Artículo de revista: Alkire, Sabina; Nogales, Ricardo, Quinn, Natalie Naïri; Suppa, Nicolai (2023). “On track or not? Projecting the global Multidimensional Poverty Index”. Journal of Development Economics, 165, 103150 (ISSN 1872-6089) https://doi.org/10.1016/j.jdeveco.2023.103150 Journal of Development Economics 165 (2023) 103150 Contents lists available at ScienceDirect Journal of Development Economics journal homepage: www.elsevier.com/locate/devec Regular article On track or not? Projecting the global Multidimensional Poverty Index Sabina Alkire a , Ricardo Nogales a,b , Natalie Naïri Quinn c , Nicolai Suppa a,d ,∗ a Oxford Poverty and Human Development Initiative (OPHI), University of Oxford, UK b Universidad Privada Boliviana, Bolivia c Department of Economics, University of Oxford, UK d Centre for Demographic Studies, CERCA, Autonomous University of Barcelona, Spain ARTICLE INFO ABSTRACT Dataset link: https://ophi.org.uk/data-tables-d This paper proposes a framework for modelling projections of multidimensional poverty. We use recently o-files-2020-archive/ published repeated observations of multidimensional poverty, based on time-consistent indicators, for 75 Keywords: countries. We consider and evaluate different approaches to model these countries’ trajectories of poverty Multidimensional poverty reduction. Our preferred model respects theoretical bounds, is supported by empirical evidence, and ensures Projection consistency of our main measure with its subindices. In our empirical analysis we first use this approach to Poverty dynamics assess whether countries are on track to halve poverty incidence between 2015 and 2030 if recent trends SDG continue – 51 are – before assessing the reasonableness of this target. Subsequently, we discuss implications of our modelling framework for computing projections under sustained efforts, setting poverty reduction targets, and the evaluation of trajectory changes. These implications mainly follow from the bounded nature of our outcome variables and are, therefore, applicable to a wide array of development indicators. 1. Introduction rates of change during the period of observation. Deeper analyses are challenging, mainly due to data scarcity. For instance, little can be said In the report of the World Bank Commission on Monitoring Global about (i) the most recent levels of poverty at the national or global Poverty, its chair Sir Tony Atkinson observed that the exercise of mea- level, (ii) expected slow-downs or accelerations in poverty reduction, suring global poverty is ‘‘highly controversial’’. While acknowledging or (iii) whether countries are on track to meet their poverty reduction that some might even regard the exercise as futile, he argued that targets. A more detailed account of the evolution of poverty, therefore, ‘‘estimates of global poverty are flawed but not useless. By focusing directs attention towards the poverty trajectories that countries follow. on changes over time, we can learn [...] about the evolution of global To overcome these limitations, in this paper we develop a frame- poverty’’ (World Bank, 2017, p. xvi). In this report, and also in his last work based on a logistic dynamic model for computing projections of book (Atkinson, 2019), he stressed that poverty must be considered ac- global multidimensional poverty at the country level. This framework cording to both national and international definitions and be measured respects theoretical bounds, is supported by empirical evidence, and in both monetary and multidimensional spaces. ensures consistency of our main measure with its subindices. An internationally comparable and widely recognised multidimen- Our work relates to similar exercises that have been conducted sional poverty measure is the global Multidimensional Poverty Index with other development indicators, in particular for extreme mone- (MPI) developed by Alkire and Santos (2014), which uses the method tary poverty, health, and educational outcomes. While we build on proposed by Alkire and Foster (2011). In their analysis of changes over these exercises, none of their approaches is immediately applicable to time, Alkire et al. (2020c) report extensive progress in global poverty our context. The approach of the World Bank (2018), for instance, reduction. During the period of observation, 65 of the 75 countries made significant progress in reducing multidimensional poverty, while relies on covariates of monetary poverty that are more frequently over 50 reduced the number of people in poverty. Yet the trends story is observed (e.g., per capita GDP) and on assumptions about direct pass- incomplete: the time periods covered range from three to twelve years through of economic growth to household income or consumption and the period of analysis spans 2000–2019, with the initial observa- expenditure. Current practices to monitor and project health out- tion for some countries well after the final observation for others. While comes utilise several rounds of data. This permits implementation of informative, their analysis is limited to annualised absolute or relative non-parametric methods and regression-based approaches, applied to ∗ Corresponding author at: Centre for Demographic Studies, CERCA, Autonomous University of Barcelona, Spain. E-mail address: nsuppa@ced.uab.es (N. Suppa). https://doi.org/10.1016/j.jdeveco.2023.103150 Received 19 May 2022; Received in revised form 8 July 2023; Accepted 12 July 2023 Available online 23 July 2023 0304-3878/© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). S. Alkire et al. Journal of Development Economics 165 (2023) 103150 child mortality (UN IGME, 2019; Alkema and New, 2014) and child track to meet development goals, such as the SDGs, (ii) setting sensible nutrition outcomes (WHO-UNICEF, 2017; JCME, 2020), respectively. development goals in the first place, and (iii) identifying changes in the Our current data, however, does not allow us to incorporate further indicators’ trajectories. covariates to project multidimensional poverty.1 More importantly, Our results have consequences for applications beyond the global evidence from cross-country analyses suggests that multidimensional MPI. First, while our empirical analysis is of poverty as measured by poverty has a less direct relationship with, for example, economic the global MPI, the same modelling framework is directly applicable to growth than does monetary poverty (Santos et al., 2019). Finally, many national multidimensional poverty measures, which are increas- previously applied methods typically assume an exponential or constant ingly implemented as official poverty statistics in practice.3 Second, our relative change dynamic model, which only accounts for the theoretical framework naturally extends to most development indicators, as those lower bound of zero. As well as applications to health outcomes, this that are incidence measures must respect lower and upper bounds. The approach has been applied by Ram (2020) who, like us, explores implications of our study therefore apply not only to the trajectories multidimensional poverty as measured by the global MPI, but without followed by most development indicators, but to their policy targets considering its theoretical upper bound. A notable exception is research as well. And yet, most studies of the MDGs and SDGs have been on schooling and other education indicators in which the underlying implicitly or explicitly based on an exponential or constant relative trajectories have been shown to be S-shaped (Meyer et al., 1992). change dynamic model (e.g., Fukuda-Parr et al., 2013; French, 2015; This strand of the literature, to which our study most closely relates, McArthur and Rasmussen, 2018; Ahimbisibwe and Ram, 2018). Similar routinely uses a logistic model to represent indicator dynamics (e.g., assessments by practitioners, such as whether countries are on track Permanyer and Boertien, 2019; Friedman et al., 2020). to meet their development targets, are also frequently based on the For our empirical analysis we obtain projections from our modelling implicit adoption of an exponential model (e.g., WHO-UNICEF, 2017). framework using recently observed trends. First, we analyse whether Our study demonstrates why choosing an exponential over a logistic dy- countries would be on track to meet the poverty reduction target of namic model can lead to inaccurate description and misinterpretation halving multidimensional poverty incidence by 2030 – a target closely of development indicator dynamics. related to Sustainable Development Goal (SDG) 1.2 – if recent progress The paper proceeds as follows. In Section 2, we describe the data continues.2 We find that 51 of 75 countries analysed would, indeed, used in the study. In Section 3, we explore alternative models for reduce poverty incidence at least by half by 2030. Taking sampling country-level multidimensional poverty dynamics, identifying logistic error into account, 36 countries would significantly exceed the target, trajectories as our preferred model on the basis of both theoretical reducing poverty incidence by significantly more than half. We also adequacy and compelling cross-country evidence. We calibrate this show that for many countries the target was hardly feasible from model to obtain trajectory projections for each country. In Section 4 the outset, while at the same time, it was entirely unambitious for we analyse whether countries would meet a poverty reduction target, many others. These results resonate with research that criticises the based on both obtained and plausible counterfactual trajectories. In Millennium Development Goals (MDGs) and SDGs for following a one- Section 5 we discuss implications for setting poverty targets and for size-fits-all approach to set targets and further for being unfair and ways to test for changes of trajectory. In Section 6 we conclude. unfeasible in many instances (Clemens et al., 2007; Vandemoortele, 2009; Easterly, 2009; Allwine et al., 2015; Lange and Klasen, 2017). 2. Data Moreover, our results also suggest that headcount ratios which measure incidence and are frequently used to assess progress towards devel- The primary data source for this study is the global MPI Changes opment goals may conceal achieved progress, in particular for the over Time dataset, constructed by Alkire et al. (2020b), which contains poorer countries. We argue that refined measures such as the adjusted intertemporally harmonised estimates of aggregate measures of multi- headcount ratio (Alkire and Foster, 2011), which also takes intensity of dimensional poverty for 80 countries in the developing world based poverty into account, may help to overcome this limitation. on the global MPI. Of those, results reported in this paper focus on While some studies argue that such development goals should rather the 75 countries that are jointly analysed by OPHI and UNDP’s Human be country-specific and take historical progress and other peculiarities Development Report Office (UNDP, OPHI, 2020).4 into account (e.g., Clemens et al., 2007; Ranganathan et al., 2017), the First published in the 2010 UNDP Human Development Report, the question of how to arrive at feasible and ambitious targets has received global MPI is a poverty index that aggregates 10 indicators, which are little attention so far. In this paper we argue that projections based on grouped under the dimensions of Health, Education and Living Stan- recent trends offer a useful reference point for a scenario of sustained dards (Alkire and Santos, 2014). As an adjusted headcount ratio (Alkire efforts or ‘business-as-usual’, and that initial levels, country-specific and Foster, 2011), the global MPI is based on the joint distribution historical progress, and an appropriate model are all essential to arrive of these indicators at household level, capturing intensity as well as at sensible projections. Our modelling framework also makes possible incidence of multidimensional poverty. The definitions of the indicators meaningful performance comparisons across geographical units and were revised in 2018 to better align to the SDGs (Alkire et al., 2022; over time, thereby helping to identify feasible and ambitious targets. Alkire and Kanagaratnam, 2021). Table 1 shows the current structure Finally, we also discuss how to test for changes of trajectory, which of the global MPI. Each dimension is assigned an equal weight (1/3), is important to establish, for instance, the efficacy of recent policy and indicators are also assigned equal weights within dimensions. A reforms or the impact of adverse shocks induced by conflict, political person is identified as being multidimensionally poor if they live in a unrest and natural disasters. We conclude that for bounded outcome household that simultaneously experiences 1∕3 or more of the weighted variables like multidimensional poverty, applying a logistic dynamic deprivations. model is in general crucial for (i) establishing whether countries are on 3 Currently, around 30 countries use a multidimensional poverty measure 1 With only two observations for each country, we have insufficient degrees as an official statistic. 4 of freedom to estimate a longitudinal model with country-specific trends. To align with the OPHI-UNDP collaboration, results reported in this 2 SDG target 1.2 is to, ‘‘by 2030, reduce at least by half the proportion of study omit five countries that (i) dropped a health or education indicator men, women and children of all ages living in poverty in all its dimensions in the harmonisation process, or (ii) experienced large absolute or relative according to national definitions’’ (UN, 2015, our emphasis). We use the global changes in the harmonised MPI value in comparison with the non-harmonised MPI rather than national MPIs for reasons of comparability and because not value (Alkire et al., 2020b, pp. 8–9). The omitted countries are Afghanistan, all countries have defined national MPIs. Montenegro, Trinidad & Tobago, Viet Nam, and Yemen. 2 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Table 1 Global MPI. Dimension of poverty Indicator Deprived if ... SDG area Weight 1 Nutrition Any person under 70 years of age for whom there is nutritional SDG 2 Health 6 information is undernourished. 1 Child mortality A child under 18 has died in the household in the five-year period SDG 3 6 preceding the survey. 1 Years of schooling No eligible household member has completed six years of schooling. SDG 4 Education 6 1 School attendance Any school-aged child is not attending school up to the age at which SDG 4 6 he/she would complete class 8. 1 Cooking fuel A household cooks using solid fuel, such as dung, agricultural crop, shrubs, SDG 7 18 wood, charcoal or coal. 1 Sanitation The household has unimproved or no sanitation facility or it is improved SDG 6 Living Standards 18 but shared with other households. 1 Drinking water The household’s source of drinking water is not safe or safe drinking water SDG 6 18 is a 30-minute walk or longer walk from home, roundtrip. 1 Electricity The household has no electricity. SDG 7 18 1 Housing The household has inadequate housing materials in any of the three SDG 11 18 components: floor, roof, or walls. 1 Assets The household does not own more than one of these assets: radio, TV, SDG 1 18 telephone, computer, animal cart, bicycle, motorbike, or refrigerator, and does not own a car or truck. Notes: This is a simplified version, for more details on global MPI and Changes over Time data, see UNDP, OPHI (2020) and Alkire et al. (2020c), respectively. Fig. 1. Survey dates. Notes: Only a few selected observations are labelled for reasons of readability. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) Two observations are available for each country in the Changes over Fig. 2 provides a first view of the data at hand. Each dot represents Time dataset. The timing of these observations depends on microdata the observed MPI value for a specific country at a specific point in time, survey availability; the earlier observation for each country is dated and they are connected by a line when they correspond to the same between 2000 and 2014 (median 2010) while the later observation country. Several insights emerge at first glance. There is considerable is dated between 2008 and 2019 (median 2014). The elapsed time heterogeneity of the first and second observed MPI values for each between observations is between 3 and 12 years (median 5 years). The country, as well as of the elapsed period between observations. Most distribution of survey dates is illustrated in Fig. 1, where one can see countries experienced poverty reductions over the period between ob- that for most countries, including Ethiopia and Sierra Leone, the period servations, albeit to varying extents and with some visible exceptions, between the survey dates is between 3 to 6 years, and in many cases, such as Benin or Serbia, where poverty has increased. Also, note that both surveys are at least as recent as 2010. For some other countries, such as India and Madagascar, the first observation is slightly older in some cases, changes in multidimensional poverty over time are (2008 and 2006, respectively) and the period between observations is minimal, such as in Benin and Serbia, and also Cameroon, Togo, Chad 9 years or more. In fewer cases, such as Mozambique and Bolivia, we and Niger, among others. Finally, some countries achieved substantial can only draw on data prior to 2005 for the first observation. For a poverty reduction in absolute terms, for example Sierra Leone and complete list of countries, datasets, and survey years, see Table A.1 of India. For a more detailed descriptive analysis of these data see Alkire the appendix. et al. (2020c). 3 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Fig. 2. Changes over time data: The adjusted headcount ratio (MPI). Notes: Dots represent point estimates based on micro data; Countries are colour-coded by world region: Arab States; East Asia and the Pacific; Europe and Central Asia; Latin America and the Caribbean; South Asia; Sub-Saharan Africa. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) 3. Modelling multidimensional poverty dynamics Given also a poverty cutoff 𝑘 (which Alkire and Santos, 2014, set to 1 3 for the global MPI), the individual is considered multidimensionally With multidimensional poverty estimates available at just two points poor if 𝑐𝑖 ≥ 𝑘; their censored deprivation score 𝑐𝑖 (𝑘) = 𝑐𝑖 I(𝑐𝑖 ≥ 𝑘). The level of multidimensional poverty (MPI level) in a population in time for each country, we cannot precisely estimate nor forecast ( ) individual countries’ multidimensional poverty trajectories.5 Given our is then 𝑀 = E 𝑐𝑖 (𝑘) , the average censored deprivation score in that 6 data constraints, in this section we explore alternative dynamic models population. Applying the law of iterated expectations, ( ) ( ) that we may use to implement projections of countries’ trajectories. 𝑀 = E 𝑐𝑖 (𝑘)|𝑐𝑖 ≥ 𝑘 P(𝑐𝑖 ≥ 𝑘) + E 𝑐𝑖 (𝑘)|𝑐𝑖 < 𝑘 P(𝑐𝑖 < 𝑘) We identify preferred models, which respect theoretical bounds on ( ) = E 𝑐𝑖 (𝑘)|𝑐𝑖 ≥ 𝑘 P(𝑐𝑖 ≥ 𝑘) + 0 × P(𝑐𝑖 < 𝑘) multidimensional poverty levels and are strongly supported by cross- ( ) country evidence on countries’ trajectories. We conclude the section by = E 𝑐𝑖 |𝑐𝑖 ≥ 𝑘 P(𝑐𝑖 ≥ 𝑘) implementing country-specific calibrations of these models. =𝐴×𝐻 (2) ( ) where 𝐴 is the intensity of multidimensional poverty E 𝑐𝑖 |𝑐𝑖 ≥ 𝑘 , the 3.1. Analytical framework and notation average deprivation score among the poor, and 𝐻 is its incidence or headcount ratio P(𝑐𝑖 ≥ 𝑘) = E(I(𝑐𝑖 ≥ 𝑘)). As the product of 𝐴 and 3.1.1. Multidimensional poverty 𝐻, 𝑀 can thus be described as an adjusted headcount ratio, where the Multidimensional poverty is measured following the method estab- adjustment factor 𝐴 captures intensity of poverty. By construction, all lished by Alkire and Foster (2011), in which the poor are identified three indices are bounded; 𝐻 ∈ [0, 1], 𝐴 ∈ [𝑘, 1] and 𝑀 ∈ [0, 1]. through simultaneous deprivations in multiple indicators; the index thus depends on the joint distribution of the several deprivation indi- 3.1.2. Trajectories cators. Specifically, we focus on the global MPI described in Section 2, Countries are indexed 𝑠 = 1, 2, … , 𝑆. Our objects of interest are which is one particular implementation of this method. The standard countries’ time-paths (or trajectories) of multidimensional poverty, exposition (see, for example, Alkire et al., 2015) develops sample 𝑀𝑠 (𝑡), its intensity 𝐴𝑠 (𝑡) and incidence 𝐻𝑠 (𝑡). We may write 𝑦𝑠 (𝑡) to estimators appropriate for a simple random sample; we establish the represent any of these outcomes of interest, or 𝑦(𝑡) when we do not refer to a specific country. Time-derivatives are notated with dots, so population analogue here. ̇ = d𝑦 𝑦(𝑡) . Given achievements 𝑥𝑖𝑗 in 𝑑 indicators 𝑗 = 1, 2, … , 𝑑, each of which d𝑡 ∑ An estimate obtained from microdata will be labelled with a hat, so is assigned a deprivation cutoff 𝑧𝑗 and weight 𝑤𝑗 such that 𝑑𝑗=1 𝑤𝑗 = 1, ̂ 𝑠 (𝑡𝑠𝜏 ) is the estimated level of multidimensional poverty in country 𝑠 𝑀 an individual 𝑖’s deprivation score is at time 𝑡𝑠𝜏 . (As discussed in Section 2, we have poverty estimates at ∑ 𝑑 two discrete points in time for each country, so 𝜏 = 1, 2, but these 𝑐𝑖 = 𝑤𝑗 I(𝑥𝑖𝑗 < 𝑧𝑗 ). (1) points in time are different for different countries, thus 𝑡 must be 𝑗=1 labelled by 𝑠 as well as 𝜏.) An alternative, abbreviated notation for such estimates, which constitute the observations for our empirical 5 Note that we do not rely on any external additional information, thus our 6 projections are not necessarily predictions of poverty levels in the statistical Following Alkire and Foster (2011), the usual notation is 𝑀0 . To simplify sense. Rather, we determine how poverty levels would change over time if notation, we drop the subscript 0 in this paper as we do not use any other observed trends continue. members of Alkire and Foster’s class of multidimensional indices. 4 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 models, is 𝑀̂ 𝑠𝜏 or even 𝑀𝑠𝜏 (respectively, 𝐻𝑠𝜏 , 𝐴𝑠𝜏 and in general 𝑦𝑠𝜏 ). Table 2 Dynamic models and their rates of change. A projection obtained from a projection model is labelled with a wedge, 𝑦(𝑡) ̇ so, for example, 𝐻̌ 𝑠 (𝑡) is the projected incidence in country 𝑠 at time 𝑡 Model 𝑦(𝑡) 𝑦(𝑡) ̇ 𝑦(𝑡)∕𝑦(𝑡) ̇ 𝑦(𝑡)(1−𝑦(𝑡)) (continuous). lin lin lin lin −𝛽 lin Linear 𝛼 −𝛽 𝑡 −𝛽 −𝛽 ∕𝑦 𝑦(1−𝑦) We observed above that multidimensional poverty may be decom- Exponential 𝑒𝛼 crc −𝛽 crc 𝑡 −𝛽 crc 𝑦 −𝛽 crc −𝛽 crc ∕(1 − 𝑦) posed as the product of intensity and incidence; at most two of 𝑀𝑠 (𝑡), 1 Logistic −𝛼 log +𝛽 log 𝑡 −𝛽 log 𝑦(1 − 𝑦) −𝛽 log (1 − 𝑦) −𝛽 log 𝐴𝑠 (𝑡) and 𝐻𝑠 (𝑡) can vary independently. Therefore, to ensure consis- 1+𝑒 tency, we shall model 𝑀𝑠 (𝑡) indirectly as 𝑀𝑠 (𝑡) = 𝐻𝑠 (𝑡)𝐴𝑠 (𝑡) (3) model ignores the theoretical upper and lower bounds, the exponential throughout this paper. model does not respect the upper bound, whereas the logistic model respects both. By implication, this also means that the logistic model 3.1.3. Canonical dynamic models features an initial period of accelerating poverty reduction, and a slow- A very simple dynamic model for the trajectory of outcome 𝑦(𝑡) is down in poverty reduction when approaching the zero lower bound. In the linear model contrast, according to the exponential model, the entire trajectory is characterised by a slowdown of poverty reduction, whereas according 𝑦(𝑡) = 𝛼 lin − 𝛽 lin 𝑡 (4) to the linear model, neither acceleration nor slowdown occur. in which the rate of change 𝑦(𝑡)̇ = −𝛽 lin is constant. While simple, It is important to note that each dynamic model implies specific linear models are rarely used for projections of development indicators; profiles of absolute and relative rates of change in the underlying an exception is Nicolai et al. (2015) who implemented simple linear variable of interest, as summarised in Table 2. In the linear and projections for many SDG indicators. In the case of multidimensional the exponential models, the 𝛽s coincide with absolute and relative poverty, a linear model does not respect the bounded nature of all changes, respectively. According to the logistic model, however, ab- outcomes of interest. solute changes follow a u-shaped pattern (reflecting acceleration and Another simple dynamic model for the trajectory of outcome 𝑦(𝑡), slowdown), whereas relative changes decrease over the entire domain which respects a lower bound at 0, is the exponential or constant to converge towards 𝛽 log as 𝑦 → 0. It follows that at very low levels of relative change dynamic model7 poverty the exponential model is a good approximation for the logistic. crc −𝛽 crc 𝑡 Finally, it is instructive to note that in the logistic dynamic model, the 𝑦(𝑡) = 𝑒𝛼 (5) 𝑦̇ (constant) logit rate of change −𝛽 log = 𝑦(1−𝑦) captures adjusted changes in the outcome variable. in which the relative rate of change 𝑦(𝑡) ̇ 𝑦(𝑡) = −𝛽 crc is constant; equiv- alently, the rate of change 𝑦(𝑡) ̇ crc = −𝛽 𝑦 is proportional to 𝑦. Note 3.2. Cross-country evidence on poverty dynamics ̃ = ln(𝑦(𝑡)) = 𝛼 crc − 𝛽 crc 𝑡 is linear in the that the log-transformation 𝑦(𝑡) parameters. This is the dynamic model implemented to assess progress As noted above, with observations at only two time-points for each in child malnutrition (WHO-UNICEF, 2017) and by Ram (2020) to country, we cannot estimate statistically or forecast trajectories at project the incidence of multidimensional poverty. It is a plausible country level. We therefore utilise cross-country evidence to inform our candidate for trajectories of 𝐻(𝑡) and 𝑀(𝑡), which are bounded below choice of projection models. at 0, and could be adapted for 𝐴(𝑡), which is bounded below at 𝑘. For each outcome of interest, Fig. 4 illustrates the cross-country A slightly more complex dynamic model for the trajectory of out- relationship between average annual change 𝛥𝑦, a proxy for 𝑦(𝑡), ̇ and come 𝑦(𝑡), which respects both lower and upper bounds, is the logistic its level. In the case of 𝐻 and 𝑀, the relationship is clearly nonlinear; model in the case of 𝐴 the pattern is less clear. While countries from different 1 𝑦(𝑡) = log log (6) world regions tend to cluster at different levels of the headcount ratio, 1 + 𝑒−𝛼 +𝛽 𝑡 the emerging relationship between levels and changes is not driven by in which the rate of change 𝑦(𝑡) ̇ = −𝛽 log 𝑦(1 − 𝑦) is quadratic in 𝑦, one particular world region, but rather supported by countries across passing through (0, 0) and (1, 0). Note that the logit-transformation the globe. There is, of course, substantial variation across different 𝑦(𝑡) ̃ = ln (𝑦(𝑡)∕(1 − 𝑦(𝑡))) = 𝛼 log − 𝛽 log 𝑡 is linear in the parameters; countries. 𝑦(𝑡) ̇ we will refer to the constant 𝑦(𝑡)(1−𝑦(𝑡)) = −𝛽 log as the logit rate of To explore the dynamics more systematically, we estimate cross- change. Logistic models are routinely implemented to model education country models of average annual changes in the outcomes of interest indicator dynamics (e.g., Permanyer and Boertien, 2019). The logistic 𝛥𝑦 as polynomial functions of levels of the outcomes. As the average dynamic model is a plausible candidate for trajectories of 𝐻(𝑡) and annual change is a proxy for 𝑦(𝑡) ̇ at both 𝑡1 and 𝑡2 , we retain both 𝑀(𝑡), which are bounded between 0 and 1, and could be adapted for observations for each country. We are relaxed about any artificial 𝐴(𝑡), which is bounded between 𝑘 and 1. inflation of sample size, as this exercise is purely for model selection Several observations are pertinent. First, for each of these models, purposes and any effect will have a similar impact on all models. there is a transformation 𝑦(𝑡)̃ = 𝛼 − 𝛽𝑡 that is linear in the parameters; Table 3 reports results for 𝐻, the incidence of poverty. Direct that shall prove useful in the subsequent empirical modelling. More- estimation of a constant rate of change (model 1), and linear functions over, in each case, the parameter 𝛽 represents the rate of change or for 𝛥𝐻 (models 2 and 3) are strongly rejected in favour of a quadratic speed of transition. Finally, each dynamic model is characterised by a (model 4). Adding a cubic term (model 5) slightly improves the fit, first-order ordinary differential equation (ODE): 𝑦(𝑡) ̇ is constant in the but the cubic coefficient is not significantly different from zero and linear model, linear in 𝑦(𝑡) in the exponential (constant relative change) the improvement in fit is too slight to justify the considerable extra model and quadratic in 𝑦(𝑡) in the logistic model. complexity in the dynamic model.8 Interestingly, a model with 𝐻, 𝐴 Fig. 3 illustrates linear (4), exponential (5) and logistic (6) trajecto- and their product (model 7) gives equally good fit, and adding 𝐻 2 ries calibrated to pass through a particular point with a particular rate (model 8) even better, but again, the slight gain does not outweigh of change. Note that in all three models, the respective 𝛽 determines the considerable extra modelling complexity. We conclude that the the entire trajectory as illustrated in Fig. 3. It is evident that the linear quadratic model 𝐻(𝑡)̇ = 𝑎𝐻 2 + 𝑏𝐻 + 𝑐 is most appropriate. 7 8 Described as proportional in Alkire et al. (2020a). The corresponding ODE may not have a closed-form solution. 5 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Fig. 3. Canonical dynamic models. Notes: Linear model ; exponential model ; logistic model ; all calibrated to pass through with a particular rate of change. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) Fig. 4. Cross-country evidence. Notes: Own calculations, changes on vertical axis are average annual changes, black lines show quadratic fit, grey lines linear fit. Countries are colour-coded by world region: Arab States; East Asia and the Pacific; Europe and Central Asia; Latin America and the Caribbean; South Asia; Sub-Saharan Africa. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) Table 3 Dynamic model selection for 𝐻(𝑡). (1) (2) (3) (4) (5) (6) (7) (8) 𝛥𝐻 𝛥𝐻 𝛥𝐻 𝛥𝐻 𝛥𝐻 𝛥𝐻 𝛥𝐻 𝛥𝐻 𝐻 −0.0287∗∗∗ −0.0148∗∗∗ −0.0918∗∗∗ −0.1271∗∗∗ −0.0391∗∗∗ −0.1611∗∗∗ −0.1336∗∗∗ [−11.62] [−5.03] [−13.57] [−6.79] [−5.77] [−11.27] [−8.40] 𝐻2 0.0966∗∗∗ 0.2034∗∗∗ 0.0523∗∗ [12.05] [3.53] [2.70] 𝐻3 −0.0810 [−1.87] 𝐴 0.0931∗∗∗ −0.0497∗ [3.83] [−2.11] 𝐻𝐴 0.2688∗∗∗ 0.1301∗∗ [9.88] [2.74] Constant −0.0134∗∗∗ −0.0082∗∗∗ −0.0011 0.0005 −0.0435∗∗∗ 0.0163 −0.0016∗∗ [−15.20] [−8.00] [−1.84] [0.67] [−4.66] [1.83] [−2.65] N 160 160 160 160 160 160 160 160 𝑅2 0.000 0.148 0.492 0.504 0.213 0.505 0.524 adj. 𝑅2 0.000 0.143 0.486 0.494 0.203 0.496 0.515 AIC −984.0 −973.3 −1007.6 −1088.4 −1090.2 −1018.3 −1090.6 −1096.7 BIC −980.9 −970.2 −1001.5 −1079.1 −1077.9 −1009.1 −1078.3 −1084.4 ∗ ∗∗ ∗∗∗ Notes: Own calculations, t -statistics in brackets, indicated levels of significance are 𝑝 < 0.05, 𝑝 < 0.01, 𝑝 < 0.001. See A.1 for the list of datasets underlying these results. 6 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Table 4 Dynamic model selection for 𝐴(𝑡). (1) (2) (3) (4) (5) (6) (7) 𝛥𝐴 𝛥𝐴 𝛥𝐴 𝛥𝐴 𝛥𝐴 𝛥𝐴 𝛥𝐴 𝐴 −0.0088∗∗∗ −0.0184∗∗∗ −0.1045∗∗ −0.0276∗∗ [−16.69] [−6.29] [−3.23] [−3.23] 𝐴2 0.0871∗∗ [2.67] 𝐻 −0.0052∗∗∗ −0.0131∗∗∗ −0.0175∗∗ [−6.32] [−4.22] [−3.23] 𝐻2 0.0099∗∗ [2.76] 𝐻𝐴 0.0330∗∗ [3.11] Constant −0.0040∗∗∗ 0.0047∗∗ 0.0253∗∗ −0.0022∗∗∗ −0.0015∗∗ 0.0090∗∗ [−14.59] [3.33] [3.23] [−5.75] [−3.24] [2.61] 𝑁 160 160 160 160 160 160 160 𝑅2 0.000 0.184 0.223 0.182 0.219 0.235 adj. 𝑅2 0.000 0.179 0.213 0.177 0.209 0.220 AIC −1354.3 −1377.1 −1384.8 −1390.7 −1384.5 −1389.9 −1391.1 BIC −1351.2 −1374.0 −1378.7 −1381.5 −1378.4 −1380.7 −1378.8 ∗ ∗∗ ∗∗∗ Notes: Own calculations, t -statistics in brackets, indicated levels of significance are 𝑝 < 0.05, 𝑝 < 0.01, 𝑝 < 0.001. See A.1 for the list of datasets underlying these results. Recalling that the characteristic ODE for the logistic dynamic model function of 𝑀 itself; model 6 does indeed explain more of the variation is 𝑦(𝑡) ̇ = −𝛽𝑦(1 − 𝑦), this is equivalent to a quadratic with coefficients in 𝛥𝑀 than polynomial functions of 𝑀 (models 3–5). This reassures us satisfying 𝑎 + 𝑏 = 0 and 𝑐 = 0. The 𝑝-value for a Wald test of the joint that our approach to modelling 𝑀(𝑡) is appropriate. hypothesis is 0.2078; we fail to reject it and so conclude that the logistic We therefore conclude (i) that a logistic dynamic model is the most model (Eq. (6)) is most appropriate for 𝐻(𝑡). appropriate for trajectories of incidence of poverty 𝐻(𝑡), (ii) that for We perform the same analysis also for our other outcome variables. the intensity of poverty 𝐴(𝑡) a modified logistic model (which accounts Table 4 reports results for 𝐴, the intensity of poverty; we regress 𝛥𝐴 for the lower bound of 𝐴 being 13 ) is the most appropriate and (iii) that on polynomial functions of 𝐴 and 𝐻. A linear function of 𝐴 (model 𝑀(𝑡) is best modelled as the product of 𝐴(𝑡) and 𝐻(𝑡). 3) explains only 18% of the variation in 𝛥𝐴 and adding the quadratic term (model 4) increases that to just 21%. Interestingly, 𝐻 has similar 3.3. Calibration of projection model parameters explanatory power (models 5 and 6), and a model with 𝐴, 𝐻 and their product (model 7) slightly more, but the gain is marginal over model The dynamic models developed above account for a remarkable 4. Despite the small difference in explanatory power between models proportion of the changes in multidimensional poverty across the coun- 3 and 4, we prefer the quadratic function, which corresponds to a tries in our dataset. Of course, there remains significant unexplained logistic trajectory, over the linear function, which corresponds to an variation across countries, so in order to implement projections for exponential trajectory. Several countries actually experience increases individual countries we calibrate the model parameters separately for in intensity of poverty, so it is important that our projection model each country. As each of our models is a two-parameter model that may respects the upper bound on 𝐴. be linearised in the parameters, this is straightforward. ̇ = 𝑎𝐴2 + 𝑏𝐴 + 𝑐 (model 4), the 𝑝-value for the Wald test Given 𝐴(𝑡) Given incidence and intensity estimates 𝐻𝑠1 , 𝐴𝑠1 and 𝐻𝑠2 , 𝐴𝑠2 for log log of the joint hypothesis 𝑎 + 𝑏 = 0 and 𝑐 = 0 is indistinguishable from country 𝑠 at 𝑡𝑠1 and 𝑡𝑠2 , its calibrated parameters 𝛼̌ , 𝛽̌ , 𝛼̌ ml and 𝛽̌mlℎ𝑠 ℎ𝑠 𝑎𝑠 𝑎𝑠 zero; we strongly reject the hypothesis and so cannot adopt the simple solve logistic model (Eq. (6)) for 𝐴(𝑡). This is natural, as 𝐴 is bounded below log log 1 𝐻 log (𝑡𝑠𝜏 ; 𝛼̌ ℎ𝑠 , 𝛽̌ℎ𝑠 ) = = 𝐻𝑠𝜏 , 𝜏 = 1, 2, at 13 rather than 0. The modified (three-parameter) logistic function that log ̌log 1 + 𝑒−𝛼̌ ℎ𝑠 +𝛽ℎ𝑠 𝑡𝑠𝜏 respects the bounds at 13 and 1 is characterised by the ODE 𝐴(𝑡) ̇ = 1 and 𝛽(3𝐴 2 − 4𝐴 + 1), which is equivalent to the quadratic coefficients 2 ml ̌ml 1 + 3𝑒𝛼̌ 𝑎𝑠 −𝛽𝑎𝑠 𝑡𝑠𝜏 satisfying 𝑎 = 3𝑐 and 𝑏 + 4𝑐 = 0. The 𝑝-value for a Wald test of the 𝐴ml (𝑡𝑠𝜏 ; 𝛼̌ 𝑎𝑠 ml ̌ml , 𝛽𝑎𝑠 ) = ml ml = 𝐴𝑠𝜏 , 𝜏 = 1, 2. joint hypothesis is 0.4990; we fail to reject it and conclude that the 3(1 + 𝑒𝛼̌ 𝑎𝑠 −𝛽̌𝑎𝑠 𝑡𝑠𝜏 ) modified logistic dynamic model The calibrated projection models for incidence, intensity and level of ml ml multidimensional poverty are then 1 + 3𝑒𝛼 −𝛽 𝑡 𝑦(𝑡) = , (7) log log log 3(1 + 𝑒𝛼ml −𝛽 ml 𝑡 ) 𝐻̌ 𝑠 (𝑡) =𝐻 log (𝑡; 𝛼̌ ℎ𝑠 , 𝛽̌ℎ𝑠 ), ml ml ml ml is most appropriate for 𝐴(𝑡). In this case, the modified logit- 𝐴̌ 𝑠 (𝑡) =𝐴 (𝑡; 𝛼̌ 𝑎𝑠 , 𝛽̌𝑎𝑠 ), and transformation 𝑦(𝑡) ̃ = ln ((3𝑦(𝑡) − 1)∕3(1 − 𝑦(𝑡))) = 𝛼 ml − 𝛽 ml 𝑡 is linear ̌ 𝑠log (𝑡) =𝐻̌ 𝑠log (𝑡)𝐴̌ ml (𝑡). 𝑀 𝑠 in the parameters. Finally, we turn to dynamic model selection for 𝑀(𝑡), the adjusted Parameter calibration and trajectory projection for all three outcomes headcount ratio or level of multidimensional poverty, reported in Ta- of interest are illustrated for a hypothetical country in Fig. 5. The ble 5. As noted above, 𝑀(𝑡) = 𝐻(𝑡)𝐴(𝑡), so 𝑀(𝑡) ̇ ̇ = 𝐻(𝑡)𝐴(𝑡) ̇ + 𝐻(𝑡)𝐴(𝑡). coloured dots represent the observations of 𝐻, 𝐴 and 𝑀, while the Given our preferred models for 𝐻(𝑡) and 𝐴(𝑡), we thus expect 𝑀(𝑡) ̇ to coloured lines represent the calibrated projection models. log be a polynomial function of 𝐻, 𝐻𝐴, 𝐻 2 𝐴 and 𝐻𝐴2 . Such a function Fig. 6 shows the distribution of the calibrated parameters 𝛽̌ℎ𝑠 and (model 6) explains 54% of the variation in 𝛥𝑀. We may ask whether ̌ml 𝛽𝑎𝑠 , which represent the speed of reduction in poverty incidence and ̇ modelling 𝑀(𝑡) as a function of 𝐻 and 𝐴 performs any better than as a intensity, respectively. Short dashed lines indicate the global median 7 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Table 5 Dynamic model selection for 𝑀(𝑡). (1) (2) (3) (4) (5) (6) 𝛥𝑀 𝛥𝑀 𝛥𝑀 𝛥𝑀 𝛥𝑀 𝛥𝑀 𝑀 −0.0339∗∗∗ −0.0213∗∗∗ −0.0750∗∗∗ −0.1242∗∗∗ [−12.38] [−7.11] [−10.23] [−8.84] 𝑀2 0.1076∗∗∗ 0.3427∗∗∗ [7.71] [4.88] 𝑀3 −0.2711∗∗ [−3.32] 𝐻 −0.2098∗∗∗ [−3.54] 𝐻𝐴 0.5814∗∗ [2.87] 𝐻2𝐴 0.0309 [1.38] 𝐻𝐴2 −0.4643∗∗ [−2.86] Constant −0.0084∗∗∗ −0.0044∗∗∗ −0.0013∗∗∗ −0.0001 0.0000 [−15.06] [−8.05] [−3.52] [−0.19] [0.13] 𝑁 160 160 160 160 160 160 𝑅2 0.000 0.274 0.489 0.523 0.550 adj. 𝑅2 0.000 0.269 0.482 0.514 0.539 AIC −1129.8 −1145.6 −1179.0 −1233.2 −1242.4 −1249.7 BIC −1126.7 −1142.6 −1172.8 −1224.0 −1230.1 −1234.3 ∗ ∗∗ ∗∗∗ Notes: Own calculations, t -statistics in brackets, indicated levels of significance are 𝑝 < 0.05, 𝑝 < 0.01, 𝑝 < 0.001. See A.1 for the list of datasets underlying these results. Fig. 5. Parameter calibration and trajectory projection. Notes: Illustration of parameter calibration and trajectory projections given hypothetical observations 𝐻𝑠1 , 𝐻𝑠2 , 𝐴𝑠1 and 𝐴𝑠2 . (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) for each set of parameters, and long dashed lines represent the 25% find both parameters to be positively correlated in general. In some and 75% quantiles, respectively. Countries are colour-coded by world instances, however, we also observe that poverty intensity can increase region. In terms of poverty incidence reduction, regional top performers while incidence decreases (which may occur when largely the least include Sierra Leone in Sub-Saharan Africa, Honduras in Latin America poor leave poverty). and the Caribbean, and China in East Asia and the Pacific. At the Fig. 7(a) illustrates the Changes over Time data for 𝐻 by translating log global level, we observe a median 𝛽̌ℎ𝑠 of approximately 0.1 and an the time variable for each country 𝑠 by its calibrated logistic dynamic log log log interquartile range of approximately 0.07–0.13, which provides a sense model parameters, 𝑡̃𝑠𝜏 = 𝑡𝑠𝜏 − 𝛼̌ ℎ𝑠 ∕𝛽̌ℎ𝑠 . This lines up each country’s log of magnitude and variation of poverty incidence reduction globally. calibrated trajectory such that the point of inflection occurs at 𝑡̃𝑠𝜏 = 0, In terms of poverty intensity, we find a median of about 0.04 and allowing easier comparison across countries. Each country’s two obser- an interquartile range of 0.02–0.06 at the global level. Moreover, we vations are illustrated (connected by a straight line) but its calibrated 8 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 log Fig. 6. Distribution of 𝛽̌ℎ𝑠 and 𝛽̌𝑎𝑠 ml for logistic dynamic model. log Notes: Short dashed lines indicate the global median for the logit rates of change 𝛽̌ℎ𝑠 and 𝛽̌𝑎𝑠ml . Long dashed lines represent the respective 25% and 75% quantiles. Countries are colour-coded by world region: Arab States; East Asia and the Pacific; Europe and Central Asia; Latin America and the Caribbean; South Asia; Sub-Saharan Africa; only selected countries are labelled for reasons of readability. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) trajectory is not. From the figure, it is clear that many countries are 4. Projection results following similar trajectories, but are at different points along those trajectories. There is some variation, with some countries making faster 4.1. On track analysis progress (following a steeper curve) while others make slower progress (following a shallower curve). In this section we perform an on track analysis in which we explore Moreover, Fig. 7(a) also depicts trajectories for selected quantiles whether countries would meet a particular poverty reduction target of the empirical distribution of our calibrated logit rates of change in a given year of reference under the assumption that recent trends log 𝛽̌ℎ𝑠 , which provides a sense of the implications of the value of 𝛽 for continue. Our poverty reduction target is closely related to SDG 1.2 and the shape of the trajectory. Specifically, the trajectory of the median calls for reducing the proportion of people living in multidimensional 𝛽 (𝛽𝑄50 = 0.098) is depicted as a turquoise line, the trajectory of the poverty at least by half by 2030 compared with the reference value in upper quartile 𝛽 (𝛽𝑄75 = 0.128) is depicted in blue, and the one of the 2015.9 As the target refers to the proportion of people living in poverty, lower quartile 𝛽 (𝛽𝑄25 = 0.065) is depicted in olive-green. that is, its incidence, we focus this section on the headcount ratio 𝐻. The lower charts in Fig. 7 show the implied time profiles of abso- We discuss related findings for the adjusted headcount ratio or level of lute changes (b) and relative changes (c) over the same timeline and multidimensional poverty, 𝑀, as well. for the same quantiles of the observed distribution of betas. Several Fig. 8 presents the results for all 75 countries and contains the observations are salient: First, both graphs neatly expose phases of projected poverty reduction from 2015 to 2030 according the logistic acceleration and slow-down in poverty reduction as captured by the dynamic model (red arrows), the confidence intervals for the 2030 logistic dynamic model. Moreover, the quantile trajectories in graph projection of the logistic dynamic model (grey bars), the projected (c) also illustrate the convergence of the logistic dynamic model to headcount ratio in 2030 according to the exponential or constant an exponential (constant relative change) model as 𝑡 → ∞. Finally, relative change model (purple dot), and the poverty target value in one can also observe how challenging and potentially deceptive cross 2030 (black line). If the red arrow reaches below the black line for country comparisons using simple absolute or relative changes can be. any country, then we can state that it is on track to halve poverty For example, the same absolute change may reflect a point either on – as measured by the headcount ratio of the global MPI – between the accelerating or decelerating side of the trajectory. Furthermore, 2015 and 2030. According to this analysis 51 countries would be on the same absolute change may be associated with entirely different track (shown on the right of the figure), whereas 24 countries are trajectories and, likewise, the same relative change may be associated off track (shown on the left of the figure). Visual inspection reveals with entirely different trajectories. that in many, but not all cases, the exponential model suggests similar While the logistic dynamic model for 𝐻(𝑦), the modified logistic model for 𝐴(𝑦) and their product for 𝑀(𝑦) are our strongly preferred models, for comparison purposes we also implement projections using 9 The actual SDG target 1.2 refers to national definitions. Besides our lack the linear and exponential models for 𝐻. For further details on these of appropriate data and measures for most countries, using national definitions models see Appendix B. would further complicate the cross-country analyses that we offer. 9 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Fig. 7. Trajectory and change profiles of calibrated logistic models for 𝐻. log log log Notes: Years are translated as follows: 𝑡̃𝑠𝜏 = 𝑡𝑠𝜏 − 𝛼̌ ℎ𝑠 ∕𝛽̌ℎ𝑠 . (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) Fig. 8. Projections for incidence of poverty 2015–2030 (if observed trends continue). Notes: Authors’ calculations; target and progress are obtained using the logistic model; confidence intervals refer to logistic model;‡ indicates that estimated change is not significantly different from 0. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) 10 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Table 6 Results summary of on-track analysis (proportion of countries). World # countries Headcount ratio (𝐻) Adjusted headcount region ratio (𝑀) meeting target same outcome projections projections by model both models (logistic) (logistic) log. exp. target target signif. target in different meeting signif. met not met exceeds 95% CI from target exceeds target of proj. exp. target (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) AS 5 .6 .6 .6 .4 .4 .8 0 .6 .4 EAP 8 1 1 1 0 .88 .25 .13 1 .75 ECA 11 .91 .91 .91 .09 .55 .55 0 .91 .55 LAC 12 .75 .67 .67 .25 .5 .58 0 .75 .5 SA 4 .75 .75 .75 .25 .75 .25 0 .75 .75 SSA 35 .51 .31 .31 .49 .34 .34 .51 .57 .43 Total 75 .68 .57 .57 .32 .48 .43 .25 .71 .51 Notes: Authors’ calculation; cells contain proportions of countries and all projections are for 2030; in column (6), (7) and (10) statistical tests and 95-% confidence intervals are computed as detailed in appendix C; in column (8) models are considered different if projections differ by 5%-points or more; world regions are Arab States (AS), East Asia and the Pacific (EAP), Europe and Central Asia (ECA), Latin America and the Caribbean (LAC), South Asia (SA), Sub-Saharan Africa (SSA). outcomes. We note that the exponential and logistic projections diverge (18 countries) in sub-Saharan Africa. The reason for this finding is most substantially for countries with relatively high poverty levels simply that for low levels of poverty, the exponential model may serve and relatively high projected poverty reduction. Additionally, visual as reasonable approximation of the logistic model. For high poverty inspection of the confidence intervals demonstrates the importance of countries, however, the two models project substantially different tra- sampling errors for the conclusions drawn. jectories – crucially, the logistic model anticipates both acceleration These findings are also summarised in Table 6, which shows results and deceleration periods, whereas the exponential model does not. by world region and for the entire set of countries in our analysis; We conclude this section by highlighting the value that the adjusted for country-specific results see Table A.2. More specifically, we find headcount ratio can add to the evaluation of poverty reduction goals. that according to the logistic dynamic model (column 2), 51 countries Like most development targets, SDG 1.2 is formulated in terms of a around the world (68% of our countries) are on track to meet the target headcount ratio, that is, the proportion of people experiencing a cer- to cut poverty by half by 2030. The proportions vary from 51% of the tain condition. Headcount ratios suffer from well-known limitations as countries being on track in sub-Saharan Africa to 100% in East Asia and poverty measures (Sen, 1976), specifically that they obscure improve- the Pacific. As column (3) shows, we find that fewer countries are on ment (or deterioration) of conditions among those who remain poor. track according to the exponential model (57%), which is theoretically In the context of multidimensional poverty measurement, Alkire and expected because this model does not anticipate acceleration and decel- Foster’s (2011) adjusted headcount ratio, discussed in Section 3.1.1, eration phases for high-poverty countries’ trajectories. Countries where addresses this by adjusting for the intensity of poverty among the poor. each model suggests a different conclusion are either located in Latin Our on-track analysis for the adjusted headcount ratio 𝑀 in columns America and the Caribbean or in sub-Saharan Africa. Considering both (9) and (10) of Table 6 suggests very similar findings to those for the models together, columns (4) and (5) demonstrate that for 89% of the headcount ratio (for country-level findings see Fig. A.1). This is not countries both give the same assessment of on track or not, whereas for surprising as, for most countries, the change in intensity of poverty 11% of the countries results are model-dependent, with discrepancies between observations is smaller than the change in incidence. There again only found in Latin America & the Caribbean and sub-Saharan are, however, notable exceptions. For example, we observe that Guinea Africa. and Ghana are on track to significantly exceed the 𝑀-target but not the We now assess the extent to which sampling errors affect conclu- 𝐻-target. Both exhibit significant and substantial intensity reductions, sions about whether a country is on track to meet its target. As the which are among the three largest that we observe in our data. This poverty observations for each country are in fact mean point estimates example demonstrates how intensity reductions play a critical role in from survey data, they are subject to sampling error and thus so are our explaining how countries may meet the 𝑀-target despite missing the calibrated parameter values and projections. Our approach to inference 𝐻-target. Thus, formulating the poverty target in terms of 𝑀 – instead for the projections respects the bounded nature of the variables and of or alongside 𝐻 – can make visible progress among the poor (even is detailed in Appendix C. Column 6 of Table 6 shows that only 36 if they are still poor) and, moreover, entails the possibility to meet the countries (48% of our countries) are projected to achieve a poverty poverty reduction target via reductions in poverty intensity. We show headcount ratio in 2030 significantly below (exceeding) their target here that these reductions can be substantial, especially for some of the (see Appendix C for more detail on this one-tailed test). Relatedly, the globally poorest countries, and that they will not be recognised while share of countries with projected headcount ratios significantly below the focus remains exclusively on poverty incidence reductions. their targets is also smaller in most world regions. In order to assess the importance of sampling errors for such an on track analysis more 4.2. Is cutting poverty by half a reasonable target? generally, Table 6 also reports the share of countries for which the interval estimates include the targets in column (7). For 43% of the In this section we revisit the previous results, taking a more reflec- countries we find the target to be within the 95% confidence interval tive stance. In particular, we address the question of whether our by of their projected value (irrespective of the projection itself being above SDG 1.2-inspired poverty reduction target was feasible for all countries, or below the target). and how far away countries are from meeting or failing to meet the Finally, to provide a sense of the magnitude by which projections target. To do so, we perform two exercises, focusing on the headcount by different models may diverge from one another, column (8) of ratio and the logistic dynamic model to facilitate the presentation. Table 6 reports the proportions of countries for which the projected Our first analysis focuses on those countries that would miss the incidence of multidimensional poverty in 2030 differs by 5 percent- target if recent trends continue. We ask whether, under an improved age points or more between the logistic and the exponential models. performance in poverty reduction, they would manage to meet the While the two models suggest substantially different projections only target. Our second analysis focuses on countries that are expected to for 27% of all countries (20 countries), this share increases to 51% meet targets under recent trends, and we ask whether falling back to 11 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Fig. 9. Projections of the headcount ratio for 2015–2030 for alternative scenarios. Notes: Authors’ calculations; projected 2015–2030 progress is based on logistic model; ‡ indicates that the estimated change for a country is not significantly different from 0. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) log a poorer performance may result in missing the target. These analyses the targets, and to obtain a sense for the magnitude of 𝛽̌ℎ𝑠 . Fig. 10 log provide a sense of (i) what is feasible, in principle, (ii) the sensitivity shows the actual and the needed 𝛽̌ℎ𝑠 to achieve the target in 2030, of the target-achievement outcome to changes in observed pace of log log and the years needed to achieve the target under the actual 𝛽̌ℎ𝑠 . poverty reduction, and (iii) what changes in the magnitude of the 𝛽̌ℎ𝑠 Several interesting insights emerge. First, we observe that a dozen or mean. To choose meaningful counterfactual performances, we draw on so countries, shown on the left of the figure, would need 10 or more the observed distribution of estimated coefficients. This ensures that additional years to reach the target, given their recent performance. counterfactual performances have been observed in other countries and Seven of those countries would, in fact, need some 20 years or more. thus are not entirely unfeasible. Specifically, we identify the median Conversely, 10 countries appear to trivially achieve the targets about log performance using the empirical distribution of 𝛽̌ℎ𝑠 across countries. 10 years earlier, and another 8 countries 9 years earlier. Given that Similarly, we also define ‘poor’ and ‘good’ performance based on the the SDG target period for halving poverty incidence is 15 years, we lower and upper quartiles of this distribution. conclude that setting a uniform ‘cutting poverty by half’ target results Fig. 9 shows the results of this counterfactual analysis for all 75 in unrealistically ambitious targets for many countries and entirely countries. We posit ‘median’ performance (green dots) as a meaningful unambitious targets for many others. We note that both overly ambi- benchmark for countries that are found to be off track according to tious and unambitious targets are problematic. Over-ambitious targets projections obtained from the logistic dynamic model. This typically may be discouraging from the outset and even remarkable progress implies a moderate (i.e. realistic) improvement in their observed perfor- (including substantial accelerations in poverty reduction) may appear mance. For countries that are found to be on track according to logistic as entire failures. Meanwhile, unambitious targets would neither direct model-based projections, it is important to note that, typically, they any attention to the underlying problem, nor induce policy changes and already perform at a level corresponding to ‘median’ performance or are, therefore, essentially ineffective and irrelevant. log better. Thus we consider ‘lower quartile’ performance as an appropriate Turning to the actual and needed 𝛽̌ℎ𝑠 , we observe that among those counterfactual benchmark (yellow dots) for these countries. countries that would need some additional 10 or more years to achieve In Fig. 9 we observe that more than a dozen of the 24 countries the target if recent trends continue, several countries would require a log which are not on track to meet the target, would manage to achieve this 𝛽̌ℎ𝑠 of 0.7–0.8 (e.g., Cameroon, Nigeria or Senegal), which is in fact less log goal if they boost their poverty reduction performance to a ‘median’ than the global median performance (𝛽̌𝑄 = 0.098). This observation 50 level (e.g., Senegal or Pakistan). Since slow trajectories of poverty is important as it suggests that the required empirical performance reduction are not predetermined, the results show that there is a chance is feasible, at least in the mid-run, because it has been observed in to increase the likelihood of halving poverty by 2030 if some of the many other countries. In other cases, however, such as Chad or Niger, log countries shifted gear. However, Fig. 9 also reveals that the challenge the required 𝛽̌ℎ𝑠 suggest that these countries would need to deliver is greater for some of the poorest countries, which are all in Africa, log better than the upper quartile performance (𝛽̌𝑄 = 0.128). While all 75 as these would not halve poverty by 2030, even if recent performance required performances are at least empirically observed, the needed were boosted. performance would, however, make those countries top-performing in Conversely, we also find that several countries among the 51 that poverty reduction by global standards, which appears very unrealistic would be expected to meet the target according to the logistic model- again. Finally, how unrealistic these targets may be is also evidenced based projections if observed trends continue are actually at risk of by the case of Ethiopia, which delivers a relatively good performance log failing to halve poverty if their poverty reduction dynamics are in- within the group of high poverty countries (𝛽̌ℎ𝑠 = 0.081), which is terrupted (e.g., Malawi or Rwanda). These results are as important as also close to global median performance. And yet Ethiopia would fail the previous ones in that they offer compelling evidence of the need to meet the target of halving poverty by some 10 years. of continuity and sustainability of policy efforts in the quest to halve In summary, three main conclusions emerge from this section. First, log poverty by 2030. our results suggest that plausible variation of the 𝛽̌ℎ𝑠 may affect target In our second exercise, we calculate the required performance to achievement for both better or worse: a change of policy in either meet the goal in 2030 for each country and the additional years needed direction can make a difference. Second, for many countries, their to achieve the target under the actually observed performance. This targets appear to be either too ambitious or too unambitious and, analysis provides additional information to assess the feasibility of thereby, cast doubt on a one-size fits all approach to setting poverty 12 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Fig. 10. Actual and required performance to meet targets. Notes: Authors’ calculations; countries with observed increases of poverty omitted (Benin and Serbia); dashed lines indicate upper quartile (𝛽𝑄75 = 0.128), median (𝛽𝑄50 = 0.098), and lower quartile (𝛽𝑄25 = 0.065) performance; countries are sorted by difference between actual and needed 𝛽s. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) log log reduction targets in relative terms. Third, the 𝛽̌ℎ𝑠 offer a convenient observed value of 𝛽̌ℎ𝑠 may vary widely. For instance, we find that way to assess performance in poverty reduction, as they are comparable several countries with relatively high poverty headcount ratios of, say across countries and prevailing levels of poverty. 60%, exhibit a performance level that is outside of the interquartile range on both sides (e.g., Madagascar or Togo on the one side with 5. Discussion Liberia or Rwanda on the other). This means that some of those countries are truly ‘top’- and ‘bottom’ performers by global standards. 5.1. Implications for setting poverty targets In fact, the very same observation also applies for poverty rates below, say 10%, too. Even for countries with very high poverty rates (80%+), The previous section casts doubt on the usefulness of a one-size fits we find performance differences in the same magnitude of the inter- all approach to set poverty reduction targets, such as cutting poverty quartile range (e.g., Chad and Ethiopia), even though all of them by half on the national level. Given this, how can poverty reduction show below-median performance.10 Therefore, for identifying feasible targets be reasonably set? Indeed, many other development targets poverty targets both current levels and recent changes have to be are set in a similar way, and this practice has been criticised from taken into account. Only together do they reflect the initial conditions various directions. More specifically, previous research emphasises that experienced by a country to the extent they are already embodied in in order to obtain feasible targets, initial conditions of a country have the available data. to be taken into account. Initial levels of the indicator are prominently While accounting for initial conditions (including recent trends) is discussed in the literature (Easterly, 2009), as is initial inequality in critical for arriving at a reasonable business-as-usual projection, it is the case of (monetary) poverty targets (Allwine et al., 2015) and also, also important to do so using the appropriate model. The projections less prominently, historical trajectories or recent trends (Ranganathan for 2030 as discussed in Section 4.1 may also be interpreted as such et al., 2017). business-as-usual projections. Recall that according to Table 6, for 17% Besides having different initial conditions among countries, political of the countries around the world logistic and exponential (constant priorities may vary, too. In fact, even the SDG framework states that tar- relative change) dynamic models result in significantly different pro- gets are universally applicable ‘‘taking into account different national jections for 2030. This share even increases to 26% for sub-Saharan realities, capacities and levels of development and respecting national policies and priorities’’ (UN, 2015, p. 3). Consequently, there is no Africa, where most of the highest poverty countries are found. In line purely technical way to derive definitive poverty targets. Nonetheless, with our theoretical discussion, the logistic and exponential models our analysis entails several implications for the process to arrive at imply different projections particularly for countries with higher levels feasible and yet ambitious poverty reduction targets. In this context, of poverty. In contrast, for lower levels of poverty, the exponential business-as-usual projections may serve a useful reference, as they model may well be viewed as a reasonable approximation of the logistic suggest a feasible outcome given the recently observed trends. We begin model. Naturally, whether the choice of projection model makes a real our discussion with two important considerations for these business-as- difference in a particular case depends on several factors, including usual projections, followed by a suggestion about how to arrive at more sample size, the country’s position on its trajectory, and the period over ambitious goals. which the target applies. First, our results suggest that in order to appropriately account Country-specific projections based on the logistic dynamic model for the initial conditions of a country, one has to consider both initial and recently observed trends may be understood as the outcome under levels and recent trends of poverty (or development indicators more continued efforts or business as usual. They can thus be deemed feasible generally)—initial levels alone, which have been emphasised in pre- given the currently available evidence. However, identifying more vious research, are insufficient. ambitious targets may be desirable to mobilise additional efforts and Intuitively, this insight follows from the fact countries may actually follow very different trajectories despite exhibiting a similar headcount ratio at one particular point of time. Fig. 11 provides empirical evi- 10 That we do not observe top-performers for very high levels of poverty dence in support of this idea (cf. also Fig. 7 above). The important follows from the fact such performance results in quickly graduating from such observation is that for every given level of the headcount ratio, the very high incidences. 13 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 log Fig. 11. Headcount ratios and calibrated 𝛽̌ℎ𝑠 . Notes: Authors’ calculations; dashed lines indicates lower quartile, median and upper quartile performance. (For accurate rendering of coloured elements of this figure, the reader is referred to the web version of this article.) resources, calling for policy adjustments. In contrast, both overly am- 5.2. Testing for changes of trajectory bitious and unassuming targets entail non-negligible costs. The question is, therefore, how to identify more ambitious and yet feasible targets. A country’s projected trajectory of poverty reduction, which is One promising way forward is to revert to the observed logit rates summarised by the rate of change parameter 𝛽 of the selected dynamic of change and screen which other countries or sub-national regions log log models (the 𝛽̌ ) is not irrevocably determined. It can change for sev- ℎ𝑠 of the same country were able to achieve targets. Recall that the 𝛽̌ℎ𝑠 eral reasons, including shocks and policy reforms. Therefore, one may describe the entire trajectory of a country, including acceleration and also be interested in establishing whether a change of trajectory has slow down. Therefore, valid comparisons can be made for countries at effectively taken place. We argue that, in general, an appropriate mod- different points in their trajectory; comparisons need not be limited to elling of the outcome variable is critical to test for the existence of such countries with similar poverty levels. For example, our results suggest changes. Let us then explain what, exactly, adopting an exponential that Mozambique and Kenya are following a very similar trajectory (constant relative change) or a logistic dynamic model entails. in poverty reduction, despite having a difference in the headcount For the sake of the argument we consider the case of a simple ratio of around 30 percentage points in their first year of observation. before-after comparison for a single country, for which we have an Similar comparisons of simple absolute or relative changes would be extensive time series of multidimensional poverty observations. If the misleading, since both are expected to vary over the entire trajectory trajectory of an outcome 𝑦𝑡 follows the exponential dynamic model, we for theoretical reasons, as shown and discussed in Section 3.1.3. In log short, our argument is that comparing 𝛽̌ℎ𝑠 across different countries may formulate the econometric model to test for a trajectory change as or subnational regions provides critical evidence for feasibility assess- follows ments. Finally, we note that while arriving at sensible targets requires ln 𝑦𝑡 = 𝛼 crc + 𝛿1crc 𝐷𝑡 + 𝛽 crc 𝑡 + 𝛿2crc 𝐷𝑡 × 𝑡 + 𝑒𝑡 , (8) log a careful analysis and choice of the 𝛽̌ℎ𝑠 , the actual formulation of the target may well be translated into an absolute or relative change, or where 𝐷𝑡 = 0 for all periods before the assumed change and 𝐷𝑡 = 1 for even a level-end target to be achieved in a given year, which is easier all periods after it. to communicate to the interested public. If we were able to reject 𝐻0 ∶ 𝛿2crc = 0, we may conclude that the Finally, in many cases it may be preferable to formulate the target country effectively experienced a change of trajectory. Moreover, in the in terms of the adjusted headcount ratio (𝑀). As discussed in Sec- exponential model it holds that 𝛽 crc = 𝑦𝑦̇ |𝐷=0 and 𝛽 crc + 𝛿 crc = 𝑦𝑦̇ |𝐷=1 . tion 4.1, progress among the poor may go unnoticed due to well-known Therefore, in the exponential model, an equivalent approach to test for limitations of the headcount ratio (𝐻), in particular in countries with changes in the trajectory is to calculate the average relative rates of relatively high poverty levels. A natural follow up question is then how change for both periods (before and after the potential structural break) reasonable poverty targets in terms of the adjusted headcount ratio and subsequently test the null of equality between them: 𝐻0 ∶ 𝑦𝑦̇ |𝐷=0 = may be set. One option is to choose 𝛽̌𝑎𝑠 ml in a similar way to 𝛽̌log , to 𝑡 𝑦̇ ℎ𝑠 | 𝑦𝑡 𝐷=1 . represent realistic ambitions in terms of reductions in poverty intensity as well as incidence, and then derive the respective target value of If the trajectory of an outcome 𝑦𝑡 follows instead the logistic dy- 𝑀 as the product of the implied target values of 𝐻 and 𝐴. Naturally, namic model, we may formulate the econometric model to test for a the resulting 𝑀 target should be carefully formulated and explained trajectory change as follows ( ) in the most communicable way. Alternatively, targeting a particular 𝑦𝑡 log log incidence reduction together with an unchanged intensity may also ln = 𝛼 log + 𝛿1 𝐷𝑡 + 𝛽 log 𝑡 + 𝛿2 𝐷𝑡 × 𝑡 + 𝑒𝑡 . 1 − 𝑦𝑡 be sensible. Indeed, this approach implicitly mandates policymakers to log promote improvements among both the least poor and the poorest of We infer a change of trajectory if we are able to reject 𝐻0 ∶ 𝛿2 = 0. the poor. The reason for this is that if only the least poor were to exit In the logistic model, however, testing for differences in the relative poverty, intensity (which is average deprivation among the poor) would rates of change before and after the potential change of trajectory is increase. no longer an equivalent approach. Specifically, even if the country 14 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 log remains on the exact same trajectory (𝛿2 = 0), we actually expect the improving, it remains hard to quantify all relevant implications for relative rates of change to decrease from a theoretical perspective, as updated trajectories for several reasons. These include the fact that 𝑦̇ 𝑦 = −𝛽 log (1 − 𝑦𝑡 ). Therefore, testing for changes in the average relative reliable and fully comparable data on how the pandemic and related change may be highly misleading if the true modelling framework is policy-responses have affected the lives of the poor remain scarce. A actually a logistic one. first attempt to assess the potential increase of global multidimensional In principle, this conclusion applies to all bounded variables, and poverty due to COVID-19 was conducted by Alkire et al. (2021). That thus some of the most popular development indicators. And yet, the study uses simulation techniques which are informed by assessments exponential model is most often applied in the academic literature (e.g., of UN agencies about food insecurity and school closures. Their results Fukuda-Parr et al., 2013; French, 2015; Jacob, 2017; McArthur and suggest that a decade of progress in poverty reduction might be undone. Rasmussen, 2018; Ahimbisibwe and Ram, 2018), and it implicitly However, the study is not an ex-post evaluation, and it is unclear to underlies assessments by practitioners (e.g., WHO-UNICEF, 2017). The what extent this increase translates into persistent deprivation (policy- extent to which this makes a difference in practice depends on the makers still seek to attenuate the effects). Accordingly, it also remains empirical distribution of the underlying indicator. More specifically, for unclear whether the pandemic and related responses affect only the an indicator such as a poverty headcount ratio, where lower values levels or also the slopes of poverty reduction trajectories around the are better, the exponential model may be considered a reasonable world. approximation of the logistic model only for low levels of poverty. This This paper represents a step forward towards a comprehensive is not true for high indicator levels. analysis of multidimensional poverty trends, and it manages to ac- commodate important current data limitations. However, future re- 6. Concluding remarks search in this direction will require more and better data, which In this paper, we develop a modelling framework for computing pro- would allow the introduction of complementary statistical approaches, jections of global multidimensional poverty at the country level to offer including non-parametric techniques, to assess the role played by co- a more detailed account of the evolution of multidimensional poverty variates of multidimensional poverty for its trajectories. Having several over time. Thereby, we seek to lay the foundation for better methods to rounds of data or subnational data, would permit the estimation of address highly relevant questions in practice, such as whether countries longitudinal models and allow, for instance, accounting for projection are on track to meet their poverty reduction targets, how to set such errors. Future research may also seek to identify and analyse differ- targets in the first place, and how to test for changes in the underlying ent profiles underlying the overall trajectories (e.g., in terms of the trajectory. underlying subindices of incidence and intensity of poverty or the Our empirical analysis of the global MPI for 75 countries suggests underlying indicator trajectories), and to incorporate shock simulations that 51 countries were on track to reduce the proportion people living into projections. in multidimensional poverty by half between 2015 and 2030 (before In his last book, Measuring Poverty Around the World, Atkinson aimed the outbreak of the COVID-19 pandemic). Accounting for sampling to ‘‘provide the evidence about the extent and nature of poverty that error, 36 of those countries were on track to significantly exceed the is necessary to spur action and to design effective policies’’ (Atkinson, target, reducing poverty incidence by significantly more than half. Our 2019). Recognising that exercises related to global poverty may be results also indicate that for projection purposes, choosing a logistic highly controversial, and flawed, but nonetheless useful, this paper has instead of an exponential dynamic model makes a difference—in par- sought to complement efforts to improve the analysis of global poverty, ticular for countries with high levels of poverty, most of which are by developing a modelling framework to project multidimensional located in sub-Saharan Africa. Taking a more reflective stance, our poverty trajectories. Given the relevance and level of complexity as well subsequent analysis questions the appropriateness of a one-size-fits-all as current data limitations, this paper is a first rather than a last word approach to target setting, as it results in both hardly feasible targets on the subject. The aim, through critical exchange, is to strengthen the for some countries and entirely unambitious targets for many others. methods available in ways that might spur both discussion and action. More specifically, we find around a dozen countries that require 10 or more (and sometimes more than 20) additional years to meet the target if they continue their observed trajectories. At the same time, Funding around 20 countries would reach the target about 10 years earlier than the deadline. Our results also suggest that using the headcount ratio This work was supported by the John Fell Oxford University Press to measure progress towards a poverty reduction target may obscure (OUP) Research Fund [VGD00340]; the Economic and Social Research improvements achieved among the poor, in particular for the poorer Council (ESRC), UK and the Department for International Develop- countries. Refined measures such as the adjusted headcount ratio are ment (DfID), UK [ESRC-DFID ES/N01457X/1, DfID project 300706]; one way to overcome this limitation. the Swedish International Development Cooperation Agency (Sida) Furthermore, we argue that business-as-usual projections may play [projects 11141, 14547]; and the Spanish Ministry of Science, Inno- an important role in setting feasible and yet ambitious targets. In this vation and Universities through its Juan de la Cierva Research Grant context, we emphasise that our modelling framework makes possible Program [IJCI-2017-33950 to N.S.]. The authors are grateful to all meaningful international or subnational performance comparisons, as funders for their support. well as comparisons over time, providing additional guidance on the feasibility of more ambitious targets. As the discussed implications es- CRediT authorship contribution statement sentially follow from the bounded nature of our outcome variables, they are also applicable to most other development indicators. In particular, the practice of inferring different trajectories from different relative Sabina Alkire: Conceptualisation, Writing – original draft, Writing change rates turns out to be potentially problematic. Future research – review & editing, Funding acquisition. Ricardo Nogales: Conceptu- should take this into account and revisit related empirical analyses, alisation, Methodology, Software, Formal analysis, Writing – original paying due attention to the double-bounded nature of the underlying draft, Writing – review & editing. Natalie Naïri Quinn: Conceptualisa- outcome variables. tion, Methodology, Software, Formal analysis, Writing – original draft, Projections based on observed recent trends naturally ignore the Writing – review & editing, Visualisation. Nicolai Suppa: Conceptu- manifold effects of shocks, such as the recently unfolding COVID-19 alisation, Methodology, Software, Formal analysis, Writing – original pandemic, for global poverty trends. Even though data availability is draft, Writing – review & editing, Visualisation, Funding acquisition. 15 S. Alkire et al. Journal of Development Economics 165 (2023) 103150 Data availability Alkire, S., Santos, M.E., 2014. Measuring acute poverty in the developing world: Robustness and scope of the multidimensional poverty index. World Dev. 59, 251–274. The data underlying this article can be freely downloaded from Allwine, M., Rigolini, J., López-Calva, L.F., 2015. The unfairness of (poverty) targets. the website of the Oxford Poverty and Human Development Initiative Oxf. Econ. Pap. 68 (2), 379–397. (OPHI) under https://ophi.org.uk/data-tables-do-files-2020-archive/ Atkinson, A.B., 2019. Measuring Poverty Around the World. Princeton University Press. as ‘‘Table 6: Changes over Time’’. 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