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Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

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  • Robert Lund
  • I. V. Basawa

Abstract

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving‐average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one‐step‐ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.

Suggested Citation

  • Robert Lund & I. V. Basawa, 2000. "Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(1), pages 75-93, January.
    • Handle: RePEc:bla:jtsera:v:21:y:2000:i:1:p:75-93
    DOI: 10.1111/1467-9892.00174
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    Cited by:

    1. Yorghos Tripodis & Jeremy Penzer, 2009. "Modelling time series with season-dependent autocorrelation structure," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 28(7), pages 559-574.
    2. Anderson, Paul L. & Kavalieris, Laimonis & Meerschaert, Mark M., 2008. "Innovations algorithm asymptotics for periodically stationary time series with heavy tails," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 94-116, January.
    3. Caporin, Massimiliano & Preś, Juliusz, 2012. "Modelling and forecasting wind speed intensity for weather risk management," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3459-3476.
    4. Daniel Dzikowski & Carsten Jentsch, 2024. "Structural Periodic Vector Autoregressions," Papers 2401.14545, arXiv.org.
    5. Basawa, I. V. & Lund, Robert & Shao, Qin, 2004. "First-order seasonal autoregressive processes with periodically varying parameters," Statistics & Probability Letters, Elsevier, vol. 67(4), pages 299-306, May.
    6. Roy, Roch & Saidi, Abdessamad, 2008. "Aggregation and systematic sampling of periodic ARMA processes," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4287-4304, May.
    7. Abdelouahab Bibi & Christian Francq, 2003. "Consistent and asymptotically normal estimators for cyclically time-dependent linear models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(1), pages 41-68, March.
    8. Paul L. Anderson & Mark M. Meerschaert, 2005. "Parameter Estimation for Periodically Stationary Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(4), pages 489-518, July.
    9. Domenico Cucina & Manuel Rizzo & Eugen Ursu, 2018. "Identification of multiregime periodic autotregressive models by genetic algorithms," Post-Print hal-03187870, HAL.
    10. Christian Francq & Roch Roy & Abdessamad Saidi, 2011. "Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(6), pages 699-723, November.
    11. Koopman, Siem Jan & Ooms, Marius & Carnero, M. Angeles, 2007. "Periodic Seasonal Reg-ARFIMAGARCH Models for Daily Electricity Spot Prices," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 16-27, March.
    12. Shao, Q. & Ni, P.P., 2004. "Least-squares estimation and ANOVA for periodic autoregressive time series," Statistics & Probability Letters, Elsevier, vol. 69(3), pages 287-297, September.
    13. T. Manouchehri & A. R. Nematollahi, 2019. "Periodic autoregressive models with closed skew-normal innovations," Computational Statistics, Springer, vol. 34(3), pages 1183-1213, September.
    14. Aleksandra Grzesiek & Prashant Giri & S. Sundar & Agnieszka WyŁomańska, 2020. "Measures of Cross‐Dependence for Bidimensional Periodic AR(1) Model with α‐Stable Distribution," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(6), pages 785-807, November.
    15. Amaral, Luiz Felipe & Souza, Reinaldo Castro & Stevenson, Maxwell, 2008. "A smooth transition periodic autoregressive (STPAR) model for short-term load forecasting," International Journal of Forecasting, Elsevier, vol. 24(4), pages 603-615.
    16. Aknouche, Abdelhakim & Almohaimeed, Bader & Dimitrakopoulos, Stefanos, 2020. "Periodic autoregressive conditional duration," MPRA Paper 101696, University Library of Munich, Germany, revised 08 Jul 2020.
    17. Paul L. Anderson & Farzad Sabzikar & Mark M. Meerschaert, 2021. "Parsimonious time series modeling for high frequency climate data," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(4), pages 442-470, July.
    18. Amaal Elsayed Mubarak & Ehab Mohamed Almetwally, 2024. "Modelling and Forecasting of Covid-19 Using Periodical ARIMA Models," Annals of Data Science, Springer, vol. 11(4), pages 1483-1502, August.
    19. Hindrayanto, Irma & Koopman, Siem Jan & Ooms, Marius, 2010. "Exact maximum likelihood estimation for non-stationary periodic time series models," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2641-2654, November.
    20. Jiajie Kong & Robert Lund, 2023. "Seasonal count time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(1), pages 93-124, January.
    21. Sarnaglia, A.J.Q. & Reisen, V.A. & Lévy-Leduc, C., 2010. "Robust estimation of periodic autoregressive processes in the presence of additive outliers," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2168-2183, October.
    22. Abdelhakim Aknouche & Bader Almohaimeed & Stefanos Dimitrakopoulos, 2022. "Periodic autoregressive conditional duration," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(1), pages 5-29, January.
    23. Hurd, H. & Makagon, A. & Miamee, A. G., 0. "On AR(1) models with periodic and almost periodic coefficients," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 167-185, July.
    24. Qin Shao & Robert Lund, 2004. "Computation and Characterization of Autocorrelations and Partial Autocorrelations in Periodic ARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(3), pages 359-372, May.

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