We consider the problem of testing for the martingale difference hypothesis for univariate strictly stationary time series by implementing a novel test for conditional mean independence based on the concept of martingale difference divergence. The martingale difference divergence function allows us to measure the degree to which a certain variable is conditionally mean dependent upon its past values: in particular, it does so by computing the regularized norm of the covariance between the current value of the variable and the characteristic function of its past values. In this paper, we make use of such a concept, along with the theoretical framework of generalized spectral density, to construct a Ljung-Box type test for the martingale difference hypothesis. In addition to the results obtained with the implementation of the test statistic, we proceed to show some asymptotics for martingale difference divergence in the time series framework.

翻译：本文考虑通过实施一种基于鞅差散度概念的条件均值独立性新检验方法，检验单变量严格平稳时间序列的鞅差假设问题。鞅差散度函数能够度量某变量与其历史值之间的条件均值依赖程度：具体而言，它通过计算该变量当前值与历史值特征函数之间协方差的正则化范数来实现。本文利用该概念及广义谱密度的理论框架，构建了一种用于鞅差假设的Ljung-Box型检验。除检验统计量的应用结果外，我们还在时间序列框架下展示了鞅差散度的渐近性质。