Testing for the Markov Property in Time Series via Deep Conditional
  Generative Learning
Yunzhe Zhou Chengchun Shi Lexin Li Qiwei Yao
The Markov property is widely imposed in analysis of time series data.
Correspondingly, testing the Markov property, and relatedly, inferring the
order of a Markov model, are of paramount importance. In this article, we
propose a nonparametric test for the Markov property in high-dimensional time
series via deep conditional generative learning. We also apply the test
sequentially to determine the order of the Markov model. We show that the test
controls the type-I error asymptotically, and has the power approaching one.
Our proposal makes novel contributions in several ways. We utilize and extend
state-of-the-art deep generative learning to estimate the conditional density
functions, and establish a sharp upper bound on the approximation error of the
estimators. We derive a doubly robust test statistic, which employs a
nonparametric estimation but achieves a parametric convergence rate. We further
adopt sample splitting and cross-fitting to minimize the conditions required to
ensure the consistency of the test. We demonstrate the efficacy of the test
through both simulations and the three data applications.