An Approximate Inverse for the Covariance Matrix of Moving Average and Autoregressive Processes

Abstract

Let $\mathbf{\Sigma}$ denote the convariance matrix of a vector $\mathbf{x} = (x_1, \cdots, x_T)'$ of $T$ successive observations from a stationary process $\{ x_t\}$ with continuous positive spectral density $f(\lambda)$. Let $\mathbf{\Gamma}$ be the $T \times T$ matrix with elements $\gamma(s, t) = (2\pi)^{-2} \int^\pi_{-\pi} e^{i\lambda (s-t)} f^{-1}(\lambda) d\lambda$. The properties of $\mathbf{\Gamma}$ considered as an approximate inverse of $\mathbf{\Sigma}$ are studied. When $\{ x_t\}$ is a$(n)$ moving average (autoregressive) process of order $q$, rows (columns) $q + 1, \cdots, T - q$ of $\mathbf{\Sigma\Gamma} - \mathbf{I}$ are zero vectors. In this case $\mathbf{\Sigma\Gamma} - \mathbf{I}$ has $2q$ positive characteristic roots which approach paired positive limiting values as $T \rightarrow \infty$ if the roots of $\sum^q_{j=0} \beta_j z^{q-j} = 0$ are less than 1 in absolute value, where $\beta_1, \cdots, \beta_q$ are the coefficients of the process. Statistical properties of $\mathbf{x'Tx} - \mathbf{x'\Sigma}^{-1} \mathbf{x}$ and $\mathbf{x'\Gamma x}/ \mathbf{x'\Sigma}^{-1} \mathbf{x}$ are also discussed.