1 Introduction

The 2030 Agenda for Sustainable Development is a global plan adopted by the United Nations in 2015. It includes 17 Sustainable Development Goals (SDGs) that cover a wide range of issues whose main objective is the achievement of a sustainable and inclusive development. The 17 SDGs cover three macro areas which are

  • Social area: combating poverty, reducing economic and social inequalities and access to basic education and healthcare.

  • Economic area: promoting innovation and economic growth, creating decent jobs and increasing the resilience of economies.

  • Environmental Area: conserving biodiversity, combating climate change and protecting ecosystems.

The 2030 Agenda is designed to be an universal and global action plan to address the challenges of the 21st century and aims to achieve a series of objectives by 2030. It has also been adopted by all member countries of the United Nations and represents a global commitment to promoting sustainable and fair development for all.

The definition of sustainable and inclusive development is highly debated in literature. The classical definition of sustainable development was first given by the Brundtland Commission in 1987 which says that the sustainable development is “the development that meets the needs of the present without compromising the ability of future generations to meet their own needs” (Brundtland et al., 1987). Inclusive development, on the other hand, refers to the economic growth that ensures fair distribution across society and creates opportunities for all (OECD, 2018). For a detailed description of the topic refer to Kamran et al. (2023).

Therefore, the sustainable and inclusive development aims to create a growth model that is fair when guaranteeing the access to essential resources and opportunities for everyone, while improving the quality of life and preserving natural resources for future generations (Griessler & Littig, 2005).

In literature (Smith, 1937), inequalities have been defined as “differences in the distribution of income and wealth, as well as economic and social opportunities, among different groups within a society”.

Reducing economic and social inequalities is one of the most pressing challenges that countries around the world should address for the promotion of the sustainable and inclusive development.

Inequalities can be based on a variety of factors, including race, ethnicity, gender, sexuality, class, or disability. Inequalities can have a negative impact on people’s well-being, social cohesion and stability of societies. Specifically, economic inequalities refers to differences in wealth and income between individuals or groups. For example, they can be measured through the gap between the highest and lowest incomes within a society (Lanza, 2015). Social inequalities, instead, refers to differences in access to rights, opportunities, and services between individuals or groups. They may be caused by factors such as gender, ethnicity, sexual orientation, or physical ability; for a comprehensive review of social inequalities, see Neckerman (2004).

In brief, economic and social inequalities are a complex and multi-faceted phenomenon which can affect people’s lives, their health, education, work, and overall well-being. They can also have negative consequences on social cohesion and stability of societies and can limit a society’s potential for development, preventing many individuals from reaching their full potential.

According to a quote by Nelson Mandela

There can be no justice when opportunities are not equally distributed.

However, despite efforts to promote equality, such inequalities persist in all societies and represent an impending challenge. Therefore, the issue of reducing economic and social inequalities is an important matter that should be addressed at a global, national and/or local level.

At a local level, various organizations and groups, such as local authorities, non-governmental organizations, local communities, and citizens can promote initiatives and programs to reduce economic and social inequalities.

At a national level, the governments of the world have adopted various initiatives and programs to reduce economic and social inequalities within their regions.

At a global level, the 2030 Agenda for Sustainable Development represents one of the main initiatives to reduce economic and social inequalities.

Here, our aim is to study at a global level the main determinants of the socio-economic inequalities in Italy, highlighting similarities and differences between north, centre and south regions, in order to provide a decision-making tool for policy-makers. Therefore, we briefly describe some of the Sustainable Development Goals on which the 2030 Agenda is based. In this paper, among the various goals of the 2030 Agenda, we focus our attention on Goal 10, namely “reducing inequalities within and among regions”, and on Goals 3, 4 and 8 which refer to the socio-economic determinants of inequality.

Every country in the world is required to contribute to addressing these major challenges towards a sustainable path, by developing its own National Strategy for Sustainable Development.

In Italy, the National Sustainable Development Strategy (NSDS) was presented to the Council of Ministers and officially approved by the Inter-ministerial Committee for Economic Programming in 2017. It was prepared by the Italian Ministry of Environment Land and Sea in consultation with the Ministry of Foreign Affairs and International Cooperation and all line Ministries, including other national authorities (National Statistical Institute—ISTAT, National Institute for Environmental Research—ISPRA, etc.). The NSDS aims to be the strategic framework for guiding the implementation of the 2030 Agenda in Italy and represents the national reference framework for planning and programming new policies. It represents a step forward in equipping Italy with a governance structure for the 2030 Agenda, a tool that will enable the government to promote equitable and sustainable well-being through the definition of new approaches and policies (ATC, 2015).

To study how the different determinants impact economic inequality between the different Italian regions, we considered the data produced by ISTAT regarding the 2030 Agenda. The data consists of simple indicators related to the seventeen goals.

To know what are the most important socio-economic variables/indicators which influence inequality, we propose to consider a suitable dependence model, i.e. Partial Least Squares (PLS) regression. PLS regression is a statistical technique that is used to predict one or more response variable based on one or more predictor variables. It is similar to multiple linear regression, but it is particularly useful in cases where the number of predictor variables is large with respect to the number of observations.

The paper is structured as follows. Section 2 concerns a brief overview of the literature on the measurement of economic inequalities and how they are measured in Goal 10 of the 2030 Agenda. Section 3 involves an analysis of the socio-economic determinants of inequalities, measured in Goals 3, 4 and 8. Section 4 describes the statistical methodology. Section 5 presents the model for evaluating the inequalities and some conclusion remarks are made in Sect. 6.

2 Inequalities in Sustainability

Reducing socio-economic inequalities is a crucial factor for the sustainable and inclusive development. In this section, we give a brief literature review on the main methodologies for measuring and predicting socio-economic inequalities.

To measure inequality, the Gini index (Gini, 1912) is still one of the most widely used indicators in the literature. Despite its limitations in obtaining estimates that vary depending on individual aversion to inequality (Atkinson, 1970), several authors such as Kakwani (1980), Yitzhaki (1983), and Donaldson and Weymark (1983) have developed extended versions of the Gini index. For a complete discussion of the Gini index, see Farris (2010). Apart from Gini’s index, one may consider the Theil index (Theil, 1967), the Gamma index (Lorenz, 1905), the Gini-Simpson index (Simpson, 1949), the Generalized Entropy Index (Shorrocks, 1984, 1994, 1999), and the Oaxaca-Blinder decomposition (Oaxaca, 1973; Blinder, 1973).

To predict inequality many researchers have used a quantile regression model. For example, Piketty and Saez (2003) used quantile regression to study the distribution of income in the United States. Lynch et al. (2004) used quantile regression to study the relationship between income and self-rated health in the United States.

However, quantile regression like any other ordinary least squares model can suffer from multicollinearity which occurs when two or more predictor variables in a model are highly correlated with each other. Therefore, it is important to check for multicollinearity before fitting a regression model, and to use appropriate techniques if it is present.

Here, we consider partial least squares regression which can handle correlated predictor variables appropriately. We study the dependence relationship between the simple indicators of the Goal 10 and the indicators of the socio-economic determinants (which refer to Goals 3, 4 and 8, and are described in Sect. 3), collected on the 20 Italian regions. The response variable, i.e. the Goal 10 “Reduce inequality within and among regions" is composed of 10 targets, each measured by one or more indicators. However, note that we take into consideration only those indicators whose data are available in the most recent year, i.e. 2019. Table 1 shows the considered target and the associated indicator, while the first two rows of Table 2 concern their label and a brief description. For further information on the construction and composition of indicators, see “SDGs 2021 Report: Statistical information for the 2030 Agenda in Italy" published by ISTAT (https://www.istat.it/it/benessere-e-sostenibilit%C3%A0/obiettivi-di-sviluppo-sostenibile/gli-indicatori-istat).

Table 1 Description of Goal 10 and its indicators, data available for 2019
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3 Inequalities Determinants

Socio-economic inequalities can be determined by a large number of variables/indicators, such as income, education, employment, family composition, gender, ethnicity, and access to quality healthcare services (Mackenbach et al., 2002). For example, people with lower incomes are more likely to suffer from chronic diseases such as diabetes and cardiovascular disease due to poor diet, sedentary lifestyle, and lack of access to quality healthcare services (Timmis et al., 2022).

To address socio-economic inequalities, it is crucial to implement strategies that target the reduction of income and education disparities and enhance the availability of high-quality healthcare services for all individuals. This can be accomplished through various measures, such as: (a) elevating minimum wages; (b) investing in education and training; (c) improving access to quality healthcare services; (d) promoting healthy lifestyles. Additionally, it is important to raise public awareness about the importance of reducing health inequalities and to promote community participation in the formulation and implementation of healthcare policies.

Indeed, education is a crucial factor in reducing inequality globally. Over the past few decades there have been advances in increasing access to education worldwide, but significant inequalities persist in access to quality education among social, economic and geographic groups. Educational inequality can have negative long-term consequences on individuals’ and communities’ personal, social and economic development. To reduce these inequalities globally, systemic and targeted interventions are needed at the social, economic and political levels (Schmidt et al., 2015).

The issue of decreasing joblessness among young people is a relatively new topic on the global stage, but it is a crucial problem for many nations around the world. The number of people in the age group 15–24 who are not working, studying or receiving training has become alarmingly high and this has led to the creation of a new term, “NEET” (Not Education Employment Training). This term refers not just to a social category but also to the individuals themselves who fit into that category.

According to the analysis conducted by Caroleo et al. (2020), countries with lower economic development tend to have higher rates of NEET. Furthermore, in countries where the school-to-work transition institutions are more developed, the proportion of NEET individuals is lower.

The International Monetary Fund (IMF) argues that economic inequality can have negative effects on long-term economic growth and financial stability, as well as on social cohesion and on democracy quality.

Three instrumental reasons for pursuing economic policies that engender less income inequality are (Birdsall, 2001)

  • Inequality can inhibit growth and slow poverty reduction.

  • Inequality often undermines the political process: this may lead to an inadequate social contract and may trigger bad economic policies-with ill effects on growth, human development, and poverty reduction.

  • Inequality can undermine civil, social, and political life and inhibit certain forms of collective decision-making.

In this paper, to study the determinants of socio-economic inequalities we emphasize the indicators of three SDGs of the Italian NSDS, in particular

  • Goal 3: “Ensure healthy lives and promote well-being for all at all ages”. It is composed of 10 targets and 14 indicators which aim to measure progress towards the Goal of ensuring healthy lives and promoting well-being for all. In particular, it aims to ensure that everywhere people can enjoy good health and physical and mental well-being. It is expected to be achieved through a range of actions, expanding healthcare systems and promoting healthy lifestyles, such as regular physical activity and healthy diet. It is also expected to face emerging challenges such as the obesity epidemic and non-communicable diseases, and to protect people from epidemics and pandemics;

  • Goal 4: “Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all”. It is composed of 8 targets and 12 indicators which aim to measure progress towards the goal of ensuring quality education for all. It aims to ensure that everyone, regardless of their background or circumstances, can access to high-quality education and lifelong learning opportunities. So, this means promoting literacy, at primary and secondary schooling, as well as at tertiary education and vocational training. It also aims to promote gender equality and reduce the education gap between countries and within countries. Achieving this goal will contribute to creating more inclusive and sustainable societies, as well as promoting economic growth and sustainable development;

  • Goal 8: “Promote sustained, inclusive and sustainable economic growth, full and productive employment and decent work for all, and an enhanced productive capacity for least developed regions”. It is composed of 8 targets and 8 indicators which aim to measure progress towards the goal of promoting inclusive and sustainable economic growth. This goal aims to promote inclusive and sustainable economic growth that creates decent and dignified employment opportunities for all. It is expected to be achieved by supporting entrepreneurship and innovation, strengthening entrepreneurial capacities, improving access to credit and infrastructure, and by promoting fair and sustainable international trade. Achieving this goal will contribute to create a stronger and more stable global economy, reducing poverty and inequality, and promoting sustainable development.

The selected indicators of Goals 3, 4 and 8 of the 2030 Agenda, available for the year 2019, are listed in Table 2 of Sect. 5.

4 Methodology

Ordinary least squares (OLS) model allows explaining one or more quantitative response variables in terms of predictors. The model is widely used in many fields, however, it can be affected by the multicollinearity problem, which occurs when two or more of the predictors are highly correlated. This can lead to unreliable and unstable estimates of the regression coefficients, as well as large standard errors on these estimates. It can also make hard to determine the unique effect of each predictor on the response variable (Camminatiello et al., 2017).

In presence of multicollinearity, the stepwise selection of the predictors could be performed. However, the stepwise procedure will tend to choose the predictors that best match the data sample, and thus to favor variables that, by the luck of the draw of the sample, happen to have high-magnitude coefficients for that sample but not necessarily for the underlying population. Therefore, Ridge regression estimator (Hoerl & Kennard, 1970) and its modifications, continuum regression (Stone & Brooks, 1990), least absolute shrinkage regression (Tibshirani, 1996), partial least squares (PLS) regression (Wold, 1966, 1975, 1985, 1978), principal components regression and latent root regression (Webster et al., 1974) have been proposed as alternative tools for facing multicollinearity.

Among these methods we consider PLS for its mathematical and statistical properties Tenenhaus (1998); Frank et al. (1993).

4.1 PLS Regression

PLS regression has been originally developed by Wold (1966) and is also known as Nonlinear Iterative Partial Least Squares (NIPALS). There are several different variants of the initial PLS algorithm (Wold, 1966, 1975), including one called SIMPLS (De Jong, 1993).

PLS regression has several advantages over traditional OLS regression (Durand, 2001), indeed it can handle

  • a large number, p, of predictor variables and a small number, n, of observations;

  • correlated predictor variables;

  • missing data;

  • both continuous and categorical response variables;

  • both continuous and categorical predictor variables.

In PLS regression, the columns of the matrices \(\textbf{X}\) and \(\textbf{Y}\) representing the p independent and q dependent variables, respectively, are centered and scaled to have zero mean and unit length. The first step in the PLS procedure involves carrying out uncorrelated latent variables. These latent variables, namely PLS components, are linear combinations of the original independent variables, which maximize the covariance between the independent and dependent variables (Helland, 1988).

After creating the latent variables, a least squares regression is performed using a subset of the extracted variables. The PLS regression model can be written as

$$\begin{aligned} \textbf{Y}=\textbf{T}\mathbf {\Gamma }+\textbf{F} \end{aligned}$$
(1)

where \(\textbf{T}\) is the \({n \times a}\) matrix of the PLS components, \(\mathbf {\Gamma }\) is the \(a\times q\) matrix of the PLS regression coefficients, and \(\textbf{F}\) is the \(n\times q\) matrix of the disturbances.

PLS regression leads to biased but lower variance estimates of the regression coefficients compared to OLS regression. Goutis (1996) showed that the estimates of PLS coefficients are always shrunken compared to OLS ones, that is

$$\begin{aligned} \Vert \hat{\mathbf{\Gamma }}_1\Vert \le \Vert \hat{\mathbf{\Gamma }}_2\Vert \le \dots \le \Vert \hat{\mathbf{\Gamma }}_a\Vert \le \dots \Vert \hat{\textbf{B}}\Vert \end{aligned}$$
(2)

where \(\hat{\mathbf{\Gamma }}_a\) is the PLS estimator obtained after the extraction of the a components and \(\hat{\textbf{B}}\) is the OLS matrix of coefficient estimators.

For assessing the goodness of fitting of the PLS model, the determination coefficient \(R^2\) is commonly used, while for measuring the predictive ability of the model, among different criteria, we consider the PRESS (PRediction Error Sum of Squares) statistic.

4.2 Model Calibration and Validation

The number a of latent variables to be retained in the model can be selected according to different criteria. The most common ones have been based upon test set and Cross Validation (CV) or its generalization (Lombardo et al., 2009). By using a test set, the criterion is given by the Residual Sum of Squares (RSS) related to each PLS component. Although this approach is fast, as we run the PLS algorithm once for each component, it is not often considered, because a test set, independent from the training set, is only exceptionally available. Differently, when using cross-validation (Wold, 1978) the criterion being used is the PRESS which evaluates the accuracy and predictive power of a model. The cross-validation involves dividing a dataset into multiple subsets, say l, where each subset (or fold) is used as both a training set and a validation set. To determine the optimal number of latent variables using PRESS, start with a minimum number of latent variables, such as 1. Use cross-validation (e.g., l-fold cross-validation) to fit the model with the chosen number of latent variables. Calculate the PRESS for each omitted subset, and then restore the omitted subset. The procedure is repeated for the number h (for \(h=1,2,...a\)) of latent variables until each subset has been left out once. The l individual PRESS’s are summed up to give the \(PRESS_k\) of each response variable. Finally, the \(PRESS_{total}\) is calculated by summing up the \(PRESS_k\), for \(k=1,..,q\). The number a of latent variables that corresponds to the lowest \(PRESS_{total}\) represents the optimal choice in terms of minimizing prediction errors.

Here we perform a leave-one-out-cross validation, therefore \(l=n\), and for each h the \(PRESS_{k}\) related to the response k can be computed as

$$\begin{aligned} PRESS_k=\sum _{i=1}^{n}(y_{ik}-\hat{y}_{-ik(h)})^2 \end{aligned}$$
(3)

where the predicted values \(\hat{y}_{-ik(h)}\) are based on the PLS estimates (with h components) obtained by removing the ith subset.

4.3 Bootstrap Confidence Interval

Bootstrapping is a useful method for estimating the sampling distribution of the coefficients in partial least squares regression model when the assumption of normality is not met or when the sample size is small.

In partial least squares regression, a sample of predictor and response variables is used to fit the model. The model is then used to predict the response for each predictor in the sample, and the prediction errors are used to estimate the coefficients of the model. The bootstrap procedure involves sampling with replacement from the original sample and fitting the PLS model to the resampled data (Tenenhaus & De Jong, 2003; De Jong & Wieringa, 2002). This is repeated many times to create a distribution of estimates for the coefficients, which can be used to compute standard errors and confidence intervals (Magnanensi et al., 2017; Knecht & Hautle, 2013; Naes & Naes, 2005).

In the programming environment R, there are numerous packages and functions (for example, see www.jf-durand-pls.com) that perform PLS. We consider the bootpls and pls packages to perform bootstrapped partial least squares regression.

For constructing confidence intervals from bootstrapped samples, the percentile method has been used. It involves computing the percentiles of the bootstrapped samples that correspond to the desired confidence level. For example, to compute a 95% confidence interval, the 2.5th and 97.5th percentiles of the bootstrapped samples are taken as the lower and upper limits of the interval.

Table 2 The SDGs indicators concerning the Goals 10, 3, 5 and 8 of the 2030 Agenda
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5 Evaluating Inequalities in Sustainability through PLS Regression

The 2030 Agenda for Sustainable Development, adopted by all United Nations Member States in 2015, provides a shared blueprint for peace and prosperity for people and the planet, now and for the future. At its heart are the 17 Sustainable Development Goals (SDGs), which represent an urgent call for action by all countries—developed and developing—in a global partnership. They recognize that ending poverty and other deprivations must go hand-in-hand with strategies that improve health and education, reduce inequality, and spur economic growth—all while tackling climate change and working to preserve our oceans and forests (https://sdgs.un.org/Goals).

As highlighted in Sect. 3, here we aim to evaluate the most important determinants of inequalities across the Italian regions, looking at the ISTAT indicators of the SDGs (https://www.istat.it/it/benessere-e-sostenibilit%C3%A0/obiettivi-di-sviluppo-sostenibile/gli-indicatori-istat). For the sake of data explanation, we consider the variables listed in Table 2 related to Goals 10, 3, 5 and 8 of the 2030 Agenda. The variables/indicators of Table 2 have been collected on the 20 Italian regions, i.e. Piemonte, Valle d’Aosta, Liguria, Lombardia, Trentino-Alto Adige, Veneto, Friuli-Venezia Giulia, Emilia-Romagna, Toscana, Umbria, Marche, Lazio, Abruzzo, Molise, Campania, Puglia, Basilicata, Calabria, Sicilia, Sardegna. Some of them are simple indicators and others are composite indicators (Lafortune et al., 2018; Alaimo et al., 2021a; Alaimo & Maggino, 2020; OECD, 2008). For each composite indicator, we checked the reliability using the Cronbach’s Alpha and they all were consistent, showing an alpha index greater than 0.7 (Table 3). By definition, Cronbach’s alpha is used to evaluate internal consistency for indicators measured by more than one variable. Therefore, no Cronbach’s alpha was computed for the indicators HealthWellB1, QualityEdu2, QualityEdu3, and QualityEdu4 as they are single-item indicators. These simple indicators have been normalized.

Table 3 Cronbach’s Alpha for composite Indicators
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5.1 The PLS Model

A very desirable condition in a set of regression data is that there is no multicollinearity among the predictors included in the model. For this reason we calculate the condition index (CI) that allows us to check the level of collinearity. The condition index is given by

$$CI = \sqrt{\lambda _{max}/\lambda _{min}}$$

where \(\lambda _{max}\) and \(\lambda _{min}\) are the maximum and minimum eigenvalues, respectively, of the correlation matrix among predictors.

For our predictors listed in Table 2, the CI is equal to 79.36, which indicates a very strong level of collinearity (Lucadamo et al., 2021). Therefore the use of the PLS regression is surely more suitable for getting a reliable parameter estimation.

We consider the multivariate PLS model where the responses are the two simple indicators of Goal 10 listed in the first two rows of Table 2 (see also Table 1), related to Income and Poverty condition. The correlation between them is negative and very high equal to \(-0.89\).

The choice of the suitable number of PLS components has been made by using the PRESS criterion (Sect. 4.2). It results that the predictive ability of the model is excellent for each dependent variable, we get that \(PRESS_{total}=0.22\), and in detail \(PRESS_{Income}=0.16\) and \(PRESS_{Poverty} =0.06\), see also Fig. 1. After determining the suitable number of PLS components, we perform the PLS model with three components which explain 65.23% of \(\textbf{X}\) variability and the 94.21% of \(\textbf{Y}\) variability. Table 4 shows the percentage of Y variability with respect to each model dimension. Furthermore, it results that model fits each response very well, i.e. \(R^2_{Income}=0.91\) and \(R^2_{Poverty}=0.97\).

Table 4 PLS regression performance
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Fig. 1
figure 1

The PRESS criterion for the choice of the number of PLS components

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When performing a PLS regression, it is possible to visualize the relationship between the responses, the predictors and the latent variables by using a graph called the PLS loading plot which is a correlation circle and allows to study the correlation among original and latent variables. Also, to portray the relationship between the observations and the PLS components one can consider the PLS score plot. Being the percentage of explained variance by the first two components very high, equal to 92.28 (see Table 4), we consider only two components for visualizing the results in Fig. 2. The left side of Fig. 2 shows the PLS loading plot on the plane \(t_1,t_2\). While the right side of Fig. 2 shows the PLS score plot. Looking at the PLS loading plot of Fig. 2, we can observe a high positive correlation among:

  • the response Poverty (risk of poverty) and the predictors QualityEdu1 (inadequate literacy, numerical and English skills), WorkGrowth2 (unemployment/employment rate) and WorkGrowth3 (young people who do not work and not study-NEET).

  • the predictors QualityEdu2 (early exit from the education and training system) QualityEdu4 (rate of participation in educational activities for 5-year-olds), and HealthWellB1 (under-5 mortality rate);

  • the response Income (gross disposable income per capita) and the predictors QualityEdu3 (places authorized in socio-educational services per 100 children aged 0–2) and QualityEdu6 (digital skills);

  • the predictors QualityEdu7 (graduates and other tertiary qualifications) and HealthWellB4 (dentists, pharmacists, nurses and midwives, doctors).

Moreover the correlation circle of Fig. 2 shows a high negative correlation between the two responses—Income (gross disposable income per capita) and Poverty (risk of poverty). Note that the response Poverty is negatively correlated with QualityEdu3 (places authorized in socio-educational services per 100 children aged 0–2) and QualityEdu6 (digital skills), while the response Income is inversely related to QualityEdu1 (inadequate literacy, numerical and English skills) WorkGrowth2 (unemployment/employment rate), and WorkGrowth3 (young people who do not work and do not study-NEET).

Furthermore, QualityEdu2 (early exit from the education and training system), QualityEdu4 (rate of participation in educational activities for 5-year-olds) and HealthWellB1 (under-5 mortality rate) are negatively correlated with QualityEdu7 (graduates and other tertiary qualifications) and HealthWellB4 (dentists, pharmacists, nurses and midwives, doctors) (Fig. 2).

Looking at the observation or score plot of the right side of the Fig.  2, we can observe that there are many differences across regions with respect to the response variables, i.e. Income and Poverty. In particular, we can see that

  • Campania and Basilicata are characterised by Poverty (risk of poverty), mainly due to QualityEdu1 (inadequate literacy, numerical and English skills), to WorkGrowth2 (unemployment/employment rate), and to WorkGrowth3 (young people who do not work and do not study- NEET);

  • Puglia, Calabria, Sicilia and Sardegna are characterised by QualityEdu2 (early exit from the education and training system), by QualityEdu4 (the high rate of participation in educational activities for 5-year-olds), and by HealthWellB1 (under-5 mortality rate);

  • Emilia-Romagna, Friuli-Venezia Giulia, Lombardia, Piemonte, Trentino-Alto Adige, Veneto and Valle d’Aosta present high values of Income, QualityEdu3 (number of places authorized in socio-educational services per 100 children aged 0–2), and of QualityEdu6 (good digital skills);

  • Lazio, Liguria, Marche, Umbria, Molise, Abruzzo and Toscana are principally characterised by QualityEdu7 (graduates and other tertiary qualifications) and by HealthWellB4 (dentists, pharmacists, nurses and midwives, doctors).

Fig. 2
figure 2

The PLS plots on the components \(t_1,t_2\). On the left side, the PLS loading plot. On the right side, the PLS score plot

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Fig. 3
figure 3

The PLS coefficients for the Income response

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To assess the relevance of these predictors we look at Table 5, Figs. 3 and 4 that show the corresponding PLS coefficients. They are computed from three extracted components. As they relate to the centered and scaled data, they express the strength of the relationship between the two dependent variables and all the significant variables in the model. It results that the most important predictor of Income is QualityEdu6 (digital skills) which is related positively (i.e. the higher QualityEdu6 is, the higher the Income is). The second, third and fourth predictor in order of importance are QualityEdu1 (inadequate skills), WorkGrowth3 (young people who do not work and do not study- NEET) and WorkGrowth2 (unemployment/employment rate), respectively, but the estimates of their regression coefficients are negative indicating an inverse relationship with the response (i.e. the higher the indicators QualityEdu1, WorkGrowth3 and WorkGrowth2 are, the lower the Income is). It follows QualityEdu3 (places authorized in socio-educational services) that has a positive effect on Income.

After bootstrapping the model, it results that all these predictors are statistically significant within a 95% confidence interval. In Table 6, the coefficient values of the significant predictors are highlighted in bold, while the coefficient values which do not result statistically significant are highlighted in italics.

Looking at the most important and significant predictors, broadly speaking we can conclude that education and employment play a very important role in increasing income across the Italian regions.

Fig. 4
figure 4

The PLS coefficients for the Poverty response

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Differently, the two most important predictors of the response Poverty are QualityEdu3 (places authorized in socio-educational services) and QualityEdu6 (digital skills) which are related negatively (i.e. the higher the indicators QualityEdu3 and QualityEdu6 are, the lower the Poverty is). The next five most important predictors of Poverty are HealthWellB2 (excess weight, healthy life expectancy at birth), WorkGrowth3 (young people who do not work and do not study-NEET), QualityEdu1 (inadequate literacy, numerical and English skills), WorkGrowth2 (unemployment/employment rate) and QualityEdu2 (early exit from the education and training system). All these predictors are statistically significant within a 95% confidence interval. Indeed, Table 6 shows the quantile confidence intervals of the PLS regression coefficients from bootstrapped samples of the responses and predictor variables It is evident that for the two responses of Goal 10 the statistically significant predictors are not the same, however in common between them there are some indicators belonging to Goal 4, i.e. QualityEdu1 (inadequate skills), QualityEdu3 (places authorized in socio-educational services), QualityEdu6 (digital skills), and some belonging to Goal 8, i.e. WorkGrowth2 (unemployment/employment rate) and WorkGrowth3 (young people who do not work and do not study-NEET). These predictors principally characterise two Italian south regions for high values, i.e. Basilicata and Campania and for low values Friuli-Venenzia-Giulia, Toscana, Marche, Lazio, Liguria and Umbria. Note that only for the response Poverty the predictor HealthWellB2 (excess weight, healthy life expectancy at birth) related to Goal 3, results statistically significant. This predictor variable well describes Basilicata and Campania regions.

Table 5 Estimates of the PLS regression coefficients for the two responses
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Table 6 Confidence intervals of the PLS regression coefficients for the two responses
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6 Conclusion

Socio-economic inequalities are present in various levels in all European countries and available data suggest that the income and health gap is increasing. Many studies have been conducted to explain socio-economic inequalities and much has been learned about the various factors that underlie them.

In this paper, we have investigated inequalities across Italian regions modeling the dependence relationship through the multivariate Partial Least Squares regression model which can be suitable to analyse a data set of simple and composite indicators (predictors highly correlated). As in literature, a great debate concerns the construction of such composite indicators (Fattore, 2017; Alaimo et al., 2021a, b), a comprehensive discussion of the variants of aggregation of simple indicators and their effects when modeling shall be left for future consideration.

Here the inequality has been described by the two simple indicators related to Income and Poverty (responses of the multivariate model) and different determinants have been considered to explain and predict such indicators. Doing so, it results very relevant to improve literacy, numerical and English skills (QualityEdu1), to increase the places authorized in socio-educational services for children aged 0–2 (QualityEdu3), to improve digital skills (QualityEdu6), to increase the employment rate of young people (WorkGrowth2) and decrease the number of young people that do not study and do not work (WorkGrowth3), specially for Campania, Basilicata, Puglia, Calabria, Sicilia and Sardegna.

In general, knowing how different (or not) the inequality is across Italian regions can help policy-makers in deciding what could be a suitable reform to introduce in each region of Italy. Such reforms should require focusing on the reorganisation of the education system or on social operators and their incentives.

Also this study highlights that Italy can be divided into three distinct macroregions, each exhibiting distinctive characteristics. Northern Italy and some regions of the Centre show higher values than the national average across many indicators of equality, including income, employment, educational opportunities, and health. Some central regions have indicators that fall around the national average. While most of the southern regions experience living conditions significantly below the national average, with limited employment and educational opportunities.

The regional divide is a phenomenon that joins several other European countries besides Italy (like Sweden, Finland, Romania, and Estonia).

The severity of regional disparities in Italy, particularly about the labor market, necessitates the implementation of territorial development policies which focus on providing opportunities to people residing in the southern regions and leveraging untapped potential (Fina et al., 2023). In this study, we propose a reorientation of regional policies to achieve the following objectives:

  • Increase public investment in health and education to stimulate short-term economic activity and enhance the potential for long-term economic growth.

  • Enhance employment support measures to alleviate the challenges faced by individuals in the southern regions. It is essential to emphasize that regional equality not only contributes to social and political cohesion but also serves as a means to face social inequalities and foster sustainable and inclusive economic development.