This article proposes a novel test for the martingale difference hypothesis based on the martingale difference divergence function, a recently developed dependence measure suitable for measuring the degree of conditional mean dependence of a random variable with respect to another. First, we discuss the use of martingale difference divergence in a time series framework as an alternative to the autocovariance function for detecting the existence of forms of nonlinear serial dependence. In particular, the measure equals zero if and only if the considered time-series components are conditionally mean-independent. This characteristic makes it suitable for studying the behavior of white noise processes characterized by non-null mean conditional on the past. We discuss the asymptotic properties of sample martingale difference divergence in a univariate time series framework, refining some of the results existing in the literature. Doing this allows us to build a Ljung-Box-type test statistic by summing the sample martingale difference divergence function over a finite number of lags. Under suitable conditions, the asymptotic null distribution of our test statistic is also established. The finite sample performance is discussed via a Monte Carlo study as we demonstrate its consistency against uncorrelated non-martingale processes. Finally, we show an empirical application for our methodology in analyzing the properties of the Standard and Poor's 500 stock index.

翻译：本文提出了一种基于鞅差散度函数（一种最近开发的、用于衡量随机变量对另一个随机变量的条件均值依赖程度的依赖性度量）的鞅差假设检验新方法。首先，我们讨论了在时间序列框架中使用鞅差散度作为自协方差函数的替代方法，以检测非线性序列依赖形式的存在性。特别地，该度量在且仅当所考虑的时间序列分量条件均值独立时为零。这一特性使其适用于研究以过去为条件具有非零均值的白噪声过程的行为。我们分析了单变量时间序列框架下样本鞅差散度的渐近性质，改进了文献中的一些现有结果。在此基础上，我们通过将有限滞后阶数的样本鞅差散度函数求和，构建了一个Ljung-Box型检验统计量。在适当条件下，我们建立了该检验统计量的渐近零分布。通过蒙特卡洛模拟研究了有限样本性能，证明了其对非鞅不相关过程的一致性。最后，我们展示了该方法在分析标准普尔500股指特性时的实证应用。