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A Class of Non‐Embeddable ARMA Processes
Author(s) -
Brockwell A. E.,
Brockwell P. J.
Publication year - 1999
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 - 10.1111/1467-9892.00151
Subject(s) - mathematics , autocovariance , autoregressive–moving average model , moving average , unit circle , autoregressive model , unit root , combinatorics , statistics , mathematical analysis , fourier transform
We show that a stationary ARMA( p , q ) process { X n = 0, 1, 2, ...} whose moving‐average polynomial has a root on the unit circle cannot be embedded in any continuous‐time autoregressive moving‐average (ARMA) process { Y }( t ), t ≥ 0}, i.e. we show that it is impossible to find a continuous‐time ARMA process { Y }( t )} whose autocovariance function at integer lags coincides with that of { X n }. This provides an answer to the previously unresolved question raised in the papers of Chan and Tong ( J. Time Ser. Anal. 8 (1987), 277–81), He and Wang ( J. Time Ser. Anal. 10 (1989), 315–23) and Brockwell ( J. Time Ser. Anal. 16 (1995), 451–60).