2.1.1 Data sources.

The data source for this study is the Global Database on Intergenerational Mobility (GDIM), which provides information on cohorts born in the 1940s, 1950s, 1960s, 1970s, and 1980s [33]. In the GDIM, each observation represents a distinct combination of code- cohort-parent-child. The variable code denotes the country code. The variable cohort designates the generation born within a specified decade. The variable parent refers to the parents of individuals in this cohort and is disaggregated by the type of parental educational attainment (Mothers/Fathers/Average/Max). The variable child pertains to the educational attainment of the child and is disaggregated by the type of child’s educational attainment (Sons/Daughters/All). Hence, the GDIM has 12 estimates by each country and cohort (by type of parent and by type of child) or 12 units of analysis for each survey (uniquely identified by code-cohort combinations). The study includes a total of 153 countries classified according to their economic development, fragility, and region (see S1 Appendix for data summary). Of the 153 countries, 115 are considered developing economies, and they are distributed across various regions such as South Asia, Sub-Saharan Africa, Latin America and the Caribbean, Europe and Central Asia, and East Asia and the Pacific. Fragility classification (or World Bank Fragile and Conflict-affected Situations) is used to indicate the level of vulnerability and instability of a country, and it is assessed based on factors such as conflict, political instability, and weak governance. The study also provides information on the region in which each country is located, such as the Middle East and North Africa, Sub-Saharan Africa, Europe and Central Asia, South Asia, and East Asia and the Pacific. This comprehensive classification of countries allows for a nuanced analysis of intergenerational mobility across different types of countries and regions, providing valuable insights into the factors that affect social mobility in different contexts.

2.1.2 Variable definitions.

Upward mobility, the main concept of this study, refers to the degree of absolute mobility in the GDIM. This indicator shows the probability that a child achieves a higher level of education than their parent. Education is a suitable proxy for upward mobility because people value equal opportunities (e.g., access to education) more than equal outcomes (e.g., income or wealth) [34]. Additionally, income is a problematic variable to compare across countries and generations due to different standards of living and inflation. Education, by contrast, is a more stable and comparable measure, and a crucial predictor of income [35].

Education expansion, as referenced in the study of Deirdre Bloome et al. [13], represents the difference between the mean years of schooling of children and the mean years of schooling of parents. It signifies the magnitude of educational expansion within a specific population. Higher values of education expansion indicate a greater increase in educational attainment among younger generations, potentially contributing to upward mobility. It should be noted that the expansion of education can potentially elevate the educational levels of a nation as a whole, but it does not necessarily imply a transformation in the internal social structure of that nation.

Education inequality, as referenced in the study by Blanden and Macmillan [36], is quantified by the disparity between the standard deviation of educational years among children and that among parents. This metric encapsulates the variations in educational achievement within a given population. A larger value of education inequality implies a more significant divergence in educational outcomes, which can potentially impact the opportunities for upward mobility among individuals from diverse socio-economic backgrounds. It is worth noting that within the analytical framework in Fig 1, education inequality could conceivably supplant the role of social inequality as education serves as a key indicator for computing intergenerational mobility measures in this study.

Parental dependency, as referenced in the study of Deirdre Bloome et al. [13], on the other hand, is captured by the correlation coefficient between children’s years of schooling and their parents’ years of schooling, as defined by GDIM. The use of the correlation coefficient allows for an accurate measurement of parental dependency, especially in the context of educational expansion which refers to the general trend of improving education in subsequent generations. A higher correlation coefficient, therefore, suggests a stronger association between parental education and children’s educational attainment. This is a more accurate interpretation of parental dependency than maintaining the same level of education across generations, which may not be sufficient in the presence of educational expansion.

For the gender dimension, this research uses the child variable in GDIM which represents the gender of the child whose intergenerational mobility is being measured. It includes three categories: ‘sons’ representing sons’ intergenerational mobility, ‘daughters’ representing daughters’ intergenerational mobility, and ‘all’ representing the intergenerational mobility of both sons and daughters combined. To assess gender inequality, this variable is transformed into a binary value, with 1 assigned if the parent category is ‘daughters’ and 0 assigned for other categories. To obtain more nuanced results, the parent variable in GDIM is used to categorize the type of parent’s education for examining mobility. This variable is transformed into a binary value, with 1 assigned if the parent category is ‘mothers’ and 0 assigned for other categories.

To account for the indirect impacts of gender inequality via contextual factors, I have included three variables in my analysis: ‘fragile,’ ‘developing,’ and ‘region.’ In the context of the GDIM, the variable ‘fragile’ assesses a country’s fragility or instability based on the World Bank’s classification of countries deemed fragile or conflict-affected. The variable ‘developing’ categorizes countries into two groups based on their gross national income per capita: developing economies, and high-income economies. The variable ‘region’ captures the geographic, political, cultural, and social distinctions among countries. It refers to the seven regions classified by the World Bank, namely East Asia and Pacific, Europe and Central Asia, Latin America and Caribbean, Middle East and North Africa, North America, South Asia, and Sub-Saharan Africa. By incorporating these factor variables, the analysis can capture the multidimensional aspects of gender inequality and its relationship to intergenerational mobility, considering factors such as a country’s fragility, economic development, and regional characteristics.

To assess the significance of these contextual variables, I conducted a preliminary analysis by generating violin charts (see Fig 2), which combine a box plot and a kernel density plot to visually represent data distributions and identify data peaks. These charts allow for a comparison of means across different levels of the variables, and an analysis of variance (ANOVA) test was performed. The null hypothesis for the ANOVA test assumes that all population means are equal, while the alternative hypothesis suggests that at least one population mean is different from the others. A p-value below the chosen significance level (typically p<0.05) indicates that there is significant evidence to reject the null hypothesis, indicating differences in mean values among groups. The results of the analysis indicate that there are significant differences in mobility levels between developing countries and high-income economies, with developing countries exhibiting lower mobility. Additionally, fragile countries show significantly lower mobility compared to non-fragile countries. Moreover, the analysis reveals variations in mobility levels across different regions. Based on these findings, it is evident that these variables demonstrate significant differences in relation to intergenerational mobility. As a result, they meet the criteria for inclusion in the subsequent modelling process.

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Fig 2. Upward mobility around the world.

Notes: The data for this estimation comes from the pre-modelled dataset (see S2 Appendix). The method used for comparing means is ANOVA and p is its p-value.

https://doi.org/10.1371/journal.pone.0302173.g002

2.2.1 Data pre-processing.

To compare effect sizes, the dataset (see S2 Appendix) was standardized using log-transformation. However, some numeric variables in the pre-modelled data contained negative values, making log transformation impossible. To address this issue, an offset was applied to these negative values by taking the absolute value of the minimum value in the entire dataset and adding 1. It should be noted that traditional standardization (Z-score normalization), which scales variables to have zero mean and unit variance, did not work for this study because it created a very high coefficient of variation due to the small mean value of the original data. Categorical variables were dummy encoded. In dummy coding, for N categories in a variable, N-1 binary variables are created. Each binary variable represents one category and takes the value 1 if the observation belongs to that category and 0 otherwise. The category that is not represented by a binary variable is called the reference category. In this study, the reference category for ‘cohort’ (which decade individuals were born in) was 1940 and for ‘region’ it was East Asia & Pacific. The choice of the reference category was set to default as the first category. Before standardization, ‘mobility’ was country-centred by subtracting the country mean from each observation within that country. This technique effectively removes fixed effects (or country-specific effects). The final dataset for modelling is summarized in S3 Appendix.

2.2.2 Empirical strategy.

It is evident that the analysis framework reveals potential confounding issues that may influence the outcome of interest, are related to the investigated factors and may introduce bias, thus making it difficult to determine the true relationship between the variables under consideration. Traditional methods of panel data analysis are inadequate to fully address this problem. To overcome this challenge, I employed Causal Machine Learning methods [37] (see Fig 3 for more details). Consequently, the estimation model with θ₀ being the target effect takes the form: where Y represents the outcome variable, or ‘mobility,’ D denotes the treatment variable, or target variable (with estimates for each of the following: education expansion, education inequality, and parental dependency), X signifies the confounding variables which affect both Y and D, while ϵ and ρ signify the error terms. This modelling framework captures the causal relationship between D, X, and Y, recognizing that X exerts influence on both Y and D. Within the context of this study, it is essential to acknowledge the intricate interconnections among significant factors, such as educational expansion, educational inequality, and parental dependency, as they collectively shape mobility outcomes [38]. When the treatment variable (D) is set to represent education inequality, for example, the inclusion of confounding variables (X), which consist of variables like educational expansion and parental dependency, leads to a multicollinearity problem. The coefficient estimates of the model can fluctuate significantly based on which other predictor variables are included in the model. The precision of the coefficient estimates is reduced, which makes the p-values unreliable [39]. Thus, the presence of multicollinearity poses challenges in disentangling the individual effects of each factor on intergenerational mobility. To address this estimation predicament, orthogonalization techniques are employed to eliminate the influence of control variables on both the outcome and treatment variables. Specifically, two orthogonalization techniques are utilized: Partialling-out Lasso [40] and Partial Linear Regression (PLR) [32]. The former assumes linear associations X↦Y, D (or l₀(X) = Xβ₀ and m₀(X) = Xπ₀), with the imposition of Rigorous Lasso effects, while the latter encompasses more complex, unseen relationships X↦Y, D, employing Double Machine Learning (DML). The concurrent implementation of these orthogonalization techniques serves as a robustness check, ensuring the validity and reliability of the analysis.

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Fig 3. How do causal machine-learning techniques address confounding problems?.

Notes: In the given figure, the primary effect of interest (denoted as 1) is the impact of D on the outcome variable Y. However, this effect is confounded by variables X that influence both D and Y (denoted as 2 and 3). To address this problem, causal machine learning is utilized, fundamentally consisting of two parts. The machine learning component used to generate D’ and Y’–these are the corresponding parts of D and Y that are explained by the confounders X, and the parameters of this process are referred to as the nuisance parameters (such as the indices representing the impact of X on D and Y). The causal inference component will perform estimations from the residuals of Y and D after removing D’ and Y’. The orthogonalization technique used in the process ensures that the target effect is not influenced by confounders (in other words, it is invariant to the nuisance parameters). It is also noted that these causal machine-learning techniques only report the target effect.

https://doi.org/10.1371/journal.pone.0302173.g003

The orthogonality principle underlying these methods [32, 40] can be described as follows. I seek a score function ψ(W, θ, η), where W = (Y, D, X) and η represent the nuisance parameters, in order to satisfy the following conditions: . Here, ψ(W; θ, η) is defined as , and η is composed of the functions l and m, with η₀ = (l₀, m₀). Specifically, l₀(X) denotes , and m₀(X) denotes . The score function ψ, with W = (Y, D, X), θ₀ as the parameter of interest, and η representing nuisance functions with a population value of η₀, plays a key role in the inference procedure. It satisfies the moment condition , where θ₀ (the target effect) is the unique solution, and it also adheres to the Neyman orthogonality condition . This condition ensures that the moment condition used for identifying and estimating θ₀ remains unaffected by small perturbations of the nuisance function η around η₀.

Machine learning techniques often lead to overfitting bias, which is where cross-validation comes into play [32]. The fundamental principle of cross-validation is to address bias in parameter estimation by dividing the data into folds and estimating nuisance functions and the parameters of interest in separate samples. This approach helps alleviate overfitting and misspecification problems by minimizing the impact of any particular subset of data. The cross-validation for the double machine learning procedure operates as follows. To begin, an assumption is made regarding a full sample , where W = (Y, D, X), and a Neyman-orthogonal score function ψ(W; θ, η) is employed. Subsequently, a K-fold random partition , obtained by folding the observation indices {1,…,N} is utilized, with each fold I_k having a size of N/K. For every I_k, a machine learning estimator of η₀, denoted as , is constructed using the out-of-sample data . Finally, the estimator for the causal parameter is constructed as the solution to the equation:

However, the issue of gender effects on mobility in question Q2 cannot be fully resolved by the aforementioned techniques due to the inherent challenge of disentangling the effects of X and D on Y, particularly when gender bias factors are intertwined with socioeconomic factors [15]. In a manner akin to the research conducted by Philipp Bach et al. [18] on the multidimensional gender effects with socio-economic characteristics on the gender wage gap in the US, a rigorous methodology is employed in this instance to tackle this problem. Specifically, the Partialling-out Lasso (with interactions) [40] and the Interactive Model Regression (IRM) [32] (as alternatives to the PLR) are utilized. In the context of Partialling-out Lasso with interactions, the treatment variable D encompasses a set of gender variables, including itself and interaction terms with other socioeconomic variables, while X represents the control variables and their corresponding interaction terms. This analytical approach remains consistent. In the meanwhile, in the IRM framework, the term Dθ₀+l₀(X) is replaced by g₀(D, X), indicating the inseparability of the effects of D and X. The estimation of θ₀ is equivalent to determining the average treatment effect (ATE), using a novel score function:

Here, W denotes the variables (Y, D, X), while g(D, X) represents , and m(X) signifies ℙ[D = 1|X]. The parameters η correspond to (g, m), with η₀ = (g₀, m₀). This advanced methodology is employed to address the intricate nature of gender effects and ensure a more robust analysis.