pith. sign in

arxiv: 2605.19154 · v1 · pith:GKIK7CTOnew · submitted 2026-05-18 · 💰 econ.GN · q-fin.EC

Indirect Estimators of Intergenerational Mobility

Andrea Del Pizzo , Martin Nybom , Jan Stuhler This is my paper

Pith reviewed 2026-05-20 07:06 UTC · model grok-4.3

classification 💰 econ.GN q-fin.EC
keywords intergenerational mobilityindirect estimatorsinstrumental variablesimputation methodssurname-based estimatorsmultigenerational linkagestransmission channelspersistence rates
0
0 comments X

The pith

Indirect estimators of intergenerational mobility weight different transmission pathways depending on the chosen instrument or imputation strategy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reviews indirect approaches to measuring intergenerational mobility, including instrumental variables, imputation from education or occupation, surname-based methods, and multigenerational links, when direct income data are missing. It introduces a stylized framework in which socioeconomic status passes through several pathways, each with its own rate of persistence across generations. Within the framework, every estimator—direct or indirect—emerges as a weighted average of those pathways. The weights are fixed by the instrument or the set of observable characteristics used for imputation, so each method emphasizes a different slice of the overall transmission process. A reader should care because this shows that indirect estimates do not necessarily reproduce the usual parent-child correlation and instead supply complementary information about long-run persistence and the channels that sustain inequality.

Core claim

In a stylized model where socioeconomic status is transmitted through multiple pathways with heterogeneous persistence rates, both direct and indirect estimators of mobility emerge as weighted averages across those channels; the particular weights are set by the instrument or imputation rule employed, so that different indirect approaches illuminate different parts of the overall transmission process rather than necessarily reproducing the conventional parent-child correlation.

What carries the argument

The stylized framework of multiple transmission pathways with heterogeneous persistence rates, which lets every estimator be interpreted as a weighted average of those channels.

Load-bearing premise

Socioeconomic status is transmitted through multiple distinct pathways that persist at different rates across generations.

What would settle it

Finding that estimates from every indirect method converge to the same numerical value no matter which instrument or imputation variables are chosen would contradict the weighted-average interpretation.

read the original abstract

This chapter reviews indirect estimators of intergenerational mobility, focusing on approaches that infer parent-child or other family associations when direct income data are incomplete or unavailable. We synthesize methods based on instrumental variables, imputation using observable characteristics such as education and occupation, surname-based estimators, and multigenerational linkages. To unify these approaches, we introduce a stylized framework in which socioeconomic status is transmitted through multiple pathways with heterogeneous persistence rates. Within this framework, both direct and indirect estimators can be interpreted as weighted averages of these underlying transmission channels. A central insight is that the choice of instrument or imputation strategy determines these weights, leading different methods to capture distinct aspects of the transmission process. We highlight implications for interpretation, showing that indirect estimators need not recover conventional parent-child correlations but can instead provide complementary evidence on long-run persistence and the mechanisms underlying persistent inequalities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper reviews indirect estimators of intergenerational mobility, focusing on approaches that infer parent-child or other family associations when direct income data are incomplete or unavailable. It synthesizes methods based on instrumental variables, imputation using observable characteristics such as education and occupation, surname-based estimators, and multigenerational linkages. To unify these approaches, the paper introduces a stylized framework in which socioeconomic status is transmitted through multiple pathways with heterogeneous persistence rates. Within this framework, both direct and indirect estimators can be interpreted as weighted averages of these underlying transmission channels, with the choice of instrument or imputation strategy determining the weights and thus the aspects of the transmission process captured.

Significance. If the framework holds, this synthesis is significant for the field of intergenerational mobility research in labor economics. It provides a coherent interpretive model that explains why different indirect methods produce varying estimates and positions them as tools for studying long-run persistence and mechanisms of inequality rather than solely recovering conventional parent-child correlations. The paper's strength is its logical synthesis of disparate methods into a multi-channel transmission model without relying on circular definitions or self-referential fitting, offering guidance for empirical applications with incomplete data.

major comments (1)
  1. [Framework section] Framework section (around the multi-channel model): The central claim that indirect estimators emerge as weighted averages whose weights are set by the instrument or imputation rule is load-bearing for the interpretive synthesis. Please provide the explicit derivation or equation mapping a specific instrument (e.g., the IV case) to the resulting weights on each persistence parameter to confirm the representation holds generally rather than under unstated restrictions on channel independence or linearity.
minor comments (2)
  1. [Abstract] Abstract: The text refers to 'this chapter'; if the manuscript is submitted as a standalone journal article rather than a book chapter, revise to 'this paper' for consistency with journal format.
  2. [Implications section] Implications discussion: Consider adding one or two concrete examples from the existing literature showing how the weighted-average interpretation reconciles differing estimates across methods (e.g., IV vs. surname-based) to strengthen the applied relevance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments and positive overall assessment of the manuscript. We address the single major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [Framework section] Framework section (around the multi-channel model): The central claim that indirect estimators emerge as weighted averages whose weights are set by the instrument or imputation rule is load-bearing for the interpretive synthesis. Please provide the explicit derivation or equation mapping a specific instrument (e.g., the IV case) to the resulting weights on each persistence parameter to confirm the representation holds generally rather than under unstated restrictions on channel independence or linearity.

    Authors: We agree that an explicit derivation strengthens the central claim. In the revised manuscript we will add a new subsection (or appendix) that derives the IV case under the multi-channel model. Let status be transmitted as y = sum_k rho_k * c_k + epsilon, where c_k are the latent channels with heterogeneous persistence rates rho_k. For an instrument Z, the IV estimator equals sum_k w_k rho_k, where the weights w_k = cov(Z, c_k) / cov(Z, y) (normalized appropriately). The derivation relies on the additive linear structure but does not require full channel independence; we will explicitly state the maintained assumptions and note where linearity is used. This will confirm that the weighted-average representation holds generally within the stylized framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; interpretive synthesis is self-contained

full rationale

The paper presents a stylized multi-channel transmission framework solely as an interpretive device to unify existing indirect estimators (IV, imputation, surname, multigenerational). Within this model, estimators emerge as weighted averages whose weights are set by the instrument or imputation rule; this representation follows directly from the framework's own assumptions about heterogeneous persistence rates and does not reduce to any fitted parameter, self-citation chain, or renamed empirical pattern. No equations or derivations in the provided text equate a claimed prediction to its inputs by construction, and the central insight is offered as complementary evidence rather than a forced result. The analysis remains self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper's main contribution is the interpretive framework rather than new parameters or entities.

axioms (1)
  • domain assumption Socioeconomic status is transmitted through multiple pathways with heterogeneous persistence rates.
    This forms the basis of the stylized framework introduced to unify direct and indirect estimators.

pith-pipeline@v0.9.0 · 5668 in / 1073 out tokens · 72214 ms · 2026-05-20T07:06:38.327256+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    socioeconomic status is transmitted through multiple pathways with heterogeneous persistence rates... both direct and indirect estimators can be interpreted as weighted averages of these underlying transmission channels

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    And Yet It Moves

    Aaronson, D. and B. Mazumder (2008). Intergenerational Economic Mobility in the United States, 1940 to 2000.Journal of Human Resources 43(1), 139–172. Abramitzky, R., L. Boustan, K. Eriksson, J. Feigenbaum, and S. P ´erez (2021). Automated Linking of Historical Data.Journal of Economic Literature 59(3), 865–918. Abramitzky, R., L. Boustan, E. J ´acome, an...

    work page 2008

  2. [2]

    Aigner, D. J., C. Hsiao, A. Kapteyn, and T. Wansbeek (1984). Latent Variable Models in Econometrics.Handbook of Econometrics 2, 1321–1393. Althoff, L. and H. Reichardt (2024). Jim Crow and Black Economic Progress after Slavery.The Quarterly Journal of Economics 139(4), 2279–2330. ´Alvarez, A., J. Jaramillo-Echeverri, et al. (2023). The Persistence of Segr...

    work page 1984

  3. [3]

    Anderson, L. R., P. Sheppard, and C. W. Monden (2018). Grandparent Effects on Educational Outcomes: A Systematic Review.Sociological Science 5, 114–142. Angrist, J. D. and A. B. Krueger (1992). The Effect of Age at School Entry on Educational Attainment: An Application of Instrumental Variables with Moments from Two Samples. Journal of the American Statis...

    work page 2018

  4. [4]

    Celhay, P

    Princeton University Press. Celhay, P. and S. Gallegos (2025). Schooling Mobility across Three Generations in Six Latin American Countries.Journal of Population Economics 38(1),

    work page 2025

  5. [5]

    Cervini-Pl´a, M. (2015). Intergenerational Earnings and Income Mobility in Spain.Review of Income and Wealth 61(4), 812–828. Chan, T. W. and V . Boliver (2013). The Grandparents Effect in Social Mobility: Evidence from British Birth Cohort Studies.American Sociological Review 78(4), 662–678. Chang, Y ., S. N. Durlauf, B. Hu, and J. Y . Park (2025). Accoun...

    work page 2015

  6. [6]

    Collins, W. J. and M. H. Wanamaker (2022). African American Intergenerational Economic Mobility Since 1880.American Economic Journal: Applied Economics 14(3), 84–117. Connolly, M., M. Corak, and C. Haeck (2019). Intergenerational Mobility between and within Canada and the United States.Journal of Labor Economics 37(S2), S595–S641. Connor, D. S. and M. Sto...

    work page 2022

  7. [7]

    Kenedi, G. and L. Sirugue (2023). Intergenerational Income Mobility in France: A Comparative and Geographic Analysis.Journal of Public Economics 226, 104974. Klevmarken, A. (1982). Missing Variables and Two-Stage Least-Squares Estimation from More Than One Data Set. Technical report, IUI Working Paper. Lefranc, A., F. Ojima, and T. Yoshida (2010). The Int...

    work page 2023

  8. [8]

    Lundberg, I. (2020). Does Opportunity Skip Generations? Reassessing Evidence from Sibling and Cousin Correlations.Demography 57(4), 1193–1213. Mare, R. D. (2011). A Multigenerational View of Inequality.Demography 48(1), 1–23. Mazumder, B. (2014). Black-White Differences in Intergenerational Economic Mobility in the U.S.Economic Perspectives, Federal Reser...

    work page 2020

  9. [9]

    Modalsli, J. (2023). Multigenerational Persistence: Evidence from 146 Years of Administrative Data.Journal of Human Resources 58(3), 929–961. Modalsli, J. and K. V osters (2024). Spillover bias in multigenerational income regressions. Journal of Human Resources 59(3), 743–776. Mu˜noz, E. and R. van der Weide (2025, July). Intergenerational Income Mobility...

    work page 2023

  10. [10]

    Nybom, M

    Elsevier. Nybom, M. and J. Stuhler (2019). Steady-state assumptions in intergenerational mobility re- search.The Journal of Economic Inequality. Nybom, M. and J. Stuhler (2025). Geographic variation in multigenerational mobility.Socio- logical Methods & Research 54(4), 1532–1575. Olivetti, C. and M. D. Paserman (2015). In the Name of the Son (and the Daug...

    work page arXiv 2019

  11. [11]

    non-cognitive

    We bottom-code all non-missing annual observations at 10,000 SEK (roughly 1,000 USD) to decrease the influence of very low incomes on IGE estimates. We drop fathers with fewer than seven annual earnings observations and children with fewer than two earnings observations. We then construct residualized log earnings and earnings ranks for both fathers and c...

    work page 1990

  12. [12]

    from the 1920 Census. Our main sample for intergenerational analysis consists of male children who were between 10 and 20 years old in 1920 and who could be successfully linked to their 1940 Census records, where they appear aged 30 to

    work page 1920

  13. [13]

    46 2023), achieving a match rate exceeding 50%

    We use links provided by the CensusTree project (Buckles et al. 46 2023), achieving a match rate exceeding 50%

    work page 2023

  14. [14]

    occ- score

    We further restrict the sample to individuals for whom both individual-level and surname-level measures of paternal outcomes are observed. Surname-level averages are computed using the full working-age male population in the 1920 Census and are assigned to children through their linked fathers. This assignment through the father’s observed surname also av...

    work page 1920

  15. [15]

    In addition to the baseline sample, we construct two subsamples to assess the sensitivity of surname-based estimators to sampling variation and overlap

    In addition to occupational scores, our dataset includes several covariates: educational attainment from the 1940 Census, state and county of residence in 1940, birthplace, and an indicator for urban versus rural status. In addition to the baseline sample, we construct two subsamples to assess the sensitivity of surname-based estimators to sampling variat...

    work page 1940

  16. [16]

    Second, we draw a separate random 5% sample of working-age males from the 1920 Census and use this sample to construct surname averages

    Within this subsample, surname averages are constructed using only the fathers observed in the same sample, implying full overlap between individual outcomes and surname-level averages. Second, we draw a separate random 5% sample of working-age males from the 1920 Census and use this sample to construct surname averages. These av- erages are then merged t...

    work page 1920

  17. [17]

    These averages are then assigned to children through their linked fathers

    Specifically, we interact surnames with geographic identifiers in the 1920 Census and compute average outcomes within each surname–area cell (e.g., individuals with a given surname within a state or county). These averages are then assigned to children through their linked fathers. A.2 ADDITIONALDERIVATIONS A.2.1 DERIVATION OF EQUATION(12) In this section...

    work page 1920

  18. [18]

    IQ sample

    We iterate this procedure until we encounter the common 53During the linking process, some individuals are matched with inconsistent ages across censuses. We exclude cases where the implied age difference exceeds five years, while retaining smaller discrepancies to account for potential reporting errors in age. 47 ancestor for surnamesin generationτ s, wh...

    work page 1920