We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in a generalized version of the new setting in Jin, Wang and Wang (2015), who exploit Walsh (1923) functions in order to compare sub-sample covariances with the full sample counterpart. They impose strict stationarity under the null, only consider linear processes under either hypothesis in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonormal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet and Walsh functions; and we allow for linear or nonlinear processes with possibly non-iid innovations. This is important in macroeconomics and finance where nonlinear feedback and random volatility occur in many settings. We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation difference statistic, where the maximum is taken over the correlation lag $h$ and basis generated sub-sample counter $k$ (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter $\mathcal{H}_{T}$ and $\mathcal{K}_{T}$. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.

翻译：我们提出了一种适用于可能依赖且全局非平稳时间序列的协方差平稳性检验。我们在 Jin、Wang 和 Wang（2015）新框架的广义版本中开展工作，他们利用 Walsh（1923）函数将子样本协方差与全样本协方差进行比较。在原假设下他们强加了严格平稳性，仅在线性过程假设下采用参数估计器来求解高维渐近协方差矩阵的逆矩阵，且未考虑其他正交基。相反，我们在包含 Haar 小波和 Walsh 函数的温和条件下使用一般正交基；并允许使用可能具有非独立同分布新息的线性或非线性过程。这在宏观经济和金融领域至关重要，因为非线性反馈和随机波动常出现在多种场景中。我们通过自助法最大化相关系数差异统计量，完全规避了渐近协方差矩阵的估计与求逆问题，其中最大值取自相关延迟 \(h\) 和基生成的子样本计数器 \(k\)（系统样本数）。我们实现了最大延迟和计数器 \(\mathcal{H}_{T}\) 与 \(\mathcal{K}_{T}\) 更高的可行增长率。特别值得注意的是，我们的检验能够检测方差突变以及远离平稳性的遥远或极轻微偏离。