6.5 A regression interpretation of the moment estimator for a CAR model. Let D be a finite domain in ℤd and let  be a finite symmetric neighborhood of the origin, with half-neighborhood †. Let D = {t ∈ D ∶ t + s ∈ D for all s ∈ }, and let y denote the vector {xt ∶ t ∈ D } with elements arranged in lexicographic order.

Next define a “design matrix” X of size |D | × |†| with entries xts = xt−s + xt+s, t ∈ D , s ∈ †.

A regression of y on X yields estimates for regression coefficient vector ????

and the residual variance ????2

???? given by a sample version of the moment identity (6.42), where the elements of the matrix A and the vector g and ????0 are estimated by areg;h1h2 = 1

|D |

∑

t∈D

(xt−h1 + xt+h1

)(xt−h2 + xt+h2

), greg;h = 1

|D |

∑

t∈D

xt(xt+h + xt−h),