Warp Speed Price Moves — Intraday Jump Test

An R implementation of the noise-robust intraday jump test developed in:

Christensen, K., Timmermann, A., & Veliyev, B. (2025). Warp speed price moves: Jumps after earnings announcements. Journal of Financial Economics, 167, 104010. https://doi.org/10.1016/j.jfineco.2025.104010

Overview

Corporate earnings announcements should, in efficient markets, trigger near-instantaneous jumps in stock prices. Testing this at the tick-by-tick level requires a jump test that is robust to the high levels of microstructure noise characteristic of after-hours trading sessions. This package implements the pre-averaging, subsampling-based bipower variation jump test of Christensen et al. (2025), which generalizes the classical Barndorff-Nielsen and Shephard (2006) test.

The key innovations are:

• Pre-averaging observed noisy log-returns using a tent kernel g(x) = min(x, 1 - x) to attenuate microstructure noise

• A subsampling estimator for the asymptotic variance matrix, valid under heteroscedasticity and serial dependence in the noise process

• Bias correction for the long-run noise variance following Jacod, Li, and Zheng (2019)

• Truncation of the bipower covariance estimator to achieve robustness to large jumps

Functions

intradayJumpTest(pData, theta, p, L, mQ, q, varpi)

Main function. Computes the noise-robust jump test statistic for a single intraday price series.

Arguments:

Argument	Description
pData	Data frame with a column PRICE containing tick-by-tick transaction prices
theta	Pre-averaging window parameter. The bandwidth is kn = round(theta * sqrt(n)) (rounded to even). Typical value: 0.5
p	Block size multiplier for the subsampler (number of pre-averaging windows per block). Typical value: 10
L	Number of subsamples used to estimate the asymptotic variance. Typical value: 10
mQ	A 2 × 2 matrix of power exponents. Each row (q1, q2) defines a variation measure. Row 1 should be c(1, 1) (bipower variation) and row 2 c(2, 0) (realized variance)
q	Quantile multiplier for the truncation threshold. Typical value: 5
varpi	Truncation rate exponent. The threshold is q * sqrt(BV) * n^(0.25 - varpi). Typical value: 0.24

Returns: A named list:

Element	Description
RV	Pre-averaged realized variance, bias-corrected and annualized as 100 * sqrt(252 * RV) (%)
BV	Pre-averaged bipower variation, bias-corrected and annualized as 100 * sqrt(252 * BV) (%)
JV	Jump proportion: 100 * (1 - BV_raw / RV_raw) (%)
JF	Jump indicator: 1 if the test rejects at the 1% Bonferroni level, 0 otherwise

Internal functions (C++)

Implemented in src/subsampler.cpp via RcppArmadillo:

Function	Description
aux_preavgk(logp, K)	Pre-averaged return series (tent kernel hard-coded)
f_mu(vpow)	Computes E[|Z|^r] for Z ~ N(0,1)
f_subsampler(logp, K, p, L, mQ, q, varpi)	Subsampled power variation estimates and covariance matrix Sigma*
f_psi(K)	Kernel constants psi1, psi2 for bias correction (tent kernel)
omega2(logp)	Long-run noise variance (Jacod, Li & Zheng 2019)

The Test Statistic

The jump test statistic (equation (21) in the paper) is:

J_n = n^(1/4) * (RV*_n - BV*_n(1,1)) / sqrt(v' * Sigma*_n * v)

where v = [1, -1]', RV*_n is the pre-averaged realized variance, BV*_n(1,1) is the truncated pre-averaged bipower variation, and Sigma*_n is the subsampled covariance estimator. Under the null of no jumps, J_n -> N(0,1).

The null is rejected at level alpha when J_n > Phi^{-1}(1 - alpha/n), where the Bonferroni correction accounts for the n observation-level hypotheses.

Usage

Load from the project root (compiles C++ and sources R/jump_test.R):

source("load.R")

Run the test on the bundled AAPL data:

mQ <- matrix(c(1, 2, 1, 0), nrow = 2, ncol = 2)

# Extended trading hours
pData  <- readr::read_csv("data/AAPL_20080609_e.csv")
result <- intradayJumpTest(pData, theta = 0.5, p = 10, L = 10, mQ = mQ, q = 5, varpi = 0.24)

# Regular trading hours
pData  <- readr::read_csv("data/AAPL_20080609_o.csv")
result <- intradayJumpTest(pData, theta = 0.5, p = 10, L = 10, mQ = mQ, q = 5, varpi = 0.24)

result$JF  # 1 = jump detected, 0 = no jump
result$RV  # annualized realized volatility (%)
result$BV  # annualized bipower variation (%)
result$JV  # jump proportion (%)

Notes

• The function expects tick-time sampled data (all price changes retained). Observations with identical timestamps should be aggregated to an average price before calling the function, consistent with Griffin and Oomen (2008).

• The pre-averaging bandwidth kn is forced to be even, as required by the theory.

• If the data are too sparse relative to the chosen p and L (i.e., c = floor(floor(n / (p * kn)) / L) == 0), the function automatically reduces p or L to ensure at least one block is available.

• RV and BV in the output are bias-corrected for microstructure noise and floored at zero before annualization.

References

• Barndorff-Nielsen, O.E., & Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics, 4(1), 1–30.

• Christensen, K., Oomen, R., & Podolskij, M. (2018). Fact or friction: Jumps at ultra high frequency. Journal of Financial Economics, 114(3), 576–599.

• Griffin, J.E., & Oomen, R.C.A. (2008). Sampling returns for realized variance calculations: Tick time or transaction time? Econometric Reviews, 27(1–3), 230–253.

• Jacod, J., Li, Y., & Zheng, X. (2019). Estimating the integrated volatility with tick observations. Journal of Econometrics, 208(1), 80–100.

• Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., & Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Processes and their Applications, 119(7), 2249–2276.

• Podolskij, M., & Vetter, M. (2009). Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli, 15(3), 634–658.