A NOTE ON THE EMBEDDING OF DISCRETE-TIME ARMA PROCESSES

Abstract. 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.