IDEAS home Printed from https://ideas.repec.org/a/spr/sistpr/v17y2014i3p221-244.html
   My bibliography  Save this article

On stationarity and second-order properties of bilinear random fields

Author

Listed:
  • Abdelouahab Bibi
  • Karima Kimouche

Abstract

One-dimensional indexed bilinear (BL) models are widely used for modeling non Gaussian time series. Extending BL models to multidimensional indexed (spatial) SBL one, yields a novel class of models which are capable of taking into account the important characteristic of non Gaussianity and spatiality behavior. Our main contribution here is to extend various results on BL processes to SBL one. Our attention is thus focussed on the probabilistic structure of some SBLmodels. So, we establish necessary and sufficient conditions for the existence of regular stationary and ergodic solutions in term of their transfer functions. As a consequence, we observe that the second order structure is similar to a spatial ARMA model with some uncorrelated white noise. So, it is necessary to look into higher-order moments in order to distinguish between linear and nonlinear random fields. Our study may be applied to GARCH or ARMA random fields. Copyright Springer Science+Business Media Dordrecht 2014

Suggested Citation

  • Abdelouahab Bibi & Karima Kimouche, 2014. "On stationarity and second-order properties of bilinear random fields," Statistical Inference for Stochastic Processes, Springer, vol. 17(3), pages 221-244, October.
    • Handle: RePEc:spr:sistpr:v:17:y:2014:i:3:p:221-244
    DOI: 10.1007/s11203-014-9102-9
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11203-014-9102-9
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11203-014-9102-9?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Dimitriou-Fakalou, Chrysoula, 2009. "Modified Gaussian likelihood estimators for ARMA models on," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4149-4175, December.
    2. Jiin‐Huarng Guo & L. Billard, 1998. "Some Inference Results for Causal Autoregressive Processes on a Plane," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(6), pages 681-691, November.
    3. Yao, Qiwei & Brockwell, Peter J, 2006. "Gaussian maximum likelihood estimation for ARMA models II: spatial processes," LSE Research Online Documents on Economics 5416, London School of Economics and Political Science, LSE Library.
    4. Qiwei Yao & Peter J. Brockwell, 2006. "Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 857-875, November.
    5. Korezlioglu, Hayri & Loubaton, Philippe, 1986. "Spectral factorization of wide sense stationary processes on 2," Journal of Multivariate Analysis, Elsevier, vol. 19(1), pages 24-47, June.
    6. Yao, Qiwei & Brockwell, Peter J., 2006. "Gaussian maximum likelihood estimation for ARMA models I: time series," LSE Research Online Documents on Economics 5825, London School of Economics and Political Science, LSE Library.
    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. repec:esx:essedp:767 is not listed on IDEAS
    2. Dimitriou-Fakalou, Chrysoula, 2019. "On accepting the edge-effect (for the inference of ARMA-type processes in Z2)," Econometrics and Statistics, Elsevier, vol. 10(C), pages 53-70.
    3. Yong Bao, 2018. "The asymptotic covariance matrix of the QMLE in ARMA models," Econometric Reviews, Taylor & Francis Journals, vol. 37(4), pages 309-324, April.
    4. Rosa Espejo & Nikolai Leonenko & Andriy Olenko & María Ruiz-Medina, 2015. "On a class of minimum contrast estimators for Gegenbauer random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 657-680, December.
    5. Zheng, Tingguo & Xiao, Han & Chen, Rong, 2015. "Generalized ARMA models with martingale difference errors," Journal of Econometrics, Elsevier, vol. 189(2), pages 492-506.
    6. Qin Shao & Lijian Yang, 2017. "Oracally efficient estimation and consistent model selection for auto-regressive moving average time series with trend," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 507-524, March.
    7. Tianhao Wang & Yingcun Xia, 2015. "Whittle Likelihood Estimation of Nonlinear Autoregressive Models With Moving Average Residuals," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1083-1099, September.
    8. Robinson, Peter M., 2011. "Inference on power law spatial trends (Running Title: Power Law Trends)," LSE Research Online Documents on Economics 58100, London School of Economics and Political Science, LSE Library.
    9. Norkutė, Milda & Westerlund, Joakim, 2019. "The factor analytical method for interactive effects dynamic panel models with moving average errors," Econometrics and Statistics, Elsevier, vol. 11(C), pages 83-104.
    10. Abdelkamel Alj & Rajae Azrak & Christophe Ley & Guy Mélard, 2017. "Asymptotic Properties of QML Estimators for VARMA Models with Time-dependent Coefficients," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 617-635, September.
    11. Zheng, Tingguo & Chen, Rong, 2017. "Dirichlet ARMA models for compositional time series," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 31-46.
    12. Bastian Schäfer, 2021. "Bandwidth selection for the Local Polynomial Double Conditional Smoothing under Spatial ARMA Errors," Working Papers CIE 146, Paderborn University, CIE Center for International Economics.
    13. Sheena Yu-Hsien Kao & Anil K. Bera, 2018. "Testing spatial regression models under nonregular conditions," Empirical Economics, Springer, vol. 55(1), pages 85-111, August.
    14. Ke Zhu & Shiqing Ling, 2015. "LADE-Based Inference for ARMA Models With Unspecified and Heavy-Tailed Heteroscedastic Noises," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 784-794, June.
    15. Ke Zhu, 2018. "Statistical inference for autoregressive models under heteroscedasticity of unknown form," Papers 1804.02348, arXiv.org, revised Aug 2018.
    16. Yang, Yaxing & Ling, Shiqing & Wang, Qiying, 2022. "Consistency of global LSE for MA(1) models," Statistics & Probability Letters, Elsevier, vol. 182(C).
    17. Hernández, Juan R., 2016. "Unit Root Testing in ARMA Models: A Likelihood Ratio Approach," MPRA Paper 100857, University Library of Munich, Germany.
    18. Robinson, P.M., 2011. "Asymptotic theory for nonparametric regression with spatial data," Journal of Econometrics, Elsevier, vol. 165(1), pages 5-19.
    19. Peter M Robinson, 2011. "Inference on Power Law Spatial Trends (Running Title: Power Law Trends)," STICERD - Econometrics Paper Series 556, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    20. Huang, Lei & Jiang, Hui & Wang, Huixia, 2019. "A novel partial-linear single-index model for time series data," Computational Statistics & Data Analysis, Elsevier, vol. 134(C), pages 110-122.
    21. Dimitriou-Fakalou, Chrysoula, 2009. "Modified Gaussian likelihood estimators for ARMA models on," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4149-4175, December.

    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:spr:sistpr:v:17:y:2014:i:3:p:221-244. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.