Adaptive Test for Jump

Technical Summary: Adaptive Test for Jump

Problem Statement
The detection of jumps from discretely observed high-frequency financial data remains a central challenge in financial econometrics. Existing literature has produced numerous nonparametric and semiparametric tests, generally falling into two dominant categories with distinct strengths and weaknesses. The first category, represented by the Aït-Sahalia–Jacod (AJ) ratio statistic, aggregates power variation across sampling frequencies. It is naturally effective when jump evidence is spread over many increments (dense alternatives) but may lack sensitivity to isolated, pronounced jumps. The second category, represented by the Lee–Mykland (LM) standardized maximum statistic, focuses on the largest local excursion. It excels when jumps are sparse and concentrated but may underperform when jump signals are diffuse. Empirical evidence suggests no single test dominates uniformly across dense and sparse regimes or varying sampling frequencies. Furthermore, at ultra-high frequencies, market microstructure noise alters the marginal asymptotics of these statistics, complicating inference.

Methodology
The authors propose an adaptive jump test that combines the AJ sum-type statistic and the LM max-type statistic using the Cauchy combination rule. The methodology is developed in two stages: first for latent continuous-path processes and second for processes observed with additive microstructure noise.

1. Latent Process Framework:

   • Model: The underlying price process $X_t$ is modeled as a general Itô semimartingale with stochastic drift, stochastic volatility, and leverage effects.
   • Statistics:
     • The AJ statistic ($Z^{AJ}_n$) is constructed from the ratio of fine-scale to coarse-scale power variations. Under the continuous-path null, it converges to a standard normal distribution.
     • The LM statistic ($\xi_n$) is constructed from the maximum of locally standardized increments. Under the continuous-path null, it converges to a standard Gumbel distribution.
   • Combination: The p-values from both statistics are transformed via the tangent function and averaged to form a Cauchy combination statistic ($T^C_n$).

2. Noisy Extension:

   • Noise Model: The observed process $Y_{t_i}$ includes additive, i.i.d. microstructure noise.
   • Robustification:
     • The AJ component is adapted using pre-averaging techniques (Aït-Sahalia et al., 2012) to mitigate noise, resulting in a pre-averaged ratio statistic ($Z^{PA}_n$).
     • The LM component is adapted using local averaging and a feasible spot-volatility estimator based on the two-scale realized spot variance (TSRSV) of Zu and Boswijk (2014). This yields a local-average max statistic ($\xi^{LA}_n$).
   • Calibration: To address the slow convergence of extreme-value statistics in finite samples under noise, a double bootstrap procedure is implemented for the LM component to control size accurately.

Key Theoretical Contributions
The paper's primary theoretical contribution is the proof of asymptotic independence between the sum-type (AJ) and max-type (LM) statistics under both the continuous-path null and various alternative hypotheses.

• Independence under Null: The authors demonstrate that, after appropriate marginal normalization, the AJ statistic and the LM statistic are asymptotically independent under the continuous-path null. This result is non-trivial because both statistics are nonlinear functionals of the same discretely observed Itô semimartingale with overlapping increments, stochastic volatility, and leverage. Unlike high-dimensional settings where independence often arises from coordinate-wise independence, here it must be derived directly from the primitive continuous-time model.
• Independence under Alternatives: The asymptotic independence is shown to hold under:
  • Dense local alternatives: Where jump evidence is spread over many increments.
  • Fixed finite-activity jumps: Where jumps are isolated and of fixed size.
• Noisy Asymptotics: The independence result is extended to the noisy setting. The authors prove that the pre-averaged AJ statistic and the new feasible LM-type statistic remain asymptotically independent under the noisy null and noisy dense alternatives.

Results

• Theoretical Properties:
  • The proposed Cauchy combination test is analytically calibrated; under the null, the combined statistic follows a standard Cauchy distribution, allowing for a closed-form critical value.
  • Under dense local alternatives, the test maintains asymptotic independence, and the power function is derived analytically.
  • Under fixed finite-activity jumps, the test is consistent (power approaches 1) because the LM component drives the rejection, even if the AJ component is less sensitive.
• Simulation Evidence:
  • Size: The test maintains correct size under the continuous-path null, both in noise-free and noisy environments (with the double bootstrap for the noisy LM component).
  • Power:
    • In dense regimes (many small jumps), the AJ statistic is generally more powerful, and the combined test inherits this strength.
    • In sparse regimes (few large jumps), the LM statistic is more powerful, and the combined test tracks this performance closely.
    • The Cauchy combination consistently delivers the most robust overall performance, adapting to the nature of the jump alternative without prior knowledge of the regime.
  • Noise: The noisy extension performs well across Gaussian and heavy-tailed (t-distributed) noise, with the combined test outperforming individual components in most configurations.
• Empirical Application:
  • Applied to intraday data for six major equity ETFs and stocks (SPY, QQQ, IWM, AAPL, MSFT, NVDA) at a 3-second frequency.
  • The LM test detected jumps in approximately 51–61% of daily sessions, while the AJ test detected jumps in less than 2% of sessions.
  • The combined test yielded rejection rates between the two, closer to LM, suggesting that in real market data, jump evidence is often concentrated in sparse, sharp events rather than diffuse.
  • At coarser sampling frequencies (1, 2, 5 minutes), the relative performance shifted: the AJ test became stronger at 2 and 5 minutes, while the combined test remained the most stable procedure across frequencies.

Significance and Claims
The paper claims to provide a "simple and robust default" for jump detection when the form of the jump alternative (dense vs. sparse) is unknown. By establishing the asymptotic independence of sum-type and max-type statistics in a general Itô semimartingale framework (including leverage and noise), the authors enable an analytically tractable combination test. The significance lies in:

1. Adaptivity: The test automatically adapts to the jump regime, inheriting the strengths of both the AJ and LM procedures.
2. Theoretical Rigor: It provides the first proof of asymptotic independence between these specific nonlinear functionals in the presence of stochastic volatility, leverage, and microstructure noise, moving beyond classical i.i.d. high-dimensional arguments.
3. Practical Utility: The inclusion of a noise-robust extension with a double bootstrap calibration addresses the practical difficulties of ultra-high-frequency data analysis, offering a procedure that performs well across a wide range of empirical configurations.

The authors conclude that while future work could address serially dependent noise or multivariate settings, the current procedure effectively synthesizes the strengths of existing leading tests to deliver a reliable, adaptive solution for high-frequency jump detection.