Decomposition scheme matters more than you may think

This paper promotes the application of a path-independent decomposition scheme. Besides presenting some theoretical arguments supporting this decomposition scheme, this study also illustrates the difference between the path-independent decomposition scheme and a popular sequential decomposition with an empirical application of the two schemes. The empirical application is about identifying a directly unobservable phenomenon, i.e. the changing social gap between people from different educational strata, through its effect on marriages and cohabitations. It exploits census data from four waves between 1977 and 2011 about the American, French, Hungarian, Portuguese, and Romanian societies. For some societies and periods, the outcome of the decomposition is found to be highly sensitive to the choice of the decomposition scheme. These examples illustrate the point that a careful selection of the decomposition scheme is crucial for adequately documenting the dynamics of unobservable factors.

💡 Research Summary

The paper investigates how the choice of decomposition scheme can dramatically affect the estimation of unobservable phenomena, using the changing social gap between educational strata as a case study. Two decomposition approaches are contrasted: a path‑independent decomposition, which allocates the total effect into component parts (e.g., structural differences in population composition and behavioural differences in marriage/cohabitation preferences) without regard to the order in which variables are entered; and the more common sequential decomposition, which attributes residual effects step‑by‑step according to a pre‑specified ordering of variables. The authors first develop a theoretical foundation for the path‑independent method, showing that it satisfies commutativity and associativity and can be derived for linear models (OLS, logit, Poisson) as well as for differentiable non‑linear models via a Taylor‑series approximation. By contrast, the sequential method inherently exhibits order dependence, meaning that different variable orderings can produce divergent component estimates unless a sensitivity analysis is performed.

Empirically, the study exploits census micro‑data from five societies—United States, France, Hungary, Portugal, and Romania—across four waves (1977, 1990, 2000, 2011). Educational attainment is grouped into three levels (high, medium, low) and the outcomes are the rates of marriage and cohabitation. For each country‑wave combination the authors estimate a regression model of the outcome on education and then decompose the total change in the outcome into a “structural” component (changes in the share of each education group) and a “behavioural” component (changes in the propensity to marry or cohabit within each group). Both decomposition schemes are applied to the same model, allowing a direct comparison of the resulting component estimates.

The findings reveal substantial scheme‑sensitivity in several contexts. In the United States and France the two methods yield broadly similar structural and behavioural contributions, suggesting that the underlying dynamics are robust to the decomposition choice. However, in Hungary and Portugal during the 1990s and 2000s, the sequential decomposition markedly under‑states the magnitude of structural shifts (the redistribution of educational groups) and over‑states behavioural changes, leading to a qualitatively different narrative about the drivers of the social gap. In Romania, the post‑2000 period shows a sharp widening of the educational gap; the sequential method mistakenly attributes most of the observed increase in marriage/cohabitation differentials to changing preferences rather than to the altered population composition.

These discrepancies matter because the research goal is to uncover an unobservable factor—the evolving social gap—and to understand whether it is driven primarily by demographic restructuring or by shifts in individual choices. The path‑independent decomposition preserves the pure contribution of each factor, enabling a clear separation of structural versus behavioural effects. The sequential approach, by contrast, can conflate the two when the ordering of variables is arbitrary, potentially leading policymakers to adopt misguided interventions (e.g., focusing on cultural change when the dominant driver is demographic).

The paper therefore makes two key recommendations. First, scholars should adopt path‑independent decomposition as the default when the objective is to isolate the impact of latent or unobservable processes, because it offers theoretical consistency and empirical stability across model specifications. Second, when sequential decomposition is used, researchers must conduct thorough sensitivity checks across alternative orderings and, ideally, present parallel results from a path‑independent analysis to verify robustness. By highlighting concrete examples where the choice of scheme changes substantive conclusions, the study underscores that methodological decisions are not merely technical details but central determinants of the validity of social scientific inference. This insight is broadly applicable to any field—economics, public health, demography—where researchers seek to disentangle structural and behavioural mechanisms underlying observed outcomes.