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Basic Data Analysis for Time Series with R by DeWayne Derryberry

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9 SMOOTHERS, THE BIAS-VARIANCE TRADEOFF, AND THE SMOOTHED PERIODOGRAM

9.1 WHY IS SMOOTHING REQUIRED?

It is obvious that . In other words, the expected value of the periodogram is equal to the spectrum. However, the periodogram is not a consistent estimator of the spectrum. No matter what sample size is chosen, the periodogram does not converge to the spectrum (Kitagawa, 2010, p. 38).

The reason is quite intuitive. All the values of for k = 1…n−1 are required for the estimation of at any frequency f. When k is large (close to n), only a few sample values are used to estimate this covariance, so this value cannot be estimated consistently. Since the periodogram is a linear combination of estimates that are not consistent, it is a bit optimistic to hope that the result would be consistent. However, when sample size increases, there are always ways of averaging to get consistent estimators. See Box et al. (2008, pp. 41–42) for a slightly different intuitive explanation for the inconsistency of the periodogram.

9.2 SMOOTHING, BIAS, AND VARIANCE

So, can the periodogram be used to construct a consistent estimate of the spectrum?

The periodogram can be viewed as a collection of data ...

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