IDEAS home Printed from https://ideas.repec.org/a/wly/emetrp/v92y2024i6p1837-1868.html
   My bibliography  Save this article

Sparse Network Asymptotics for Logistic Regression Under Possible Misspecification

Author

Listed:
  • Bryan S. Graham

Abstract

Consider a bipartite network where N consumers choose to buy or not to buy M different products. This paper considers the properties of the logit fit of the N × M array of “i‐buys‐j” purchase decisions, Y=[Yij]1≤i≤N,1≤j≤M, onto a vector of known functions of consumer and product attributes under asymptotic sequences where (i) both N and M grow large, (ii) the average number of products purchased per consumer is finite in the limit, (iii) there exists dependence across elements in the same row or same column of Y (i.e., dyadic dependence), and (iv) the true conditional probability of making a purchase may, or may not, take the assumed logit form. Condition (ii) implies that the limiting network of purchases is sparse: only a vanishing fraction of all possible purchases are actually made. Under sparse network asymptotics, I show that the parameter indexing the logit approximation solves a particular Kullback–Leibler Information Criterion (KLIC) minimization problem (defined with respect to a certain Poisson population). This finding provides a simple characterization of the logit pseudo‐true parameter under general misspecification (analogous to a (mean squared error (MSE) minimizing) linear predictor approximation of a general conditional expectation function (CEF)). With respect to sampling theory, sparseness implies that the first and last terms in an extended Hoeffding‐type variance decomposition of the score of the logit pseudo composite log‐likelihood are of equal order. In contrast, under dense network asymptotics, the last term is asymptotically negligible. Asymptotic normality of the logistic regression coefficients is shown using a martingale central limit theorem (CLT) for triangular arrays. Unlike in the dense case, the normality result derived here also holds under degeneracy of the network graphon. Relatedly, when there “happens to be” no dyadic dependence in the data set in hand, it specializes to recently derived results on the behavior of logistic regression with rare events and i.i.d. data. Simulation results suggest that sparse network asymptotics better approximate the finite network distribution of the logit estimator. A short empirical illustration, and additional calibrated Monte Carlo experiments, further illustrate the main theoretical ideas.

Suggested Citation

  • Bryan S. Graham, 2024. "Sparse Network Asymptotics for Logistic Regression Under Possible Misspecification," Econometrica, Econometric Society, vol. 92(6), pages 1837-1868, November.
  • Handle: RePEc:wly:emetrp:v:92:y:2024:i:6:p:1837-1868
    DOI: 10.3982/ECTA19051
    as

    Download full text from publisher

    File URL: https://doi.org/10.3982/ECTA19051
    Download Restriction: no

    File URL: https://libkey.io/10.3982/ECTA19051?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. repec:dau:papers:123456789/10840 is not listed on IDEAS
    2. Isaiah Andrews & James H. Stock & Liyang Sun, 2019. "Weak Instruments in Instrumental Variables Regression: Theory and Practice," Annual Review of Economics, Annual Reviews, vol. 11(1), pages 727-753, August.
    3. Aldous, David J., 1981. "Representations for partially exchangeable arrays of random variables," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 581-598, December.
    4. Fafchamps, Marcel & Gubert, Flore, 2007. "The formation of risk sharing networks," Journal of Development Economics, Elsevier, vol. 83(2), pages 326-350, July.
    5. Marcel Fafchamps & Flore Gubert, 2007. "Risk Sharing and Network Formation," American Economic Review, American Economic Association, vol. 97(2), pages 75-79, May.
    6. Laurent Davezies & Xavier D’haultfœuille & Yannick Guyonvarch, 2021. "Empirical process results for exchangeable arrays," Post-Print hal-04430851, HAL.
    7. Konrad Menzel, 2016. "Inference for Games with Many Players," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 83(1), pages 306-337.
    8. Cattaneo, Matias D. & Crump, Richard K. & Jansson, Michael, 2014. "Small Bandwidth Asymptotics For Density-Weighted Average Derivatives," Econometric Theory, Cambridge University Press, vol. 30(1), pages 176-200, February.
    9. Roussille, Nina & Scuderi, Benjamin, 2023. "Bidding for Talent: A Test of Conduct in a High-Wage Labor Market," IZA Discussion Papers 16352, Institute of Labor Economics (IZA).
    10. repec:dau:papers:123456789/4392 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bryan S. Graham, 2020. "Sparse network asymptotics for logistic regression," Papers 2010.04703, arXiv.org.
    2. Graham, Bryan S. & Niu, Fengshi & Powell, James L., 2024. "Kernel density estimation for undirected dyadic data," Journal of Econometrics, Elsevier, vol. 240(2).
    3. Bryan S. Graham, 2019. "Network Data," CeMMAP working papers CWP71/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Bryan S. Graham, 2019. "Network Data," Papers 1912.06346, arXiv.org.
    5. Bryan S. Graham, 2019. "Dyadic Regression," Papers 1908.09029, arXiv.org.
    6. Harold D. Chiang & Kengo Kato & Yuya Sasaki, 2023. "Inference for High-Dimensional Exchangeable Arrays," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(543), pages 1595-1605, July.
    7. Alejandro Sanchez-Becerra, 2022. "The Network Propensity Score: Spillovers, Homophily, and Selection into Treatment," Papers 2209.14391, arXiv.org.
    8. Rute M. Caeiro & Pedro C. Vicente, 2020. "Knowledge of vitamin A deficiency and crop adoption: Evidence from a field experiment in Mozambique," Agricultural Economics, International Association of Agricultural Economists, vol. 51(2), pages 175-190, March.
    9. Sylvain Dessy & Luca Tiberti & Marco Tiberti & David Zoundi, 2024. "Coping with Drought in Village Economies: The Role of Polygyny," Working Papers - Economics wp2024_13.rdf, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
    10. Konstantinos Matakos & Dimitrios Minos & Ari Perdana & Elizabeth Radin, 2022. "“Dragon boating” alone? Community ties and systemic income shocks," Journal of International Development, John Wiley & Sons, Ltd., vol. 34(1), pages 55-81, January.
    11. Cui Zhang & Dandan Zhang, 2023. "Spatial Interactions and the Spread of COVID-19: A Network Perspective," Computational Economics, Springer;Society for Computational Economics, vol. 62(1), pages 383-405, June.
    12. Fe, Hao, 2023. "Social networks and consumer behavior: Evidence from Yelp," Journal of Economic Behavior & Organization, Elsevier, vol. 209(C), pages 1-14.
    13. Sriroop Chaudhuri & Mimi Roy & Louis M. McDonald & Yves Emendack, 2021. "Reflections on farmers’ social networks: a means for sustainable agricultural development?," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 23(3), pages 2973-3008, March.
    14. Scognamillo, Antonio & Song, Chun & Ignaciuk, Adriana, 2023. "No man is an Island: A spatially explicit approach to measure development resilience," World Development, Elsevier, vol. 171(C).
    15. Rama Lionel Ngenzebuke & Bram De Rock & Philip Verwimp, 2018. "The power of the family: kinship and intra-household decision making in rural Burundi," Review of Economics of the Household, Springer, vol. 16(2), pages 323-346, June.
    16. Kerui Du & Qilin Huang & Presley K. Wesseh, 2025. "Domestic Pollution Havens: Linking Interregional Capital Flight and Water Pollution Regulation in China," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 88(1), pages 125-161, January.
    17. Chakraborty, Tanika & Pandey, Manish, 2021. "Temporary International Migration, Shocks and Informal Insurance: Analysis using panel data," GLO Discussion Paper Series 759, Global Labor Organization (GLO).
    18. Bryan S. Graham, 2017. "An econometric model of network formation with degree heterogeneity," CeMMAP working papers 08/17, Institute for Fiscal Studies.
    19. Renaud Bourlès & Yann Bramoullé & Eduardo Perez-Richet, 2021. "Altruism and Risk Sharing in Networks," Journal of the European Economic Association, European Economic Association, vol. 19(3), pages 1488-1521.
    20. Carayol, Nicolas & Bergé, Laurent & Cassi, Lorenzo & Roux, Pascale, 2019. "Unintended triadic closure in social networks: The strategic formation of research collaborations between French inventors," Journal of Economic Behavior & Organization, Elsevier, vol. 163(C), pages 218-238.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:emetrp:v:92:y:2024:i:6:p:1837-1868. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/essssea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.