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Title: Finite predictor coefficients and the inverse Yule-Walker matrix: On the extension of Akaike's identity to random fields Authors:  Carsten Jentsch - TU Dortmund University (Germany) [presenting]
Marco Meyer - TU Braunschweig (Germany)
Abstract: For univariate stationary and centered time series $(X_t)_{t\in\mathbb{Z}}$, a useful identity links the inverse of the Yule-Walker matrix $\Gamma(p)=E(\boldsymbol{X}\boldsymbol{X}^\prime)$, where $\boldsymbol{X}=(X_{t-1},\ldots,X_{t-p})^\prime$, to the corresponding finite predictor coefficients. This factorization of $\Gamma(p)^{-1}$ is employed in different areas of statistics, and it is particularly crucial to derive asymptotic theory for autoregressive spectral density estimators. We investigate the validity of a natural extension of this factorization to univariate stationary random fields on a lattice $(X_t)_{t\in\mathbb{Z}^d}$. We prove the surprising result that such a factorization holds true if and only if a certain Toeplitz condition on the autocovariance function $\gamma(h)=E(X_{t+h}X_t)$ and the shape of the fitted autoregressive models is fulfilled. We show that this condition turns out to be very restrictive. In fact, it implies that an analogue of Akaike's identity in general does not hold for many commonly used spatial autoregressive models such as half-plane or quarter-plane models. Instead, we establish an unexpected link between the entries of the inverse Yule-Walker matrix and prediction theory. We illustrate the theoretical findings for different kinds of autoregressive fits.