2.1 Composite indicators
Composite indicators are becoming more and more critical as determining tools for policy analysis. However, they need careful consideration of their construction (Joint Research Centre-European Commission 2008; Freudenberg 2003; Greco et al. 2017; Seth & McGillivray 2018; Hudrliková 2013; Gatto and Drago 2021). Composite indicators may be examined as single aggregate indexes and produce a unique measure; this measure can then be used to compare different statistical units, which can, indeed, be useful for diverse purposes (Joint Research Centre-European Commission 2008; Freudenberg 2003; Gatto and Drago 2021). Another essential feature of composite indicators is their transparency and comparability (Joint Research Centre-European Commission 2008). Therefore, it is essential that these indicators are presented in the same unit of measure and are not susceptible to any biases. Several institutions, agencies, and researchers from different nations have seen significant improvements in the composite indicators they use in recent years. However, statistics agencies are still developing indicators that are more easily understood by the general population. While having a significant impact on the message to communicate, synthetic indicators increase the possibility of criticism due to confident choices' inherent subjectivity (Greco et al. 2019; Drago 2017, 2018 and 2019; Gatto and Drago 2020b and 2021; Seri et al. 2021).
A significant feature of composite indicators is the ability to aggregate and summarize detailed measurements while still giving a broad picture of a particular (complex) phenomenon (Gatto and Drago 2021). Thus, composite indicators are helpful for a variety of reasons. However, the ability to convey the outputs of synthetic indicators and a particular rating to the general public is intuitively regarded as the most valuable feature of this system (Joint Research Centre-European Commission 2008; Saltelli 2007; Freudenberg 2003).
The article in question is concerned with the construction of a composite indicator of energy poverty via access to electricity (intended as balanced by negative polarity). We use these variables in the construction of the composite indicator as described in Table 1.
These variables contribute to the final electricity access score for the Regulatory Indicators for Sustainable Energy (RISE) index and database (RISE 2021, see Banerjee et al. 2017). The rationale was to following the last RISE database, issued in 2020, as the score is a means of support in the attainment of Sustainable Development Goal 7, which asks for "universal access to clean and modern energy" (see also Global Off-Grid Lighting Association 2015). The RISE is a collection of indicators designed to be used to assess policy and "regulatory frameworks that countries have put in place to support the achievement of Sustainable Development Goal 7".
The data considered in our study are related to 2019, the last available year. Therefore, the initial observations are excluded; all these are characterized by missing observation (N\A). In the end, we obtain a dataset based on 54 developing countries as cross-section statistical units, characterized by eight different indicators. Moreover, the different indicators (obtained by the same source, i.e. RISE 2021) are exhaustive of the phenomenon we are investigating because they are jointly at the same part of the scores of the electricity access in the RISE electricity access pillar. Therefore, one can conclude that both indicators are, in fact, a side of the same phenomenon.
This work aims to construct an innovative composite indicator of electricity access to calculate energy poverty. In this respect, we also innovate the methodology because this measure is not based on a single value but an interval of values instead. The advantage of using an interval is that we explicitly have a unique measure of the composite indicator (the center) summarizing and representing a single value for the statistical units considered. However, we also have got an interval representing the variation between a lower bound and an upper bound (Gioia & Lauro 2005; Moore et al. 2009).
Following Gioia and Lauro (2006), an interval can be considered as:
Tiny intervals, also known as degenerate intervals of the type are equal to real numbers (Gioia & Lauro 2005, 2006; Moore et al. 2009). Classical composite indicators can be seen as degenerate intervals where the relevant problem is the assumptions (weightings, the selection of the indicators to include) on which the construction of the composite indicator is based, which are to a certain extent subjective (Freudenberg 2003; Greco et al. 2019).
It does not help to construct an interval considering two or three different versions of the composite indicator based on different approaches. Here we are exploiting a continuum of different observations belonging to an interval. These types of composite indicators have rapidly growing in literature.
We start from the standardization of the single indicators (Joint Research Centre-European Commission 2008; Freudenberg 2003; Gatto and Drago 2021):
Here identify the variable q considered at a specific point t in time
is the mean for the considered variable and is the standard deviation
In order to construct the composite indicator, we identify all the variables as components. Then, the various indicators have been standardized to achieve the same scale for all indicators in use. When indications are converted to a common scale having a zero mean and a standard deviation of one, this is referred to as standardization or z-scores.
Then, to calculate the final composite indicator, the sorting components are combined by aggregating the different values. In this sense, we compute the composite indicators facing the existing uncertainties on constructing the composite indicators making use of interval data. The advantage of using an interval is that we have not been forced to adopt a unique specification for constructing our composite indicator. However, we can consider different assumptions and specifications (we have, indeed, not forced to either adopt an equal weighting specification). The final result is an interval of the different composite indicators obtained. Following Lauro and Palumbo (2000) and Moore et al. (2009), we can also calculate some parameterizations of our interval, allowing a comparison between the different intervals. In this way:
Where is the upper bound of the interval and the is the lower bound.
The center is a measure that allows summarizing the most plausible value obtained by the interval. Finally, a measure of the variability is the radius, which has the same value considering the two different radii: the radius that departs from the lower bound to the center and the radius that departs from the center and goes to the upper bound.
So the radii are:
A third measure that can be computed can be the interval range , which is simply obtained by summing the two radii or subtracting from the upper bound the lower bound:
Different rankings can be obtained by considering the center of the intervals (equivalent to rank classical composite indicators, which can be considered tiny intervals). However, it is also possible to rank the radii differently and upper and lower bound (see Mballo and Diday 2005; Song et al. 2012).
This procedure allows internalizing the sensitivity analysis, a relevant phase of constructing a classic composite indicator (Saisana et al. 2005; Saltelli et al. 2008). In this respect, the interval data allows measuring the variability induced by using different assumptions on the construction of the composite indicator. In this case, the interval equally considers all different results due to the different assumptions (for instance, weightings), and we can observe an increase of the radii of the interval-based composite indicator.
More explicitly, following Gatto and Drago (2021), we can obtain the different composite indicators, which are helpful to explore the space of the results of the original composite indicator considering the different assumptions (in our case, the weighting schemes) we vary. So we consider a Monte-Carlo simulation in which we can obtain different scenarios for our composite indicator. In this sense, we can make use of different assumptions as different weighting schemes. So following Gatto and Drago (2021, 2020b) and Drago (2017), we get four different phases on the construction of an interval-based composite indicator:
- A uniform distribution on the Monte-Carlo simulation we are performing allows computing every single weight. In this respect, we simulate a new preliminary weight for each component, which is not necessarily equal to the component's final weights. So considering the simulation, the equal weighting is a plausible case but we also cover situations in which some components can be weighted more than others.
- The composite indicator's preliminary weights should be added together to provide a value representing a theoretical total. Therefore, the various candidate weights are added together, and then each one is divided for the sum of the weighting scheme obtained to obtain the final weight for each of the eight components considered. Thus, the result comprehends a single weight for the components of the composite indicators.
- We repeat the procedure 100000 times, and for each simulation, we obtain repeating the procedure above a single (probabilistically different) weighting scheme. Hence, in the end, we obtain 100000 weighting schemes which are the baseline in the construction of the composite indicator based on an interval.
- Finally, we compute the parameters of the composite indicators considering the center, the radii, and the range.
From the diverse centers and radii, we can construct the different rankings. Then, to perform a sensitivity analysis of the different results, we compute a triplex representation based on less intense assumptions (on the triplex representations see Williamson 1989; Drago 2021 on the context of the interval-based composite indicators). In this case, the sensitivity analysis is different from the classic one because we have already considered the different assumptions in our indicator, and now we compare different representations.
A triplex representation of the interval can be defined in this way:where the corresponds to the equal weighting scenario. However, it is essential to note that the radii are not symmetric but can be different. This fact is significant in interpreting the results of the analysis in which the triplex representation is involved.
The triplex representation is used to represent the intervals the quantiles 0.95 and 0.05, excluding the observations lying outside these intervals. In this respect, we are using some more reasonable intervals that exclude some extreme scenarios (namely, some weighting schemes that are particularly favorable for a country or particularly adverse). The center of the interval on the triplex representation is based on the equal weighting scenario. Therefore, it is helpful to compare the results obtained by the center of the interval with the equal weight scenario. Furthermore, the triplex representation can be used as a sensitivity analysis to compare the results obtained with the interval composite indicators with different assumptions. In order to compare these different assumptions, we subtract the triplex representation from the interval. So we have two generic intervals:
In this way, we elicit the relevance of the extreme scenarios. It is noticeable that the higher the difference between the two intervals, the more relevant the possibility for extreme scenarios. In this sense, the results are an interval and represent the relevance of possible extreme scenarios in the interval analysis. So, this means that to analyze policies and evaluate vulnerability (the difference between lower limits) and resilience for each nation, the findings are critical (that is the difference between the upper bounds).
 For the interval data and the statistical techniques in which they can lie, see Gioia and Lauro 2005, for the interval-based composite indicators and the methodologies to construct and analyze them see Drago (2017, 2018, 2019), Drago & Gatto (2018); Drago & Ricciuti (2018); Mazziotta and Pareto (2020); Seri et al. (2021) and also with applications to energy, Gatto and Drago (2020b and 2021).
 About a review of the procedures in the construction of the composite indicators, see Joint Research Centre-European Commission 2008.
 For the algebra of the interval and particularly for the subtraction Lauro and Palumbo (2000) Moore et al. (1979).