Glossary

Bernstein polynomial

Flexible mathematical tool used to approximate the shape of any smooth distribution. The order \(m\) of the polynomial controls how closely it can follow the data, with higher \(m\) allowing for more complex shapes.

Bonferroni correction

Adjustment for running many statistical tests at once, which increases the chance that at least one appears significant purely by chance. It divides the significance level by the number of tests, so each individual test must meet a stricter threshold.

Circumstances

Factors beyond individuals’ control that may nonetheless shape their opportunity sets, such as family background, race or ethnicity, gender, and place of birth.

Conditional inference tree (Ctree)

Machine learning algorithm which splits the population into groups formed by maximizing the statistical significance of the difference in their average value along a given dimension.

Consumer price index (CPI)

Index of how the cost of a fixed representative basket of goods and services changes relative to a base period, commonly used to measure inflation.

Counterfactual distribution

Hypothetical distribution that would be observed in a different situation from the one actually occurring. For instance, removing differences in outcome between all individuals sharing the same set of characteristics generates a counterfactual distribution in which there is no inequality within these groups.

Cumulative distribution function (CDF)

Mathematical function which gives, at every point of a given distribution, the share of values below this point. It fully describes the shape of the distribution.

Ex-ante inequality of opportunity

Measure of unfair inequality based on the principle that opportunity sets available to individuals should have the same expected value regardless of predetermined characteristics.

Ex-post inequality of opportunity

Measure of unfair inequality based on the principle that individuals exerting the same degree of effort should achieve the same outcome.

Gini index

Most widespread measure of inequality. In the absence of negative values, it ranges from 0 (perfect equality) to 1 (maximum inequality).

Inequality

Degree to which a given outcome, such as income or consumption, is unequally distributed between individuals in a population. Our headline inequality measure is the Gini index of inequality in age-adjusted equivalized household disposable income.

Inequality of opportunity (IOp)

Part of inequality that can be attributed to predetermined characteristics beyond individual control such as family background, race or ethnicity, gender, and place of birth.

Intergenerational persistence

Strength of the association in an outcome between generations. Our headline measure of intergenerational persistence is the correlation coefficient between parents’ percentile income ranks and their children’s, from the bottom 1% to the top 1% of each generation’s income distribution.

M-fluctuation test

Statistical test that detects whether the parameters describing the outcome distribution change significantly across the values of a circumstance. It is analogous to testing for a structural break in a time series.

Mean logarithmic deviation (MLD)

Measure of inequality which is more sensitive than the Gini index to differences at the tails of the distribution. It equals zero in case of perfect equality but has no upper bound.

Most predictive factor

Inherited characteristic which most often has the highest contribution to inequality of opportunity across survey years. In case of ties, the most recent one is selected.

Outcome

Money-metric well-being indicator over which we compute overall inequality, and inequality of opportunity. We use age-adjusted equivalized household disposable income in most cases, and age-adjusted equivalized household consumption expenditure instead when unavailable.

Partial dependence plot (PDP)

Model-interpretation plot showing how the average prediction changes as one predictor varies, averaging over the observed values of the other predictors.

Purchasing power parity (PPP)

Rates of currency conversion that aim to equalize the purchasing power of different currencies by eliminating differences in price levels between countries.

Quantile

Values that divide an ordered distribution into equal-sized groups, or, by extension, the groups themselves. For instance, centiles divide the population into 100 groups of 1% from the bottom 1% to the top 1%, deciles are groups of 10%, etc.

Random forest

Machine learning algorithm which grows many decision trees on different sub-samples and averages their predictions, producing more stable estimates than a single tree.

Relative inequality of opportunity

Ratio of absolute inequality of opportunity to overall inequality, capturing the share of inequality that can be predicted by inherited characteristics such as family background, race or ethnicity, gender, and place of birth.

Shapley value decomposition

Method used to compute the weighted average marginal contribution of each component of a system across all possible combination orders.

Tranche

Group of people who exert the same degree of effort, regardless of their type. Since effort is not observed, it is proxied by an individual’s rank in the outcome distribution of their own type, so a tranche groups people at the same quantile of their type’s distribution.

Transformation tree (Trafotree)

Machine learning algorithm which splits the population into groups formed by maximizing the statistical significance of the difference in their distribution along a given dimension.

Tree-based machine learning

Supervised machine learning approach which generates an optimal partition of the data from iterated splits over a given set of variables.

Type

Group of people who share the same set of circumstances, i.e., inherited characteristics such as family background, race or ethnicity, gender, and place of birth. Each individual belongs to a single type, such that the type partition characterizes 100% of the population.