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A NOTE ON THE EMBEDDING OF DISCRETE‐TIME ARMA PROCESSES
Author(s) -
Brockwell Peter J.
Publication year - 1995
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1995.tb00246.x
Subject(s) - mathematics , autoregressive–moving average model , autoregressive model , autocorrelation , discrete time and continuous time , discrete time stochastic process , moving average , discrete mathematics , combinatorics , statistics , stochastic process , continuous time stochastic process
. Let { X n , n = 0, 1, 2,…} be a discrete‐time ARMA( p, q ) process with q < p whose autoregressive polynomial has r (not necessarily distinct) negative real roots. According to a recent result of He and Wang (On embedding a discrete‐parameter ARMA model in a continuous‐parameter ARMA model. J. Time Ser. Anal. 10 (1989), 315–23) there exists a continuous‐time ARMA ( p', q' ) process { Y ( t ), t ≥0} with q' < p' = p + r such that { Y ( n ), n = 0, 1, 2,…} has the same autocorrelation function as { X n }. In this paper we show that this result is false by considering the case when { X n } is a discrete‐time AR(2) process whose autoregressive polynomial has distinct complex conjugate roots. We identify the proper subset of such processes which are embeddable in a continuous‐time ARMA(2, 1) process. We show that every discrete‐time AR(2) process with distinct complex conjugate roots can be embedded in either a continuous‐tie ARMA(2, 1) process or a continuous‐time ARMA(4, 2) process, or in some cases both. We derive an expression for the spectral density of the process obtained by sampling a general continuous‐time ARMA( p, q ) process (with distinct autoregressive roots) at arbitrary equally spaced time points. The expression clearly shows that the sampled process is a discrete‐time ARMA ( p', q' ) process with q' < p .