First-order planar autoregressive model
Volume 12, Issue 1 (2025), pp. 83–121
Pub. online: 6 August 2024
Type: Research Article
Open Access
Open Access
Received
5 February 2024
5 February 2024
Revised
29 May 2024
29 May 2024
Accepted
20 July 2024
20 July 2024
Published
6 August 2024
6 August 2024
Abstract
This paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation
A stationary solution X to the equation exists if and only if $(1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)\gt 0$. The stationary solution X satisfies the causality condition with respect to the white noise ϵ if and only if $1-a-b-c\gt 0$, $1-a+b+c\gt 0$, $1+a-b+c\gt 0$ and $1+a+b-c\gt 0$. A sufficient condition for X to be purely nondeterministic is provided.
An explicit expression for the autocovariance function of X on the axes is provided. With Yule–Walker equations, this facilitates the computation of the autocovariance function everywhere, at all integer points of the plane. In addition, all situations are described where different parameters determine the same autocovariance function of X.
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