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Modern Stochastics: Theory and Applications


On model fitting and estimation of strictly stationary processes
Volume 4, Issue 4 (2017), pp. 381–406
Pub. online: 22 December 2017      Type: Research Article      Open accessOpen Access
Received
13 September 2017
Revised
22 November 2017
Accepted
25 November 2017
Published
22 December 2017

Abstract

Stationary processes have been extensively studied in the literature. Their applications include modeling and forecasting numerous real life phenomena such as natural disasters, sales and market movements. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, we challenge this conventional approach. Instead of fitting an ARMA model, we apply an AR(1) characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter.

References

[1] 
Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York (1991). MR1093459
[2] 
Davis, R., Resnick, S.: Limit theory for the sample covariance and correlation functions of moving averages. The Annals of Statistics 14(2), 533–558 (1986). MR0840513
[3] 
Francq, C., Zakoïan, J.-M.: Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10(4), 605–637 (2004)
[4] 
Francq, C., Roy, R., Zakoïan, J.-M.: Diagnostic checking in ARMA models with uncorrelated errors. Journal of the American Statistical Association 100(470), 532–544 (2005)
[5] 
Hamilton, J.D.: Time Series Analysis, 1st edn. Princeton university press, Princeton (1994). MR1278033
[6] 
Hannan, E.J.: The estimation of the order of an ARMA process. The Annals of Statistics 8(5), 1071–1081 (1980). MR0585705
[7] 
Hannan, E.J.: The asymptotic theory of linear time-series models. Journal of Applied Probability 10(1), 130–145 (1973)
[8] 
Horváth, L., Kokoszka, P.: Sample autocovariances of long-memory time series. Bernoulli 14(2), 405–418 (2008). MR2544094
[9] 
Koreisha, S., Pukkila, T.: A generalized least-squares approach for estimation of autoregressive moving-average models. Journal of Time Series Analysis 11(2), 139–151 (1990)
[10] 
Lamperti, J.: Semi-stable stochastic processes. Transactions of the American mathematical Society 104(1), 62–78 (1962)
[11] 
Lévy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S., Reisen, V.A.: Robust estimation of the scale and of the autocovariance function of Gaussian short-and long-range dependent processes. Journal of Time Series Analysis 32(2), 135–156 (2011)
[12] 
Lin, J.-W., McLeod, A.I.: Portmanteau tests for ARMA models with infinite variance. Journal of Time Series Analysis 29(3), 600–617 (2008)
[13] 
Ling, S., Li, W.: On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92(439), 1184–1194 (1997)
[14] 
Mauricio, J.A.: Exact maximum likelihood estimation of stationary vector ARMA models. Journal of the American Statistical Association 90(429), 282–291 (1995)
[15] 
McElroy, T., Jach, A.: Subsampling inference for the autocovariances and autocorrelations of long-memory heavy-tailed linear time series. Journal of Time Series Analysis 33(6), 935–953 (2012)
[16] 
Mikosch, T., Gadrich, T., Kluppelberg, C., Adler, R.J.: Parameter estimation for ARMA models with infinite variance innovations. The Annals of Statistics 23(1), 305–326 (1995)
[17] 
Viitasaari, L.: Representation of stationary and stationary increment processes via Langevin equation and self-similar processes. Statistics & Probability Letters 115, 45–53 (2016)
[18] 
Yao, Q., Brockwell, P.J.: Gaussian maximum likelihood estimation for ARMA models. I. time series. Journal of Time Series Analysis 27(6), 857–875 (2006)

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Copyright
© 2017 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
AR(1) representation asymptotic normality consistency estimation strictly stationary processes

MSC2010
60G10 62M09 62M10 60G18

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