The Annals of Statistics

Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains

T. W. Anderson

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Abstract

The autoregressive moving average model is a stationary stochastic process $\{y_t\}$ satisfying $\sum^p_{k=0} \beta_ky_{t-k} = \sum^q_{g=0} \alpha_g\nu_{t-g}$, where the (unobservable) process $\{v_t\}$ consists of independently identically distributed random variables. The coefficients in this equation and the variance of $v_t$ are to be estimated from an observed sequence $y_1, \cdots, y_T$. To apply the method of maximum likelihood under normality the model is modified (i) by setting $y_0 = y_{-1} = \cdots = y_{1-p} = 0$ and $\nu_0 = v_{-1} = \cdots = v_{1-q} = 0$ and alternatively (ii) by setting $y_0 \equiv y_T, \cdots, y_{1-p} \equiv y_{T+1-p}$ and $v_0 \equiv v_T, \cdots, v_{1-q} \equiv v_{T+1-q}$; the former lead to procedures in the time domain and the latter to procedures in the frequency domain. Matrix methods are used for a unified development of the Newton-Raphson and scoring iterative procedures; most of the procedures have been obtained previously by different methods. Estimation of the covariances of the moving average part is also treated.

Article information

Source
Ann. Statist., Volume 5, Number 5 (1977), 842-865.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343942

Digital Object Identifier
doi:10.1214/aos/1176343942

Mathematical Reviews number (MathSciNet)
MR448762

Zentralblatt MATH identifier
0368.62075

JSTOR
links.jstor.org

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62H99: None of the above, but in this section

Keywords
Maximum likelihood estimation autoregressive moving average models Newton-Raphson and scoring iterative procedures time and frequency domains time series analysis

Citation

Anderson, T. W. Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains. Ann. Statist. 5 (1977), no. 5, 842--865. doi:10.1214/aos/1176343942. https://projecteuclid.org/euclid.aos/1176343942


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