By ByoungSeon Choi (auth.)

During the final 20 years, significant development has been made in statistical time sequence research. the purpose of this ebook is to give a survey of 1 of the main energetic components during this box: the id of autoregressive moving-average types, i.e., deciding on their orders. Readers are assumed to have already taken one path on time sequence research as could be provided in a graduate direction, yet another way this account is self-contained. the most issues coated comprise: Box-Jenkins' process, inverse autocorrelation services, penalty functionality identity reminiscent of AIC, BIC recommendations and Hannan and Quinn's technique, instrumental regression, and a variety of trend id tools. instead of disguise the entire equipment intimately, the emphasis is on exploring the basic principles underlying them. huge references are given to the examine literature and consequently, all these engaged in study during this topic will locate this a useful relief to their work.

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LThe present author thanks to Eric Thurschwell at the Institute for Scientific Information for sending a copy of this part of the Current Contents. 2. Akaike's Information Criterion 51 The derivation of the AIC* has been limited to the sequence of independent and identically distributed random variables. Akaike (1973c) has commented that the same line of discussion can be extended to cover the case of finite order Markov processes by Billingsley's approach (1961, pp. 14-16). Following the suggestion, Ogata (1980) has derived the AIC procedure for AR models.

Nowadays the procedure is called 46 3. Penalty Function Methods Akaike's FPE criterion because it has been popular since his publications. The FPE procedure has been applied to more general statistical modeling than AR order selection. 9. In practice it is impossible to calculate the FPE for all orders. Usually an integer K is preassigned as an upper bound of possible orders. Clearly, the integer K should be large enough to contain the true order. Several subjective benchmarks about the upper bound K have been proposed.

Also, refer to Gutowski, Robinson, and Tl'eitel (1978), Nitzberg (1979), Thomson (1981), Fuhrmann and Liu (1986), Kay and Shaw (1988), and the references therein. 2) There are other IACF estimation methods presented by Battaglia (1983, 1986, 1988), Kanto (1987), and Subba Rao and Gabr (1989). 3 Penalty Function Methods Since the early 1970s, some estimation-type identification procedures have been proposed. They are to choose the orders k and i minimizing P(k , i) 2 . + (k + i)C(T) = lnak,. T ' where a~ i is an estimate of the white noise variance obtained by fitting the ARMA(k, i) model to the observations.

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