This repository contains the implementation for the JRSSB paper "Testing for the Markov Property in Time Series via Deep Conditional Generative Learning " in Python.

The Markov property is widely imposed in time series analysis. Correspondingly, testing the Markov property, and relatedly, inferring the order of a Markov model, is of paramount importance. In this article, we propose a nonparametric testing procedure 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 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 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 three data applications.

Figures:

• Python 3.6
  • numpy 1.18.5
  • scipy 1.5.4
  • torch 1.0.0
  • tensorflow 2.1.3
  • sklearn 0.23.2

• markov_test/: This module contains all python functions used in numerical experiments and real data analysis.
  • MDN_VAR_CV.py contains the utility functions to implement MDN based test for VAR model.
  • MDN_SHRED_CV.py contains the utility functions to implement MDN based test for Threshold model.
  • MDN_MARCH_CV.py contains the utility functions to implement MDN based test for MARCH model.
  • REAL.py contains the utility functions for the application of real dataset.
  • _utility.py and _utility_RL.py contains the basic functions used in simulations and real data analysis.
• data/: This folder where the output results and the dataset should be put.
  • 7cityMonth.dat is the Temperature data.
  • The PM2.5 data is publicly available at here.
  • The OhioT1DM dataset is publicly available at here.
• result/: This folder where the output results should be put.
  • result.ipynb is the jupyter notebook to output the results.
• command.ls contains all the python commands to replicate the results in the paper. It will run all the xxx_run_cv.py files in the main folder. You can also customize the config of hyperparameters by yourself.