IDEAS home Printed from https://ideas.repec.org/p/nbr/nberwo/20341.html
   My bibliography  Save this paper

An econometric model of link formation with degree heterogeneity

Author

Listed:
  • Bryan S. Graham

Abstract

I introduce 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). Like in fixed effects panel data analyses, the joint distribution of observed and unobserved agent-level characteristics is left unrestricted. Two estimators for the (common) homophily parameter, `beta_0`, 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 second, joint maximum likelihood (JML), estimator treats the degree heterogeneity ` {A_(i0)}_(i=1)^N` as additional (incidental) parameters to be estimated. The TL estimate is consistent under both sparse and dense graph sequences, whereas consistency of the JML estimate is shown only under dense graph sequences.

Suggested Citation

  • Bryan S. Graham, 2014. "An econometric model of link formation with degree heterogeneity," NBER Working Papers 20341, National Bureau of Economic Research, Inc.
    • Handle: RePEc:nbr:nberwo:20341
    Note: DEV IO LS TWP
    as

    Download full text from publisher

    File URL: http://www.nber.org/papers/w20341.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Fernández-Val, Iván & Weidner, Martin, 2016. "Individual and time effects in nonlinear panel models with large N, T," Journal of Econometrics, Elsevier, vol. 192(1), pages 291-312.
    2. Paul Goldsmith-Pinkham & Guido W. Imbens, 2013. "Social Networks and the Identification of Peer Effects," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(3), pages 253-264, July.
    3. Manski, Charles F, 1987. "Semiparametric Analysis of Random Effects Linear Models from Binary Panel Data," Econometrica, Econometric Society, vol. 55(2), pages 357-362, March.
    4. Orazio Attanasio & Abigail Barr & Juan Camilo Cardenas & Garance Genicot & Costas Meghir, 2012. "Risk Pooling, Risk Preferences, and Social Networks," American Economic Journal: Applied Economics, American Economic Association, vol. 4(2), pages 134-167, April.
    5. Jackson, Matthew O. & Wolinsky, Asher, 1996. "A Strategic Model of Social and Economic Networks," Journal of Economic Theory, Elsevier, vol. 71(1), pages 44-74, October.
    6. Whitney K. Newey & Frank Windmeijer, 2009. "Generalized Method of Moments With Many Weak Moment Conditions," Econometrica, Econometric Society, vol. 77(3), pages 687-719, May.
    7. Áureo de Paula & Seth Richards-Shubik & Elie Tamer, 2015. "Identification of preferences in network formation games," CeMMAP working papers CWP29/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Bloch, Francis & Jackson, Matthew O., 2007. "The formation of networks with transfers among players," Journal of Economic Theory, Elsevier, vol. 133(1), pages 83-110, March.
    9. Fafchamps, Marcel & Gubert, Flore, 2007. "The formation of risk sharing networks," Journal of Development Economics, Elsevier, vol. 83(2), pages 326-350, July.
    10. Hoff P.D. & Raftery A.E. & Handcock M.S., 2002. "Latent Space Approaches to Social Network Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1090-1098, December.
    11. Blundell,Richard & Newey,Whitney & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521692106, September.
    12. Benjamin Golub & Matthew O. Jackson, 2012. "How Homophily Affects the Speed of Learning and Best-Response Dynamics," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 127(3), pages 1287-1338.
    13. Bryan S. Graham, 2015. "Methods of Identification in Social Networks," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 465-485, August.
    14. Venkatesh Bala & Sanjeev Goyal, 2000. "A Noncooperative Model of Network Formation," Econometrica, Econometric Society, vol. 68(5), pages 1181-1230, September.
    15. repec:dau:papers:123456789/4392 is not listed on IDEAS
    16. Elie Tamer, 2003. "Incomplete Simultaneous Discrete Response Model with Multiple Equilibria," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 70(1), pages 147-165.
    17. repec:dau:papers:123456789/10840 is not listed on IDEAS
    18. Jeremy T. Fox & Patrick Bajari, 2013. "Measuring the Efficiency of an FCC Spectrum Auction," American Economic Journal: Microeconomics, American Economic Association, vol. 5(1), pages 100-146, February.
    19. Jackson, Matthew O. & López-Pintado, Dunia, 2013. "Diffusion and contagion in networks with heterogeneous agents and homophily," Network Science, Cambridge University Press, vol. 1(1), pages 49-67, April.
    20. Montgomery, James D, 1991. "Social Networks and Labor-Market Outcomes: Toward an Economic Analysis," American Economic Review, American Economic Association, vol. 81(5), pages 1407-1418, December.
    21. Jackson Matthew O. & Rogers Brian W., 2007. "Relating Network Structure to Diffusion Properties through Stochastic Dominance," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 7(1), pages 1-16, February.
    22. Federico Ciliberto & Elie Tamer, 2009. "Market Structure and Multiple Equilibria in Airline Markets," Econometrica, Econometric Society, vol. 77(6), pages 1791-1828, November.
    23. Gary Chamberlain, 1980. "Analysis of Covariance with Qualitative Data," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(1), pages 225-238.
    24. Jinyong Hahn & Whitney Newey, 2004. "Jackknife and Analytical Bias Reduction for Nonlinear Panel Models," Econometrica, Econometric Society, vol. 72(4), pages 1295-1319, July.
    25. Coren L. Apicella & Frank W. Marlowe & James H. Fowler & Nicholas A. Christakis, 2012. "Social networks and cooperation in hunter-gatherers," Nature, Nature, vol. 481(7382), pages 497-501, January.
    26. Marcel Fafchamps & Flore Gubert, 2007. "Risk Sharing and Network Formation," American Economic Review, American Economic Association, vol. 97(2), pages 75-79, May.
    27. Blundell,Richard & Newey,Whitney & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521871549, September.
    28. Abhijit Banerjee & Arun G. Chandrasekhar & Esther Duflo & Matthew O. Jackson, 2012. "The Diffusion of Microfinance," NBER Working Papers 17743, National Bureau of Economic Research, Inc.
    29. Honore, Bo E. & Powell, James L., 1994. "Pairwise difference estimators of censored and truncated regression models," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 241-278.
    30. Nicholas A. Christakis & James H. Fowler & Guido W. Imbens & Karthik Kalyanaraman, 2010. "An Empirical Model for Strategic Network Formation," NBER Working Papers 16039, National Bureau of Economic Research, Inc.
    31. Bryan S. Graham, 2008. "Identifying Social Interactions Through Conditional Variance Restrictions," Econometrica, Econometric Society, vol. 76(3), pages 643-660, May.
    32. Marijtje A. J. van Duijn & Tom A. B. Snijders & Bonne J. H. Zijlstra, 2004. "p2: a random effects model with covariates for directed graphs," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 58(2), pages 234-254, May.
    33. Arun G. Chandrasekhar & Matthew O. Jackson, 2014. "Tractable and Consistent Random Graph Models," NBER Working Papers 20276, National Bureau of Economic Research, Inc.
    34. Joachim De Weerdt & Marcel Fafchamps, 2011. "Social Identity and the Formation of Health Insurance Networks," Journal of Development Studies, Taylor & Francis Journals, vol. 47(8), pages 1152-1177, June.
    35. Blundell,Richard & Newey,Whitney K. & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521871532, September.
    36. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-681, May.
    37. Karyne B. Charbonneau, 2014. "Multiple Fixed Effects in Binary Response Panel Data Models," Staff Working Papers 14-17, Bank of Canada.
    38. J. M. C. Santos Silva & Silvana Tenreyro, 2006. "The Log of Gravity," The Review of Economics and Statistics, MIT Press, vol. 88(4), pages 641-658, November.
    39. Ting Yan & Jinfeng Xu, 2013. "A central limit theorem in the β-model for undirected random graphs with a diverging number of vertices," Biometrika, Biometrika Trust, vol. 100(2), pages 519-524.
    40. Blundell,Richard & Newey,Whitney K. & Persson,Torsten (ed.), 2007. "Advances in Economics and Econometrics," Cambridge Books, Cambridge University Press, number 9780521692090, September.
    41. Angelo Mele, 2010. "A structural model of segregation in social networks," CeMMAP working papers CWP32/10, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    42. Brian Lai & Dan Reiter, 2000. "Democracy, Political Similarity, and International Alliances, 1816-1992," Journal of Conflict Resolution, Peace Science Society (International), vol. 44(2), pages 203-227, April.
    43. Ahn, Hyungtaik & Powell, James L., 1993. "Semiparametric estimation of censored selection models with a nonparametric selection mechanism," Journal of Econometrics, Elsevier, vol. 58(1-2), pages 3-29, July.
    44. Charles F. Manski, 2004. "Social Learning from Private Experiences: The Dynamics of the Selection Problem," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 71(2), pages 443-458.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    3. 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.
    4. Koen Jochmans & Martin Weidner, 2019. "Fixed‐Effect Regressions on Network Data," Econometrica, Econometric Society, vol. 87(5), pages 1543-1560, September.
    5. Bryan S. Graham, 2015. "Methods of Identification in Social Networks," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 465-485, August.
    6. 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.
    7. repec:spo:wpecon:info:hdl:2441/dpido2upv86tqc7td18fd2mna is not listed on IDEAS
    8. 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.
    9. Boucher, Vincent, 2016. "Conformism and self-selection in social networks," Journal of Public Economics, Elsevier, vol. 136(C), pages 30-44.
    10. repec:spo:wpmain:info:hdl:2441/78vacv4udu92eq3fec89svm9uv is not listed on IDEAS
    11. Bryan S. Graham, 2016. "Homophily and transitivity in dynamic network formation," CeMMAP working papers 16/16, Institute for Fiscal Studies.
    12. repec:spo:wpecon:info:hdl:2441/75dbbb2hc596np6q8flqf6i79k is not listed on IDEAS
    13. repec:hal:wpspec:info:hdl:2441/75dbbb2hc596np6q8flqf6i79k is not listed on IDEAS
    14. repec:hal:wpspec:info:hdl:2441/dpido2upv86tqc7td18fd2mna is not listed on IDEAS
    15. 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.
    16. Zenou, Yves & picard, pierre & Patacchini, Eleonora, 2015. "Urban Social Structure, Social Capital and Spatial Proximity," CEPR Discussion Papers 10501, C.E.P.R. Discussion Papers.
    17. repec:spo:wpmain:info:hdl:2441/dpido2upv86tqc7td18fd2mna is not listed on IDEAS
    18. repec:hal:spmain:info:hdl:2441/78vacv4udu92eq3fec89svm9uv is not listed on IDEAS
    19. repec:hal:spmain:info:hdl:2441/dpido2upv86tqc7td18fd2mna is not listed on IDEAS
    20. Harsh Gupta & Mason A. Porter, 2020. "Mixed Logit Models and Network Formation," Papers 2006.16516, arXiv.org, revised Aug 2022.

    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, 2017. "An Econometric Model of Network Formation With Degree Heterogeneity," Econometrica, Econometric Society, vol. 85, pages 1033-1063, July.
    2. Bryan S. Graham, 2015. "An econometric model of link formation with degree heterogeneity," CeMMAP working papers 43/15, Institute for Fiscal Studies.
    3. Bryan S. Graham, 2017. "An econometric model of network formation with degree heterogeneity," CeMMAP working papers 08/17, Institute for Fiscal Studies.
    4. Bryan S. Graham, 2019. "Network Data," Papers 1912.06346, arXiv.org.
    5. Bryan S. Graham, 2015. "Methods of Identification in Social Networks," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 465-485, August.
    6. Bryan S. Graham, 2019. "Network Data," CeMMAP working papers CWP71/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Áureo de Paula, 2015. "Econometrics of network models," CeMMAP working papers CWP52/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Matthew O. Jackson & Brian W. Rogers & Yves Zenou, 2016. "Networks: An Economic Perspective," Papers 1608.07901, arXiv.org.
    9. Bryan S. Graham, 2016. "Homophily and Transitivity in Dynamic Network Formation," NBER Working Papers 22186, National Bureau of Economic Research, Inc.
    10. Tiziano Arduini & Eleonora Patacchini & Edoardo Rainone, 2015. "Parametric and Semiparametric IV Estimation of Network Models with Selectivity," EIEF Working Papers Series 1509, Einaudi Institute for Economics and Finance (EIEF), revised Oct 2015.
    11. Irene Botosaru & Chris Muris & Krishna Pendakur, 2020. "Intertemporal Collective Household Models: Identification in Short Panels with Unobserved Heterogeneity in Resource Shares," Department of Economics Working Papers 2020-09, McMaster University.
    12. Jackson, Matthew O. & Zenou, Yves, 2015. "Games on Networks," Handbook of Game Theory with Economic Applications,, Elsevier.
    13. Boucher, Vincent & Fortin, Bernard, 2015. "Some Challenges in the Empirics of the Effects of Networks," IZA Discussion Papers 8896, Institute of Labor Economics (IZA).
    14. Botosaru, Irene & Muris, Chris & Pendakur, Krishna, 2023. "Identification of time-varying transformation models with fixed effects, with an application to unobserved heterogeneity in resource shares," Journal of Econometrics, Elsevier, vol. 232(2), pages 576-597.
    15. Bryan S. Graham, 2016. "Homophily and transitivity in dynamic network formation," CeMMAP working papers 16/16, Institute for Fiscal Studies.
    16. Schumann, Martin & Severini, Thomas A. & Tripathi, Gautam, 2021. "Integrated likelihood based inference for nonlinear panel data models with unobserved effects," Journal of Econometrics, Elsevier, vol. 223(1), pages 73-95.
    17. Pakel, Cavit, 2019. "Bias reduction in nonlinear and dynamic panels in the presence of cross-section dependence," Journal of Econometrics, Elsevier, vol. 213(2), pages 459-492.
    18. Irene Botosaru & Chris Muris, 2017. "Binarization for panel models with fixed effects," CeMMAP working papers 31/17, Institute for Fiscal Studies.
    19. Aradillas-López, Andrés & Rosen, Adam M., 2022. "Inference in ordered response games with complete information," Journal of Econometrics, Elsevier, vol. 226(2), pages 451-476.
    20. Geert Dhaene & Koen Jochmans, 2015. "Split-panel Jackknife Estimation of Fixed-effect Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 82(3), pages 991-1030.

    More about this item

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:nbr:nberwo:20341. 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: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/nberrus.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.