We consider the problem of detecting deviations from a white noise assumption in time series. Our approach differs from the numerous methods proposed for this purpose with respect to two aspects. First, we allow for non-stationary time series. Second, we address the problem that a white noise test, for example checking the residuals of a model fit, is usually not performed because one believes in this hypothesis, but thinks that the white noise hypothesis may be approximately true, because a postulated models describes the unknown relation well. This reflects a meanwhile classical paradigm of Box(1976) that "all models are wrong but some are useful". We address this point of view by investigating if the maximum deviation of the local autocovariance functions from 0 exceeds a given threshold $\Delta$ that can either be specified by the user or chosen in a data dependent way. The formulation of the problem in this form raises several mathematical challenges, which do not appear when one is testing the classical white noise hypothesis. We use high dimensional Gaussian approximations for dependent data to furnish a bootstrap test, prove its validity and showcase its performance on both synthetic and real data, in particular we inspect log returns of stock prices and show that our approach reflects some observations of Fama(1970) regarding the efficient market hypothesis.

翻译：本文研究时间序列中白噪声假设偏离的检测问题。我们的方法在两个方面区别于该领域众多现有方法：首先，我们允许时间序列具有非平稳特性；其次，我们针对以下常见情况提出解决方案——白噪声检验（例如检查模型拟合残差）通常并非基于对该假设的完全置信，而是认为白噪声假设可能近似成立，因为所设模型能较好地描述未知关系。这体现了Box（1976）提出的经典范式："所有模型都是错误的，但有些是有用的"。我们通过研究局部自协方差函数与零的最大偏差是否超过给定阈值Δ来处理这一观点，该阈值可由用户指定或通过数据驱动方式选择。以这种形式表述问题会引发若干数学挑战，这些挑战在检验经典白噪声假设时并不存在。我们采用高维相依数据的高斯近似方法构建自助法检验，证明其有效性，并通过合成数据与真实数据展示其性能。特别地，我们检测了股票价格的对数收益率，并证明我们的方法能够反映Fama（1970）关于有效市场假说的若干观测结论。