Annals of Statistics

Consistent Autoregressive Spectral Estimates

Kenneth N. Berk

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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

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

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M15: Spectral analysis
Secondary: 62E20: Asymptotic distribution theory

Autoregression time series spectral analysis


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

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