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An econometric model of link formation with degree heterogeneity

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  • Bryan S. Graham

    (Institute for Fiscal Studies and University of California, Berkeley)

Abstract

I formulate and study a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics (homophily) as well as unrestricted agent level heterogeneity in link surplus (degree heterogeneity). Similar to fixed effects panel data analyses, the joint distribution of observed and unobserved agent-level characteristics is left unrestricted. To motivate the introduction of degree heterogeneity, as well as its fixed effect treatment, I show how its presence can bias conventional homophily measures. Two estimators for the (common) homophily parameter, beta0, are developed and their properties studied under an asymptotic sequence involving a single network growing large. The first,tetrad logit (TL), estimator conditions on a sufficient statistic for the degree heterogeneity. The TL estimator is a fourth-order U-Process minimizer. Although the fourth-order summation in the TL criterion function is over the i = 1...N agents in the network, due to a degeneracy property, the leading variance term of hat-beta_TL is of order 1/n, where n = N*(N-1)/2 equals the number of observed dyads. Using martingale theory, I show that the limiting distribution of hat-beta_TL (appropriately scaled and normalized) is normal. The second, joint maximum likelihood (JML), estimator treats the degree heterogeneity as additional (incidental) parameters to be estimated. The properties of hat-beta_JML are also non-standard due to a parameter space which grows with the size of the network. Adapting and extending recent results from random graph theory and non-linear panel data analysis (e.g., Chatterjee, Diaconis and Sly, 2011; Hahn and Newey, 2004), I show that the limit distribution of hat-beta_JML is also normal, but contains a bias term. Accurate inference necessitates bias-correction. The TL estimate is consistent under sparse graph sequences, where the number of links per agent is small relative to the total number of agents, as well as dense graphs sequences, where the number of links per agent is proportional to the total number of agents in the limit. Consistency of the JML estimate, in contrast, is shown only under dense graph sequences. The finite sample properties of hat-beta_TL and hat-beta_JML are explored in a series of Monte Carlo experiments.

Suggested Citation

  • Bryan S. Graham, 2015. "An econometric model of link formation with degree heterogeneity," CeMMAP working papers CWP43/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:43/15
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    Cited by:

    1. Koen Jochmans, 2018. "Semiparametric Analysis of Network Formation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 36(4), pages 705-713, October.
    2. Bryan S. Graham, 2016. "Homophily and transitivity in dynamic network formation," CeMMAP working papers CWP16/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Koen Jochmans, 2017. "Two-Way Models for Gravity," The Review of Economics and Statistics, MIT Press, vol. 99(3), pages 478-485, July.
    4. Patacchini, Eleonora & Picard, Pierre M & Zenou, Yves, 2015. "Urban Social Structure, Social Capital and Spatial Proximity," CEPR Discussion Papers 10501, C.E.P.R. Discussion Papers.
    5. González-Díaz, Julio & Palacios-Huerta, Ignacio, 2016. "Cognitive performance in competitive environments: Evidence from a natural experiment," Journal of Public Economics, Elsevier, vol. 139(C), pages 40-52.
    6. Matthew O. Jackson, 2014. "Networks in the Understanding of Economic Behaviors," Journal of Economic Perspectives, American Economic Association, vol. 28(4), pages 3-22, Fall.
    7. Koen Jochmans & Martin Weidner, 2019. "Fixed‐Effect Regressions on Network Data," Econometrica, Econometric Society, vol. 87(5), pages 1543-1560, September.
    8. Bryan S. Graham, 2015. "Methods of Identification in Social Networks," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 465-485, August.
    9. Marco Battaglini & Valerio Leone Sciabolazza & Eleonora Patacchini, 2020. "Effectiveness of Connected Legislators," American Journal of Political Science, John Wiley & Sons, vol. 64(4), pages 739-756, October.
    10. Chih‐Sheng Hsieh & Lung‐Fei Lee & Vincent Boucher, 2020. "Specification and estimation of network formation and network interaction models with the exponential probability distribution," Quantitative Economics, Econometric Society, vol. 11(4), pages 1349-1390, November.
    11. Koen Jochmans, 2017. "Two-Way Models for Gravity," The Review of Economics and Statistics, MIT Press, vol. 99(3), pages 478-485, July.
    12. Boucher, Vincent, 2016. "Conformism and self-selection in social networks," Journal of Public Economics, Elsevier, vol. 136(C), pages 30-44.
    13. Harsh Gupta & Mason A. Porter, 2020. "Mixed Logit Models and Network Formation," Papers 2006.16516, arXiv.org, revised Mar 2021.
    14. Yann Algan & Quoc-Anh Do & Nicolò Dalvit & Alexis Le Chapelain & Yves Zenou, 2015. "How Social Networks Shape Our Beliefs: A Natural Experiment among Future French Politicians," Sciences Po publications info:hdl:2441/78vacv4udu9, Sciences Po.

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    JEL classification:

    • C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models
    • C35 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions

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