A Representation-Theoretic Method for Analyzing Order Dependence in Decomposition Analysis: An Application to Fertility Change in Korea

Changin Jeon

No tdhcy_v1, SocArXiv from Center for Open Science

Abstract: Decomposition analysis is widely used to attribute changes in an aggregate outcome to changes in its underlying components. When several factors change simultaneously, however, sequential replacement generally yields different contributions depending on the order in which factors are replaced. The Shapley value resolves this problem by averaging marginal contributions over all possible orders, but the variation across those orders is itself potentially informative. This study proposes a decomposition method that analyzes this order dependence rather than averaging it away. For each factor, its contribution over all decomposition paths is treated as a function on the symmetric group. Harsanyi dividends are used to express path dependence in terms of interaction components, and irreducible representations of the symmetric group are then used to decompose the variance of path-specific contributions into interpretable spectral components. This framework distinguishes three properties of a decomposition result: average contribution, order instability, and the interaction complexity underlying that instability. For a decomposition with six factors, the method generates 64 counterfactual states and 720 replacement paths. Synthetic experiments confirm the predicted relationship between interaction order and admissible representation components and verify the numerical decomposition of order variance to machine precision. The empirical application examines fertility change among Korean women aged 20–49 using population size, age structure, marital composition, parity composition among married women, nonmarital fertility, and parity-specific marital fertility. Between 2000 and 2024, marital composition and parity-specific marital fertility contributed strongly to the decline in births, while changes in parity composition substantially offset that decline. The results also show that path-specific contributions can vary substantially around their Shapley means: for marital composition, parity composition and parity-specific marital fertility, the path standard deviation amounts to roughly 31–42% of the total change between 2000 and 2024. Cross-period differences in spectral structure, however, are strongly shaped by the magnitude of change over each interval. The proposed method extends decomposition analysis from the allocation of average contributions to the analysis of the full distribution and structure of path-specific contributions. It therefore provides an additional diagnostic layer for studying interaction, robustness, and structural dependence in multi-factor decomposition.

Date: 2026-10-08
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Persistent link: https://EconPapers.repec.org/RePEc:osf:socarx:tdhcy_v1

DOI: 10.31235/osf.io/tdhcy_v1

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