  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's 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 the dependent
white noise. Due to recent popularity of conditional heteroscedastic models
(e.g., 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 showed
that the asymptotic null distribution of Box-Pierce's test statistic with
general weights still holds under unknown weak dependence so long as the lag
truncation number grows at an appropriate rate with increasing sample size.
Further applications to diagnostic checking of the ARMA and FARIMA models with
dependent white noise errors are also addressed. Our results go beyond earlier
ones by allowing non-Gaussian and conditional heteroscedastic errors in the
ARMA and FARIMA models and provide theoretical support for some empirical
findings reported in the literature.
