## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 3 (1974), 489-502.

### Consistent Autoregressive Spectral Estimates

#### Abstract

We consider an autoregressive linear process $\{x_t\}$, a one-sided moving average, with summable coefficients, of independent identically distributed variables $\{e_t\}$ with zero mean and fourth moment, such that $\{e_t\}$ is expressible in terms of past values of $\{x_t\}$. The spectral density of $\{x_t\}$ is assumed bounded and bounded away from zero. Using data $x_1,\cdots, x_n$ from the process, we fit an autoregression of order $k$, where $k^3/n \rightarrow 0$ as $n \rightarrow \infty$. Assuming the order $k$ is asymptotically sufficient to overcome bias, the autoregression yields a consistent estimator of the spectral density of $\{x_t\}$. Furthermore, assuming $k$ goes to infinity so that the bias from using a finite autoregression vanishes at a sufficient rate, the autoregressive spectral estimates are asymptotically normal, uncorrelated at different fixed frequencies. The asymptotic variance is the same as for spectral estimates based on a truncated periodogram.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 3 (1974), 489-502.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342709

**Digital Object Identifier**

doi:10.1214/aos/1176342709

**Mathematical Reviews number (MathSciNet)**

MR421010

**Zentralblatt MATH identifier**

0317.62064

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M15: Spectral analysis

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Autoregression time series spectral analysis

#### Citation

Berk, Kenneth N. Consistent Autoregressive Spectral Estimates. Ann. Statist. 2 (1974), no. 3, 489--502. doi:10.1214/aos/1176342709. https://projecteuclid.org/euclid.aos/1176342709