The Annals of Statistics

Maximum Likelihood Estimation of Parameters of Autoregressive Processes with Moving Average Residuals and Other Covariance Matrices with Linear Structure

T. W. Anderson

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Abstract

The autoregressive process with moving average residuals is a stationary process $\{y_t\}$ satisfying $\sum^p_{s = 0} \beta_sy_{t - s} = \sum^q_{j = 0} \alpha_j\nu_{t - j}$, where the sequence $\{\nu_t\}$ consists of independently identically distributed (unobservable) random variables. The distribution of $y_1,\cdots, y_T$ can be approximated by the distribution of the $T$-component vector $\mathbf{y}$ satisfying $\sum^p_{s = 0} \beta_s\mathbf{K}_s\mathbf{y} = \sum^q_{j = 0} \alpha_j\mathbf{J}_j\mathbf{v}$, where $\mathbf{v}$ has covariance matrix $\sigma^2\mathbf{I}, \mathbf{K}_s = \mathbf{J}_s = \mathbf{L}^s$, and $\mathbf{L}$ is the $T \times T$ matrix with 1's immediately below the main diagonal and 0's elsewhere. Maximum likelihood estimates are obtained when $\mathbf{v}$ has a normal distribution. The method of scoring is used to find estimates defined by linear equations which are consistent, asymptotically normal, and asymptotically efficient (as $T\rightarrow \infty$). Several special cases are treated. It is shown how to calculate the estimates.

Article information

Source
Ann. Statist., Volume 3, Number 6 (1975), 1283-1304.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343285

Digital Object Identifier
doi:10.1214/aos/1176343285

Mathematical Reviews number (MathSciNet)
MR383672

Zentralblatt MATH identifier
0331.62067

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
Autoregressive processes with moving average residuals covariance matrices with linear structure maximum likelihood estimation estimation of linear transformations multivariate normal distribution

Citation

Anderson, T. W. Maximum Likelihood Estimation of Parameters of Autoregressive Processes with Moving Average Residuals and Other Covariance Matrices with Linear Structure. Ann. Statist. 3 (1975), no. 6, 1283--1304. doi:10.1214/aos/1176343285. https://projecteuclid.org/euclid.aos/1176343285


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