Adaptive testing method for ergodic diffusion processes based on high frequency data

TL;DR

This paper introduces an adaptive, high-frequency data-based testing framework for multidimensional ergodic diffusions, employing three types of test statistics with proven asymptotic properties.

Contribution

It develops a novel two-step testing method using adaptive estimators for diffusion and drift parameters, with comprehensive theoretical and simulation validation.

Findings

Test statistics converge to chi-squared distribution under null hypothesis.

Tests are consistent against alternatives.

Test statistics converge to non-central chi-squared under local alternatives.

Abstract

We consider parametric tests for multidimensional ergodic diffusions based on high frequency data. We propose two-step testing method for diffusion parameters and drift parameters. To construct test statistics of the tests, we utilize the adaptive estimator and provide three types of test statistics: likelihood ratio type test, Wald type test and Rao's score type test. It is proved that these test statistics converge in distribution to the chi-squared distribution under null hypothesis and have consistency of the tests against alternatives. Moreover, these test statistics converge in distribution to the non-central chi-squared distribution under local alternatives. We also give some simulation studies of the behavior of the three types of test statistics.