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Testing for white noise has been well studied in the literature ofeconometrics and statistics. For most of the proposed test statistics, such asthe well-known Box-Pierce's test statistic with fixed lag truncation number,the asymptotic null distributions are obtained under independent andidentically distributed assumptions and may not be valid for the dependentwhite noise. Due to recent popularity of conditional heteroscedastic models(e.g., GARCH models), which imply nonlinear dependence with zeroautocorrelation, there is a need to understand the asymptotic properties of theexisting test statistics under unknown dependence. In this paper, we showedthat the asymptotic null distribution of Box-Pierce's test statistic withgeneral weights still holds under unknown weak dependence so long as the lagtruncation number grows at an appropriate rate with increasing sample size.Further applications to diagnostic checking of the ARMA and FARIMA models withdependent white noise errors are also addressed. Our results go beyond earlierones by allowing non-Gaussian and conditional heteroscedastic errors in theARMA and FARIMA models and provide theoretical support for some empiricalfindings reported in the literature.

Testing for white noise under unknown dependence and its applications to goodness-of-fit for time series models