FX duration models can understate persistence when the data vendor rounds its clock. The supporting evidence comes from Holý's own Monte Carlo rather than an independent test. He generates data from the corrected model at n = 10,000 across 2,000 replications, then estimates the standard model on those samples. The generating parameters match his estimates: phi = 0.998, alpha = 0.25, omega = 0, gamma = 1.2, kappa = 0.8, rho = 0.2, sigma = 0.015. Every rho above zero produces downward bias in phi, with the largest gap around rho = 0.25.
The 2024 data show the same pattern. For EUR/USD, phi moves from 0.9977 under the standard generalized gamma ACD to 0.9985 under the corrected version. Across the seven majors, the standard estimates range from 0.9959 to 0.9985, against 0.9969 to 0.9985 for the corrected model. Some apparent clustering in market activity comes from somebody's clock.
High-frequency FX inter-trade durations bunch at and around 1, 2, 3 seconds. Holý plots the empirical density for the fractional part of EUR/USD durations, which spikes at 0 and 1. Snap some raw event times to a second boundary before taking differences, and this is the pattern you would expect.
The paper says the heaping appears in durations while the transaction timestamps themselves do not show it. It also describes the effect as specific to FX rather than stock markets. Those claims do most of the work behind the interpretation, yet the paper provides no figure of the raw timestamp series or stock-market comparison. The proposed source is vendor consolidation. FX vendors combine venue records that may use different timestamp precision, clock synchronization and other reporting conventions.
How the mixture reallocates mass
Holý uses a two-component mixture. With probability 1 minus rho, the duration follows a generalized gamma with time-varying scale lambda_i and shape parameters gamma and kappa. With probability rho, it comes from the second component. This component takes the generalized gamma probability mass from the interval beginning at round(x) minus one and ending at round(x), then places it within the half-second window on either side of the integer. A truncated normal centred on that integer, with standard deviation sigma, supplies the spreading kernel.
Round up, then jitter.
Values below 0.5 round to zero and receive no inflated mass, matching the data.
A score-driven recursion governs the conditional log-scale. ln(lambda_i) equals omega plus a nonparametric intraweek seasonal term and a persistent component E_i. In turn, E_i equals phi times E_{i-1} plus alpha times the log-likelihood score. The seasonal term is a smoothing spline in time-of-week, estimated from log durations and then plugged in. Holý deliberately avoids using it to pre-adjust durations because deseasonalising would smear the integer spikes. Maximum likelihood jointly estimates all seven parameters, including rho and sigma.
The score carries the consequence of misspecification. In a GAS recursion, the log-likelihood gradient updates the conditional scale. A density that is wrong near integers makes each nearby observation push the state in the wrong direction. At low and moderate rho, the standard generalized gamma score is much too high near one second and too low between integers. With larger rho, the error spreads: the standard model's score is systematically lower across all short durations.
Simulation first, ticks after
The simulation also finds a sign change in the bias of alpha around rho = 0.20. Alpha is overstated below that point and understated above it. The misspecification reaches the dynamics.
Holý's empirical panel covers a full year of 2024 Refinitiv Eikon data for seven majors. Each pair has more than 10 million durations, reported to three decimals. Estimated rho ranges from 0.0501 for USD/JPY to 0.2734 for USD/CAD, while EUR/USD comes in at 0.1861. Across all seven majors, sigma stays between 0.0145 and 0.0165 seconds. That narrow spread makes the artifact look like a feature of the reporting pipeline: every major pair produces nearly the same number, regardless of the currencies involved.
Average log-likelihood per observation improves for every pair. EUR/USD moves from -1.2817 to -1.0465, and USD/CAD from -1.4461 to -1.0455. USD/JPY records the smallest gain, +0.0363, alongside the lowest rho.
GA-ACD nests the standard model at rho = 0, which guarantees a non-negative fit gain. The objection carries little force here: two extra parameters are being estimated against ten million observations. Holý reports no standard errors because everything is significant at this sample size. He also gives no information criteria because the differences in penalization are negligible. Under GA-ACD, phi rises in five of the seven pairs, including EUR/USD from 0.9977 to 0.9985. GBP/USD and NZD/USD remain unchanged at four decimals, as Holý notes.
The exotics are less comfortable. USD/KRW produces rho = 0.8347 and sigma = 0.2166, putting most of the mass in the heaped component while spreading it an order of magnitude more widely than for the majors. Its feed is effectively reporting at second resolution. My own extension, which Holý does not make, is to compute rho as a data-quality screen before deciding whether a feed can support a microstructure study at all.
What remains unresolved?
Holý concedes two weaknesses with direct consequences. The EUR/USD density has asymmetric heaping, slightly heavier after the integer than before, while his truncated normal is symmetric. More seriously, the fitted omega, gamma and kappa in real data resemble the higher-rho simulation cases rather than the estimated-rho case. He suggests that the true data-generating process may differ from the GA-ACD specification.
Inference for omega is weak at realistic persistence. Hessian-based 95% intervals cover 83.00% at n = 10,000. Coverage reaches 93.95% only at a million observations. Every other parameter is near nominal from n = 1,000 upward. The paper also replaces zero durations by hand with 0.0005 seconds, despite citing zero-inflated duration research from which it could have borrowed.
We did not find a forecasting comparison, execution application, or economic evaluation in the paper. Its evidence consists of in-sample likelihood from one vendor over one calendar year.
We could not run this ourselves. The exercise requires individual FX trade prints with sub-second timestamps. Our finest data are one-minute OHLCV bars, which cannot represent inter-trade durations at all. They certainly cannot establish whether an observation falls within 15 milliseconds of a second boundary.
One comparison would move me from treating this as a useful diagnostic to using it: estimate rho and sigma for the same pair across two different vendors. Different parameters would confirm the artifact and make rho a feed metric. Agreement would force a rewrite of the story.