## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 4 (1975), 975-981.

### The Estimation of Arma Models

#### Abstract

In estimating a vector model, $\Sigma B(j)x(n-j)=\Sigma A(j)\epsilon(n-j), A(0)=I_r, E(\epsilon(m)\epsilon(n)')=\delta_{mn}K$ it is suggested that attention be confined to cases where $g(z) =\Sigma A(j)z^j, h(z)=\Sigma B(j)z^j$ have determinants with no zeroes inside the unit circle and have $I_r$ as greatest common left divisor and where $\1brack A(p)\vdots B(q) \rbrack$ is of rank r, where p, q are the degrees of g, h, respectively. It is shown that these conditions ensure that a certain estimation procedure gives strongly consistent estimates and the last of the conditions is probably necessary for this to be so, when the first two are satisfied. The strongly consistent estimation procedure may serve to initiate an iterative maximisation of a likelihood.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 4 (1975), 975-981.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343200

**Digital Object Identifier**

doi:10.1214/aos/1176343200

**Mathematical Reviews number (MathSciNet)**

MR391446

**Zentralblatt MATH identifier**

0311.62056

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 62M15: Spectral analysis 62E20: Asymptotic distribution theory

**Keywords**

Autoregressive-moving average process identification strongly consistent estimation

#### Citation

Hannan, E. J. The Estimation of Arma Models. Ann. Statist. 3 (1975), no. 4, 975--981. doi:10.1214/aos/1176343200. https://projecteuclid.org/euclid.aos/1176343200