The Annals of Statistics

Linear Prediction by Autoregressive Model Fitting in the Time Domain

R. J. Bhansali

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Let $\{x_t\}$ be a purely nondeterministic stationary process satisfying all the assumptions made by Berk (1974), and $\{y_t\}$ be another purely nondeterministic stationary process. Assume that $y_t$ is independent of $x_t$ but has exactly the same statistical properties as that of $x_t$. Consider the linear prediction of future values of $y_t$ on the basis of past values, using prediction constants estimated from a realisation of $T$ observations of $x_t$ by least-squares fitting of an autoregression of order $k$. By assuming that $k \rightarrow \infty, k^3/T \rightarrow 0$ as $T \rightarrow \infty$, the effect on the mean square error of prediction of estimating the autoregressive coefficients is determined. This effect is the same as for the case when the prediction constants are estimated by factorising a "windowed" estimate of the spectral density function of $x_t$.

Article information

Ann. Statist., Volume 6, Number 1 (1978), 224-231.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Autoregressive process time domain Yule-Walker equations time series


Bhansali, R. J. Linear Prediction by Autoregressive Model Fitting in the Time Domain. Ann. Statist. 6 (1978), no. 1, 224--231. doi:10.1214/aos/1176344081.

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