero mean and homoscedastic or identical variance conditional on the exogenous variables, that is, 
. Due to the relationship between and 
, the dependent variable is also normally distributed with constant variance. In other words, in a LRM, the distribution of the dependent variable is derived from the distribution of the disturbance. As explained in section 10.2.2, in a GLM the distribution of the dependent variable is specified directly.
The GLM family of models involves predicting a function of the conditional mean of a dependent variable as a linear combination of a set of explanatory variables. Classic linear regression is a specific case of a GLM in which the conditional expectation of the dependent variable is modelled by the identity function. GLMs extend the domain of applicability of classic linear regression to contexts where the dependent variable is not continuous or normally distributed. GLMs also permit us to model continuous dependent variables that have positively skewed distributions.
Generalized linear models relax the assumption of additive error in equation (10.1). The random component is now attributed to the dependent variable itself. Thus, for GLMs we need to specify the conditional distribution of the dependent variable given the values of the explanatory variables, denoted as 
. These distributions often belong to the linear exponential family, such as the Gaussian, binomial, poisson, and gamma, among others—although recently have been extended to non-exponential families (McCullagh and Nelder 1989).
A generalized linear model is one that takes the form:
|
|
(10.7) |
where the systematic part or linear predictor () is now a function (
) of the conditional expectation of the dependent variable 
; 
is a one-to-one differentiable function referred to as the link function, and 
is referred to as the linear predictor. The link function transforms the conditional expectation of the dependent variable to the linear predictor, which is a linear function of the explanatory variables that could be of any nature. This allows the linear predictor to include continuous or categorical variables, a combination of both, or interactions—as well as transformations of continuous variables. Note that when the link function 
is the identity function, we have an LRM.
In most applications, as in the regression analysis with AF measures, the primary interest is the conditional mean . This could be easily retrieved from equation (10.7) by inverting the link function; hence we can write
|
|
(10.8) |
where 
is the inverse link 
also called the mean function. Equations (10.7) and (10.8) provide two alternative specifications for a GLM, either as a linear model for the transformed conditional expectation of the dependent variable—given by (10.7)—or as a non-linear model for the conditional mean—given by (10.8).
A GLM is thus composed of three components: (i) a random component resulting from the specification of the conditional distribution of the dependent variable given the values of the explanatory variables (this is implicit and cannot be seen directly); (ii) a linear predictor 
, and iii) a link function (cf. Fox 2008: ch.15).
The distribution of the dependent variable and the choice of the link function are intimately related and depend on the type of variable under study. The form of a proper link function is determined to some extent[15] by the range of the dependent variable and consequently by the range of variation of its conditional mean.
In the case of AF poverty measures, we may consider two types of dependent variables with a different range of variation and distribution. The first type is a binary indicator identifying multidimensionally poor households. This variable takes the value of one if the household is identified as multidimensionally poor and zero otherwise. The Bernoulli distribution is suitable to describe this kind of variable. A typical model in this case is the probit or logit model. As we will see, in a GLM this is equivalent to choosing a logit link. The second type of dependent variable that we could study in the AF approach is a proportion. The Adjusted Headcount Ratio and the incidence are fractions or proportions that take values in the unit interval. The binomial distribution may be suitable as a model for these proportions.
In each of these cases, the link function should map the range of variation of the dependent variable—
for the binary indicator and 
for the proportion—to the whole real line 
. The scale is chosen in such a way that the fitted values respect the range of variation of the dependent variable. Columns one to five in Table 10.1 present the two types of dependent variables with AF measures that we study in this section, along with their range of variation, type of model, level of analysis, and random variation described by the conditional distribution. The link and mean functions are explained in the examples in sections 10.3 and 10.4. Before presenting the examples, we briefly explain the estimation and goodness of fit of GLMs.
Table 10.1 Generalized Linear Regression Models with AF Measures
|
Dependent variable AF measure: |
Range of |
Regression Model |
Level |
Conditional Distribution |
Link |
Mean function |
||
|
Binary |
0,1 |
Probability |
Micro |
Bernoulli |
Logit |
|
|
|
|
|
[0,1] |
Proportion |
Macro |
Binomial |
Probit |
|
|
|
|
Note: |
||||||||
Once we have selected the particular models of our study, we need to estimate the parameters and measure their precision. For this purpose we maximize the likelihood or log likelihood[16] of the parameters of our data denoted by 
.[17] The likelihood function of a parameter is the probability distribution of the parameter given 
.
To assess goodness of fit of the possible estimates we use the scaled deviance. This statistic is formed from the logarithm of a ratio of likelihoods and measures the discrepancy, or goodness of fit, between the observed data and the fitted values generated by the model. To assess the discrepancy we use as a baseline the full or ‘saturated’ model. Given observations, the full model has parameters, one per observation. This model fits the data perfectly but is uninformative because it simply reproduces the data without any parsimony. Nonetheless it is useful for assessing discrepancy vis-à-vis a more parsimonious model that uses K parameters. Hence in the saturated model the estimated conditional mean 
= 
and the scaled deviance is zero. For intermediate models, say with K parameters, the scaled deviance is positive.
The scaled deviance statistic
|
|
(10.9) |
is twice the difference between 
which is the maximum log likelihood of a saturated model or exact fit, and 
the log likelihood of the current or reduced model.
The goodness of fit is assessed by a significance test of the null hypothesis that the current model holds against the alternative given by the saturated or full model. Under the null hypothesis, 
is approximately distributed as a 
random variable where the number of degrees of freedom equals the difference in the number of regression parameters in the full and the reduced models. However, an appropriate assessment of the goodness of fit is based on the conditional distribution of 
given 
. If is not significant, it suggests that the additional parameters in the full model are unnecessary and that a more parsimonious model with fewer parameters may be sufficient.
The scaled deviance statistic is also useful for model selection. Due to its additive property, the discrepancy between nested sets of models can be compared if maximum likelihood estimates are used. Suppose we are interested in comparing two models, A and B, that represent two different choices of explanatory variables, 
and 
, that are nested. Intuitively this means that all explanatory variables included in model A are also present in model B, a more complex or less parsimonious model. The improvement in fit may be assessed by a significance test of the null hypothesis that model A holds against the alternative given by model B. If the value of the scaled deviance statistic is found to be significant, there is an improvement in the fit of model B vis-à-vis model A, although a general conclusion on model selection should also consider the added complexity of model B.
In the case of micro regression analysis, the focal variable is the (household) censored deprivation score . This score reflects the joint deprivations characterizing a household identified as multidimensionally poor. From a policy perspective a natural question that arises consequently is how to understand the ‘causes’ that underlie the (multidimensional) poverty status of a household. The simplest model for this purpose is a probability model, which we illustrate in this section; although one could also consider modelling the vector directly. We are thus interested in assessing the probability of a household being multidimensionally poor. Within the AF framework this is equivalent to comparing the deprivation score of a household with the multidimensional poverty cutoff (). If is above the multidimensional poverty cutoff (), the household is identified as multidimensionally poor. This is represented by a binary random variable () that takes the value of one if the household is identified as multidimensionally poor and zero otherwise, as follows:
|
|
(10.10) |
The outcomes of this binary variable occur with probability 
which is a conditional probability on the explanatory variables. For a (sampled) household identified as multidimensionally poor this is represented as
|
|
(10.11) |
and thus the conditional mean equals the probability as follows:
|
|
(10.12) |
For a binary model the conditional distribution of the dependent variable, or random component in a GLM, is given by a Bernoulli distribution (Table 10.1). Thus the probability function of is
|
|
(10.13) |
To ensure that the conditional mean given by the conditional probability stays between zero and one, a GLM commonly considers two alternative link functions (). These are given by the quantile functions of the standard normal distribution function 
and the logistic distribution function 
. The former is referred to as the probit link function and the latter as the logit link function. The probit link function does not have a direct interpretation, while the logit is directly interpretable as we discuss below.[19]
The logit of 
is the natural logarithm of the odds that the binary variable 
takes a value of one rather than zero. In our context, this gives the relative chances of being multidimensionally poor. If the odds are ‘even’—that is, equal to one—the corresponding probability (
of falling into either category, poor or non-poor, is 0.5, and the logit is zero. The logit model is a linear, additive model for the logarithm of odds as in equation (10.14), but it is also a multiplicative model for the odds as in equation (10.15):
|
|
(10.14) |
|
|
|
(10.15) |
The conditional probability 
is then
|
|
(10.16) |
The partial regression coefficients 
are interpreted as marginal changes of the logit, or as multiplicative effects on the odds. Thus, the coefficient 
indicates the change in the logit due to a one-unit increase in 
, and 
is the multiplicative effect on the odds of increasing by one, while holding constant the other explanatory variables. For example, if the first explanatory variable increases by one unit, the odds ratio in equation (10.15) associated with this increase is 
, and 
. For this reason, is known as the odds ratio associated with a one-unit increase in . To see the percentage change in the odds, we need to consider the sign of the estimated parameter. If 
is negative, the change in denotes a decrease in the odds; this decrease is obtained as 
. Likewise if is positive, the change in indicates an increase in the odds. In this case, the increase is obtained as 
To illustrate the type of micro regression models that have been discussed, we use a subsample of the Indonesian Family Life Survey (IFLS) dataset. This is a dataset analysed by Ballon and Apablaza (2012) to assess multidimensional poverty in Indonesia during the period 1993–2007. The IFLS is a large-scale longitudinal survey of the socioeconomic, demographic, and health conditions of individuals, households, families, and communities in Indonesia. The sample is representative of about 83% of the population and contains over 30,000 individuals living in thirteen of the twenty-seven provinces in the country. Ballon and Apablaza (2012) measure multidimensional poverty at the household level in five equally weighted dimensions: education, housing, basic services, health issues, and material resources. For this illustration we retain a poverty cutoff of 33%. Thus a household is identified as multidimensionally poor if the sum of the weighted deprivations is greater than 33%. That is, takes the value of one if 
33% and zero otherwise. Within the GLM framework this binary dependent variable is estimated by specifying a Bernoulli distribution and a logit link function. This is equivalent to a logit regression.
The household poverty profile that we specify regresses the log of the odds of being multidimensionally poor (using 
on the demographic and socioeconomic characteristics of the household head. For this illustration we use data for West Java in 2007. West Java is a province of Indonesia located in the western part of the island of Java. It is the most populous and most densely populated of Indonesia’s provinces, which is why we selected it. The explanatory variables included in this illustration are non-indicator measurement variables and comprise:
· Education of the household head, defined as the number of years of education (not necessarily completed);
· The presence of a female household head, represented by a dummy variable taking a value of one if the household head is a female and zero if male;
· Household size, defined by the number of household members;
· The area in which the household resides, represented by a dummy variable taking a value of one if the household resides in the urban areas of West Java and zero otherwise;
· Muslim religion, represented by a dummy variable taking a value of one if the household’s main religion is Muslim and zero if not.
Table 10.2 Logistic Regression Model of Multidimensional Poverty in West Java
Table 10.2 reports the logistic regression results of this poverty profile for West Java in 2007. Columns two to five report the estimated regression parameters along with their standard errors, t ratios, and significance levels at 5%[20]. Apart from being Muslim, all other determinants are significant at the 5% level and show the expected signs. For a given household, the log of the odds of being multidimensionally poor decreases with the education of the household head and with an urban location and increases with the presence of a female household head and with household size. The odds ratio for years of education of the household head indicates that an increase of one year of education decreases the odds of being multidimensionally poor by 49%, ceteris paribus, whereas having a female household head increases the odds of being multidimensionally poor by 28%, ceteris paribus.[21] Similarly, the odds of a household of being multidimensionally poor decrease by 57% for households living in urban areas, ceteris paribus, and increase by 10% for each additional household member. Figure 10.1 shows the odds model for urban and rural areas as a function of the education of the household head, holding constant the gender status of the household head (female), assuming five household members (average), and being Muslim. The logistic curves show a decrease in the probability of a household being multidimensionally poor as the education of the household head improves. These probabilities are lower for households living in urban areas compared to rural ones.
Figure 10.1 Logistic Regression Curve—West Java
As religion turns out to be statistically insignificant, we could consider an alternative poverty profile without religion as an explanatory variable (model B). To test whether this restrained model (without religion) is as good as the current model (model A), we compare the deviance statistics of both models.[22] Formally we test the following hypothesis:
Model A is as good as model B
Model A fits better than model B.
To reject the null hypothesis we compare with the corresponding chi-square statistic 
with 
degrees of freedom. These degrees of freedom correspond to the difference in the number of parameters in model A and model B. A non-rejection of the null hypothesis indicates that both models are statistically equivalent and thus the most parsimonious model, which has the smaller number of explanatory variables, should be chosen—which is B in this context. A rejection of the null indicates a statistical justification for model A. In our case the comparison of the two nested models, A and B, gives a scaled deviance statistic 
of 0.05. We compare this value with the corresponding chi-square statistic of one degree of freedom and a 5% type I error rate; this gives a value of 3.84. As is smaller than 3.84, we cannot reject the null hypothesis; so we choose the more parsimonious model B and drop religion as an explanatory variable.
We now turn to the econometric modelling for the Adjusted Headcount Ratio
and the incidence of multidimensional poverty as endogenous or dependent variables. As and are bounded between zero and one, an econometric model for these endogenous variables must account for the shape of their distribution. and are fractional (proportional) variables bounded between zero and one with the possibility of observing values at the boundaries. This restricted range of variation also applies for the conditional mean, which is the focus of our analysis. Thus specifying a linear model, which assumes that the endogenous variable and its mean take any value in the real line, and estimating it by ordinary least squares is not the right strategy, as this ignores the shape of the distribution of these dependent variables. Clearly if the interest of the research question is not in modelling the conditional mean of the proportion but rather in modelling the absolute change (between two time periods) of or , which can take any value, standard linear regression models may apply. In what follows we describe the statistical strategy for modelling the conditional mean of or as a function of a set of explanatory variables.
Various approaches have been used in the literature to model a fraction or proportion. We can differentiate between two types of approaches—often referred to as one-step or two-step approaches. These differ in the treatment of the boundary values of the fractional dependent variable. In a one-step approach, one considers a single model for the entire distribution of the values of the proportion, where both the limiting observations and those falling inside the unit interval are modelled together. In a two-step approach, the observations at the boundaries are modelled separately from those falling inside the unit interval. In other words, in a two-step approach one considers a two-part model where the boundary observations are modelled as a multinomial model and remaining observations as a fractional one-step regression model (Wagner 2001; Ramalho, Ramalho and Murteira 2011). The decision whether a one- or a two-part model is appropriate is often based on theoretical economic arguments. Wagner (2001) illustrates this point. He models the export/sales ratio of a firm and argues that firms choose the profit-maximizing volume of exports, which can be zero, positive, or one. Thus the boundary values of zero or one may be interpreted as the result of a utility-maximizing mechanism. Following this theoretical economic argument he specifies a one-step fractional model for the exports/sales ratio. In the absence of an a priori criteria for the selection of either a one- or two-part model, Ramalho et al. (2011) propose a testing methodology that can be used for choosing between one-part and two-part models. In the case of or we consider that non-poverty and full poverty, the boundary values, as well as the positive values, are characterized by the same theoretical mechanism. This is thus represented by a one-part model. For further references on alternative estimation approaches for one-part models, see Wagner (2001) and Ramalho et al. (2011).
To model or we follow the modelling approach proposed by Papke and Wooldridge (1996). For this purpose we denote the Adjusted Headcount Ratio or the incidence by . For a given spatial aggregate, say a country, the Adjusted Headcount Ratio or the incidence is 
. Papke and Wooldridge (PW hereafter) propose a particular quasi-likelihood method to estimate a proportion. The method follows Gourieroux, Monfort, and Trognon (1984) and McCullagh and Nelder (1989) and is based on the Bernoulli log-likelihood function which is given by
where 
is a known nonlinear function satisfying 
. In the context of a GLM, is the mean function defined in equation (10.8) as the inverse link function. PW suggest as possible specifications for any cumulative distribution function, with the two most typical examples being the logistic function and the standard normal cumulative density function as described in Table 10.1.
The quasi-maximum likelihood estimator (QML) obtained from equation (10.17) is consistent and asymptotically normal, provided that the conditional mean is correctly specified. This follows the QML theory where consistency and asymptotic normality characterize all QML estimators belonging to the linear exponential family of distributions, which is the case of the Bernoulli distribution of equation (10.17).
We would like to conclude with a few recommendations for performing a macro regression with or as explained variables. First, we suggest testing for linearity before specifying a non-linear functional form. For this purpose one can apply the Ramsey RESET[23] test of functional misspecification. The test consists of evaluating the presence of nonlinear patterns in the residuals that could be explained by higher-order polynomials of the dependent variable. Second, we recommend testing for possible endogeneity using a two-stage or instrumental variable (IV) estimation. In regressions of the type of the macrodeterminants of or 
it is very likely that there will be a correlation between one or more of the explanatory variables and the error term. Let us suppose we regress the Adjusted Headcount Ratio on the logarithm of the per capita gross national income in PPP of the same year for a group of countries. This is the GNI converted to international dollars using purchasing power parity rates. This gives a contemporaneous model for the semi-elasticity between growth and poverty. In this very simple model, it is highly likely that the GNI would be correlated with the disturbance of the equation, which consists of unobserved variables affecting the poverty rate. This violates a necessary condition for the consistency of standard linear estimators. To deal with endogeneity, often one replaces the endogenous explanatory variable with a proxy assumed to be correlated with the endogenous explanatory variable but uncorrelated with the error term.
Third, one is also very likely to find measurement errors among the explanatory variables in a model for or . This issue can also be treated with the IV method by replacing the measured-with-error variable with a proxy. To minimize the loss of efficiency that may result from an IV estimation, one can complement the estimation results using the Generalized Method of Moments. Lastly, we would like to point out that although this Chapter has focused on the modelling of levels of poverty (rates of poverty: , ), it is at once straightforward and necessary to analyse changes in poverty. It suffices to estimate the model in levels and then compute the marginal effects of the expected poverty rate with respect to the explanatory variables included in the model.
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[1] Cf. Nelder and Wedderburn (1972) and McCullagh and Nelder (1989).
[2] Also referred to as models for proportions.
[3] Skrondal and Rabe-Hesketh (2004), Rabe-Hesketh and Skrondal (2012) address this extension.
[4] The term determinants shall be understood in a ‘weak’ sense and refers to ‘proximate’ causes of poverty as defined in Haughton and Khandker (2009: 147).
[5] Also called independent, exogenous, or explanatory variables. We prefer the terms ‘exogenous’ or ‘explanatory’ to refer to the right-hand-side variables of a regression. In this section we use both terms interchangeably.
[6] See, for example, Bound, Jaeger, and Baker (1995) and Stock, Wright, and Yogo (2002).
[7] These are variables with explanatory power that were not used when constructing the poverty measure. These variables are expected to be uncorrelated with the error term of the model.
[8] Small-sample econometric and statistical techniques could be envisaged in the case of aggregates with very few categories.
[9] See, for example, De Janvry and Sadoulet (2010) and Roelenand Notten (2011).
[10] In the statistical literature 
is referred to as a regressor or covariate that is exogenous when the assumptions on the disturbance term are conditional on the covariates. In our exposition, all assumptions on the disturbance term or the dependent variable are conditional on the regressors so we use the term ‘exogenous’ instead of the generic term regressor. By ‘exogenous’ we mean non-stochastic or conditionally stochastic right-hand-side variables.
[11] An example of a non-normal continuous variable is income (consumption expenditures). The distribution of income is skewed (to the right), takes on only positive values, and is often heteroscedastic.
[12] Or conditional mean. We use both terms interchangeably.
[13] An equivalent expression of the LRM is a matrix representation of the form 
, where 
is an 
vector of observations; 
is an 
vector of disturbances; 
is a 
matrix of explanatory variables, where each row refers to a different observation each column to a different explanatory variable; and 
is a 
vector of parameters. However for the expositional purposes of this Chapter we do not use the matrix representation but rather the one specified in equation (10.5).
[14] To denote a random variable as normally distributed we follow the statistical convention and denote it as 
.
[15] The range of variation of the dependent variable is a mild requirement for the choice of a proper link function. As noted by Firth (1991) this mild requirement is complemented by multiple criteria where the choice of a proper link function is made on the grounds of its fit to the data, the ease of interpretation of parameters in the linear predictor, and the existence of simple sufficient statistics.
[16] The parameters in a GLM are estimated by a numerical algorithm, namely, iterative weighted least squares (IWLS). For models with the links considered in this section, the IWLS algorithm is equivalent to the Newton–Raphson method and also coincides with Fisher scoring (McCullagh and Nelder 1989).
[17] Note we drop the subscript 
as the log likelihood depends on the full sample. For ease of exposition we also write 
as 
.
[18] Note 
and 
are the cumulative distribution functions of the standard-normal and logistic distributions, respectively.
[19] Alternative link functions include the log-log and the complementary log-log links; however, these two are not symmetric around the median.
[20] Note we can also report marginal effects if the interest is to see the effect of an explanatory variable on the change of the probability.
[21] All estimated parameters exhibiting a negative sign denote a decrease in the odds; this is obtained as (1-odds ratio)×100. Likewise, estimated parameters with a positive sign denote an increase in the odds; this is obtained as (odds ratio-1)×100. For the effect of education we have (1-0.51)×100, and for the effect of gender we have (1.28-1)×100%.
[22] The deviance statistic: 
.
[23] RESET stands for Regression Equation Specification Error Test.
9 Distribution and Dynamics
This chapter provides techniques required to measure and analyse inequality among the poor (section 9.1), to describe changes over time using repeated cross-sectional data (section 9.2), to understand changes across dynamic subgroups (section 9.3) and to measure chronic multidimensional poverty (section 9.4). Each of these sections extends the 
methodological toolkit beyond the partial indices presented in Chapter 5, to address common empirical problems such as poverty comparisons, and illustrate these with examples. We build upon and do not repeat material presented in earlier chapters, and as in other chapters, confine attention to issues that are distinctive in multidimensional poverty measures.
Given the long-standing interest in inequality among the poor, we first enquire whether 
can be extended to reflect inequality among the poor. To make a long story short, it can easily do so. But the problem is that the resulting measure loses the property of dimensional breakdown that provides critical information for policy. So, taking a step back, we consider key properties a measure should have in order to reflect inequality among the poor and be analysed in tandem with 
Our chosen measure uses the distribution of censored deprivation scores to compute a form of variance across the multidimensionally poor. We also illustrate interesting related applications of this measure: for example to assess horizontal disparities across groups.
Chapter 5 showed that the Adjusted Headcount Ratio 
can be expressed as a product of the incidence of poverty (
) and the intensity of poverty (
) among the poor. Thus, captures two very important components of poverty—incidence and intensity. But it remains silent on a third important component: inequality across the poor. Now, the ultimate objective is to eradicate poverty—not merely reduce inequality among the poor. However, the consideration of inequality is important because the same average intensity can hide widely varying levels of inequality among the poor. For this reason, following the seminal article by Sen (1976), numerous efforts were made to incorporate inequality into unidimensional and latterly multidimensional poverty measures.[1]
This section explores how inequality among the poor can be examined when poverty analyses are conducted using the measure (Alkire and Foster 2013, Seth and Alkire 2014a, b).[2]
Section 5.7.2 already presented one way of bringing inequality into multidimensional poverty measures. This was achieved by using 
or some other gap measure applied to cardinal data, where the exponent on the normalized gap is strictly greater than one. Such an approach is linked to Kolm (1977) and generalizes the notion of a progressive transfer (or more broadly a Lorenz comparison) to the multidimensional setting by applying the same bistochastic matrix to every variable to smooth out the distribution of each variable (the powered normalized gap) while preserving its mean.[3] Poverty measures that are sensitive to inequality fall (or at least do not rise) in this case.
A second form of multidimensional inequality is linked to the work of Atkinson and Bourguignon (1982) and relies on patterns of achievements across dimensions. Imagine a case where one poor person initially has more of everything than another poor person and the two persons switch achievements for a single dimension in which both are deprived. This can be interpreted as a progressive transfer that preserves the marginal distribution of each variable and lowers inequality by relaxing the positive association across variables under the assumption that the dimensions are substitutes. The resulting transfer principle specifies conditions under which this alternative form of progressive transfer among the poor should lower poverty, or at least not raise it. The transfer properties are motivated by the idea that poverty should be sensitive to the level of inequality among the poor, with greater inequality being associated with a higher (or at least not lower) level of poverty.[4] Alkire and Foster (2011a) observe that the AF class of measures can be easily adjusted to respect the strict version of the second kind of transfer (the strong deprivation rearrangement property as discussed in section 5.2.5) involving a change in association between dimensions by replacing the deprivation count or score 
with a related individual poverty function 
for some 
, and averaging across persons.[5]
Many multidimensional poverty measures that employ cardinal data, including 
satisfy one or both of these transfer principles.[6] Alkire and Foster (2013) formulate a strict version of distribution sensitivity — ‘dimensional transfer’ (defined in 5.2.5)—which is applicable to poverty measures such as that use ordinal data. This property follows the Atkinson–Bourguignon type of distribution sensitivity, in which greater inequality among the poor strictly raises poverty. Alkire and Foster (2013) also prove a general result establishing that ‘the highly desirable and practical properties of subgroup decomposability, dimensional breakdown, and symmetry prevent a poverty measure from satisfying the dimensional transfer property’. In other words, does not reflect inequality among the poor, and, furthermore, no measure that satisfies dimensional breakdown and symmetry will be found that does satisfy dimensional transfer.
Given that it is necessary to choose between measures that satisfy dimensional transfer and those that can be broken down by dimension, and given that both properties are arguably important, how should empirical studies proceed? The first option is to employ the class of measures that respect dimensional breakdown and to supplement these with associated inequality measures. The second is to employ poverty measures that are inequality-sensitive but cannot be broken down by dimension, and to supplement them with separate dimensional analyses.
While both should be explored, this book favours the first route in applied work for several reasons. Dimensional breakdown enriches the informational content of poverty measures for policy, enabling them to be used to tailor policies to the composition of poverty, to monitor changes by dimension, and to make comparisons across time and space. Poverty reduction in measures respecting dimensional breakdown can be accounted for in terms of changes in deprivations among the poor and analysed by region and dimension. This creates positive feedback loops that reward effective policies. Also, the inequality-adjusted poverty measures may lack the intuitive appeal of the measure. Some of the inequality-adjusted measures (Chakravarty and D’Ambrosio, 2006, Rippin, 2012) are broken down into different components separately capturing incidence, intensity, and inequality, but without clarifying the relative weights attached to these components.
Whether or not an inequality measure is not computed, measures can be supplemented by direct descriptions of inequality among the poor. A first descriptive but powerfully informative tool is to report subsets of poor people which have mutually exclusive and collectively exhaustive graded bands of deprivation scores. This is possible by effectively ordering all 
poor persons according to the value of their deprivation score 
and dividing them into groups. If the poverty cutoff is 30%, the analysis might then report the percentage of poor people whose deprivation scores fall in the band of 30–39.9% of deprivations, 40–49.9%, and so on to 100%. The percentage of people who experience different intensity gradients of poverty across regions and time can be compared to see how inequality among the poor is evolving.[7] Figure 9.1 presents an example of two countries—Madagascar and Rwanda—which have similar multidimensional headcount ratios () and MPIs. However, the distributions of intensities across the poor are quite different. Also, data permitting, these intensity groups can be decomposed by population subgroups such as region or ethnicity. The comparisons can be enriched by applying a dimensional breakdown to examine the dimensional composition of poverty experienced by those having different ranges of deprivation scores.
Figure 9.1 Distribution of Intensities among the Poor in Madagascar and Rwanda
Another tool is to supplement with a measure of inequality among the poor. Using the distribution of (censored) deprivation scores across the poor or some transformation of these, it is actually elementary to create an inequality measure, much in the same way that traditional inequality measures such as Atkinson, Theil, or Gini are constructed. Such measures will offer a window onto one type of multidimensional inequality—one that is oriented to the breadth of deprivations people experience. This approach is quite different from other constructions of multidimensional inequality, but it is useful, particularly when data are ordinal. Building on Chakravarty (2001), Seth and Alkire (2014a,b) propose such an inequality measure that is founded on certain properties. Note that these are properties of inequality measures, and are defined differently from those presented in Chapter 2 (despite similar names), but introduced intuitively below. Let us briefly discuss these properties before introducing the measure.
The first property, translation invariance, requires inequality not to change if the deprivation score of every poor person increases by the same amount. Implicitly, we assume that the measure reflects absolute inequality. Seth and Alkire (2014) argue that measures reflecting absolute inequality are more appropriate when each deprivation is judged to be of intrinsic importance. In addition, the use of the absolute inequality measure ensures that inequality remains the same whether poverty is measured by counting the number of deprivations or by counting the number of attainments. The use of the relative inequality measure is more common in the case of income inequality, where it is often assumed that as long as people’s relative incomes remain unchanged, inequality should not change. However, it is difficult to argue that inequality between two poor persons who are deprived in one and two dimensions respectively is the same as the inequality between two poor persons who are deprived in five and ten dimensions, respectively, if these deprivations referred to, for example, serious human rights violations. Any relative inequality measure, such as the Generalized Entropy measures (which include the Squared Coefficient of Variation associated with the FGT2 index) or Gini Coefficient, would evaluate these two situations as having identical inequality across the poor. Moreover, a relative inequality measure may provide counterintuitive conclusion while assessing inequality within a counting approach framework. In fact, no non-constant inequality measure exists that is simultaneously invariant to absolute as well as relative changes in a distribution.
The second property requires that the inequality measure should be additively decomposable so that overall inequality in any society can be broken down into within-group and between-group components. This can be quite useful for policy (Stewart 2010). We have shown in Chapter 5 that the additive structure of the indices in the AF class allows the overall poverty figure to be decomposed across various population subgroups. A country or a region with same level of overall poverty may have very different poverty levels across different subgroups, or a country may have the same level of poverty across two time periods, but the distribution of poverty across different subgroups may change over time. Furthermore, within each population subgroup, there may be different distributions of deprivation scores across poor persons living within that subgroup, thus reflecting various levels of within-group inequality.
The third property, within-group mean independence, requires that overall within-group inequality should be expressed as a weighted average of the subgroup inequalities, where the weight attached to a subgroup is equal to the population share of that subgroup. This assumption makes the interpretation and analysis of the inequality measure more intuitive.
Four additional properties are commonly satisfied when constructing any inequality measure. The anonymity property requires that a permutation of deprivation scores should not alter inequality. According to the replication invariance property, a mere replication of population leaves the inequality measure unaltered. The normalization property requires that the inequality measure should be equal to zero when the deprivation scores are equal for all. The transfer property requires that a progressive dimensional rearrangement among the poor should decrease inequality.
The proposed inequality measure, which is the only one to satisfy those properties, takes the general form
 |
(9.1) |
where 
is a vector with 
elements. Relevant applications using our familiar notation will be provided in equations (9.6) and (9.7) below, but we first present the general form and notation. As we will show, in relevant applications an element 
may be the deprivation score of a person 
or 
or the average poverty level of a region. The size of the vector 
for an entire population would be 
and for the poor it would be 
The functional form in equation (9.1) is a positive multiple (
) of the variance. The measure reflects the average squared difference between person 
’s deprivation score and the mean of the deprivation scores in . The value of parameter can be chosen in such a way that it normalizes the inequality measure to lie between 0 and 1.
The overall inequality in may be decomposed into two components: total within-group inequality and between-group inequality. Following the notation in Chapter 2, suppose there are population subgroups. The deprivation score vector of subgroup 
is denoted by 
with 
elements. The decomposition expression is given as follows:
|
Total within-group Total between-group |
where 
is the population share of subgroup in the overall population and 
is the mean of all elements in for all 
.
The between-group inequality component 
in (9.2) can be computed as
|
|
where 
is the mean of all elements in .
The within-group inequality component of subgroup can be computed using (9.1) as
|
|
(9.4) |
and thus the total within-group inequality component in (9.2) can be computed as
|
|
There are different relevant applications of this inequality framework to multidimensional poverty analyses based on . The first central case is to assess inequality among the poor. To do so we suppose that the deprivation scores are ordered in a descending order and the first persons are identified as poor. The elements are taken from the censored deprivation score vector, 
We choose vector such that it contains only the deprivation scores of the poor (
). The average of all elements in then is the intensity of poverty which for persons is 
. We can then denote the inequality measure that reflects inequality in multiple deprivations only among the poor by 
, which can be expressed as
|
|
The 
measure effectively summarizes the information underlying Figure 9.1. It goes well beyond that figure because each individual deprivation score is used, which effectively creates a much finer gradation of intensity than that figure portrays. Furthermore, it can be decomposed by subgroup, to permit comparisons of within-subgroup inequalities among the poor. It can also be used over time to show how inequality among the poor changed.
Our second central case considers inequalities in poverty levels across population subgroups. It is motivated by studies of horizontal inequalities that find group-based inequalities to predict tension and in some cases conflict (Stewart 2010). Essentially, the measure reflects population-weighted disparities in poverty levels across population subgroups.
Suppose the censored deprivation score vector of subgroup is denoted by 
with 
elements. If instead of only considering the deprivation scores of the poor, we now sum across the whole population so (
), then we realize that 
or the average of all elements in is actually the of subgroup , which for simplicity we denote by 
. The between-group component of 
shows the disparity in the national Adjusted Headcount Ratio across subgroups and is written using (9.3) as
|
|
Thus, equation (9.7) captures the disparity in s across 
population subgroups, which can be used to detect patterns in horizontal disparities over time. Naturally, the number and population share of the subgroups must be considered in such comparisons.
While studying disparity in MPIs across sub-national regions, Alkire, Roche, and Seth (2011) found that the national MPIs masked a large amount of sub-national disparity within countries, and Alkire and Seth (2013) and Alkire, Roche, and Vaz (2014) found considerable disparity in poverty trends across subnational groups. In some countries, the overall situation of the poor improved, but not all subgroups shared the equal fruit of success in poverty reduction and indeed poverty levels may have stagnated or risen in some groups. Therefore, it is also important to look at inequality or disparity in poverty across population subgroups. This separate inequality measure, elaborated in Seth and Alkire (2014), provides such framework.
Table 9.1 presents two pair-wise comparisons. For the inequality measure, we choose 
4 because the deprivation scores are bounded between 0 and 1; hence the maximum possible variance is 0.25. 4 ensures that the inequality measure lies between 0 and 1. The first pair of countries, India and Yemen, have exactly the same levels of MPI. The multidimensional headcount ratios and the intensities of poverty are also similar. However, the inequality among the poor (computed using equation (9.1)) is much higher in Yemen than in India. We also measure disparity across sub-national regions. Yemen has twenty-one sub-national regions whereas, India has twenty-nine sub-national regions. We find that, like the national MPIs, the disparities across subnational MPIs—computed using equation 9.7)—are similar. This means that the inequality in Yemen is not primarily due to regional disparities in poverty levels, but may be affected by non-geographic divides such as cultural or rural–urban.
A contrasting finding for regional disparity is obtained across Togo and Bangladesh. As before, the MPIs, headcount ratios, and intensities are quite similar across two countries—but with two differences. The inequality among the poor is very similar, but the regional disparities are stark. Even though both countries have similar number of sub-national regions, the level of sub-national disparity is much higher in Togo than that in Bangladesh.
Table 9.1: Countries with Similar Levels of MPI but Different Levels of Inequality among the Poor and Different Levels of Disparity across Regional MPIs
A strong motivation for computing multidimensional poverty is to track and analyse changes over time. Most data available to study changes over time are repeated cross-sectional data, which compare the characteristics of representative samples drawn at different periods with sampling errors, but do not track specific individuals across time. This section describes how to compare and its associated partial indices over time with repeated cross-sectional data. It offers a standard methodology of computing such changes, and an array of small examples. This section does not treat the data issues underlying poverty comparisons, and readers are expected to know standard techniques that are required for such rigorous empirical comparisons. For example, the definition of indicators, cutoffs, weights, etc. must be strictly harmonized for meaningful comparisons across time, which always requires close verification of survey questions and response structures, and may require amending or dropping indicators. The sample designs of the surveys must be such that they can be meaningfully compared, and basic issues like the representativeness and structure of the data must be thoroughly understood and respected. We presume this background in what follows. This section focuses on changes across two time periods; naturally the comparisons can be easily extended across more than two time periods.
9.2.1 Changes in M0, H and A across Two Time Periods
The basic component of poverty comparisons is the absolute pace of change across periods.[10] The absolute rate of change is the difference in levels between two periods. Changes (increases or decreases) in poverty across two time periods can also be reported as a relative rate. The relative rate of change is the difference in levels across two periods as a percentage of the initial period.
For example, if the has gone down from 0.5 to 0.4 between two consecutive years, then the absolute rate of change is (0.5 – 0.4) = 0.1. It tells us how much level of poverty () has changed: 10% of the total possible set of deprivations that poor people in that society could have experienced has been eradicated; 40% remains. The relative rate of change is (0.5 – 0.4)/0.5 = 20%, which tells us that has gone down by 20% with respect to the initial level. While absolute changes are in some sense prior, because they are easy to understand and compare, both absolute and relative rates may be important to report and analyse. The value-added of the relative changes is evident in relatively low-poverty regions. A region or country with a high initial level of poverty may be able to reduce poverty in absolute terms much more than one having a low initial level of poverty. It is however possible that although a region or country with low initial poverty levels did not show a large absolute reduction, the reduction was large relative to its initial level and thus it should not be discounted for its slower absolute reduction.[11] The analysis of both absolute and relative changes gives a clear sense of overall progress.
In expressing changes across two periods, we denote the initial period by 
and the final period by 
. This section mostly presents the expressions for , but they are equally applicable to its partial indices: incidence (), intensity (), censored headcount ratios (
), and uncensored headcount ratios (
). The achievement matrices for period and are denoted by 
and 
, respectively. As presented in Chapter 5, and its partial indices depend on a set of parameters: deprivation cutoff vector 
, weight vector 
and poverty cutoff 
. For simplicity of notation though, we present and its partial indices only as a function of the achievement matrix. For strict intertemporal comparability, it is important that the same set of parameters be used across two periods.
The absolute rate of change (
) is simply the difference in Adjusted Headcount Ratios between two periods and is computed as
|
|
Similarly, for 
and 
:
|
|
|
|
The relative rate of change (
) is the difference in Adjusted Headcount Ratios as a percentage of the initial poverty level and is computed for
and (only shown) as
|
|
If one is interested in comparing changes over time for the same reference period, the expressions (5.7) and (9.11) are appropriate. However, in cross-country exercises, one may often be interested in comparing the rates of poverty reduction across countries that have different periods of references. For example the reference period of one country may be five years; whereas the reference period for another country is three years. It is evident in Table 9.2 that the reference period of Nepal is five years (2006–2011); whereas that of Peru is only three years (2005-–2008). In such cases, it is essential to annualize the change in order to preserve strict comparison.
The annualized absolute rate of change (
) is the difference in Adjusted Headcount Ratios between two periods divided by the difference in the two time periods (
) and is computed for as
|
|
The annualized relative rate of change (
) is the compound rate of reduction in per year between the initial and the final periods, and is computed for as
|
|
As formula (9.8) has been used to compute the changes in and using formulae (9.9) and (9.10), formulae (9.11) to (9.13) can be used to compute and report annualized changes in the other partial indices, namely ,
, or 
Table 9.2 presents both the annualized absolute and annualized relative rates of change in Global MPI, as outlined in Chapter 5, and its two partial indices— and —for four countries: Nepal, Peru, Rwanda, and Senegal, drawing from Alkire, Roche and Vaz (2014). Taking the survey design into account, we also present the standard errors (in parentheses) and the levels of statistical significance of the rates of reduction, as described in the Appendix of Chapter 8. The figures in the first four columns present the values and standard errors for , 
and in both time periods. The results show that Peru had the lowest MPI with 0.085 in the initial year, while Rwanda had the highest with 0.460.
Under the heading ‘annualized change’, Table 9.2 provides the annualized absolute and annualized relative reduction for , and , which are computed using equations (9.12) and (9.13). It shows, for example, that Nepal with a much lower initial poverty level than Rwanda, has experienced a greater absolute annualized poverty reduction of ‒0.027. In relative terms Nepal outperformed Rwanda. Peru had a low initial poverty level, and reduced it in absolute terms by only ‒0.006 per year, which means that the share of all possible deprivations among poor people that were removed was only less than one-fourth that of Nepal or Rwanda. But relative to its initial level of poverty, its progress was second only to Nepal. It is thus important to report both absolute and relative changes and to understand their interpretation. The same results for and are provided in the Panel II and III of the table. We see that Nepal reduced the percentage of people who were poor by 4.1 percentage points per year—for example, if the first year 64.7% of people were poor, the next year it would be 60.6%. Peru cut the poverty incidence by 1.3 percentage points per year. Relative to their starting levels, they had similar relative rates of reduction of the headcount ratio. Note that when estimates are reported in percentages, the absolute changes are reported in ‘percentage points’ and not in ‘percentages’. Thus, Nepal’s reduction in from 64.7% to 44.2% is equivalent to an annualized absolute reduction of 4.1 percentage points and an annualized relative reduction of 6.3%.
Table 9.2: Reduction in Multidimensional Poverty Index, Headcount Ratio and Intensity of Poverty in Nepal, Peru, Rwanda and Senegal
The third column provides the results for the hypothesis tests which assess if the reduction between both years is statistically significant.[12] The reductions in in Nepal and Rwanda are significant at 
=0.01, but the same in Peru is only significant at =0.10. Interestingly, the reduction in intensity of poverty in Peru is significant at =0.05. The case of Senegal is different in that the small reduction in is not even significantly different at =0.10, preventing the null hypothesis that poverty level in both years remained unchanged from being rejected.
Besides comparing the rate of reduction in , and as in Table 9.2, one should also examine whether the number of poor people is decreasing over time. It may be possible that the population growth is large enough to offset the rate of poverty reduction. Table 9.3 uses the same four countries as Table 9.2 but adds demographic information. Nepal had an annual population growth of 1.2% between 2006 and 2011, moving from 25.6 to 27.2 million people, and reduced the headcount ratio from 64.7% to 44.2%. This means that Nepal reduced the absolute number of poor by 4.6 million between 2006 and 2011.
Table 9.3 Changes in the Number of Poor Accounting for Population Growth
In order to reduce the absolute number of poor people, the rate of reduction in the headcount ratio needs to be faster than the population growth. The largest reduction in the number of multidimensionally poor has taken place in Nepal. A moderate reduction in the number of poor has taken place in Peru and Rwanda. In contrast, there has been an increase in the total number of multidimensional poor in Senegal, from 8 million to over 9 million between 2005 and 2011.
The reductions in , , or can be broken down to reveal which dimensions have been responsible for the change in poverty. This can be seen by looking at changes in the uncensored headcount ratios () and censored headcount ratios () described in section 5.5.3. We present the uncensored and censored headcount ratios of MPI indicators for Nepal in Table 9.4 for years 2006 and 2011 and analyse their changes over time. For definitions of indicators and their deprivation cutoffs, see section 5.6. Panel I gives levels and changes in uncensored headcount ratios, i.e. the percentage of people that are deprived in each indicator irrespective of deprivations in other indicators. Panel II provides levels and changes in the censored headcount ratios, i.e. the percentage of people that are multidimensionally poor and simultaneously deprived in each indicator. By definition, the uncensored headcount ratio of an indicator is equal to or higher than the censored headcount of that indicator. The standard errors are reported in the parenthesis.
Table 9.4: Uncensored and Censored Headcount Ratios of the Global MPI, Nepal 2006–2011
As we can see in the table, Nepal made statistically significant reductions in all indicators in terms of both uncensored and censored headcount ratios. The larger reductions in censored headcount are observed in electricity, assets, cooking fuel, flooring, and sanitation; all censored headcount ratios have decreased by more than 3 percentage points. Nutrition, mortality, schooling and attendance follow with annual reductions of 3, 2.3, 1.8, and 1.5 percentage points, respectively.
The changes in censored headcount ratios depict changes in deprivations among the poor. Recall that the overall is the weighted sum of censored headcount ratios of the indicators as presented in equation (5.13) and the contribution of each indicator to the can be computed by equation (5.14). Because of this relationship, the absolute rate of reduction in in equation (9.8) and the annualized absolute rate of reduction in in equation (9.12) can be expressed as weighted averages of absolute rate of reductions in censored headcount ratios and annualized absolute rate of reductions in censored headcount ratios, respectively. When different indicators are assigned different weights, the effects of their changes on the change in reflect these weights.[13] For example, in the MPI, the nutrition indicator is assigned a three times more weight than electricity. This implies that a one percentage point reduction in nutrition ceteris paribus would lead to an absolute reduction in that is three times larger than a one percentage point reduction in the electricity indicator.
Recall that it is straightforward to compute the contribution of each indicator to using its weighted censored headcount ratio as given in equation (5.14). Note that interpreting the real on-the-ground contribution of each indicator to the change in is not so mechanical. Why? A reduction in the censored headcount ratio of an indicator is not independent of the changes in other indicators. It is possible that the reduction in the censored headcount ratio of a certain indicator 
occurred because a poor person became non-deprived in indicator 
But it is also possible that the reduction occurred because a person who had been deprived in became non-poor due to reductions in other indicators, even though they remain deprived in . In the second period, their deprivation in is now censored because they are non-poor (their deprivation score does not exceed ). The comparison between the uncensored and censored headcount distinguishes these situations. For example, we can see from Panel I of Table 9.4 that the reductions in the uncensored headcount ratios of flooring and cooking fuel are lower than the annualized reductions of the censored headcount ratios of the these two indicators. Thus some non-poor people are deprived in these indicators. In intertemporal analysis it is useful to compare the corresponding censored and uncensored headcount ratios to analyse the relation between the dimensional changes among the poor and the society-wide changes in deprivations. Of course in repeated cross-sectional data, this comparison will also be affected by migration and demographic shifts as well as changes in the deprivation profiles of the non-poor.
Panel III of Table 9.5 presents the contribution of the indicators to the for Nepal in 2006 and in 2011. The contributions of assets, electricity, and attendance have gone down; whereas the contributions of flooring, cooking fuel, sanitation, and schooling have gone up. The contributions of water, nutrition and mortality have not shown large changes. Dimensional analyses is vital and motivating because any real reduction in a dimensional deprivation will certainly reduce . Real reductions are normally those which are visible both in raw and censored headcounts.[14]
One important property that the adjusted-FGT measures satisfy is population subgroup decomposability, so that the overall can be expressed as: 
where
denotes the Adjusted Headcount Ratio and 
the population share of subgroup as in equation (5.14). It is extremely useful to analyse poverty changes by population subgroups, to see if the poorest subgroups reduced poverty faster than less poor subgroups, and to see the dimensional composition of reduction across subgroups (Alkire and Seth 2013b; Alkire and Roche 2013; Alkire, Roche and Vaz 2014). Population shares for each time period must be analysed alongside subgroup trends. For example, let us decompose the Indian population into four caste categories: Scheduled Castes (SC), Scheduled Tribes (ST), Other Backward Classes (OBC), and the General category. As Table 9.5 shows, as well as have gone down statistically significantly at the national level and across all four subgroups, which is good news. However, the reduction was slowest among STs who were the poorest as a group in 1999, and their intensity showed no significant decrease. Thus, the poorest subgroup registered slowest progress in terms of reducing poverty.
Table 9.5 Decomposition of 
, 
and 
across Castes in India
To supplement the above analysis it is useful to explore the contribution of population subgroups to the overall reduction in poverty, which not only depends on the changes in subgroups’ poverty but also on changes in the population composition. This can be seen by presenting the overall change in between two periods (
) as
Note that the overall change depends both on the changes in subgroup ’s and the changes in population shares of the subgroups.
The overall changes in 
and 
discussed thus far could have been generated in many ways. It might be desirable for policy purposes to monitor how poverty changed. In particular, one may wish to pinpoint the extent to which poverty reduction occurred due to people leaving poverty vs. a reduction of intensity among those who remained poor, and also to know the precise dimensional changes which drove each.
For example, a decrease in the headcount ratio by 10% could have been generated by an exit of 10% of the population who had been poor in the first period. Alternatively, it could have been generated by a 20% decrease in the population who had been poor, accompanied by an influx of 10% of the population who became newly poor. Furthermore, the people who exited poverty could have had high deprivation scores in the first period – that is, been among the poorest – or they could have been only barely poor. The deprivation scores of those entering and leaving poverty will affect the overall change in intensity 
as will changes among those who stay poor. In addition these entries into and exits from poverty could have been precipitated by diferent possible increases or decreases in the dimensional deprivations people experienced in the first period, which will then be reflected in the changes in uncensored and censored headcount ratios.
This section introduces more precisely these dynamics of change. We first show what can be captured with panel data, then show empirical strategies to address this situation with repeated cross-sectional data. Finally we present two approaches related to Shapley decompositions which decompose changes precisely, but rely on some crucial assumptions so their empirical accuracy is questionable.
Let us consider a fixed set of population of size 
across two periods, 
and . The achievement matrices of these periods are denoted by and . The population can be mutually exclusively and collectively exhaustively categorized into four groups that we refer to as dynamic subgroups as follows:
Subgroup  |
Contains  people who are non-poor in both periods  and  |
Subgroup  |
Contains  people who are poor in both periods and (ongoing poor) |
Subgroup  |
Contains  people who are poor in period but exit poverty in period |
Subgroup  |
Contains  people who are not poor in period but enter poverty in period |
We denote the achievement matrices of these four subgroups in period 
by 
, 
, 
and 
for all 
, . The proportion of multidimensionally poor populationin period is 
and that in period is 
. The change in the proportion of poor people between these two periods is 
=
. In other words, the change in the overall multidimensional headcount ratio is the difference between the proportion of poor entering and the proportion of poor exiting poverty. Note that, by construction, no person is poor in 
, 
, 
, and 
and thus 
. This also implies, 
. On the other hand, all persons in 
, 
, 
, and 
are poor and thus 
. Therefore, the of each of these four subgroups is equal to its intensity of poverty.
In a fixed population, the overall population and the population share of each dynamic group remains unchanged across two time periods.[15] The change in the overall can be decomposed using equation (9.14) as
Thus, the right-hand side of equation (9.15) has three components. The first component
is due to the change in the intensity of those who remain poor in both periods—the ongoing poor—weighted by the size of this dynamic subgroup. The second component 
is due to the change in the intensity of those who exit poverty (weighted by the size of this subgroup) and the third component 
is due to the population-weighted change in the intensity of those who enter poverty. Together 
.
From this point there are many interesting possible avenues for analyses. Each group can be studied separately or in different combinations. For policy, it could be interesting to know who exited poverty, and their intensity in the previous period, to see if the poorest of the poor moved out of poverty. The intensity of those who entered poverty shows whether they dipped into the barely poor group, or catapulted into high-intensity poverty, perhaps due to some shock or crisis or (if the population is not fixed) migration. Intensity changes among the ongoing poor show whether their deprivations are declining, even though they have not yet exited poverty. Dimensional analyses of changes for each dynamic subgroup, which are not covered in this book but are straightforward extensions of this material, are also both illuminating and policy relevant.
In the case of panel data with a fixed population we are able to estimate these precisely. We can thus monitor the extent to which the change in 
is due to movement into and out of poverty, and the extent to which it is due to a change in intensity among the ongoing poor population. The example in Box 9.1 may clarify.
Box 9.1 Decomposing the Change in
across Dynamic Subgroups: An Illustration
Consider the following six-person, six-dimension 
matrices, in which people enter and exit poverty, and intensity among the poor also increase and decreases.
Let us use a poverty cutoff of 33% or two out of six dimensions. Increases and decreases are depicted in bold. Below we summarize , and in two periods and their changes across two periods.
So in period two there are four kinds of changes affecting the dynamic subgroups as follows:
1) 
: persons 1 and 2 become non-poor (move out of or exit poverty)
2) 
person 6 enters poverty.
3) 
two kinds of changes occur
a. deprivations of ongoing poor persons 3 and 4 reduce by one deprivation each
b. deprivations of ongoing poor person 5 increases by one deprivation.
The descriptions and the decompositions of for the changes are in the following table.
What is particularly interesting for policy is that we can notice that, in this example, 11% of the reduction in poverty was due to changes in intensity among the 50% of the population who stayed poor, that poverty was effectively increased 33% by the new entrant, but that this was more than compensated by those who exited poverty (–122%), because they initially had very high intensities. In this dramatic example, the poorest of the poor exited poverty, while the less poor experienced smaller reductions.
The previous section explained the changes for a fixed population over time. To estimate that empirically requires panel data with data on the same persons in both periods which can be used to track their movement in and out of poverty. Yet analyses over time are often based on repeated cross-sectional data having independent samples that are statistically representative of the population under study, but that do not to track each specific observation over time. This section examines the decomposition of changes in for cross-sectional data.
With cross-sectional data, we cannot distinguish between the three groups identified above, nor can we isolate the intensity of those who move into or out of poverty. Observed values are only available for: 
and 
. Using these, it is categorically impossible to decompose with the empirical precision that panel data permits.
Nonetheless, if required one can move forward with some simplifications. Instead of three groups (
let us consider just two, which might be referred to (somewhat roughly) as movers and stayers. We define movers as the 
people who reflect the net change in poverty levels across the two periods. Stayers are ongoing poor plus the proportion of previously poor people who were replaced by ‘new poor’, and total those who are poor in period 2 
. In considering only the ‘net’ change in headcount, one effectively permits the larger of 
to dominate: if poverty rose nationally, it is the group who entered poverty who dominate; if poverty fell nationally, the group who exited poverty. The subordinate third group is allocated among the ongoing poor and the dominant group. For the remainder of this section we presume that both and decreased overall. In this case, 
. So 
, and =
. As is evident, this simplification is performed because empirical data exist in repeated cross-sections for 
and .
Example: Suppose that 37% of people are ongoing poor, 3% enter poverty, 13% exit poverty, and 47% remain non-poor. Suppose the overall headcount ratio decreased by 10 percentage points, and the headcount ratio in period 2 is 40%, whereas in period 1 it was 50% (37%+13%). We now primarily consider two numbers: the headcount ratio in period 2 of 40% (interpreted broadly as ongoing poverty) and the change in headcount ratio of 10% (interpreted broadly as moving into/out of poverty). In doing so we are effectively permitting the ‘new poverty entrants’ to be considered as among the group in ongoing poverty in period 2 (37% + 3% = 40%). To balance this, we effectively replace 3% of those who exited poverty (13% – 3% = 10% =
), and consider this slightly reduced group to be those who moved out of poverty. If poverty had increased overall, the swaps would be in the other direction.
If poverty has reduced and there has not been a large influx of people into poverty, that is, if 
is presumed to be relatively small empirically, then this strategy would be likely to shed light on the relative intensity levels of those who moved out of poverty 
, and the changes in intensity among those who remained poor 
. If empirically 
is expected (from other sources of information) to be large, or if their intensity is expected to differ greatly from the average, this strategy is not advised.[16]
Consider the intensity of the net population who exited poverty – under these simplifying assumptions reflected by the net change in headcount, denoted 
and the intensity change of the net ongoing poor, whom we will presume to be , denoted 
. The 
can be decomposed according to these two groups. These decompositions can be interpreted as showing percentage of the change in that can be attributed to those who moved out of poverty, versus the percentage of change which was mainly caused by a decrease in intensity among those who stayed poor. We use the terms movers and stayers to refer to these less precise dynamic subgroups in cross-sectional data analysis.
|
|
(9.16) |
Cross sectional data does not provide the intensity of either of those who stayed poor or of those who moved out of poverty. One way forward is to estimate these using existing data. First, identify the 
poor persons having the lowest intensity in the dataset (sampling weights applied), and use the average of these scores for 
then solve for Subsequently, identify the 
poor persons having the highest intensity in that dataset, and repeat the procedure. This will generate upper and lower estimates for and 
in a given dataset, which will provide an idea of the degree of uncertainty that different assumptions introduce. To estimate stricter upper and lower bounds it could be assumed that those moved out of poverty had an intensity score of the value of (the theoretically minimum possible), and subsequently assume that their intensity was 100% (the theoretically maximum possible).[18]
Table 9.6 provides the empirical estimations for the upper and lower bounds for the same four countries discussed above plus Ethiopia. At the upper bound those who moved out of poverty could have had average intensities ranging from 59% in Peru (the least poor country) to 99% in Ethiopia or 100% in Senegal, according to the datasets. This in itself is interesting, as it shows that Rwanda—which is the poorest country of the four—had ‘movers’ with lower average deprivations than Ethiopa. Those who stayed poor would have had, in this case, small if any increases or decreases in intensity—less than four percentage points. At the lower bound, those who moved out of poverty could have had intensities from 33% in Peru and Senegal to 38% in Nepal, and intensity reductions among the ongoing poor could have ranged from 2% in Senegal to 13% in Nepal. At the upper bound (where we assume the poorest of the poor moved out of poverty), for Nepal, Rwanda and Peru, over 100% of the poverty reduction was due to the movers, because intensity among the ongoing poor would have had to increase (to create the observed ). At the lower bound, where the least poor people moved out of poverty, movers contributed 47–67% to . Senegal did not have a statistically significant reduction in poverty. Ethiopia provides a different example where the upper and lower bound are closer together and reductions in intensity among the ongoing poor would have contributed 31% to 73%.
Table 9.6 Decomposing the Change in by Dynamic Subgroups
This empirical investigation shows that, when implemented with the mild assumptions that are required for cross-sectional data, the upper and lower bounds according to each country’s dataset are very wide apart. In reality, the relative contribution, of the ‘movers’ and ‘stayers’ to overall poverty reduction could vary anywhere in this range.
As the example shows, the empirical upper and lower bounds vary greatly across countries. In the case of Ethiopia, movers explain 27% to 69% of the changes in poverty, and stayers account for 31% to 73%. These boundaries do not permit us to assess whether the actual contribution from ‘movers’ was greater than or less than that of ‘stayers’. In Nepal and Peru the ‘movers’ probably contributed more than ‘stayers’ to poverty reduction, as in all cases their lowest effect is above 50%. Given these wide-ranging upper and lower bounds, empirically we are unable to answer questions such as whether the intensity of the ongoing poor decreased, or whether it was the barely poor or the deeply poor who moved out of poverty. While this can seem disappointing, for policy purposes, as Sen stresses, it may be better to be ‘vaguely right than precisely wrong’, and repeated cross-sectional data simply do not permit us, at this time, to bound ahead with precision.
Whereas monitoring and policy inputs must be based on empirical analyses, some research topics utilize theoretical analyses. This section introduces two theory-based approaches to decomposing changes in repeated cross-sectional data according to what we call ‘incidence’ and ‘intensity’. In each approach assumptions are made regarding the intensity of those who exit or remain poor. As we have already noted, the task implies some challenges because the empirical accuracy of the underlying assumptions is completely unknown, and as Table 9.6 showed, the actual range may be quite large. These techniques are thus offered in the spirit of academic inquiry.
For simplicity of notation, in this subsection, we denote the , , and for period by 
, 
and 
and that for period by 
, 
and 
. The first approach consists in the additive decomposition proposed by Apablaza and Yalonetzky (2013), which is illustrated in Panel A of Figure 9.2. Since 
, they propose to decompose the change in by changes in its partial indices as follows
Note that the illustration in Figure 9.2 assumes reductions in , , and over time, but the graph can be adjusted to incorporate situations where they do not necessarily fall. This approach involves two assumptions. First, the intensity among those who left poverty is assumed to be the same as the average intensity in period . Second, the intensity change among the ongoing poor is assumed to equal the simple difference in intensities of the poor across the two periods. The decomposition is completed using an interaction term, as depicted in Panel A of Figure 9.2, below. This is indeed an additive decomposition of changes in the Adjusted Headcount Ratio 
Apablaza and Yalonetzky interpret these changes as reflecting: (1) changes in the incidence of poverty 
, (2) changes in the intensity of poverty 
, and (3) a joint effect that is due to interaction between incidence and intensity (
.
Figure 9.2 Theoretical Decompositions
A second theoretical approach corresponds to a Shapley decomposition proposed by Roche (2013). This builds on Apablaza and Yalonetzky (2013) and performs a Shapley value decomposition following Shorrocks (1999).[19] It provides the marginal effect of changes in incidence and intensity as follows:
|
|
|
Panel B of Figure 9.2 illustrates Roche’s application of Shapley decompositions, which focuses on the marginal effect without the interaction effect. Roche’s proposal assumes that the intensity of those who exited poverty (our terms) is the average intensity of the two periods 
, and calls this the ‘incidence effect’. He takes the other group as comprising the average headcount ratio between the two periods, and their change in intensity as the simple difference in intensities across the periods 
and describes this as the ‘intensity effect’.
Roche’s masterly presentation systematically applies Shapley decompositions to each step of dynamic analysis using the AF method. For example, if the underlying assumptions are transparently stated and accepted, the theoretically derived marginal contribution of changes in incidence and marginal contribution of changes in intensity can be expressed as a percentage of the overall change in so they both add to 100% and can be written as follows
|
|
(9.19) |
|
|
|
(9.20) |
To address demographic shifts Roche follows a similar decomposition of change as that used in FGT unidimensional poverty measures (Ravallion and Huppi, 1991) and Shapely decompositions (Duclos and Araar 2006, Shorrocks 1999). This approach, presented in Equation (9.21), attribute demographic effects to the average population shares and subgroup ’s across time. Roche argues that, if the underlying assumptions are accepted, the overall change in poverty level can be broken down in two components: 1) changes due to intra-sectoral or within-group poverty effect, 2) changes due to demographic or inter-sectoral effect. So the overall change in the adjusted headcount between two periods respectively
, could be expressed as follows
|
|
|
It is common to express the contribution of each factor as a proportion of the overall change, in which case equation (9.21) is divided throughout by 
The last columns of Table 9.6 Panel B provide Shapley decompositions for the same five countries. We see that in all cases the Shapley decompositions lie, as anticipated, between the upper and lower bounds. The Shapley decompositions have the broad appeal of presenting point estimates that pinpoint the exact contribution of each partial index to changes in poverty, according to their underlying assumptions, and thus may be used in analyses when empirical accuracy is not required or the assumptions are independently verified. A full illustration of the Shapley decomposition methods using data on multidimensional child poverty in Bangladesh is given in Roche (2013).[20]
Panel datasets provide information on precisely the same individual or household at different periods of time. Good quality panel datasets are particularly rich and useful for analysing multidimensional poverty because their analysis provides policy-relevant insights that extend what time series data provide. For example, using panel data we can distinguish the deprivations experienced by the chronically poor from those experienced by the transitory poor and thus identify the combination of deprivations that trap people in long-term multidimensional poverty. Also, we can analyse the duration over which a person was deprived in each indicator—and the sequences by which their deprivation profile evolved. As section 9.3.1 showed, we can identify precisely the contributions to poverty reduction that were generated by changes entries and exits from poverty and by the ongoing poor.
The following section very briefly presents a counting-based class of chronic multidimensional poverty measures that use a triple-cutoff method of identifying who is poor. We give prominence here to the measure that can be estimated using ordinal data. The partial indices associated with the chronic multidimensional poverty methodology include the headcount ratio and intensity, as well as new indices related to the duration of poverty and of dimensional deprivation. We also present a linked measure of transient poverty. As in other sections, we presume that interested readers will master standard empirical and statistical techniques that are appropriate for studies using panel data, and apply these in the analyses of poverty transitions and chronic poverty here described.
The closing section on poverty transitions informally sketches revealing analyses that can be undertaken without generating a chronic poverty measure. Rather, people are identified as multidimensionally poor or non-poor in each period, then population subgroups are identified that have differing sequences of multidimensional poverty. For example, one group might include non-poor people who ‘fell’ into multidimensional poverty, a second might include multidimensionally poor people who ‘rose’ out of poverty, a third might contain those who were poor in all periods, and a fourth might contain those who ‘churned’ in and out of poverty across periods. Naturally the number of ‘dynamic subgroups’ depends upon the sample design, the number of waves of data and the precise definition of each group.
Multiple approaches to measuring chronic poverty in one dimension exist, many of which have implications for the measurement of chronic multidimensional poverty.[21] Alongside important qualitative work, multiple methodologies for measuring chronic multidimensional poverty have also been proposed.[22] This section combines the AF methodology with the counting-based approach to chronic poverty measurement proposed in Foster (2009), which has a dual-cutoff identification structure and aggregation method that are very similar to the AF method. Foster (2009) provides a methodology for measuring unidimensional chronic poverty in which each time period is equally weighted for all a
An 
matrix is constructed in which each entry takes a value of one if person is identified as poor in period and a value of 0 otherwise. A -dimensional ‘count’ vector is constructed in which each entry shows the number of periods in which person was poor. A second time cutoff
is applied such that each person is identified as chronically poor if he or she has been poor in or more periods. Associated FGT indices and partial indices are then constructed from the relevant censored matrices.
This combined chronic multidimensional poverty measure applies three sets of cutoffs: deprivation cutoffs
, a multidimensional poverty cutoff , and a duration cutoff . It is possible to analyse multidimensional poverty using panel data by combining the AF methodology and the Foster (2009) chronic poverty methods following two different orders of aggregation, which we call chronic deprivation or chronic poverty. These alternatives effectively interchange the order in which the poverty and duration cutoffs are applied. In both cases, we first apply a fixed set of deprivation cutoffs to the achievement matrix in each period.
In the chronic deprivation option ( before ), we first consider the duration of deprivation in each indicator for each person and then compute a multidimensional poverty measure which summarizes only those deprivations that have been experienced by the same person across or more periods. This approach aggregates all ‘chronic’ deprivations into a multidimensional poverty index, regardless of the period in which those deprivations were experienced. This approach would provide complementary information that could enrich analyses of multidimensional poverty, but cannot be broken down by time period, nor does it show whether the deprivations were experienced simultaneously (Alkire, Apablaza, Chakravarty, and Yalonetzky 2014).
In the chronic multidimensional poverty option ( before ), we first identify each person as multidimensionally poor or non-poor in each period using the poverty cutoff. We then count the periods in which each person experienced multidimensional poverty. We identify as chronically multidimensionally poor those persons who have experienced multidimensional poverty in or more periods.
We observe achievements across 
dimensions for a set of individuals at 
different time points. Let 
stand for the achievement in attribute of person in period , where 
Let 
denote a 
matrix whose elements reflect the dimensional achievements of the population in period . The deprivation cutoff vector 
is fixed across periods. As before, a person is deprived with respect to deprivation in periods if 
and non-deprived otherwise. By applying the deprivation cutoffs to the achievement matrix for each period, we can construct the period-specific deprivation matrices 
. For simplicity, this section uses the non-normalized or numbered weights notation across dimensions, such that 
Time periods are equally weighted. When achievement data are cardinal we can also construct normalized gap matrices 
and squared gap matrices 
or more generally, powered matrices of normalized deprivation gaps 
where 
. In a similar manner as previously, we generate the -dimensional 
column vector, which reflects the weighted sum of deprivations person experiences in periods .
To identify who is chronically multidimensionally poor we first construct an identification matrix. The same matrix can be used to identify the transient poor in each period and to create subgroups of those who exhibit distinct patterns of multidimensional poverty (for analysis of poverty transitions).
Identification Matrix
Let 
be an identification matrix whose typical element 
is one if person 
is identified as multidimensionally poor in period using the AF methodology, that is, using a poverty cutoff 
which is fixed across periods, and 0 otherwise.
The typical column 
reflects the identification status 
for the th person in period , whereas the typical row 
displays the periods in which person has been identified as multidimensionally poor (signified by an entry of 1) or non-poor (0). Thus we might equivalently consider each column of 
to be an identification column vector for period such that 
if and only if person is multidimensionally poor in period according to the deprivation cutoffs , weights 
, and poverty cutoff ; and 
otherwise.
Episodes of Poverty Count Vector 
From the matrix we construct the -dimensional column vector 
whose th element 
= 
sums the elements of the corresponding row vector of and provides the total number of periods in which person is poor, or the total episodes of poverty, as identified by poverty cutoff . Naturally, 
, that is, each person may have from 0 episodes of poverty to episodes, the latter indicating that a person was poor in each of the 
periods.
Chronic Multidimensional Poverty: Identification and Censoring
We apply the duration cutoff
where 
to the vector in order identify the status of each person as chronically multidimensionally poor or not. We identify a person to be chronically multidimensionally poor if 
. That is, if they have experienced or more periods in multidimensional poverty. A person is considered non-chronically poor if 
. We doubly censor the vector such that it takes the value of 0 (non-chronically poor) if 
and takes the value of otherwise. The notation 
indicates the censored vector of poverty episodes—just as the notation 
indicated the censored deprivation count vector. Positive entries reflect the number of periods in which chronically poor people experienced poverty; entries of 0 mean that the person is not identified as chronically poor.
Among the non-chronically poor, we could (as we will elaborate) identify two subgroups: the non-poor and the transient poor. A person is considered transiently poor if 
. And naturally a person for whom 
, that is, who is non-poor in all periods, is considered non-poor.
An alternative but useful notation for the identification of chronic multidimensional poverty uses the identification function: we apply the identification function 
to censor the matrix and the vector. The doubly censored 
matrix reflects solely those periods in poverty that are experienced by the chronically poor (censoring all periods of transient poverty) and is denoted by 
. After censoring, the typical element is defined by 
. The entry takes a value of if person is chronically multidimensionally poor and 0 otherwise.
Censored 
Deprivation Matrices and Count Vector
To identify the censored headcounts, as well as the dimensional composition of poverty in each period, we censor the sets of deprivation matrices by applying the twin identification functions 
and . We denote the censored matrices by 
and the censored deprivation count vectors for each period 
.
Deprivation Duration Matrix
Finally, to summarize the overall dimensional deprivations of the poor, as well as the duration of these deprivations, it will be useful to create a duration matrix based on the censored deprivation matrices 
. Let 
be an matrix whose typical element 
provides the number of periods in which is chronically poor and is deprived in dimension . Note that 
We can use the duration matrix to obtain the deprivation-specific duration indices, which show the percentage of periods in which, on average, poor people were deprived in each indicator.
The measure of chronic multidimensional poverty when some data are ordinal may be written as follows:
|
|
(9.22) |
Thus the Adjusted Headcount Ratio of chronic multidimensional poverty
is the mean of the set of deprivation matrices 
that have been censored of all deprivations of persons who are not chronically multidimensionally poor. Alternative notation for this measure can be found in Alkire, Apablaza, Chakravarty, and Yalonetzky (2014).
When data are cardinal, the 
class of measures are, like the AF class, the means of the respective powered matrices of normalized gaps.
|
|
(9.23) |
For chronic multidimensional poverty, as for multidimensional poverty, the specification of axioms is, in some cases, a joint restriction on the (triple-cutoff) identification and aggregation strategies and, hence, on the overall poverty methodology. The properties are now defined with respect to the chronically multidimensionally poor population. The class of measures present respects the key properties that were highlighted as providing policy relevance and practicality to the AF measures, such as subgroup consistency and decomposability, dimensional monotonicity, dimensional breakdown, and ordinality. In addition, this class of measures satisfies a form of time monotonicity as highlighted in Foster (2009) in the unidimensional case. The intuition is that if a person who is chronically poor becomes poor in an additional period, poverty rises.
A full definition of the properties that this chronic multidimensional poverty measure fulfils is provided in Alkire, Apablaza, Chakravarty, and Yalonetzky (2014). The methodology of multidimensional chronic poverty measurement fulfils the appropriately stated properties of anonymity, time anonymity, population replication invariance, chronic poverty focus, time focus, chronic normalization, chronic dimensional monotonicity, chronic weak monotonicity, time monotonicity, chronic monotonicity in thresholds, monotonicity in multidimensional poverty identifier, chronic duration monotonicity, chronic weak transfer, non-increasing chronic poverty under association-decreasing switch, and additive subgroup decomposability for all . The class of measures also satisfies chronic strong monotonicity for 
and chronic strong transfer when 
.
Like , the chronic multidimensional poverty measure 
is the product of intuitive partial indices that convey meaningful information on different features of a society’s experience of chronic multidimensional poverty. In particular, 
where:
· 
is the headcount ratio of chronic multidimensional poverty—the percentage of the population who are chronically multidimensionally poor according to and .
· 
is the average intensity of poverty among the chronically multidimensionally poor—the average share of weighted deprivations that chronically poor people experience in those periods in which they are multidimensionally poor.
· 
reflects the average duration of poverty among the chronically poor — the average share of periods in which they experience multidimensional poverty.
These partial indices can also be calculated directly. In particular,
|
|
(9.24) |
That is, the headcount ratio of chronic multidimensional poverty is the number of people who have been identified as chronically multidimensionally poor divided by the total population. We denote the number of chronically multidimensionally poor people by 
The intensity of chronic multidimensional poverty is the sum of the weighted deprivation scores of all poor people over all time periods, divided by the number of dimensions times the total number of people who are poor in each period summed across periods. Note that 
 |
(9.25) |
The average duration of chronic multidimensional poverty—the percentage of periods on average in which the chronically person was poor—can be easily assessed using the vector.
 |
(9.26) |
The duration is the sum of the total number of periods in which the chronically poor experience multidimensional poverty, divided by the number of periods and the number of chronically poor. Note that 
Box 9.2 illustrates these with a simple example.
Box 9.2 Computing Incidence and Duration of Chronic Poverty
Consider three people and four periods, with = 2.
Person 1 is multidimensionally poor in period 1
Person 2 is multidimensionally poor in periods 2, 3, and 4
Person 3 is multidimensionally poor in periods 1, 2, 3, and 4
Two people are chronically poor because they experience multidimensional poverty in = 2 or more periods. So the percentage of people identified as chronically poor 
is 67%.
In this case, the vector 
=(0,3,4); 
= 2 and our duration index is
. That is, on average, chronically poor persons are poor during 87.5% of the time periods.
For chronic multidimensional poverty, it is possible and useful to generate the standard dimensional indicators for each period: the censored headcount and percentage contribution. It is also possible and useful to generate the period-specific Adjusted Headcount Ratio (), headcount ratio (), and intensity () figures, which are different from, but can be consistently related to, the chronic poverty headcount and intensity values presented in 9.4.3. Finally, and of tremendous use, it is possible to present the average duration of deprivation in each dimension and to relate this directly to the overall duration of chronic poverty. Box 9.3 presents the intuition of this set of consistent indices; for their precise definition see Alkire, Apablaza, Chakravarty, and Yalonetzky (2014).
Box 9.3 Computing Incidence and Duration of Chronic Poverty
Cross-period indices reflecting chronic poverty:
: Adjusted Headcount Ratio of chronic multidimensional poverty
: Headcount ratio, showing the percentage of the population who are chronically poor
: Intensity, showing the average percentage of deprivations experienced by the chronically multidimensionally poor in those periods in which they are poor
: Average duration of chronic poverty, expressed as a percentage of time periods
: Average censored headcount of dimension 
among the chronically poor in all periods in which they are poor and are deprived in dimension
: Average duration of deprivation in dimension among the chronically poor, expressed as a percentage of time periods
: Percentage contribution of dimension to the deprivations of the chronically poor.
Single-period indices reflecting the profiles of the chronic poor in that particular period of poverty:
: Headcount ratio, showing the percentage of the population who are chronically poor in period 
: Intensity, showing the average percentage of deprivations experienced by the chronically multidimensionally poor in period
: Censored headcount of dimension among the chronically poor in period
: Percentage contribution of dimension to the deprivations of the chronically poor in period .
Cross-period averages of the unidimensional indices can also be constructed, such as 
and 
and analysed in conjunction with the relevant duration measure.
The censored headcount ratios for each period are constructed as the mean of the dimensional column vector for each period and represent the proportion of people who are chronically poor in time period and are deprived in dimension :
 |
(9.27) |
We can also describe the average censored headcount ratios of chronic multidimensional poverty across periods in each dimension as simply the mean of the censored headcounts in each period:
 |
(9.28) |
The Adjusted Headcount Ratio of chronic multidimensional poverty across all periods is simply the mean of the average weighted censored headcount ratios:
 |
(9.29) |
The percentage contributions show the (weighted) composition of chronic multidimensional poverty in each period and across periods.
We may seek an overview of the dimensional composition of poverty across all periods. The total percentage contribution of each dimension to chronic poverty across all periods is given by
 |
(9.30) |
We may also be interested in analysing the percentage contributions of each dimension across various periods and thus in comparing the percentage contributions of dimensions across periods. The total percentage contribution in period is
 |
(9.31) |
We are also able to construct a new set of statistics that provide more detail regarding the duration of dimensional deprivations among the chronically poor. We use the 
deprivation duration matrix , constructed in 9.4.1.3, in which each entry reflects the number of periods in which person was chronically poor (by and ) and was deprived in dimension . Recall that for the chronically poor, 
in each dimension. The value of is, naturally, 0 for non-poor persons in all dimensions. Thus the matrix will have a positive entry for persons and an entry of 0 for all persons who were never chronically poor.
For each dimension we can then define a dimensional duration index for dimension as follows
 |
(9.32) |
The value of 
provides the percentage of periods in which the chronically poor were deprived in dimension 
on average.
The relationship between the mean across all and the chronic multidimensional poverty figure provided earlier is also elementary
 |
(9.33) |
and
 |
(9.34) |
From the censored identification matrix 
we can also compute the period-specific headcounts of chronic multidimensional poverty. The headcount 
for period is the mean of the column vector of for period . The average headcount across all periods is
. The average headcount across all periods and the chronic multidimensional poverty headcount are related by the average duration of poverty thus:
 |
(9.35) |
Similarly,
 |
(9.36) |
and
 |
(9.37) |
We present an example in Table 9.7 using three variables: schooling, overcrowding, and income in Chile’s CASEN (Encuesta de Caracterización Socioeconómica Nacional) dataset for three periods: 1996, 2001, and 2006. The table reports the Adjusted Headcount Ratio of chronic multidimensional poverty () and its three partial indices: the headcount ratio (), the average chronic intensity (
) and the average duration (
) for three poverty cutoffs 
and three different duration cutoffs 
. All dimensions and periods are equally weighted for both identification and aggregation. When 
and 
, the identification follows a double union approach. In this case, 49% of people are identified as chronically multidimensionally poor. However, a double-union approach does not appear to capture people who are either chronically or multidimensionally poor in any meaningful sense.
Table 9.7 Cardinal Illustration with Relevant Values of 
and 
We thus consider the cutoffs where 
. In this situation, 5% of people are chronically multidimensionally poor. The average chronic intensity is 72%, meaning that people experience deprivations in 72% of dimensions in the periods in which they are poor. The average chronic duration is 72% also, meaning that the average poor person is deprived in 72% of the three periods. The overall chronic Adjusted Headcount Ratio of 0.028 shows that Chile’s population experiences only 2.8% of the deprivations it could possibly experience. All possible deprivations occur if all people are multidimensionally poor in all dimensions, and in all periods.
Using the identification matrix and the associated doubly censored deprivation matrices that have been constructed above, it is also possible to analyse poverty transitions. Comparisons can be undertaken – for example, between subgroups experiencing different dynamic patterns of multidimensional poverty – to ascertain different policy sequences or entry points that might have greater efficacy in eradicating multidimensional poverty. This section very briefly describes the construction of dynamic subgroups and some of the descriptive analyses that can be undertaken.
The chronic multidimensional poverty measures constructed previously respect the property of subgroup consistency and subgroup decomposability; thus, they can be decomposed by any population subgroup for which the data are arguably representative. In addition, it can be particularly useful to describe multidimensional poverty for what we earlier called ‘dynamic subgroups’ – the definition of which can be extended when panel data cover more than two periods.
By ‘dynamic subgroups’, we mean population subgroups that experience different patterns of multidimensional poverty over time. These include the groups mentioned in section 9.3.1 who exited poverty (1,0), entered poverty (0,1), or were in ongoing poverty (1,1). The possible patterns will vary according to the number of waves in the sample as well as the observed patterns in the dataset. With three waves, there are four basic groups: falling—people who were non-poor and became multidimensionally poor; rising—people who were multidimensionally poor and exited poverty; churning—people who both enter and exit multidimensional poverty in different periods, and long-term—people who remain multidimensionally poor continuously.[23]
The dynamic subgroups are formed by considering the identification matrix . Note that we use the matrix that is censored by the poverty cutoff but we do not, in this section on poverty transitions, apply the duration cutoff. Consider a matrix of four persons and three periods in which each person experiences one of the four categories mentioned above. Recall that an entry of one indicates that person is multidimensionally poor in period and a 0 indicates they are non-poor.
Falling: 0 0 1 (and 0 1 1)
Rising: 1 0 0 (and 1 1 0)
Churning: 1 0 1 (and 0 1 0)
Long-term: 1 1 1
For more than three periods, additional categories can be formed. Note that the categories can and must be mutually exhaustive. Each person who is multidimensionally poor in any period (whether chronically or transiently poor) can be categorized into one of these four groups.
Having decomposed the population into the non-poor and these (or additional) dynamic subgroups of the population, it can be useful to provide the standard partial indices for each subgroup, both per period and across all three periods:
· , , and (and standard errors);
· Percentage composition of poverty by dimension (often revealing) and censored headcounts;
· Intensity profiles across the poor (or inequality among the poor—see section 9.1).
It can also be useful to provide details regarding the sequences of evolution. For example, from the matrix, isolate the subgroup of the poor who ‘fell into’ poverty between period 1 and period 2 (that is, whose entries are 0,1 for the respective periods 
and 
.). Compare their evolution with those who stayed poor (1,1) and those who stayed non-poor (0,0), in the following ways:
· At the individual level, compare the raw headcount in period 1 with the raw headcount in period 2.
· Identify the dimensions in which deprivations were (a) experienced in both periods, (b) only experienced in period 1, and (c) only experienced in period 2.
· Summarize the results, if relevant and legitimate, further decomposing the population into relevant subgroups whose compositional changes follow different patterns.
· Repeat for each adjacent pair of periods. Analyse whether the patterns are stable or differ across different adjacent periods.
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[1] For inequality-adjusted poverty measures in the unidimensional context, see Thon (1979), Clark, Hemming, and Ulph (1981), Chakravarty (1983b), Foster, Greer, and Thorbecke (1984), and Shorrocks (1995). For inequality-adjusted multidimensional poverty measures, i.e., those that satisfy transfer and/or strict rearrangement properties, see section section 3.6.
[2] This section summarizes these two papers.
[3] In order to say that one multidimensional distribution is more equal than another, each must be smoothed using the same bistochastic matrix.
[4] See Sen (1976) and Foster and Sen (1997).
[5] Bourguignon and Chakravarty (2003) and Datt (2013) also propose a similar class of indices but using a union identification criterion.
[6] We have already shown that our multidimensional measure 
satisfies weak transfer, the first type of transfer property, for 
and the second type of transfer property, weak rearrangement, for 
. Chakravarty, Mukherjee, and Ranade (1998), Tsui (2002), Bourguignon and Chakravarty (2003), Chakravarty and D’Ambrosio (2006), Maasoumi and Lugo (2008), Aaberge and Peluso (2012), Bossert, Chakravarty, and D’Ambrosio (2013), and Silber and Yalonetzky (2014).
[7] For empirical examples, see Alkire, Roche, and Seth (2013), who compare countries across four gradients of poverty.
[8] There are many variations. For example, if data are relatively accurate one might consider inequality using the uncensored deprivation score vector 
, and alternatively if one only wishes to capture inequality within some subgroup
, then 
and 
.
[9] If one is interested in decomposing (9.6) into within and between group components, then the total within-group inequality term can be computed as 
and the total between-group inequality term can be computed as 
, where 
the intensity of each subgroup 
such that 
and 
is the share of all poor in subgroup .
[10] This section draws on Alkire, Roche, and Vaz (2014).
[11] Tables of the absolute levels and absolute rates of change make this feature visible; reporting the relative rate of change underscores this more precisely.
[12] For small samples, one needs to conduct hypothesis tests using the Student-t distribution which are very similar to the hypothesis tests described in Chapter 8 that use the Standard Normal distribution.
[13] Normative issues in assigning weights were discussed in details in Chapter 6.
[14] Comparisons of reductions in both raw and censored headcounts may be supplemented by information on migration, demographic shifts, or exogenous shocks, for example.
[15] Suitable adjustments can be made for demographic shifts when the population is not fixed across two periods.
[16]The corresponding considerations apply if poverty has increased and 
is expected to be small
[17] Naturally it is also possible to create estimates for 
where the upper bound was the overall change in intensity, and the lower bound was zero, and solve for 
. However this would not permit an increase in intensity (which would happen if the barely poor left poverty and the others stayed the same, for example), nor for an even stronger reduction in intensity. For example, in the example in Table 9.2, Nepal’s A reduced by five percentage points, whereas in our upper bound, intensity among the ‘stayers’ increased by 4% and in the lower bound it decreased by 13%.
[18] These bounds are theoretically possible lower and upper bounds. Further research using panel datasets is required to investigate the likelihood of these bounds.
[19] The Shapley value decomposition was initially applied to decomposition of income inequality by Chantreuil and Trannoy (2011, 2013) and Morduch and Sinclair (1998). Shorrocks (1999) showed that it can be applied to any function under certain assumptions.
[20] The multidimensional poverty index implemented in Roche (2013) focuses on children under 5. The choice of dimensions and indicators are similar to Gordon et al. (2003).
[21] This section draws upon Alkire, Apablaza, Chakravarty, and Yalonetzky (2014). See also McKay and Lawson (2003), Dercon and Shapiro (2007), Foster (2009), Foster and Santos (2013), Jalan and Ravallion (1998), Calvo and Dercon (2013), Hoy and Zheng (2011), Gradín, Del Rio, and Canto (2011), and Bossert, Chakravarty, and D’Ambrosio (2012).
[22]Hulme and Shepherd (2003), Chakravarty and D’Ambrosio (2006), Calvo (2008), Addison, Hulme, and Kanbur (2009), Baluch and Masset (2003), Bossert, Ceriani, Chakravarty, and D’Ambrosio (2012), Nicholas and Ray (2011), Nicholas, Ray and Sinha (2013), and Porter and Quinn (2013).
[23] See Hulme, Moore and Shepherd (2001), Hulme and Shepherd (2003), and Narayan and Petesch (2007).
8 Robustness Analysis and Statistical Inference
Chapter 5 presented the methodology for the Adjusted Headcount Ratio poverty index 
and its different partial indices; Chapter 6 discussed how to design multidimensional poverty measures using this methodology in order to advance poverty reduction; and Chapter 7 explained novel empirical techniques required during implementation. Throughout, we have discussed how the index and its partial indices may be used for policy analysis and decision-making. For example, a central government may want to allocate resources to reduce poverty across its subnational regions or may want to claim credit for strong improvement in the situation of poor people using an implementation of the Adjusted Headcount Ratio. One is, however, entitled to question how conclusive any particular poverty comparisons are for two different reasons.
One reason is that the design of a poverty measure involves the selection of a set of parameters, and one may ask how sensitive policy prescriptions are to these parameter choices. Any comparison or ranking based on a particular poverty measure may alter when a different set of parameters, such as the poverty cutoff, deprivation cutoffs or weights, is used. We define an ordering as robust with respect to a particular parameter when the order is maintained despite a change in that parameter.[1] The ordering can refer to the poverty ordering of two aggregate entities, say two countries or other geographical entities, which is a pairwise comparison, but it can also refer to the order of more than two entities, what we refer to as a ranking. Clearly, the robustness of a ranking (of several entities) depends on the robustness of all possible pairwise comparisons. Thus, the robustness of poverty comparisons should be assessed for different, but reasonable, specifications of parameters. In many circumstances, the policy-relevant comparisons should be robust to a range of plausible parameter specifications. This process is referred as robustness analysis. There are different ways in which the robustness of an ordering can be assessed. This chapter presents the most widely implemented analyses; new procedures and tests may be developed in the near future.
The second reason for questioning claimed poverty comparisons is that poverty figures in most cases are estimated from sample surveys for drawing inferences about a population. Thus, it is crucial that inferential errors are also estimated and reported. This process of drawing conclusions about the population from the data that are subject to random variation is referred as statistical inference. Inferential errors affect the degree of certainty with which two and more entities may be compared in terms of poverty for a particular set of parameters’ values. Essentially, the difference in poverty levels between two entities – states for example – may or may not be statistically significant. Statistical inference affects not only the poverty comparisons for a particular set of parameter values but also the robustness of such comparisons for a range of parameters’ values.
In general, assessments of robustness should cohere with a measure’s policy use. If the policy depends on levels of , then the robustness of the respective levels (or ranks) of poverty should be the subject of robustness tests presented here. If the policy uses information on the dimensional composition of poverty, robustness tests should assess these—which lie beyond the scope of this chapter, but see Ura et al. (2012). Recall also from Chapter 6 people’s values may generate plausible ranges of parameters. Robustness tests clarify the extent to which the same policies would be supported across that relevant range of parameters. In this way, robustness tests can be used for building consensus or for clarifying which points of dissensus have important policy implications.
This chapter is divided into two sections. Section 8.1 presents a number of useful tools for conducting different types of robustness analysis; section 8.2 presents various techniques for drawing statistical inferences and section 8.3 presents some ways in which the two types of techniques can be brought together.
In monetary poverty measures, the parameters include (a) the set of indicators (components of income or consumption); (b) the price vectors used to construct the aggregate as well as any adjustments such as for inflation or urban/rural price differentials; (c) the poverty line; and (d) equivalence scales (if applied). The parameters that influence the multidimensional poverty estimates and poverty comparisons based on the Adjusted Headcount Ratio are (i) the set of indicators (denoted by subscript 
); (ii) the set of deprivation cutoffs (denoted by vector 
); (iii) the set of weights or deprivation values (denoted by vector 
); and (iv) the poverty cutoff (denoted by 
). A change in these parameters may affect the overall poverty estimate or comparisons across regions or countries.
This section introduces tools that can be used to test the robustness of pairwise comparisons as well as the robustness of overall rankings with respect to the initial choice of the parameters. We first introduce a tool to test the robustness of pairwise comparisons with respect to the choice of the poverty cutoff. This tool tests an extreme form of robustness, borrowing from the concept of stochastic dominance in the single-dimensional context (section 3.3.1).[2] When dominance conditions are satisfied, the strongest possible results are obtained. However, as dominance conditions are highly stringent and dominance tests may not hold for a large number of the pairwise comparisons, we present additional tools for assessing the robustness of country rankings using the correlation between different rankings. This second set of tools can be used with changes in any of the other parameters too, namely, weights, indicators and deprivation cutoffs.
8.1.1 Dominance Analysis for Changes in the Poverty Cutoff
Although measurement design begins with the selection of indicators, weights, and deprivation cutoffs, we begin our robustness analysis by assessing dominance with respect to changes in the poverty cutoff, which is applied to the weighted deprivation scores constructed using other parameters. We do this because as in the unidimensional context, it is the poverty cutoff that finally identifies who is poor, thereby defining the ‘headcount ratio’ and effectively setting the level of poverty. It is arguably most visibly debated.[3] We have introduced the concept of stochastic dominance in the uni- and multidimensional context in section 3.3.1. This part of the chapter builds on that concept and technique, focusing primarily on the first-order stochastic dominance (FSD) and showing how it can be applied to identify any unambiguous comparisons with respect to the poverty cutoff for our two most widely used poverty measures—Adjusted Headcount Ratio () and Multidimensional Headcount Ratio (
). Recall from section 3.3.1 the notation of two univariate distributions of achievements 
and 
with cumulative distribution functions (CDF) 
and 
, where 
and 
are the shares of population in distributions and with achievement level less than 
. Distribution first-order stochastically dominates distribution (or FSD 
if and only if 
for all 
and 
for some . Strict FSD requires that for all .[4] Interestingly, if distribution FSD , then has no lower headcount ratio than for all poverty lines.
Let us now explain how we can apply this concept for unanimous pairwise comparisons using and between any two distributions of deprivation scores across the population. For a given deprivation cutoff vector and a given weighting vector , the FSD tool can be used to evaluate the sensitivity of any pairwise comparison to varying poverty cutoff . Following the notation introduced in Chapter 2, we denote the (uncensored) deprivation score vector by 
. Note that an element of denotes the deprivation score and a larger deprivation score implies a lower level of well-being.
The FSD tool can be applied in two different ways: one is to convert deprivations into attainments by transforming the deprivation score vector into an attainment score vector 1 
and the other option is to use the tool directly on the deprivation score vector . The first approach has been pursued in Alkire and Foster (2011a) and Lasso de la Vega (2010). In this section, because it is more direct, we present the results using the deprivation score vector and thus avoid any transformation. A person is identified as poor if the deprivation score is larger than or equal to the poverty cutoff , unlike in the attainment space where a person is identified as poor if the person’s attainment falls below a certain poverty cutoff. To do that, however, we need to introduce the complementary cumulative distribution function (CCDF)—the complement of a CDF.[5] For any distribution with CDF , the CCDF of the distribution is 
1 
, which means that for any value , the CCDF 
is the proportion of the population that has values larger than or equal to . Naturally, CCDFs are downward sloping. The first-order stochastic dominance condition in terms of the CCDFs can be stated as follows. Any distribution first order stochastically dominates distribution 
if and only if 
for all and 
for some . For strict FSD, the strict inequality must hold for all .
Now, suppose there are two distributions of deprivation scores, and 
with CCDFs 
and 
. For poverty cutoff , if 
, then distribution has no lower multidimensional headcount ratio than distribution 
at . When is it possible to say that distribution has no lower than distribution for all poverty cutoffs? The answer is when distribution first order stochastically dominates distribution . Let us provide an example in terms of two four-person vectors of deprivation scores: 
and 
. The corresponding CCDFs and are denoted by a black dotted line and a solid grey line, respectively, in Figure 8.1. No part of lies above that of and so first-order stochastically dominates and we can conclude that has unambiguously lower poverty than , in terms of the multidimensional headcount ratio.
Figure 8.1 Complementary CDFs and Poverty Dominance
Let us now try to understand dominance in terms of . In order to do so, first note that the area underneath a CCDF of a deprivation score vector is the average of its deprivation scores. Consider distribution with CCDF as in Figure 8.1. The area underneath is the sum of areas I, II, III, and IV. Area IV is equal to 
, Area III is 
, and Areas I+II is 
, so essentially each area is a score times its frequency in the population. The sum of the four areas 
, is simply the average of all elements in and it coincides with the measure for a union approach. When an intermediate or intersection approach to identification is used, then the is the average of the censored deprivation score vector 
. In other words, the deprivation scores of those who are not identified as poor are set to 0. For example, for a poverty cutoff 
, the censored deprivation score vector corresponding to is 
. Obtaining the average of censored deprivation scores is equivalent to ignoring areas III and IV in Figure 8.1. The of for 
is the sum of the remaining area I, II which is 
.[6]
We now compute the area underneath the censored CCDF for every 
and plot the area on the vertical axis for each on the horizontal axis and refer to it as an curve, depicted in Figure 8.2. We denote the curves of distributions and by 
and 
, respectively. Given that the curves are obtained by computing the areas underneath the CCDFs, the dominance of curves is referred as second-order stochastic dominance. Given that first-order stochastic dominance implies second-order dominance, if first-order dominance holds between two distributions, then dominance will also hold between them. However, the converse is not necessarily true, that is, even when there is dominance there may not be dominance. Therefore, when the CCDFs of two distributions cross—i.e. there is not first-order (
) dominance—it is worth testing dominance between pairs of distributions, to which we refer as pairwise comparisons from now on, using the curves. Batana (2013) has used the curves for the purpose of robustness analysis while comparing multidimensional poverty among women in fourteen African countries.
Figure 8.2 The Adjusted Headcount Ratio Dominance Curves
The dominance requirement for all possible poverty cutoffs may be an excessively stringent requirement. Practically, one may seek to verify the unambiguity of comparison with respect to a limited variation the poverty cutoff, which can be referred to as restricted dominance analysis. For example, when making international comparisons in terms of the MPI, Alkire and Santos (2010, 2014) tested the robustness of pairwise comparisons for all poverty cutoffs 
[0.2, 0.4], in addition to the poverty cutoff of 
1/3. In this case, if the restricted FSD holds between any two distributions, then dominance holds for the relevant particular range of poverty cutoffs for both and .
In situations in which dominance tests are too stringent, we may explore a milder form of robustness, which assesses the extent to which a ranking, that is, an ordering of more than two entities, obtained under a specific set of parameters’ values, is preserved when the value of some parameter is modified. How should we assess the robustness of a ranking? One first intuitive measure is to compute the percentage of pairwise comparisons that are robust to changes in parameters – that is the proportion of pairwise comparisons that have the same ordering. As we shall see in section 8.3, whenever poverty computations are performed using a survey, the statistical inference tools need to be incorporated into the robustness analysis.
Another useful way to assess the robustness of a ranking is by computing a rank correlation coefficient between the original ranking of entities and the alternative rankings (i.e. those obtained with alternative parameters’ values). There are various choices for a rank correlation coefficient. The two most commonly used rank correlation coefficients are the Spearman rank correlation coefficient (
) and the Kendall rank correlation coefficient (
).[7]
Suppose, for a particular parametric specification, the set of ranks across 
population subgroups is denoted by 
, where 
is the rank attributed to subgroup 
. The subgroups may be ranked by their level of multidimensional headcount ratio, the Adjusted Headcount Ratio, or any other partial indices. We present the rank correlation measures using population subgroups, but they apply to ranking across countries as well. We denote the set of ranks for an alternative specification of parameters by 
, where 
is the rank attributed to subgroup . The alternative specification may be a different poverty cutoff, a different set of deprivation cutoffs, a different set of weights, or a combination of all three. If the initial and the alternative specification yield exactly the same set of rankings across subgroups, then 
for all 
. In this case, we state that the two sets of rankings are perfectly positively associated and the association is highest across the two specifications. In terms of the previous approach, 100% of the pairwise comparisons are robust to changes in one or more parameters’ values. On the other hand, if the two specifications yield completely opposite sets of rankings, then 
for all . In this case, we state that the two sets of rankings are perfectly negatively associated and the association is lowest across the two specifications. In terms of the previous approach, 0% of the pairwise comparisons are robust to changes in one or more parameters’ values.
The Spearman rank correlation coefficient can be expressed as
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Intuitively, for the Spearman rank correlation coefficient, the square of the difference in the two ranks for each subgroup is computed and an average is taken across all subgroups. The 
is bounded between 
and 
. The lowest value of is obtained when two rankings are perfectly negatively associated with each other whereas the largest value of is obtained when two rankings are perfectly positively associated with each other.
The Kendall rank correlation coefficient is based on the number of concordant pairs and discordant pairs. A pair (
) is concordant if the comparisons between two objects are the same in both the initial and alternative specification, i.e. 
and 
. In terms of the previously used terms, a concordant pair is equivalent to a robust pairwise comparison. A pair, on the other hand, is discordant if the comparisons between two objects are altered between the initial and the alternative specification such that but 
. In terms of the previously used terms, a discordant pair is equivalent to a non-robust pairwise comparison. The is the difference in the number of concordant and discordant pairs divided by the total number of pairwise comparisons. The Kendall rank correlation coefficient can be expressed as
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Like , also lies between and . The lowest value of is obtained when two rankings are perfectly negatively associated with each other whereas the largest value of is obtained when two rankings are perfectly positively associated with each other. Although both and are used to assess rank robustness the Kendall rank correlation coefficient has an intuitive interpretation. Suppose the Kendall Tau correlation coefficient is 0.90, from equation (8.2), it can be deduced that this means that 95% of the pairwise comparisons are concordant (i.e. robust) and only 5% are discordant. Equations (8.1) and (8.2) are based on the assumption that there are no ties in the rankings. In other words, both expressions are applicable when no two entities have equal values. When there are ties, Kendall (1970) offers two adjustments in the denominator of both rank correlation coefficients ( and ) to correct for tied ranks; these adjusted Kendall coefficients are commonly known as tau-b and tau-c.
Table 8.1 Correlation among Country Ranks for Different Weights[8]
Let us present one empirical illustration showing how rank robustness tools may be used in practice. The first illustration presents the correlation between 2011 MPI rankings across 109 countries and the rankings for three alternative weighting vectors (Alkire et al. 2011). The MPI attaches equal weights across three dimensions: health, education, and standard of living. However, it is hard to argue with perfect confidence that the initial weight is the correct choice. Therefore, three alternative weighting schemes were considered. The first alternative assigns a 50% weight to the health dimension and then a 25% weight to each of the other two dimensions. Similarly, the second alternative assigns a 50% weight to the education dimension and then distributes the rest of the weight equally across the other two dimensions. The third alternative specification attaches a 50% weight to the standard of living dimension and then 25% weights to each of the other two dimensions. Thus, we now have four different rankings of 109 countries, each involving 5,356 pairwise comparisons. Table 8.1 presents the rank correlation coefficient and 
between the initial ranking and the ranking for each alternative specification. It can be seen that the Spearman coefficient is around 0.98 for all three alternatives. The Kendall coefficient is around 0.9 for each of the three cases, implying that around 80% of the comparisons are concordant in each case.
The same type of analysis has been done to changes in other parameters’ values, such as the indicators used and deprivation cutoffs (Alkire and Santos 2014).
The last section showed how the robustness of claims made using the Adjusted Headcount Ratio and its partial indices may be assessed. Such assessments apply to changes in a country’s performance over time, comparisons between different countries, and comparisons of different population subgroups within a country. Most frequently, the indices are estimated from sample surveys with the objective of estimating the unknown population parameters as accurately as possible. A sample survey, unlike a census that covers the entire population, consists of a representative fraction of the population.[9] Different sample surveys, even when conducted at the same time and despite having the same design, would most likely provide a different set of estimates for the same population parameters. Thus, it is crucial to compute a measure of confidence or reliability for each estimate from a sample survey. This is done by computing the standard deviation of an estimate. The standard deviation of an estimate is referred to as its standard error. The lower the magnitude of a standard error, the larger the reliability of the corresponding estimate. Standard errors are key for hypothesis testing and for the construction of confidence intervals, both of which are very helpful for robustness analysis and more generally for drawing policy conclusions. In what follows we briefly explain each of these statistical terms.
There are different approaches to estimating standard errors. Two approaches are commonly followed:
The Appendix to this chapter presents the formulas for computing standard errors with the analytical approach depending on the survey design.
The analytical approach is based on two assumptions. Such assumptions are based on the premise that the sample surveys used for estimating the population parameters are significantly smaller in size compared to the population size under consideration.[11] For example, the sample size of the Demographic and Health Survey of India in 2006 was only 0.04% of the Indian population. The first assumption is that the samples are drawn from a population that is infinitely large, so that even the finite population under study is a sample of an infinitely large superpopulation. This philosophical assumption is based on the superpopulation approach, which is different from the finite population approach (for further discussion see Deaton 1997). A finite population approach requires that a finite population correction factor should be used to deflate the standard error if the sample size is large relative to the population. However, if the sample size is significantly smaller than the finite population size, the finite population correction factor is approximately equal to one. In this case, the standard errors based on both approaches are almost the same.
The second assumption is that we treat each sample as drawn from the population with replacement. The practical motivation behind the assumption is the size of the sample survey compared to the population. The sample surveys are commonly conducted without replacement because, once a household is visited and interviewed, the same household is not visited again on purpose. When samples are drawn with replacement, the observations are independent of each other. However, if the samples are drawn without replacement, then the samples are not independent of each other. It can be shown that in the absence of multistage sampling, a sampling without replacements needs a Finite Population Correction (FPC) factor for computing the sampling variance. The FPC factor is of the order 
, where 
is the sample size and 
is the size of the population. The use of an FPC factor allows us to get a better estimate of the true population variance. However, when the sample size is small with respect to the population, i.e. 
, the use of an FPC factor will not make much difference to the estimation of the sampling variance as the FPC factor is closer to one (Duclos and Araar 2006: 276). These assumptions would be required in order to justify our assumption that each sample is independently and identically distributed.
We now illustrate relevant methods using the Adjusted Headcount Ratio () denoting its sample estimate by 
and standard error of the estimate by 
. However, the methods are equally applicable to inferences for the multidimensional headcount ratio, the intensity, and the censored headcount ratios as long the standard errors are appropriately computed, as outlined in the Appendix of this chapter.
A confidence interval of a point estimate is an interval that contains the true population parameter with some probability that is known as its confidence level. A significance level that is used is the complement of the confidence level. Let us denote the significance level[12] by 
, which by definition ranges between 0 and 100%. The level of confidence is 
percent. Thus, for a given estimate, if one wants to be 95% confident about the range within which the true population parameter lies, then the significance is 5%. Similarly, if one wants to be 99% confident, then the significance level is 1%.
By the central limit theorem, we can say that the difference between the population parameter and the corresponding sample average divided by the standard error approximates the standard normal distribution (i.e. the normal distribution with a mean of 0 and a standard deviation of 1). Using the standard normal distribution one can determine the critical value associated with that significance level, which is given by the inverse of the standard normal distribution at 
. In other words, the critical value is the value at which the probability that the statistic is higher than that is precisely .[13] The critical values to be used when one is interested in computing a 95% confidence interval are: 
. If instead one is interested in computing a 99% or a 90% confidence interval, the corresponding critical values are 
and 
, respectively.
For example, Table 8.2 presents the sample estimate of the Adjusted Headcount Ratio (), the multidimensional headcount ratio (
), and the average deprivation share among the poor (
) from the Demographic and Health Survey of 2005–2006. India’s sample estimate of the population-Adjusted Headcount Ratio is 
, with a standard error 
. The 
% confidence interval is then 
. This means that with 95% confidence, the true population lies between 
and 
. Similarly, the 99% confidence interval of India’s is (
). The more one wants to be confident about the range within which the true population parameter lies, the larger the confidence interval will be.
Confidence Intervals for 
, 
, and 
Similar to , the confidence interval for is 
, for is 
, and for 
is 
for all . It can be seen from the table that the standard error of is 0.41%, whereas that of is 0.20%.
Confidence intervals are useful for judging the statistical reliability of a point estimate when the population parameter is unknown. However, suppose that, somehow, we have a hypothesis about what the population parameter is. For example, suppose the government hypothesizes that the Adjusted Headcount Ratio in India is 0.26. Thus, the null hypothesis is 
. This has to be tested against any of the three alternatives 
or 
or 
.[14] This is a one-sample test. Note that the first alternative requires a so-called two-tailed test, and each of the other two alternatives requires a so-called one-tailed test. Now, suppose a sample (either simple random or multistage stratified) 
of size is collected. We denote the estimated Adjusted Headcount Ratio by . By the law of large numbers and by the central limit theorem, as 
, 
, where 
is the population variance of . The standard error of can be estimated using either equation (8.11) or (8.30) in the Appendix, whichever is applicable.
In a two-tail test, the null hypothesis can be rejected against the alternative with (
) percent confidence if 
.; in words, if the absolute value of the statistic is greater than the absolute value of the critical value. An equivalent procedure to reject or not the null hypothesis entails, rather than comparing the test statistic against the critical value, comparing the significance level against the so-called 
-value. The -value is defined as the actual probability that the test statistic assumes a value greater than the value observed, i.e. it is the probability of rejecting the null hypothesis when it is true.
Let us consider the example of India’s Adjusted Headcount Ratio, reported in Table 8.2, where and . Now, 
. Thus, with 
% confidence, the null hypothesis can be rejected with respect to the alternative 
and the corresponding -value is 
, where 
stands for the cumulative standard normal distribution. Similarly, in a one-tail test to the right, the null hypothesis can be rejected against the alternative with () percent confidence if 
. The corresponding -value is 
. Finally, in a one tail test to the left, the null hypothesis can be rejected against the alternative with () percent confidence, if 
, where the relevant -value is 
.[15]
Note that the conclusions based on the confidence intervals and the one-sample tests are identical. If the value at the null hypothesis lies outside of the confidence interval, then the test will also show that null hypothesis is rejected. On the other hand, if the value at the null hypothesis lies inside the confidence interval, then the test cannot reject the null hypothesis.
Formal tests are also required in order to understand whether a change in the estimate over time—or a difference between the estimates of two countries—has been statistically significant. The difference is that this is a two-sample test. We assume that the two estimates whose difference is of interest are estimated from two independent samples.[16] For example, when we are interested in testing the difference in across two countries, across rural and urban areas, or across population subgroups, it is safe to assume that the samples are drawn independently. A somewhat different situation may arise with a change over time. It is possible that the samples are drawn independently of each other or that the samples are drawn from the same population in order to track changes over time, as, for example, in panel datasets. This section restricts its attention to assessments in which we can assume independent samples.
Suppose there are two countries, Country I and Country II. The population achievement matrices are denoted by 
and 
respectively, and the population Adjusted Headcount Ratios are denoted by 
and 
, respectively. We seek to test the null hypothesis 
, which implies that poverty in country I is not significantly different from poverty in country II in any of the three alternatives: 
which means that one of the two countries is significantly poorer than the other; or 
, which means that country I is significantly poorer than country II; or 
, which means the opposite. For the first alternative, we need to conduct a two-tailed test, and for the other two alternatives, we need to conduct a one-tailed test.
Now, suppose a sample (either simple random or multistage stratified) 
of size 
is collected from and a sample 
of size 
is collected from 
, where samples in and are assumed to have been drawn independently of each other. We denote the estimated Adjusted Headcount Ratios from the samples by 
and 
, respectively. By the law of large numbers and the central limit theorem, 
and 
. The difference of two normal distributions is a normal distribution as well. Thus,
 |
(8.3) |
where 
. Note that, as we have assumed independent samples, the covariance between the two Adjusted Headcount Ratios is zero. Hence, the standard error of 
, denoted by 
, may be estimated using equations (8.11) or (8.30) in the Appendix, whichever is applicable, as:
|
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(8.4) |
where 
is the standard error of and 
is the standard error of . Like the one-sample test discussed above, in the two-tail test, the null hypothesis can be rejected against the alternative with () percent confidence, if 
. Given that at the null hypothesis 
, this implies requiring 
. Similarly, in order to reject the null hypothesis against , we require 
and against , we require 
The corresponding -values can be computed as discussed in the one-sample test.
Table 8.3 presents an example of an estimation of MPI (an adaptation of ) in four Indian states: Goa, Punjab, Andhra Pradesh, and Tripura, with their corresponding standard errors, confidence intervals and hypothesis tests.[17] These results are computed from the Demographic and Health Survey of India for the years 2005–2006. In the table we can see that the MPI point estimate for Goa is 0.057, and with 95% confidence, we can say that the MPI estimate of Goa lies somewhere between 0.045 and 0.069. Similarly, we can say with 95% confidence that Punjab’s MPI is not larger than 0.103 and no less than 0.073, although the point estimate of MPI is 0.088. We can also state, after doing the corresponding hypothesis test, that Punjab is significantly poorer than Goa. However, we cannot draw the same kind of conclusion for the comparison between Andhra Pradesh and Tripura, although the difference between the MPI estimates of these two states (0.032) is similar to the difference between Goa and Punjab.
Table 8.3 Comparison of Indian States Using Standard Errors
It is vital to understand that in two sample tests, conclusions about the statistical significance obtained with confidence intervals do not necessarily coincide with conclusions obtained using hypothesis testing. Let us formally examine the situation. Suppose, 
. If the confidence intervals do not overlap, then the lower bound of is larger than the upper bound of , i.e. 
or 
. Given that for two independent samples, 
, if the confidence intervals do not cross, a statistically significant comparison can be made. However, if the confidence intervals overlap, it does not necessarily mean that the comparison is not statistically significant at the same level of significance. It is thus essential to conduct statistical tests on differences when the confidence intervals overlap.
In practice, the robustness analyses discussed in section 8.1 are typically performed with estimates from sample surveys. In at least two cases, it is necessary to combine the robustness analyses with the statistical inference tools just described. This section describes how to do so in practice.
The dominance analysis presented in section 8.1.1 assesses dominance between two CCDFs or two curves in order to conclude whether a pairwise ordering is robust to the choice of all poverty cutoffs. But it is also crucial to examine if the pairwise dominance of the CCDFs or curves are statistically significant. For two entities in a pairwise ordering, one should perform one-tailed hypothesis tests of the difference in the two 
estimates for each possible value, as described in section 8.2.3. This will determine whether the two countries’ poverty estimates are not significantly different or whether one is significantly poorer than the other regardless of the poverty cutoff.[18] One may also construct confidence interval curves around each CCDF curve (or curve) and examine whether two corresponding confidence interval curves overlap or not, in order to conclude dominance. More specifically, if the lower confidence interval curve of a unit does not overlap with the upper confidence interval curve of another unit, then one may conclude that statistically significant dominance holds between two entities. However, as explained at the end of section 8.2.3, no conclusion on statistical significance can be made when the confidence intervals overlap. Thus a hypothesis test for dominance should be preferred.[19]
This need to combine methods also applies to the other type of robustness analysis presented in section 8.1.2, in the sense that one can implement this analysis to a ranking of entities and report the proportion of robust pairwise comparisons across the different values. Moreover, the analysis described in section 8.2.3 (hypothesis testing or comparison of confidence intervals by pairs of entities) can be implemented not only with respect to the poverty cutoff but also with respect to changes in the other parameters, such as weights, deprivation cutoffs or alternative indicators.
As Alkire and Santos observe (2014: 260), the number of robust pairwise comparisons may be expressed in two ways. One may report the proportion of the total possible pairwise comparisons that are robust. A somewhat more precise option is to express it as a proportion of the number of significant pair-wise comparisons in the baseline measure, because a pairwise comparison that was not significant in the baseline cannot, by definition, be a robust pairwise comparison.
To interpret results meaningfully, it can be helpful to observe that the proportion of robust pairwise comparisons of alternative specifications is influenced by: the number of possible pairwise comparisons, the number of significant pairwise comparisons in the baseline distribution, and the number of alternative parameter specifications. Interpretation of the percentage of robust pairwise comparisons in light of these three factors illuminates the degree to which the poverty estimates and the policy recommendations they generate are valid across alternative plausible design specifications.
Alkire and Santos (2014) perform both types of robustness analysis with the global MPI (2010 estimates) for every possible pair of countries with respect to: (a) a restricted range of values, namely, 20% to 40%; (b) four alternative sets of plausible weights; and (c) to subgroup-level MPI values.[20] The country rankings seem highly robust to alternative parameters’ specifications.[21]
Chapter 9 further develops the techniques of multidimensional poverty measurement and analysis. Specifically, we present techniques for analysing poverty over time (with and without panel data) and for exploring distributional issues such as inequality among the poor.
This appendix provides a technical outline of how standard errors may be computed. We first present the analytical approach and then the bootstrap method using the notation in Method I presented in Box 5.7. For the multidimensional and censored headcount ratios, we use the notation in Box 5.4. The and its partial indices are written as
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(8.6) |
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(8.7) |
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Note that 
is the logical ‘and’ operator. The standard errors of the subgroups’ s and partial indices may be computed in the same way and so we only outline the standard errors of equations (8.5)–(8.8).
Simple Random Sampling with Analytical Approach
Suppose samples have been collected through simple random sampling from the population. We denote the dataset by and its 
th element by 
for all 
and . We denote the deprivation status score for by 
. For statistical inferences, our analysis focuses on the censored deprivation scores. The score, defined at the population level, becomes a random variable while performing statistical inference. We assume that a random sample (of size ) of censored deprivation scores {
is a sequence of independently and identically distributed random variables with an expected value 
and Var
. Then as approaches infinity, the random variable 
converges in distribution to 
, where 
. That is
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(8.9) |
The unbiased sample estimate of 
is
|
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and the standard error of the Adjusted Headcount Ratio is
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The analytical approach based on the central limit theorem (CLT) also applies to the calculation of the standard errors of , which leads to
|
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(8.12) |
where 
and 
. Note that unlike , is an average across 0s and 1s, i.e. the mean is a proportion and 
is estimated as
|
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and so the unbiased standard error is
|
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(8.14) |
With the same logic, the standard error for 
, can be estimated as
|
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(8.15) |
where, 
.
The formulation of 
is analogous to the formulation of , and so the standard error of ) is computed as
|
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where 
and 
is the number multidimensionally poor in the sample.
Note that if the number of multidimensionally poor is extremely low, the sample size for estimating 
may not be large enough. This may affect the precision of using (8.16). It may then be accurate to treat as a ratio of and for computing . By the Taylor series expansion (see the discussion in Casella and Berger 1990: 240–245), can be approximated as 
and 
can be estimated as
|
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where 
and 
are based on (8.10) and (8.13), respectively, and 
can be estimated as
|
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By combining (8.17) and (8.18), the alternative formulation becomes
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(8.19) |
Stratified Sampling with an Analytical Approach
We next discuss the estimation of standard errors when samples are collected through two-stage stratification.[22] Using information on the population characteristics, the population is partitioned into several strata. The first stage, from each stratum, draws a sample of PSUs with or without replacement. The second stage draws samples either with or without replacement, from each PSU.
We suppose that the population is partitioned into 
strata and there are 
PSUs in the 
th strata for all 
. The population size of the 
th PSU in the th stratum is 
so that 
. We denote the total number of poor by 
and the number of poor in the th PSU in the th strata by 
. The population measure and its partial indices are presented in (8.20)–(8.23) with the same notation for the identity function as in (8.5)–(8.8).
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(8.21) |
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(8.22) |
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Note that 
if the 
th person from the th PSU in the th stratum is deprived in the 
th dimension and 
otherwise; and 
and 
are the deprivation score and the censored deprivation score of the th person from the th PSU in the th stratum, respectively. Thus, 
; and 
if 
and 
otherwise.
Now, suppose a sample of size is collected through a two-stage stratified sampling. The first stage selects 
PSUs from the th stratum for all . The second stage selects 
samples from the th PSU in thstratum . So, 
. Each sample in the th PSU in the th stratum is assigned a sampling weight 
, which are summarized by an -dimensional vector 
. The achievements are summarized by matrix , which is a typical sample dataset.
In order to estimate the measure from the sample, first, the total population and the total number of poor should be estimated from the sample. We denote the estimates of the population by 
and the estimate of the poor population by 
. Then,
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(8.24) |
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(8.25) |
The sample estimates of the population averages in (8.20)–(8.23) are presented in (8.26)–(8.29).
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(8.28) |
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As each sample estimate is a ratio of two estimators, their standard errors are approximated using (8.17) and using equations (1.31) and (1.63) in Deaton (1997). The standard error for in (8.26) is
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where 
, 
and 
.
The standard errors of and 
are
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(8.32)
where
and 
. Terms 
and 
are the same as in (8.30).
Finally, we present the standard error for in (8.27), where the denominator is instead of 
as
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where 
, 
and 
. Intuitively, 
is the estimated average intensity for stratum , 
is the average of sampling weights in stratum across the poor, and 
is the sum of all sampling weights in PSU of stratum also across the poor.
As a reasonably smaller sample size may affect the precision of the standard error of the variance 
can be approximated as in (8.17), but using (8.30) and (8.31) as
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where
|
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(8.35)
Hence, combining (8.34) and (8.35), we have
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(8.36) |
Note that the analytical standard errors and confidence intervals may not serve too well when the sample sizes are small or when the estimates are too close to the natural upper or lower bounds.[23] In these cases, resampling non-parametric methods, such as bootstrap, may be more suitable for computing standard errors and confidence intervals.
The Bootstrap Method
An alternative approach for statistical inference is the ‘bootstrap’, which is a data-based-simulation method for assessing statistical accuracy. Introduced in 1979, it provides an estimate of the sampling distribution of a given statistic 
, such as the standard error, by resampling from the original sample (cf. Efron 1979; Efron and Tibshirani 1993). It has certain advantages over the analytical approach. First, the inference on summary statistics does not rely on CLT as the analytical approach. In particular, for reasonably small sample size, standard errors/confidence intervals computed through the CLT-based asymptotic approximation may be inaccurate. Second, the bootstrap can automatically take into account the natural bounds of the measure. Confidence intervals using the analytical approach can lie outside natural bounds, which can be prevented when the bootstrap re-sampling distribution of the statistic is directly used.
Third, the computation of standard errors may become complex when the estimator and its standard error have a complicated form or have a no-closed expression. These types of complexities are common both in the context of statistical inference of inequality or poverty measurement and in tests where comparisons of group inequality or poverty (across gender or region) are of particular interest (Biewen 2002). Although, the delta-method can handle these analytical standard errors from stochastic dependencies, but when the number of time periods or groups increases, computing the standard errors analytically can easily become cumbersome (cf. Cowell 1989, Nygard and Sandström 1989). In practice, Monte Carlo evidence suggests that bootstrap methods are preferred for these analyses and shows that the simplest bootstrap procedure achieves the same accuracy as the delta-method (Biewen 2002; Davidson and Flachaire 2007). In developing economics, bootstrap has been used to draw statistical inferences for poverty and inequality measurement (Mills and Zandvakili 1997; Biewen 2002).
Here we briefly illustrate the use of the bootstrap for computing standard errors. Readers interested in using the bootstrap for confidence interval estimation and hypothesis testing can refer to Efron and Tibshirani (1993), chapters 12 and 16, respectively.
The bootstrap algorithm can be described as a resampling technique, which is conducted 
number of times by generating a random artificial sample each time, with replacement from the original sample, which is our dataset . The 
th resample produces an estimate 
for all 
. Thus, we have a set of resample estimates of 
: 
. If the artificial samples are independent and identically distributed (i.i.d.), the bootstrap standard error estimator of , denoted 
, is defined as
|
|
(8.37) |
where 
stands for the arithmetic mean over the artificial samples. Even if the artificial sample is drawn from a more complex but known sampling framework, the bootstrap standard error can be easily estimated from standard formulas (cf. Efron 1979; Efron and Tibshirani 1993). If the resampling is conducted on an empirical distribution of a given dataset , then it is referred to as a non-parametric bootstrap. In this case, each observation is sampled (with replacement) from the empirical distribution, with probability inversely proportional to the original sample size. However, the resampling can also be selected from a known distribution chosen on an empirical or theoretical basis. In this case, it is referred to as a parametric bootstrap.
Box 8.1 illustrates the use of the bootstrap for computing standard errors of the and its partial indices. In this case, the statistic comprises , , , and . Thus, the estimate includes , , 
or . To obtain the bootstrap standard errors, we need to pursue the following steps.
bootstrap resamples from the empirical distribution function. relevant bootstrap estimates of 
, 
, 
or 
from each bootstrap sample. replications: 
, 
, 
, or 
(cf. Efron and Tibshirani 1993: 47).We have already discussed that the bootstrap approach has certain advantages—especially that it does not rely on the central limit theorem. Although the non-parametric bootstrap approach does not depend on any parametric assumptions, it does involve certain choices. The first is the number of replications. Indeed a larger number of replications increas the precision of the estimates, but is costly in terms of time. There are different approaches for selecting the appropriate number of replications (see Poi 2004). The second involves the choice of the bootstrap sample size being selected from the original sample. The third involves the choice of the resampling method. The bootstrap sample size in Efron’s traditional bootstrap is equal to the number of observations in the original sample, but the use of smaller sample sizes has also been studied (for further theoretical discussions; see Swanepoel (1986) and Chung and Lee (2001).
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[1] This chapter is confined to assessing the robustness of rank ordering across groups. Naturally it is essential also to assess the sensitivity of key values (such as the values of 
and dimensional contributions) to parameter changes, in situations in which policies use these cardinal values.
[2] There is a well-developed literature on robustness and sensitivity analyses for composite indices rankings with respect to relative weights, normalization methods, aggregation methods, and measurement errors. See Nardo et al. (2005), Saisana et al. (2005), Cherchye et al. (2007), Cherchye et al. (2008), Foster, McGillivray and Seth (2009, 2013), Permanyer (2011, 2012), Wolff et al. (2011), and Høyland et al. (2012). These techniques may require adaptation to apply to normative, counting-based measures using ordinal data.
[3] More elaborative dominance analysis can be conducted with respect to the deprivation cutoffs and weights. For multivariate stochastic dominance analysis using ordinal variables, see Yalonetzky (2014).
[4] In empirical applications, some statistical tests cannot discern between weak and strong dominance and thus assume 
first order stochastically dominates distribution 
, if 
for all 
. See, for example, Davidson and Duclos (2012: 88–89).
[5] This is also variously known as survival function or reliability function in other branches of studies.
[6] Technically, for poverty cutoff 
can be expressed as 
. In our example, Area I is computed as 
and Area II as 
.
[7] In this book, we only focus on bivariate rank correlation coefficients, but there are various methods to measure multivariate rank concordance that we do not cover. For such examples, see Boland and Proschan (1988), Joe (1990), and Kendall and Gibbons (1990). For an application of some of the multivariate concordance methods to examine multivariate concordance of MPI rankings, see Alkire et al. (2010).
[8] The computations of the Spearman and Kendall coefficients in the table have been adjusted for ties. For the exact formulation of tie-adjusted coefficients, see Kendall and Gibbons (1990).
[9] Various sampling methods, such as simple random sampling, systematic sampling, stratified sampling, and proportional sampling, are used to conduct a sampling survey.
[10] Yalonetzky (2010).
[11] If the particular condition under study does not justify the assumptions made here, then these assumptions need to be relaxed and the standard error formulations are adjusted accordingly.
[12] The significance level is also referred to as the Type I error, which is the probability of rejecting the null hypothesis when it is true. See section 8.2.3 for the notion of null hypothesis. By statistical convention, the significance level is denoted with 
. However, to avoid confusion with the use of this symbol for other purposes in this book, we denote it 
.
[13] Whenever the population standard deviation is unknown or when the sample size is small, one needs to use the Student-t distribution to compute the critical values rather than the standard normal distribution.
[14] We present the tests for country-level estimates but they are equally applicable to other population subgroups. Also, we only present the tests in terms of the measure, but again they are also applicable to 
, 
and 
for all 
and so we have chosen not to repeat the results.
[15] See Bennett and Mitra (2013) for an exposition of hypothesis testing of and other AF partial sub-indices using a minimum p-value approach.
[16] See chapters 14 and 16 of Duclos and Araar (2006) for further discussion on non-independent samples for panel data analysis.
[17] Alkire and Seth (2013b) use an MPI harmonized for strict comparability of indicator definitions across time.
[18] For formal tests on stochastic dominance in unidimensional poverty and welfare analysis, see Anderson (1996), Davidson and Duclos (2000), and Barrett and Donald (2003).
[19] Other new ways of testing robustness may be developed in the near future.
[20] They compute the MPI for four population subgroups: children 0–14 years of age, women 15–49 years of age, women aged 50 years and older, and men 15 years and older, and test the rankings of subgroup MPIs across countries.
[21] Further methodological work is needed to propose overall robustness standards for measures that will be used for policy.
[22] Appendix D of Seth (2013) gives an example of standard error estimation for one-stage sample stratification in the multidimensional welfare framework; for consumption/expenditure see Deaton (1997).
[23] When the estimate is too close to the natural upper and lower bounds (0 and 1), the confidence intervals using analytical standard error may fall outside these bounds. Different methods for adjustments are available. For a discussion of such methods, see Newcombe (1998).
14th February 2011
In an interview with The Economic Times on poverty in India, the renowned economist Angus S. Deaton commented on the MPI stating ‘Income is only one dimension of poverty , though clearly an important one. So I very much support the development of multidimensional indexes, and think the OPHI indexes are important. Clearly, they are to some extent still experimental, but this is clearly the right direction in which to go.’ Read more in The Economic Times
.
)
and 
are the cumulative distribution functions of the standard-normal and logistic distributions, respectively. For the binary model, the conditional mean 
is the conditional probability 
.
.
.
.
