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TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS APPLICATIONS TO DIAGNOSTIC CHECKING FOR TIME SERIES MODELS

Published online by Cambridge University Press:  27 August 2010

Xiaofeng Shao
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Abstract

Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box–Pierce test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for dependent white noise. Because of recent popularity of conditional heteroskedastic models (e.g., generalized autoregressive conditional heteroskedastic [GARCH] models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing test statistics under unknown dependence. In this paper, we show that the asymptotic null distribution of the Box–Pierce test statistic with general weights still holds under unknown weak dependence as long as the lag truncation number grows at an appropriate rate with increasing sample size. Further applications to diagnostic checking of the autoregressive moving average (ARMA) and fractional autoregressive integrated moving average (FARIMA) models with dependent white noise errors are also addressed. Our results go beyond earlier ones by allowing non-Gaussian and conditional heteroskedastic errors in the ARMA and FARIMA models and provide theoretical support for some empirical findings reported in the literature.

Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 2 , April 2011 , pp. 312 - 343
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

I thank Professor Pentti Saikkonen and two referees for constructive comments that led to improvement of the paper. The work was supported in part by NSF grant DMS-0804937.

References

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