The Evolution of Inequality of Opportunity in Germany: A Machine Learning Approach

We show that measures of inequality of opportunity fully consistent with Roemer (1998)’s inequality of opportunity theory can be straightforwardly estimated adopting a machine learning approach. Following Roemer, inequality of opportunity is generally defined as inequality between individuals exerting the same degree of effort but characterized by different exogenous circumstances. Due to difficulties of measuring effort, most empirical contributions so far identified groups of individuals sharing same circumstances, and then measured inequality of opportunity as between-group inequality, without considering the effort exerted. Our approach uses regression trees to identify groups of individuals characterized by identical circumstances, and a polynomial approximation to estimate the degree of effort exerted. To apply our method, we take advantage of information contained in 25 waves of the German Socio-Economic Panel. We show that in Germany inequality of opportunity declined immediately after the reunification, surged in the first decade of the century, and slightly declined again after 2010. The level of estimated unequal opportunity is today just above the level recorded in 1992.


Introduction
The ideal of equality of opportunity has fascinated mankind since centuries. Its popularity among people from both sides of the political spectrum probably derives from the fact that it encompasses and balances two aspects, equality of outcomes and freedom of choice, that build the fundaments of modern democratic societies. In addition, while nobody would argue against equality of opportunity as an important goal, the political rhetoric demanding for it might be set su ciently vague to allow for di↵erent interpretations.
For a long time moral philosophers and welfare economists defined and conceptualized the notion of equality of opportunity as well as its implications from a normative point of view. Rawls (1971) proposed a theory of social justice in which redistribution of outcomes and social roles was somehow limited by the need to take into consideration individuals' responsibility. Dworkin (1981) went a step further focusing on the distinction between preferences and resources. In his view inequality in final conditions are morally objectionable, and call for redistribution, to the extent they arise from unequal resources, while they should not be equalized when arising from preferences and choices. Among economists the most influential formalization of the principle of equal opportunity is due to Roemer (1998). Roemer's definition of inequality of opportunity comprises the interplay between circumstances that individuals are exposed to and the degree of e↵ort they exert; circumstances are exogenous factors outside individual control, while e↵ort indicates the result of choices for which the society wishes to hold individuals responsible.
Roemer's theory of equal opportunity has triggered a lively methodological and empirical literature, mainly focused on the measurement of inequality of opportunity. These contributions have proposed metrics to measure how far a given distribution of individual outcomes is from equal opportunity. Among others Lefranc et al. (2009), for example, focus on the possibility to perform statistical tests to reject the null hypothesis of equal opportunity. Checchi and Peragine (2010) and Almås et al. (2011) show how violations of di↵erent definitions of equal opportunity can be quantified by manipulating the original distribution to obtain counterfactual distributions of unfair inequality. A popular approach is also the regression-based method proposed by Ferreira and Gignoux (2011) to quantify the share of total inequality due to opportunity.
Two common issues of these empirical approaches are: i. the need to identify the set of circumstances beyond individual control and ii. the need to assume how these circumstances interact with e↵ort in determining individual outcomes. Both choices have been shown to crucially a↵ect the estimated level of inequality of opportunity (see Brunori et al. (2018b) for a discussion).
Recent contributions have proposed approaches to improve the empirical specification of the underlying models finding consistent econometric methods to identify the relevant circumstances and eventually estimate inequality of opportunity. Within this literature, Li Donni et al. (2015) and Brunori et al. (2018a) propose data-driven approaches to identify Roemerian types (i.e. sets of individuals characterized by identical circumstances). However, as argued by Ramos and va de Gaer (2019) in the vast majority of empirical contributions inequality of opportunity is estimated without taking into consideration the role of e↵ort, which is key in Roemer's theory.
In this study, we propose a method that builds on a data-driven approach and exploits two machine learning algorithms (namely, regression trees and polynomial approximation) to estimate inequality of opportunity consistent with Roemer's original theory. We apply this approach to estimate the evolution of inequality of opportunity in Germany from shortly after the fall of the Berlin wall to the present time.
In a first step, we follow Brunori et al. (2018a) to identify types and estimate the opportunity trees for Germany in each single year between 1992 and 2016. Then, we develop a new approach to estimate the degree of e↵ort exerted by individuals through a polynomial approximation of the conditional distribution of household income for each type. This novel method enables us to precisely estimate the relationship between e↵ort and outcome for types of di↵erent sample size.
Germany is an interesting case study for our analysis because of the societal changes that the country underwent during the last decades. Our application on the German Socio-Economic Panel (SOEP) shows that the opportunity structure of the German society is much more complex today than it was back in the nineties. The number of types identifiable in SOEP has increased substantially over time; a pattern that is not fully explained just by changes in survey characteristics. Besides the increasing societal diversity, having being located in the Eastern or Western part of Germany in 1989 is, more than two decades after reunification, constantly a significant circumstance defining the subdivision in types over the course of time.
Our analysis uncovers another interesting particularity. While usually societies characterized by a large number of types tend to have much higher levels of inequality of opportunity, this is not the case for Germany over time. Controlling for the sample size, albeit the number of types increases by roughly 75%, the level of inequality of opportunity in 2016 is just around 7% higher than in 1992. Generally, during this period Germany experienced first a slow decrease in inequality of opportunity after reunification, and then a sudden rise coinciding with rising income inequality and the implementation of the Hartz-reforms, a set of substantial changes to the German labour market and welfare benefit system that had persistent repercussions on the German society. Thenceforth, inequality of opportunity stayed at this relatively high level and followed a stable trend, with a slight decrease from 2010 on.
2 Inequality of Opportunity Roemer (1998) represents the seminal contribution for the empirical literature on inequality of opportunity (IOP). In his book, Roemer did not explicitly write down a definition of inequality of opportunity. His theory proposes a criterion to select the redistributive policy that would equalize opportunity in a society. This theory has been translated into more than one definition of inequality of opportunity.
Roemer's theory distinguishes between two categories of factors that determine individual outcomes: factors over which individuals have control, which he calls e↵ort, and factors for which individuals cannot be held responsible, which he calls circumstances. He defines a situation of equal opportunity in the distribution of a certain desirable outcome as the situation in which individuals are compensated for the di↵erence in their circumstances, insofar as those di↵erences a↵ect the advantage they attain. To realize equal opportunity, Roemer proposes to identify the partition of the population into types. A type is a set of individuals characterised by exactly the same circumstances (gender, race, socioeconomic background,...). When exerting e↵ort, individuals in the same type have the same ability to transform resources into outcomes. Therefore, an equal opportunity policy prescribes to ignore within-type variability in outcomes, which by definition is due to individual e↵ort, and requires to remove any between-type inequality .
Roemer's definition of equal opportunity can be formalized in a simple model. In a population of 1, ..., N individuals, individual i obtains an outcome of interest, y i , as the result of two sets of traits: a set of circumstances beyond her control, C i ,and a responsibility variable, e i , called e↵ort. y i = g(C i , e i ), 8i = 1, ...N C i contains J > 0 circumstances, each circumstance, C j 2 C, is characterized by a total of x j possible realizations. All the possible combinations of realizations taken one at time from C define a partition of the population into types. This partition is made of a maximum of K = ⇧ J j=1 x j non-empty subsets, where every individual is included in one and only one of the subsets. Note that this simple model does not introduce any random component or uncertainty.
Equality of opportunity is realized when individuals exerting same e↵ort obtain the same valuable outcome, independently from the type they belong to. In order to measure to what extent this principle is violated one has to compare the outcome of individuals belonging to di↵erent types but exerting same e↵ort. Because e↵ort is typically unobservable Roemer, for one, proposes a method for identifying e↵ort. His method is based on three assumptions. First, we fully observe relevant circumstances, that is we correctly assign individuals to types. Second, the outcome is assumed to be monotonically increasing in e↵ort: in every types higher e↵ort implies higher outcome. y k (e i ) y k (e j ) () e k i e k j , 8k = 1, ..., K, 8e i , e j 2 R Third, the degree of e↵ort exerted is by definition a variable orthogonal to circumstances. In Roemer's view, if individuals belonging to di↵erent types face di↵erent incentives and constraints in exerting e↵ort, this is to be considered a characteristic of the type and therefore included among circumstances beyond individual control.
For example, a student with well educated parents may find it much easier to spend hours sitting at her desk, while a student growing up in a less favourable environment may find it harder to study. Roemer believes that the distribution of e↵ort is, indeed, a characteristic of the type: thus, in comparing e↵orts of individuals in di↵erent types, we should somehow adjust for the fact that those e↵orts are drawn from distributions which are di↵erent, a di↵erence for which individuals should not be held responsible. Roemer (2002) p. 458.
Roemer therefore distinguishes between the 'level of e↵ort' and the 'degree of e↵ort' exerted by an individual. The latter is the morally relevant variable of e↵ort and is identified with the quantile of the e↵ort distribution for the type to which the individual belongs. We denote with G k (e) the distribution of e↵ort within type k and with ⇡ 2 [0, 1] its quantiles.
If e↵ort is not observable, but the outcome is monotonically increasing in e, Roemer suggests to identify the degree of e↵ort exerted by a given individual with the quantile of the type-specific outcome distribution she sits at: y k (G k (e)) = y k (⇡). This definition of e↵ort is insensitive to di↵erences in the absolute level of e↵ort exerted that, in Roemer's view, are due to circumstances beyond individual control and it permits to compare the e↵ort exerted by individuals in di↵erent types.
Then the requirement of same outcome for individuals exerting same e↵ort can be rewritten in terms of type-specific outcome distributions: y k (⇡) = y l (⇡) () F k (y) = F l (y), 8⇡ 2 [0, 1]; k, l = 1, ..., K Where F k (y) is the type-specific cumulative distribution of outcome in type k.
A measure of inequality of opportunity quantifies to what extent this equality of opportunity principle is violated. This is done measuring the variability of the outcome distribution within individuals exerting same e↵ort (Almås et al., 2011;Checchi and Peragine, 2010;Ferreira and Gignoux, 2011;Lefranc et al., 2009). By construction, these measures take value zero when equation (2) is satisfied and all individuals exerting the same e↵ort obtain the same outcome, and increase the larger di↵erences in outcomes obtained by individuals exerting the same degree of e↵ort.
For example, the ex-post measure of inequality of opportunity proposed by Checchi and Peragine (2010) evaluates inequality in the standardized distributionỸ EP obtained replacing individual outcome with: µ µ ⇡ , 8i = 1, ..., N; k = 1, ..., K; 8⇡ 2 [0, 1]. (3) Where y k i (⇡) is the outcome of individual i belonging to type k and sitting at quantile ⇡ of the type specific e↵ort distribution, and µ ⇡ is the average outcome of individuals sitting at quantile ⇡ across all types, and µ is the population mean outcome. Note that in the standardized distribution the average value for individuals sitting at all quantiles is the same, that is betweenquantile inequality has been removed. On the contrary, the within-quantile relative distance of outcome is preserved. Inequality of opportunity, IOP EP , is then inequality in the standardized distribution: Where I is any inequality measure satisfying the typical properties, including scale invariance. 1 Ex-post measures of inequality of opportunity are not frequently implemented in empirical analysis. The majority of applied studies focus on a second, less demanding, definition of equal opportunity. The ex-ante equality of opportunity is a 'weak equality of opportunity' criterion that allows some inequality within groups of individuals exerting the same e↵ort but requires that that mean advantage levels should be the same across types (Ferreira and Gignoux, 2011).
The ex-ante measure of inequality of opportunity first proposed by Van de gaer (1993) is a measure based on this weaker definition. The approach interprets the type-specific outcome distribution as the opportunity set of individuals belonging to each type. The (utilitarian) value of the opportunity set of each type is the mean outcome of the type. Therefore inequality of opportunity in this case is simply between-type inequality, the counterfactual distributionỸ EA is obtained replacing individual outcome with: y k i (⇡) = µ k , 8i = 1, ..., N; 8k = 1, ..., K; 8⇡ 2 [0, 1] Where µ k is the mean outcome of type k.
Adopting the ex-ante approach simplifies the measurement of inequality of opportunity, which becomes equivalent to measure between-group inequality. Furthermore, IOP EA is by far the most popular measure of inequality of opportunity. 2 However, this approach implies a lost of consistency with the principle of compensation in its original formulation, the fundamental ethical principle of Roemer's theory of equal opportunity, that states that individuals exerting same e↵ort should obtain same outcome (see Fleurbaey and Peragine (2013) for a discussion of this incompatibility). 3

Machine Learning Estimation of IOP
The estimation of IOP EP is based on two fundamental tasks: the identification of Romerian types and the measurement of the degree of e↵ort exerted. We adopt a machine learning approach to accomplish both. The partition into types is obtained estimating regression trees; the degree of e↵ort is measured estimating the type specific outcome distribution by a polynomial approximation.

Identification of types
The first step to estimate both equation (4) and equation (6) is the identification of circumstances beyond individual control that define types. 4 The selection of circumstances is a key aspects of any empirical analysis of IOP because estimates have been shown to be sensitive to the number of types considered (Brunori et al., 2018b;Ferreira and Gignoux, 2011;Rodríguez, 2008).
In principle, one should include all variables beyond individual control that can a↵ect the outcome. However, this is not a realistic option. First, surveys typically contain only a subset of all the exogenous determinants of individual outcome. Second, even when a rich dataset is available, the sample size constrains the number of circumstances that can be considered if one has to reliably estimate the counterfactual distributions.
The so-called non-parametric approach proposed by Checchi and Peragine (2010) attempts to exactly implement Roemer's definition of types: the partition in types is obtained interacting all circumstances, what typically results in a partition with a large number of types. However, many of these types are sparsely populated making it impossible to estimate with accuracy the type-specific cumulative distribution of outcome.
Empirical exercises generally address this issue by limiting the number of circumstances used to define types. Also, the categories that describe circumstances are recoded reducing their variability. For instance, districts of birth are aggregated into macro-region; parental occupations become a binary variable for white or blue collar workers; ethnicity becomes a dummy for minorities. The resulting types have large sample sizes but are based on arbitrary choices. This ad-hoc methods to identify types severely undermines the interpretability and comparability of estimates of inequality of opportunity (Brunori et al., 2018b).
Two papers have proposed data-driven criteria to identify Romerian types. Li Donni et al. (2015) specify that types are inherently unobservable and suggest to group individuals in types using latent class models. Latent class models assign individuals to types based on observed circumstances, interpreted as observable manifestations of the underlying latent types. Types membership is determined in the attempt to maximize local independence. Local independence means that, conditional on class membership, observed items are conditionally independent of each other. Once latent types are identified the researcher can move to the second step of the analysis which consists in the identification of the e↵ort exerted.
Latent types are an appealing theoretical construct, however their implementation has two problematic aspects. First, in latent class models the number of classes is exogenously given. Li Donni et al. (2015) suggest to select the number of latent types guided by the Bayesian information criterion (BIC). BIC evaluates the likelihood of the model introducing a penalty term for the number of parameters estimated. The BIC selects the most appropriate model balancing between choosing a model able to closely fit the data in the sample, and choosing a model with the lowest possible number of parameters. Estimating models based on increasing number of latent types BIC will first rise, indicating that additional classes substantially improve model fitting, and then, when the e↵ect of the penalty will become predominant, BIC will start to decline. According to Li Donni et al. (2015) the most appropriate partition is obtained choosing the number of classes that produces the highest BIC.
A perfect fit is obtained when the distribution of manifest variables is orthogonal to classes, that is when local independence is fully satisfied. The BIC of a latent class model therefore evaluates the capacity of the model to explain the correlation of manifest variables in the sample. However, when estimating inequality of opportunity, the aim is not to explain covariance of circumstances, but to identify the partition in types that best explains the outcome variability. A criterion such as the BIC may not be the most appropriate to select the number of latent types. This is a specific case of the more general problem of using latent class membership as predictor for a distal dependent variable. As discussed by Lanza et al. (2013) such an approach is likely to produce attenuated estimates of the e↵ect of the latent class membership on the outcome.
Another issue concerns the choice of observable circumstances considered in the latent class model. As discussed above, the problem of arbitrary selection of circumstances severely undermine the non-parametric estimation of inequality of opportunity; this problem is attenuated but not completely solved when using latent class models. The number of parameters one needs to estimate when applying a latent class model is in fact a function of the number of classes, the number of circumstances considered, and the number of values each circumstance can take. This implies that the choice of circumstances considered, which is arbitrary, will a↵ect the result.
This weakness of the latent types approach highlights that a proper method that aims to estimate inequality of opportunity needs to comprise, both, the identification of types based on observed circumstances, and a variable selection criterion that would select the most appropriate set of the (possibly) many observable circumstances.
In this light Brunori et al. (2018a) have proposed to use a machine learning algorithm, known as conditional inference regression trees, to identify Romerian types. Regression trees are prediction algorithms introduced by Morgan and Sonquist (1963) and popularized by Breiman et al. (1984) almost 20 years later. The algorithm aims at predicting an outcome out of sample based on a number of covariates. This is done partitioning the space of the regressors in nonoverlapping regions. The name tree comes from the way this algorithm can be graphically represented as an upside-down tree. Figure 1 shows an example of a regression tree to predict income based on two regressors: parental education and parental occupation. The predicted income is simply the average outcome of individuals assigned to each terminal node (ovals at the bottom of the tree). Regression trees are generally grown in the attempt to maximize the ability of the model to predict out-of-sample. That is, trees aim at maximizing the variability of the dependent variable that can be explained by between-node variability, without overfitting the model. A too deep tree, that is an overfitted tree, would result in a very low in-sample error but would poorly perform out-of-sample.
Various methods to grow trees avoiding overfitting exist. Weakest link pruning estimates the mean squared error out-of-sample for all trees obtained replacing a sub-tree with a terminal node. This method of pruning is computationally costly in cases with a large set of regressors. Other methods such as cost complexity pruning or conditional inference trees can be used to prevent trees from growing too deep (James et al., 2013)

Identification of e↵ort
Once types are identified, the second step consists in estimating e↵ort. Adopting Roemer's identification strategy this is done by estimating the shape of the type-specific outcome distribution in all types. Previous contributions select an arbitrary number of quantiles, generally not larger than 10, and estimate equation (4) setting µ ⇡ equal to the average outcome across all individuals belonging to the j th quantile of their type specific outcome distribution.
Although not explicitly discussed by these contributions, the need to estimate the distribution of outcomes for each type has a clear impact on the empirical exercise. If in the ex-ante approach the main constraint when identifying types is the need to reliably estimate their mean, following the ex-post approach the need to estimate the type-specific outcome distribution for each type imposes a more severe trade-o↵. On the one hand, a precise description of the data generating process requires to consider a su ciently large number of types; on the other, the estimation of the type-specific outcome distribution requires a large number of observations in each type. Much larger than the sample size required for estimating a single parameter for each type. Especially because Roemer's strategy imposes no restriction on the type-specific outcome distribution, the researcher has to estimate a number of parameters equal to the number of quantiles.
As Luongo (2011) clarifies, inequality of opportunity estimates are sensitive to the selected number of quantiles. However, if one can imagine that -theoretically-a precise number of Roemerian types does exist, it is clear that quantiles are used to approximate an intrinsically unknown continuous distribution function. Hence, there is not a true number of quantiles.
This paper proposes a non-arbitrary criterion to approximate the type-specific outcome distribution based on a procedure suggested by Hothorn (2018). Such a criterion makes measures of inequality of opportunityà la Roemer less dependent on discretionary methodological choices and more easily comparable across time and space.
Moreover, our method explicitly addresses the problem of balancing the need to precisely estimate the distribution of outcome in each type and the data constraints, constraints that will typically di↵er across types of di↵erent sample size.
We approximate the shape of the type-specific outcome distribution F k (y) using the Bernstein polynomial, that is a linear combination of Bernstein basis polynomials. The Bernstein basis polynomial of degree m for some positive continuous variable t 2 [a, b] is defined as the set of polynomials: A linear combination of Bernstein basis polynomial has the form: (folds); 2. for every b = 1, ..., 10; a. for every fold f = 1, ..., 10; I. obtain the training sample by excluding from the sample the f -th fold that will be later used as test sample; II. estimate the shape of the type-specific outcome distribution with a monotone increasing Bernstein polynomial of order b on the training set; III. predict the cumulative distribution of the type based on the Bernstein coe cientŝ 4. b ⇤ indicates most appropriate order for the Bernstein polynomial approximating the typesspecific distribution function of type k.
The algorithm, repeated for all types, produces parametric approximations of the typespecific outcome distribution functions based on the coe cients of Bernstein polynomials of di↵erent order. Under Roemer's assumption, the quantiles of such distributions measure the degrees of e↵ort individuals exerted. The estimation of equation (4) can then be conceptualized as inequality between a set of weighted, type-specific outcome distributions, and will depend on population weights and the Bernstein coe cients. Equation (4) can then be precisely approximated using a su ciently high number of points to approximate each type-specific outcome distribution.
Equation (3) is then estimated by multiplying the outcome of each individual i = 1, ..., N belonging to each type k = 1, ..., K by the average outcome in the population (µ) divided by the expected outcome of an individual exerting the same degree of e↵ort (µ ⇡ ), independently from the type she belongs to.

The German Socio-Economic Panel
Our analysis on the evolution of inequality of opportunity in Germany applying the methods explained above is based on the Socio-Economic Panel (SOEP) (Goebel et al., 2019). The SOEP is a representative longitudinal survey of private households in Germany that is conducted annually since 1984, including the East German population since 1991. It is also the largest regular survey of immigrants and their o↵spring in Germany.
The SOEP is one of the main data sources for distributional analysis in Germany, from a national perspective and in cross-country comparisons. Furthermore, it includes a remarkable number of retrospective information on individual characteristics that shall be indicative for the circumstances faced in childhood. For instance, questions on the education and occupation of parents, region of birth, migration background, and country of origin are included in the questionnaire and made comparable across the survey years.
We use the v33 version of SOEP including all subsamples apart from the two newly added refugee samples. We restrict the sample to individuals in the age range 30 to 60 with available information on household income and all circumstances that we include in our analysis. Note, that the reported incomes in a survey year refer to the year before the survey was conducted.
Descriptive statistics for the 25 waves used are reported in Appendix A. In the same Appendix, we show for each survey wave the share of individuals with missing information for potential circumstances that we identify based on the past literature on IOP and the specific German context. In order to ensure the maximum possible comparability of estimates across time, we choose the set of circumstances with the lowest number of missing information across all survey waves. The circumstances that we include are: sex, migration background, resident in East or West Germany in 1989, Father's and Mother's education and training, Father's occupation measured by the ISCO-code (one digit), the number of siblings, and an indicator of whether the individual is disabled.
We set 1991 as the first year because information on household income for people in East Germany is available from that year on, hence from the survey wave 1992. To warrant the representability of our analysis at the national level in every year, each observation is weighted by the inverse probability of selection. Household income is displayed in Euro at 2011 prices. Our outcome of interest is household disposable equivalent income. Income is equivalized dividing household income by the square root of the household size. To avoid variability due to the life cycle to bias our IOP estimates, for every year, we compute the deviation of individual income from its expected value given the respondent's age. Our outcome of interest is therefore the deviation from what is predicted by a regression in which disposable household equivalent income declared by the respondent is regressed on her age and her age squared. Therefor, an individual with outcome higher than one is an individual who, given her age, has a larger equivalized income than the average resident in Germany. 7 Total inequality trends for this residual outcome measure as well as total equivalized household income can be found in Appendix A.

Sample size
A key challenge for our analysis is that the sample size varies considerably across waves of the SOEP, ranging between 2,868 and 13,160. Conditional inference regression trees have been shown to be rather sensitive to the sample size because the splitting points are conditioned on a sequence of statistical tests and, when sample size is small, p-values of the tests tend to be higher. Other things held constant, the larger the sample size the deeper might be the resulting tree. Deeper trees, made of many terminal nodes, usually tend to produce larger IOP estimates.
In order to overcome this problem and maximize the comparability of estimates over time, we proceed in two steps. First, we show the partition in types obtained using the original samples. Then, we proceed as follows: for each year a random sub-sample of size 2,868 is drawn from the original data (sample size in 1992; the smallest recorded sample size over all survey waves that we use). Then, the opportunity tree and the resulting IOP are estimated and stored. This procedure is repeated 200 times. Doing so, the variables used, the splitting points, and the number of terminal nodes might di↵er not only across waves, but also across iterations for the same year. Finally, we report the average number of types and IOP values for each year.
Iterating the estimation of regression trees we get close to what in machine learning is known as random forest; a large collection of trees obtained using a subset of the available information. Contrary to the typically applied random forest procedure, we do not use a subset of regressors for each possible split of each tree. In contrast, the use of a subset of observations determine a certain level of heterogeneity of the tree structure across iterations. This reflects the level of uncertainty we have about the real data generating process, that in this case is our uncertainty about the real Romerian types existing in the German society. The level of IOP becomes therefore the average of 200 possible levels of IOP under alternative assumptions about the type partition.
5 The Evolution of IOP in Germany, 1992-2016 5.1 Development of the Opportunity Tree Figure 2 and 3 show the partition in types in 1992 and 2016, respectively. Both trees are obtained using the full sample and show a rather di↵erent structure of opportunities of the Germany society in the two points in time. 8 The just unified Germany is a polarized society with one main driver of between-type inequality: place of residence in 1989. For Germans originally from the East the second significant circumstance is father's occupation. Those having a father with a high level of occupation, or employed in the armed forces, show a higher level of expected outcome than those with a father in an unskilled occupation. Both East-types have an expected outcome below 80% of the national average of their age-group. For people residing in West-Germany, the splitting node is defined by Fathers' education instead. The average level of outcome in both western types is way above the national average.
The structure of opportunity is much more complex in 2016; almost the entire set of circumstances, excluding the number of siblings and migration background, determine at least one splitting point. Interestingly, the place of residence in 1989 is still a fundamental driver of inequality and the second most correlated circumstance with the outcome. The first splitting point is determined by father's occupation; with armed forces occupations now together with unskilled occupations at the lower bottom of the distribution.
It is evident that in 2016 the German society appears to be made of a much larger number of types in comparison with the simpler opportunity structure suggested by the data for 1992. The vector of circumstances used in 1992 is location in 1989, father occupation and education. In addition, in 2016 disability, sex, mother education, and father training play a role in splitting the sample. Furthermore, people who were resident abroad in 1989 and migrated to Germany afterwards form the lower types together with individuals located in the East. However, among these sub-types, migration background does not contribute to explaining further the opportunity structure beyond other circumstances, like parental background and disability. Figure 4 shows the number of terminal nodes (types) for all waves. The trend sees a gradual increase until early 2000s with dramatic rises in 2000 and 2002, when the number of types jumps first to 13 and then to 20. Thereafter, the trend is characterized by ups and downs within this higher range.
A closer look at the development of the opportunity trees from 1992 to 2016 reveals some striking patterns. Appendix B shows the tree for all years over this time period. The first, most compelling evidence is that, albeit the appearance of some characteristics on the Opportunity Tree and the rising complexity of the German society, the interaction of circumstances that defines the highest and the lowest type in the income distribution is rather constant. Indeed, at the top of the distribution we always find individuals that resided in West-Germany before the fall of the Berlin Wall, whose parents had a high occupational position, and whose mothers had a high educational degree, while East-Germans with low educated parents persistently qualify at the lowest bottom.
Up to 2001, location in 1989 is the first terminal node of the Tree, and for the rest of the time it is a circumstance that persistently splits the German society. To give an example, Source: SOEP, 1992SOEP, -2016 in 2014 people from East-Germany with high parental occupational background (Managers and Professionals) have lower average income levels than their West-German counterparts with middle parental occupation (Technicians and Clerks). Having a disability is, particularly from 2002 on, a circumstance that constantly explains inequality in Germany. This does not depend on the relative size of this group in the survey, since the share of respondents reporting disability oscillates without much variation around nine percent. Characteristics like the month of birth, the predominant place in childhood (rural or urban), and the presence of siblings does not seem too relevant to explain the distribution of outcomes. Migration background appears as relevant circumstance to split the West-German population in sub-types from 1999 to 2013. This applies to people with own migration experience (direct migration background) that moved to West-Germany before 1989. Their in Germany born children (indirect migration background) instead does not seem distinguishable in their level of outcome to the rest of the population; this circumstance mostly does not play a major role after having controlled for parental occupation and education, confirming past findings on the topic (e.g. Bönke and Neidhöfer, 2018;Krause et al., 2015). New migrants, i.e. people that were resident Abroad in 1989, mostly belong to the lower types together with people that resided in East-Germany. However, only in one year, 2001, their average age-adjusted incomes are significantly di↵erent from the outcomes of East Germans. In this year, new migrants with low parental education form the type with the lowest outcomes.  Table A6 in Appendix A) shows the development of the number of types and IOP, measured by the Gini coe cient, in Germany from 1992 to 2016. The trend is characterized by declining inequality in the nineties and a subsequent rise in IOP, leading to a rather stable trend with little variation from 2003 onwards. Interestingly, despite the 2016 partition has four times the number of types than the 1992 partition, the level of IOP is only slightly higher in 2016 (0.1046) than in 1992 (0.0959). Germany appears a much more complex society, with a complicated interaction of circumstances in producing opportunity. But the level of such inequality is only slightly higher.

IOP estimates
This becomes even more evident looking at how the type-specific empirical cumulative distribution functions (ECDFs) change over time. Figure 5 shows the type-specific ECDFs in 1992 and 2016; graphs for all the other years can be found in Appendix B. The dots are observed  , 1992 and 2016. distributions while dashed lines show the interpolation of the distribution obtained with the Bernstein polynomial.
The comparison of the ECDFs in 1992 and 2016 is an illustrative example for the spectacular changing of IOP in Germany. In 1992, the two compressed distributions of East-residents in 1992 lie rather close to each other, while the more dispersed Western distributions lie far to the right. This polarization is no longer evident in 2016. The 16 type-specific distributions lie close to each other and cross in several points. However, the distance between the highest and the lowest type is remarkable and remains stable over the entire period. These two facts together explain why the huge increase in the number of types has not been accompanied by a drastic rise in IOP.

Fixed Sample Size
As already discussed, opportunity trees and corresponding type-specific outcome distributions might not be strictly comparable due to changes in sample size. We therefore proceed fixing the number of observations at 2,868 (the minimum sample size) and repeating the entire estimation procedure 200 times. Figure 6 shows the average number of terminal nodes and the level of IOP obtained by an iterative procedure with 200 repetitions. The reported bounds show the 0.975 and 0.025 quantiles of the distribution of the estimates. 9 In comparison to the main analysis, in this sensitivity test the number of terminal nodes is reduced. The maximum number of types over the 200 iterations is on average 9.25 against the 20 obtained using the entire sample. Nevertheless, the trends in IOP and the number of types are similar and show an increase in the level of complexity over time, and an opportunity structure in 2016 markedly more complex than in 1992. The trend in IOP is also close to what is obtained with the full sample with a decline during the '90s and a steep increase in the early '00s. Again, the level of IOP in 2016 appears slightly above the level of 1992.
Estimating 200 trees with di↵erent samples makes it impossible to show opportunity trees for this application. Instead, for each circumstance we can have a closer look at the share of trees in which the variable determines at least one split (we report this shares in Table A5 in the Appendix). The first striking result is that location in 1989 is used in all 5,000 trees estimated for at least one split. Interesting trends can be observed for other circumstances. In Figure 7  we report how often father's education, disability and sex determine at least one split in the opportunity three. The first circumstance shows a clear decline over time, while the second and the third, almost absent in the opportunity structure of the '90s, appear respectively in 80% and 20% of the trees in most recent waves. While the increasing role of sex is likely to be connected with an increasing number of single parent households (from 13% in 1992 to 23% in 2016), the increasing role of disability is hard to understand since the share of respondents reporting disability is not growing over time (constantly between 7.5% and 10%). 10

Discussion
Past studies report a rise in net income inequality in Germany in the 1990s until 2005/2006 and a subsequent stagnation characterized by small ups and downs (e.g. Biewen and Juhasz, 2012;Biewen et al., 2017;Jessen, 2019;Peichl et al., 2012). These studies identify the rising dispersion of labor market incomes mainly due to skill-biased technological change (see Dustmann et al., 2009), changes in employment driven by part-time and marginal part-time work, and changes in the tax system as the mayor driver of this development. Changes in the household size and structure, as well as reforms of the transfer system have been identified as minor influencing factors. 11 The rising wage inequality in the nineties, evidenced e.g. by Fuchs-Schündeln et al. (2010), was not accompanied by rising IOP, as our analysis shows. Instead IOP drops, especially from 1995 to 1996 and then slowly rises until 2001. Then, it experiences a sharp rise in 2002 that brings it to a new, higher level which stays rather constant, with ups and downs, until 2016. 12 Part of the mechanisms described above could also explain first the decrease, then the sudden rise, and finally rather stagnant development of IOP in Germany from 1992 to 2016. 10 The shares of single parents and disabled individuals is based on our own estimates using SOEPv33. 11 Besides the economic literature on wage and income inequality in Germany, less attention has been dedicated to inequality of opportunity. Two exceptions are Peichl and Ungerer (2017) measuring East-West disparities in IOP, and Niehues and Peichl (2014) comparing IOP levels in Germany to the US.
12 Recall that the survey year refers to the incomes in the year before.
The first sharp increase is contemporaneous to major reforms of the tax and transfer system and of the unemployment benefit schemes. Especially the changes to the social benefit system also know as Hartz-reforms that were inacted in 2003, 2004, and 2005 had long lasting and controversially discussed e↵ects on the German society. Past studies have shown an overall small income inequality-reducing e↵ect of the reform, but with di↵erent impact on the middle and bottom of the distribution of income (Biewen and Juhasz, 2012). Particularly the incomes of longer-term unemployed were negatively a↵ected, while social assistance receiver slightly gained from the reforms.
As also studies dedicated to income inequality has shown, in Germany we do not observe a sizeable e↵ect of the 2008-2009 financial crisis on IOP. If any, we observe a downfall of IOP by 2 percentage points from 2009 to 2010. Bargain et al. (2017) find that in this period and until 2010 in Germany, policy changes induced a rise in poverty rates; mainly due to the slow adaption of social assistance, decreasing tax allowances and changes in the taxation of capital income. However, the tax reforms produced also lower marginal tax rates at both ends of the distribution, and particularly for low levels of gross labor incomes the budget constraint rose from 2002 to 2011 (Jessen, 2019). This possible o↵set of mechanisms could explain, why the small rises and falls in IOP in this period are of minor magnitude. Another possible reason for this, is that several elements that explained the income inequality increase before 2005 became less strong over time. For instance, the rise in wage inequality became less steep, employment opportunities increased, and the middle and upper part of the distribution benefited from the employment boom after 2006 (Biewen et al., 2017).

Conclusions
We have suggested a novel approach to estimate IOP consistent with the theory proposed by John Roemer. In Roemer's view IOP is inequality due to circumstances beyond individual control. Outcome variability due to variables of choices instead is not part of IOP. The implementation of a measure consistent with this theory is complex because it necessitates both to identify relevant circumstances beyond individual control and to measure responsibility variables.
Our approach borrows from machine learning methods to solve both issues. Types are obtained estimating conditional inference regression trees. The algorithm selects a partition in Romerian types in the attempt to maximize the outcome variability that can be correctly explained by between-type inequality. E↵ort identification relies on a polynomial approximation of the empirical cumulative distribution function of outcomes in each type. The degree of the Bernstein polynomial is selected by a 10-fold cross-validation in order to maximize its out-ofsample log-likelihood. The two solutions represent a data-driven approach to the estimation of IOP which has the main advantage of minimizing arbitrary assumptions (such as the number of types and shape of the type-specific outcome distributions).
We implement our method to 25 waves of the German Socio-Economic Panel. Our aim is to describe the evolution of IOP between 1992 and 2016. We show that the structure of opportunities has markedly evolved over time. The partition in types detected by our approach in 2016 is much more complex than what is observed in 1992. We show that this di↵erence is not driven by changes in the survey's structure; it persists also controlling for the sample size.
Despite of the increase in the complexity describing the partition of the German society, the level of inequality of opportunity we measure in 2016 is only slightly higher than in 1992. The trend we observe sees a decline in IOP in the nineties, a sharp, sustained rise from 1999 to 2003 followed by another, smaller, sudden increase in 2006, and a subsequent stagnation with small ups and downs. The increase in IOP is suggestively contemporaneous with rising income inequality in the period 1999-2005 and the introduction of the Hartz-reforms. In contrast, a clear interconnection with the rising wage inequality in the nineties and the financial crisis is not visible. Mechanisms that could have o↵set these developments to cause a stronger role of circumstances to determine outcomes are a topic of great research interest for future studies.
Several further interesting suggestions arise from our analysis of the evolution of inequality of opportunity in Germany. The most compelling fact is that the East-West divide characterizes, more than two decades after reunification, most of the disparities in the German society. It is remarkable that even in recent times this circumstance divides the society in sub-types at the lower and at the higher bottom of the income distribution. Hence, over the entire observation period the types with the highest average level of outcome always consist of individuals that resided in West Germany in 1989. Besides, individuals with high educated mothers, often in combination with fathers in high occupational positions, are constantly part of the type with the highest level of outcome.
To sum up, over the last two decades, the type with the highest level of outcome in the entire German income distribution consists of people with highly educated mothers, fathers in high occupational positions, and that resided in West-Germany before reunification. At the bottom of the distribution, we constantly find individuals with low educated fathers that resided in the former German Democratic Republic until the fall of the Berlin wall.        Year 1992Year 1993Year 1994Year 1995Year 1996Year 1997Year 1998Year 1999Year 2000Year 2001Year 2002Year 2003Year 2004Year 2005Year 2006Year 2007Year 2008

Appendix A Data and Estimates
Father.train p < 0.001