Nonparametric Estimation of the Innovation Variance and Judging the Fit of ARMA Models

Authored by: P. Kohli , M. Pourahmadi

Economic Time Series

Print publication date:  March  2012
Online publication date:  March  2012

Print ISBN: 9781439846575
eBook ISBN: 9781439846582
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The innovation (intrinsic, one-step-ahead prediction error) variance, σ2, of a stationary process is of central importance in the theory and practice of time series analysis, and there are several time-domain parametric methods available for its estimation, cf. Brockwell and Davis (1991, §8.7) and Pourahmadi (2001). These estimators are useful in several statistical tasks such as constructing prediction intervals for the unknown future values and developing order selection criteria, such as the Akaike’s information criteria (AIC). It is also a powerful tool for understanding the deeper aspects of time series models and data. For instance, Davis and Jones (1968) introduced a statistics based on the difference between the estimate of log σ2 and log of the estimated process variance, σ X 2 , for testing white noise. They showed equivalence of this test to the Bartlett’s test for homogeneity of variances. Hannan and Nicholls (1977) suggested that a nonparametric estimator of σ2 could provide useful information to judge the fits of various parametric models fitted to a time series data. Motivated by these findings, it is clearly of interest to estimate σ2 subject to as few constraints on the time series or its spectrum as possible. In this chapter, we review various nonparametric estimators of σ2 in the spectral domain using raw, smoothed, tapered, and multitapered periodograms for complete and incomplete time series data. Following Hannan and Nicholls’ (1977) suggestion, we examine the role of these nonparametric estimators in judging the fits of eight autoregressive moving average (ARMA) models fitted to the well-known Wolfer’s sunspot numbers.

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