## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 2 (1975), 532-538.

### 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.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 2 (1975), 532-538.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343085

**Digital Object Identifier**

doi:10.1214/aos/1176343085

**Mathematical Reviews number (MathSciNet)**

MR362784

**Zentralblatt MATH identifier**

0312.62066

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 15A09: Matrix inversion, generalized inverses

**Keywords**

Stationary process covariance matrix approximate inverse autoregressive-moving average (ARMA) process characteristic roots

#### Citation

Shaman, Paul. An Approximate Inverse for the Covariance Matrix of Moving Average and Autoregressive Processes. Ann. Statist. 3 (1975), no. 2, 532--538. doi:10.1214/aos/1176343085. https://projecteuclid.org/euclid.aos/1176343085