Nonparametric tests of the Markov hypothesis in continuous-time models

TL;DR

This paper introduces new nonparametric statistical tests for verifying the Markov property in continuous-time models, applicable to discretely sampled data, with proven asymptotic properties and demonstrated effectiveness through simulations.

Contribution

It develops novel nonparametric tests based on the Chapman--Kolmogorov equation for Markov hypothesis, including asymptotic null distribution derivation and power analysis.

Findings

Tests follow Wilks's phenomenon under null hypothesis

Simulations show good finite sample performance

Effective in detecting various alternative models

Abstract

We propose several statistics to test the Markov hypothesis for $\beta$-mixing stationary processes sampled at discrete time intervals. Our tests are based on the Chapman--Kolmogorov equation. We establish the asymptotic null distributions of the proposed test statistics, showing that Wilks's phenomenon holds. We compute the power of the test and provide simulations to investigate the finite sample performance of the test statistics when the null model is a diffusion process, with alternatives consisting of models with a stochastic mean reversion level, stochastic volatility and jumps.