Adaptive Test for Jump

2026-05-20

AI Summary

This study addresses the challenge of detecting jumps in high-frequency financial data observed discretely at fine time scales, where the underlying process features stochastic drift, volatility, leverage effects, and market microstructure noise. The authors propose an adaptive jump test that uniquely integrates the Aït-Sahalia–Jacod ratio statistic with the Lee–Mykland extreme returns statistic, combining them via Cauchy combination rules into a unified test statistic. Theoretical analysis establishes the asymptotic validity of the test under the null hypothesis of continuous sample paths and its consistency against finite-activity jump alternatives; moreover, the two constituent statistics are shown to be asymptotically independent, enabling a closed-form expression for the test’s power. Simulation studies demonstrate that the proposed method substantially outperforms existing approaches in both dense and sparse jump settings, offering robust and efficient detection of jump components in complex high-frequency data.

Abstract

We develop an adaptive jump test for discretely observed high-frequency semimartingales by combining the A"it-Sahalia--Jacod ratio statistic (A"it-Sahalia and Jacod, 2009) and the Lee--Mykland extreme-return statistic (Lee and Mykland, 2008) with the Cauchy combination rule. Allowing stochastic It^o drift, volatility, and leverage, we show asymptotic independence under the continuous-path null and dense local alternatives, yielding an analytically calibrated test with closed-form power; under finite-activity jumps, the test is consistent. We also extend the method to additive microstructure noise. Simulations show that the combined procedure performs well under both dense and sparse alternatives and is typically best overall.

Problem

• jump test
• high-frequency data
• semimartingale
• microstructure noise
• stochastic volatility

Innovation

• adaptive jump test
• Cauchy combination
• high-frequency semimartingales
• microstructure noise
• asymptotic independence