On the validity of Akaike's identity for random fields

摘要

For univariate stationary and centered time series (X-t) t is an element of Z, Akaike's identity links the inverse of the Yule-Walker matrix Gamma(p) = E(XX'), where X = (Xt-1,..., Xt-p)', to the corresponding finite predictor coefficients. It reads as a Cholesky-type factorization Gamma(p)(-1) = L(p)'Sigma(p)L-1(p), where L(p) is lower-triangular and Sigma(p)(-1) is diagonal. Whereas this Cholesky-type factorization exists whenever G (p) is positive definite, Akaike derived a meaningful interpretation of Gamma(p) and Sigma(p)(-1) in terms of finite predictor coefficients. It is useful in many applications and is particularly crucial to derive asymptotic theory for Berk's spectral density estimator.