Abstract
We review some applications of continuous-time ARMA processes in the modelling of time series observed at discrete times t 1,…, t N . The problem of finding a continuous-time ARMA process in which a given discrete-time ARMA process can be “embedded” is discussed and some of the properties of a family of continuous-time analogues of Tong’s SETARMA models are considered. An approach to inference for such models is described and illustrated with reference to a variety of data sets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brockwell, P.J. and R.A. Davis, 1991, Time Series: Theory and Methods, 2nd edition ( Springer-Verlag, New York).
Brockwell, P.J. and R.J. Hyndman, 1992, “On continuous-time threshold autoregression”, Int. J. Forecasting 8, 157–173.
Brockwell, P.J., 1994, “On continuous-time threshold ARMA processes”, J. Stat. Planning and Inference 39, 291–303.
Brockwell, P.J., 1995, “A note on the embedding of discrete-time ARMA processes”, J. Time Ser. Anal. 16, 451–460.
Brockwell, P.J. and O.Stramer, 1995, “On the approximation of continuous-time threshold ARMA processes”, Annals Inst. Stat. Mathematics 47, 1–20.
Brockwell, P.J. and R.J. Williams, 1997, “On the existence and application of continuous-time threshold autoregressions of order two”, Adv. Appl. Prob. 19, to appear.
Chan, K.S. and H. Tong, 1987, “A Note on embedding a discrete parameter ARMA model in a continuous parameter ARMA model”, J. Time Ser. Anal. 8, 277–281.
Doob, J.L., 1944, “The elementary Gaussian processes”, Ann. Math. Statist. 25, 229–282.
He, S.W. and J.G. Wang, 1989, “On embedding a discrete-parameter ARMA model in a continuous-parameter ARMA model”, J. Time Ser. Anal. 10, 315–323.
Jones, R.H., 1981, “Fitting a continuous time autoregression to discrete data”, Applied Time Series Analysis II ed. D.F. Findley ( Academic Press, New York ), 651–682.
Jones, R.H., 1985, “Time Series Analysis with Unequally Spaced Data”, in Time Series in the Time Domain, Handbook of Statistics 5, eds. E.J. Hannan, P.R. Krishnaiah and M.M. Rao ( North Holland, Amsterdam ), 157–178.
Jones, R.H. and L.M. Ackerson, 1990, “Serial correlation in unequally spaced longitudinal data”, Biometrika 77, 721–732.
Pemberton, R.H., 1989, “Forecasting accuracy of non-linear time series models”, Tech. Report, Dept. of Mathematics, University of Salford.
Phillips, A.W., 1959, “The estimation of parameters in systems of stochastic differential equations”, Biometrika 46, 67–76.
Stramer, O., Brockwell, P.J. and R.L. Tweedie, 1996, “Continuous-time threshold AR(1) processes”, J. Appl. Prob. 33, to appear.
Stramer, O., Tweedie, R.L. and P.J. Brockwell, 1996, “Existence and stability of continuous-time threshold ARMA processes”, Statistica Sinica, 6, to appear.
Tong, H., 1983, Threshold Models in Non-linear Time Series Analysis, Springer Lecture Notes in Statistics 21 ( Springer-Verlag, New York ).
Tong, H., 1990, Non-linear Time Series: A Dynamical System Approach ( Clarendon Press, Oxford).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Brockwell, P.J. (1996). On the Use of Continuous-time ARMA Models in Time Series Analysis. In: Robinson, P.M., Rosenblatt, M. (eds) Athens Conference on Applied Probability and Time Series Analysis. Lecture Notes in Statistics, vol 115. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2412-9_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2412-9_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94787-7
Online ISBN: 978-1-4612-2412-9
eBook Packages: Springer Book Archive