On tapering to improve Yule-Walker estimation in autoregressive processes
The most commonly used method for estimating the time domain parameters of an autoregressive process is to use the Yule-Walker equations. The Yule-Walker estimates of the parameters of an autoregressive process of order p, or AR(p), are known to often be highly biased. This can lead to inappropriate order selection and very poor forecasting. There is a Fourier transform relationship between the autocovariance sequence for an autoregressive process (which are used in the Yule-Walker equations to determine the time domain parameters) and the spectrum, a frequency domain representation of the autoregressive process. Tapering has been shown to reduce the bias of the periodogram, a naive estimator of the spectrum, and so has also been used to reduce the bias of the time domain parameter estimates, although without theoretical support. A new method, multitapering, has had great success in spectral estimation. We use the Fourier transform relationship between the frequency domain and the time domain to define multitaper Yule-Walker estimates of the time domain parameters of the autoregressive process, and evaluate their asymptotic distribution as well as their order 1/T bias, where T is the length of the series under consideration. We use the fact that the Yule-Walker estimates correspond to a very specific least squares problem to contribute to the previously wanting theoretical underpinnings for the need to taper in Yule-Walker estimation of the time domain parameters. This least squares structure is exploited, with the help of matrix calculus, to find a process-based or data-based taper or transformation of the data which will provide better parameter estimates. This least squares structure also leads to a better understanding of a long used approximation to the inverse of the autoregressive covariance matrix. We also detail some complications with previous work in the literature, including the areas of asymptotic bias calculation, mean estimation, non-symmetric tapers, and defining tapers in terms of the proportion of the data to be tapered.
|Year of publication:||
|Authors:||Crunk, Steven M|
|Type of publication:||Other|
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