The Evaluation of Certain Quadratic Forms Occurring in Autoregressive Model Fitting

Abstract

Let $\mathbf{R}$ be an infinite dimensional stationary covariance matrix, let $\mathbf{R}(k)$ and $\mathbf{W}(k)$ denote the top $k \times k$ left hand corners of $\mathbf{R}$ and $\mathbf{R}^{-1}$ respectively and let $\mathbf{\Sigma}(k)$ and $\mathbf{\Gamma}(k)$ denote the approximations for $\mathbf{R}(k)^{-1}$ suggested by Whittle (1951) and Shaman (1976) respectively. We consider quadratic forms of the type $Q(k) = \beta(k)' \mathbf{R}(k)^{-1}\alpha (k)$, when the vectors $\beta(k)$ and $\alpha(k)$ constitute the first $k$ elements of the infinite absolutely summable sequences $\{\beta_j\}$ and $\{\alpha_j\}$. If $\chi_1(k) = \beta (k)' \mathbf{W}(k) \mathbf{\alpha}(k)$ and $\chi_2(k) = \beta (k)' \mathbf{\Sigma(k)}\mathbf{\alpha}(k)$, then, as $k \rightarrow \infty, Q(k)$ and $\chi_1(k)$ converge to the same limiting value for all such $\alpha (k)$ and $\beta(k)$, but $\chi_2(k)$ does not necessarily do so. Further, if $\tilde\mathbf{\alpha}(k) = (\alpha_k, \cdots, \alpha_1)'$ and $\tilde\mathbf{\beta}(k) = (\beta_k, \cdots, \beta_1)'$ then $\chi_1(k) \equiv \tilde\mathbf{\beta}(k)'\mathbf{\Gamma}(k)\tilde\mathbf{\alpha}(k)$. We discuss the use of $\mathbf{W}(k)$ for evaluating the asymptotic covariance structure of the autoregressive estimates of the inverse covariance function and the moving average parameters.