Skip to main content
SpringerLink
Log in

Bootstrapping Network Autoregressive Models for Testing Linearity

  • Chapter
  • First Online:
Data Science in Applications

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1084))

Abstract

We develop methodology for network data with special attention to epidemic network spatio-temporal structures. We provide estimation methodology for linear network autoregressive models for both continuous and count multivariate time series. A study of non-linear models for inference under the assumption of known network structure is provided. We propose a family of test statistics for testing linearity of the imposed model. In particular, we compare empirically two bootstrap versions of a supremum-type quasi-score test. Synthetic data are employed to demonstrate the validity of the methodological results. Finally, an epidemic application of the proposed methodology to daily COVID-19 cases detected on province-level geographical network in Italy complements the work.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Dataset available at https://github.com/pcm-dpc/COVID-19/blob/master/dati-province/dpc-covid19-ita-province.csv.

References

  1. Ahmad, A., Francq, C.: Poisson QMLE of count time series models. J. Time Ser. Anal. 37, 291–314 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, D.W.K., Ploberger, W.: Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Armillotta, M., Fokianos, K.: Poisson network autoregression (2021). arXiv:2104.06296

  4. Armillotta, M., Fokianos, K.: Testing linearity for network autoregressive models (2022). arXiv:2202.03852

  5. Boos, D.D.: On generalized score tests. Amer. Stat. 46, 327–333 (1992)

    Google Scholar 

  6. Bracher, J., Held, L.: Endemic-epidemic models with discrete-time serial interval distributions for infectious disease prediction. Inter. J. Forecast. in press (2020)

    Google Scholar 

  7. Christou, V., Fokianos, K.: Quasi-likelihood inference for negative binomial time series models. J. Time Ser. Anal. 35, 55–78 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Christou, V., Fokianos, K.: Estimation and testing linearity for non-linear mixed Poisson autoregressions. Electron. J. Stat. 9, 1357–1377 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cliff, A., Ord, J.K.: Space-time modelling with an application to regional forecasting. Trans. Inst. Brit. Geograph. 119–128 (1975)

    Google Scholar 

  10. Csardi, G., Nepusz, T.: The igraph software package for complex network research. Inter. J. Complex Syst. 1695 (2006). https://igraph.org

  11. Davies, R.B.: Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43 (1987)

    MathSciNet  MATH  Google Scholar 

  12. Davis, R.A., Liu, H.: Theory and inference for a class of nonlinear models with application to time series of counts. Stat. Sinica 26, 1673–1707 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Davis, R.A., Fokianos, K., Holan, S.H., Joe, H., Livsey, J., Lund, R., Pipiras, V., Ravishanker, N.: Count time series: a methodological review. J. Amer. Stat. Assoc. 116, 1533–1547 (2021)

    Article  MathSciNet  Google Scholar 

  14. Douc, R., Fokianos, K., Moulines, E.: Asymptotic properties of quasi-maximum likelihood estimators in observation-driven time series models. Electr. J. Stat. 11, 2707–2740 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Doukhan, P.: Mixing. Lecture Notes in Statistics, vol. 85. Springer, New York (1994)

    Google Scholar 

  16. Fan, J., Yao, Q.: Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York (2003)

    Google Scholar 

  17. Fokianos, K.: Multivariate count time series modelling. To appear in Econometrics and Statistics (2022)

    Google Scholar 

  18. Fokianos, K., Neumann, M.H.: A goodness-of-fit test for Poisson count processes. Electr. J. Stat. 7, 793–819 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Fokianos, K., Tjøstheim, D.: Nonlinear Poisson autoregression. Ann. Inst. Stat. Math. 64, 1205–1225 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fokianos, K., Rahbek, A., Tjøstheim, D.: Poisson autoregression. J. Amer. Stat. Assoc. 104, 1430–1439 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fokianos, K., Støve, B., Tjøstheim, D., Doukhan, P.: Multivariate count autoregression. Bernoulli 26, 471–499 (2020)

    Google Scholar 

  22. Francq, C., Horvath, L., Zakoïan, J.M.: Sup-tests for linearity in a general nonlinear AR(1) model. Econom. Theory 26, 965–993 (2010)

    Google Scholar 

  23. Gao, J.: Nonlinear Time Series: Semiparametric and Nonparametric Methods. CRC Press, Boca Raton (2007)

    Book  MATH  Google Scholar 

  24. Gao, J., King, M., Lu, Z., Tjøstheim, D.: Specification testing in nonlinear and nonstationary time series autoregression. Ann. Stat. 37, 3893–3928 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gorgi, P.: Beta-negative binomial auto-regressions for modelling integer-valued time series with extreme observations. J. R. Stat. Soc.: Ser. B 82, 1325–1347 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gourieroux, C., Monfort, A., Trognon, A.: Pseudo maximum likelihood methods: theory. Econometrica 681–700 (1984)

    Google Scholar 

  27. Haggan, V., Ozaki, T.: Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika 68(1), 189–196 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hansen, B.E.: Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Heyde, C.C.: Quasi-likelihood and its Application. A General Approach to Optimal Parameter Estimation. Springer Series in Statistics. Springer, New York (1997)

    Google Scholar 

  30. Knight, M., Nunes, M., Nason, G.: Modelling, detrending and decorrelation of network time series (2016). arXiv:1603.03221

  31. Knight, M., Leeming, K., Nason, G., Nunes, M.: Generalized network autoregressive processes and the GNAR package. J. Stat. Softw. 96, 1–36 (2020). https://www.jstatsoft.org/v096/i05

  32. Kolaczyk, E.D., Csárdi, G.: Statistical Analysis of Network Data with R, vol. 65. Springer, Berlin (2014)

    Google Scholar 

  33. Li, G., Li, W.K.: Testing a linear time series model against its threshold extension. Biometrika 98, 243–250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lim, K., Tong, H.: Threshold autoregressions, limit cycles, and data. J. R. Stat. Soc. Ser B 42, 245–92 (1980)

    MATH  Google Scholar 

  35. Luukkonen, R., Saikkonen, P., Teräsvirta, T.: Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Martin, R.L., Oeppen, J.: The identification of regional forecasting models using space: time correlation functions. Trans. Inst. Brit. Geograph. 95–118 (1975)

    Google Scholar 

  37. McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman & Hall, London (1989)

    Book  MATH  Google Scholar 

  38. Neumann, M.: Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17, 1268–1284 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nowicki, K., Snijders, T.A.B.: Estimation and prediction for stochastic blockstructures. J. Amer. Stat. Assoc. 96, 1077–1087 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pedeli, X., Karlis, D.: A bivariate INAR (1) process with application. Stat. Modell. 11, 325–349 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pedeli, X., Karlis, D.: On composite likelihood estimation of a multivariate INAR (1) model. J. Time Ser. Anal. 34, 206–220 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pedeli, X., Karlis, D.: Some properties of multivariate INAR (1) processes. Comput. Stat. & Data Anal. 67, 213–225 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rosenblatt, M.: A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. U. S. A. 42, 43–47 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  44. Teräsvirta, T.: Specification, estimation, and evaluation of smooth transition autoregressive models. J. Amer. Stat. Assoc. 89, 208–218 (1994)

    MATH  Google Scholar 

  45. Teräsvirta, T., Tjøstheim, D., Granger, C.W.J.: Modelling Nonlinear Economic Time Series. Oxford University Press, Oxford (2010)

    Google Scholar 

  46. Tong, H.: Non-linear Time Series: A Dynamical System Approach. Oxford University Press, Oxford (1990)

    Google Scholar 

  47. Wang, C., Liu, H., Yao, J.F., Davis, R.A., Li, W.K.: Self-excited threshold Poisson autoregression. J. Amer. Stat. Assoc. 109, 777–787 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wang, Y.J., Wong, G.Y.: Stochastic blockmodels for directed graphs. J. Amer. Stat. Assoc. 82, 8–19 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wasserman, S., Faust, K., et al.: Social Network Analysis: Methods and Applications, vol. 8. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  50. Wedderburn, R.W.: Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61(3), 439–447 (1974)

    MathSciNet  MATH  Google Scholar 

  51. Zhao, Y., Levina, E., Zhu, J., et al.: Consistency of community detection in networks under degree-corrected stochastic block models. Ann. Stat. 40(4), 2266–2292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhou, J., Li, D., Pan, R., Wang, H.: Network GARCH model. Stat. Sin. 30, 1–18 (2020)

    MathSciNet  MATH  Google Scholar 

  53. Zhu, X., Pan, R.: Grouped network vector autoregression. Stat. Sin. 30, 1437–1462 (2020)

    MathSciNet  MATH  Google Scholar 

  54. Zhu, X., Pan, R., Li, G., Liu, Y., Wang, H.: Network vector autoregression. Ann. Stat. 45, 1096–1123 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhu, X., Wang, W., Wang, H., Härdle, W.K.: Network quantile autoregression. J. Econometr. 212, 345–358 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been co-financed by the European Regional Development Fund and the Republic of Cyprus through the Research and Innovation Foundation, under the project INFRASTRUCTURES/1216/0017 (IRIDA).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Mirko Armillotta & Konstantinos Fokianos

  2. Department of Electrical and Computer Engineering, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Ioannis Krikidis

Authors
  1. Mirko Armillotta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Konstantinos Fokianos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ioannis Krikidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mirko Armillotta .

Editor information

Editors and Affiliations

  1. Institute of Data Science and Digital Technologies, Vilnius University, Vilnius, Lithuania

    Gintautas Dzemyda

  2. Institute of Data Science and Digital Technologies, Vilnius University, Vilnius, Lithuania

    Jolita Bernatavičienė

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Armillotta, M., Fokianos, K., Krikidis, I. (2023). Bootstrapping Network Autoregressive Models for Testing Linearity. In: Dzemyda, G., Bernatavičienė, J., Kacprzyk, J. (eds) Data Science in Applications. Studies in Computational Intelligence, vol 1084. Springer, Cham. https://doi.org/10.1007/978-3-031-24453-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation