Annals of Statistics

Nonparametric tests of the Markov hypothesis in continuous-time models

Yacine Aït-Sahalia, Jianqing Fan, and Jiancheng Jiang

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We propose several statistics to test the Markov hypothesis for β-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.

Article information

Ann. Statist., Volume 38, Number 5 (2010), 3129-3163.

First available in Project Euclid: 13 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 60J60: Diffusion processes [See also 58J65]
Secondary: 62G20: Asymptotic properties

Markov hypothesis Chapman–Kolmogorov equation locally linear smoother transition density diffusion


Aït-Sahalia, Yacine; Fan, Jianqing; Jiang, Jiancheng. Nonparametric tests of the Markov hypothesis in continuous-time models. Ann. Statist. 38 (2010), no. 5, 3129--3163. doi:10.1214/09-AOS763.

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Supplemental materials

  • Supplementary material: Additional technical details. We provide detailed proofs for Lemmas 1–7 and Theorems 5–6. Modern nonparametric smoothing techniques and theory of U-statistics are used.