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May, 1974 Consistent Autoregressive Spectral Estimates
Kenneth N. Berk
Ann. Statist. 2(3): 489-502 (May, 1974). DOI: 10.1214/aos/1176342709


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.


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Kenneth N. Berk. "Consistent Autoregressive Spectral Estimates." Ann. Statist. 2 (3) 489 - 502, May, 1974.


Published: May, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0317.62064
MathSciNet: MR421010
Digital Object Identifier: 10.1214/aos/1176342709

Primary: 62M15
Secondary: 62E20

Keywords: Autoregression , spectral analysis , time series

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 3 • May, 1974
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