, Volume 84, Issue 4, pp 1068–1096 | Cite as

Contemporaneous Statistics for Estimation in Stochastic Actor-Oriented Co-evolution Models

  • Viviana AmatiEmail author
  • Felix Schönenberger
  • Tom A. B. Snijders


Stochastic actor-oriented models (SAOMs) can be used to analyse dynamic network data, collected by observing a network and a behaviour in a panel design. The parameters of SAOMs are usually estimated by the method of moments (MoM) implemented by a stochastic approximation algorithm, where statistics defining the moment conditions correspond in a natural way to the parameters. Here, we propose to apply the generalized method of moments (GMoM), using more statistics than parameters. We concentrate on statistics depending jointly on the network and the behaviour, because of the importance of their interdependence, and propose to add contemporaneous statistics to the usual cross-lagged statistics. We describe the stochastic algorithm developed to approximate the GMoM solution. A small simulation study supports the greater statistical efficiency of the GMoM estimator compared to the MoM.


generalized method of moments networks behaviour panel data stochastic actor-oriented model stochastic approximation 



  1. Amati, V., Schönenberger, F., & Snijders, T. A. (2015). Estimation of stochastic actor-oriented models for the evolution of networks by generalized method of moments. Journal de la Société Française de Statistique, 156(3), 140–165.Google Scholar
  2. Block, P. (2015). Reciprocity, transitivity, and the mysterious three-cycle. Social Networks, 40, 163–173.Google Scholar
  3. Bollen, K. A., Kolenikov, S., & Bauldry, S. (2014). Model-implied instrumental variable—generalized method of moments (MIIV-GMM) estimators for latent variable models. Psychometrika, 79(1), 20–50.PubMedGoogle Scholar
  4. Breusch, T., Qian, H., Schmidt, P., & Wyhowski, D. (1999). Redundancy of moment conditions. Journal of Econometrics, 91(1), 89–111.Google Scholar
  5. Burguete, J. F., Ronald Gallant, A., & Souza, G. (1982). On unification of the asymptotic theory of nonlinear econometric models. Econometric Reviews, 1(2), 151–190.Google Scholar
  6. Burk, W. J., Kerr, M., & Stattin, H. (2008). The co-evolution of early adolescent friendship networks, school involvement, and delinquent behaviors. Revue française de sociologie, 49(3), 499–522.Google Scholar
  7. Ebbers, J. J., & Wijnberg, N. M. (2010). Disentangling the effects of reputation and network position on the evolution of alliance networks. Strategic Organization, 8(3), 255–275.Google Scholar
  8. Gallant, A. R., Hsieh, D., & Tauchen, G. (1997). Estimation of stochastic volatility models with diagnostics. Journal of econometrics, 81(1), 159–192.Google Scholar
  9. Hall, A. R. (2005). Generalized method of moments. Oxford: Oxford University Press.Google Scholar
  10. Hansen, L. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054.Google Scholar
  11. Hansen, L. P., & Singleton, K. J. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica, 50(5), 1269–1286.Google Scholar
  12. Haynie, D. L., Doogan, N. J., & Soller, B. (2014). Gender, friendship networks, and delinquency: A dynamic network approach. Criminology, 52(4), 688–722.PubMedPubMedCentralGoogle Scholar
  13. Holland, P. W., & Leinhardt, S. (1977). A dynamic model for social networks. Journal of Mathematical Sociology, 5(1), 5–20.Google Scholar
  14. Hunter, D. R. (2007). Curved exponential family models for social networks. Social Networks, 29, 216–230.PubMedPubMedCentralGoogle Scholar
  15. Kim, J.-S., & Frees, E. W. (2007). Multilevel modeling with correlated effects. Psychometrika, 72(4), 505–533.Google Scholar
  16. Koskinen, J. H., & Snijders, T. A. B. (2007). Bayesian inference for dynamic social network data. Journal of Statistical Planning and Inference, 13, 3930–3938.Google Scholar
  17. Luce, R., & Suppes, P. (1965). Preference, utility, and subjective probability. Handbook of Mathematical Psychology, 3, 249–410.Google Scholar
  18. Mátyás, L. (1999). Generalized method of moments estimation. Cambridge: Cambridge University Press.Google Scholar
  19. McFadden, D. (1973). Conditional logit analysis of qualitative choice behavior. Oakland: Institute of Urban and Regional Development, University of California.Google Scholar
  20. McPherson, M., Smith-Lovin, L., & Cook, J. M. (2001). Birds of a feather: Homophily in social networks. Annual Review of Sociology, 27(1), 415–444.Google Scholar
  21. Meyer, C. D. (2000). Matrix analysis and applied linear algebra. Philadelphia: SIAM.Google Scholar
  22. Michell, L., & West, P. (1996). Peer pressure to smoke: The meaning depends on the method. Health Education Research, 11(1), 39–49.Google Scholar
  23. Newey, W., & Windmeijer, F. (2009). Generalized method of moments with many weak moment conditions. Econometrica, 77(3), 687–719.Google Scholar
  24. Neyman, J., & Pearson, E. S. (1928). On the use and interpretation of certain test criteria for purposes of statistical inference: Part II. Biometrika, 20, 263–294.Google Scholar
  25. Niezink, N. M. D., & Snijders, T. A. B. (2017). Co-evolution of social networks and continuous actor attributes. The Annals of Applied Statistics, 11(4), 1948–1973.Google Scholar
  26. Niezink, N. M. D., Snijders, T. A. B., & van Duijn, M. A. J. (2019). No longer discrete: Modeling the dynamics of social networks and continuous behavior. Sociological Methodology. Google Scholar
  27. Norris, J. R. (1997). Markov chains. Cambridge: Cambridge University Press.Google Scholar
  28. Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 50(302), 157–175.Google Scholar
  29. Pflug, G. C. (1990). Non-asymptotic confidence bounds for stochastic approximation algorithms with constant step size. Monatshefte für Mathematik, 110(3–4), 297–314.Google Scholar
  30. Polyak, B. T. (1990). A new method of stochastic approximation type. Automation and Remote Control, 51, 937–946.Google Scholar
  31. Ripley, R. M., Snijders, T. A. B., Boda, Z., András, V., & Paulina, P. (2019). Manual for RSiena. Groningen: ICS, Department of Sociology, University of Groningen.Google Scholar
  32. Robbins, H., & Monro, S. (1951). A stochastic approximation method. The Annals of Mathematical Statistics, 22, 400–407.Google Scholar
  33. Ruppert, D. (1988). Efficient estimations from a slowly convergent Robbins–Monro process. Technical report, Cornell University Operations Research and Industrial Engineering.Google Scholar
  34. Schulte, M., Cohen, N. A., & Klein, K. J. (2012). The coevolution of network ties and perceptions of team psychological safety. Organization Science, 23(2), 564–581.Google Scholar
  35. Schweinberger, M., & Snijders, T. A. B. (2007). Markov models for digraph panel data: Monte Carlo-based derivative estimation. Computational Statistics & Data Analysis, 51(9), 4465–4483.Google Scholar
  36. Snijders, T. A. B. (1996). Stochastic actor-oriented models for network change. Journal of Mathematical Sociology, 21(1–2), 149–172.Google Scholar
  37. Snijders, T. A. B. (2001). The statistical evaluation of social network dynamics. Sociological Methodology, 31(1), 361–395.Google Scholar
  38. Snijders, T. A. B. (2005). Models for longitudinal network data. In P. J. C. Conte, J. Scott, & S. Wasserman (Eds.), Models and methods in social network analysis (pp. 215–247). Cambridge: Cambridge University Press.Google Scholar
  39. Snijders, T. A. B. (2017a). Stochastic actor-oriented models for network dynamics. Annual Review of Statistics and Its Application, 4, 343–363.Google Scholar
  40. Snijders, T. A. B. (2017b). Siena algorithms. Technical report, University of Groningen, University of Oxford.
  41. Snijders, T. A. B., Koskinen, J., & Schweinberger, M. (2010a). Maximum likelihood estimation for social network dynamics. The Annals of Applied Statistics, 4(2), 567–588.PubMedPubMedCentralGoogle Scholar
  42. Snijders, T. A. B., & Lomi, A. (2019). Beyond homophily: Incorporating actor variables in statistical network models. Network Science, 7(1), 1–19.Google Scholar
  43. Snijders, T. A. B., Steglich, C. E. G., & Schweinberger, M. (2007). Modeling the co-evolution of networks and behavior. In K. van Montfort, H. Oud, & A. Satorra (Eds.), Longitudinal models in the behavioral and related sciences (pp. 41–71). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  44. Snijders, T. A. B., Van de Bunt, G. G., & Steglich, C. E. G. (2010b). Introduction to stochastic actor-based models for network dynamics. Social Networks, 32(1), 44–60.Google Scholar
  45. Snijders, T. A. B., & van Duijn, M. A. J. (1997). Simulation for statistical inference in dynamic network models. In R. Conte, R. Hegselmann, & P. Terna (Eds.), Simulating social phenomena (pp. 493–512). Berlin: Springer.Google Scholar
  46. Steglich, C. E. G., Snijders, T. A. B., & Pearson, M. (2010). Dynamic networks and behavior: Separating selection from influence. Sociological Methodology, 40(1), 329–393.Google Scholar
  47. Strang, G. (1976). Linear algebra and its applications. New York: Academic Press.Google Scholar
  48. Train, K. E. (2009). Discrete choice methods with simulation. Cambridge: Cambridge University Press.Google Scholar

Copyright information

© The Psychometric Society 2019

Authors and Affiliations

  1. 1.Social Networks Lab, Department of Humanities, Social and Political SciencesETH ZurichZurichSwitzerland
  2. 2.Department of Sociology, Faculty of Behavioral and Social SciencesUniversity of GroningenGroningenThe Netherlands
  3. 3.Nuffield CollegeUniversity of OxfordOxfordUK
  4. 4.Department of StatisticsUniversity of OxfordOxfordUK

Personalised recommendations