Fitting Hawkes models to raw consolidated NBBO mid changes loads about three quarters of one-second excitation into the first millisecond. Across 112 stock-months, the median is 0.46 of 0.63, concentrated where locks and depleted quotes resolve. Lee's paper argues that these feed events drive the result. We find the case convincing.

Follow the quote

Take a stock at a one-cent spread: bid 20.00, ask 20.01, mid 20.005. A buyer wants to bid 20.01. That order would trade on the venue holding the ask, so it is posted elsewhere. Its bid update reaches the SIP ahead of the ask venue's update. The tape then shows a locked market at 20.01/20.01 for about 0.6 ms, with a mid of 20.010. When the ask moves to 20.02, the mid reaches 20.015.

The feed has logged two half-tick events, 0.6 ms apart.

Price moves in these names usually follow this path. About half of all NBBO changes are locks, whose median life is 0.4-0.7 ms. Only 4% of INTC price moves on May 17, 2017 avoided a lock. Lee calls an NBBO state that resolves within about a millisecond a brief state. Locks are one kind. Depletion produces the other: a trade removes the only quote, and within a millisecond the level refills or the other side follows.

Matching these states to the trade file finds that 86-99% begin with a trade at the vanished price within 3 ms. The chance level is 2-6%. IBM barely has locks, at 0.6% of states. Yet a third of its raw gaps last under a millisecond, all arising from brief positive-spread states. In May 2017, gaps shorter than 1 ms account for 41-50% of raw gaps across the eight stocks. The paper states the consequence plainly: "A Hawkes model has only one way to explain two events a millisecond apart: excitation."

What goes into the fits?

Lee uses Millisecond TAQ for INTC, MU, MDLZ, BAC, WFC, T, TGT and ATVI. The sample covers November 2016 to December 2017, from 10:00 to 15:30. It contains 112 stock-months, including 80 large-tick stock-months.

The same NBBO states supply three event series. E0 retains every raw mid change. E1 merges locks and crosses under 5 ms. E1m additionally merges any state under 1 ms, retaining an event only when its mid level persists at least that long. Lee fits each series with a bivariate up/down Hawkes model using four exponentials, except for BAC, which uses three exponentials and a 2 ms threshold. The baseline has one level per 30-minute bin each day. Each event has a 1 ms dead time, and each month has its own kernel.

The main measure is kernel mass within one second: n(1 s), the moves directly triggered by one move inside that window. Raw fits yield a median 3.1 times the clean mass, with an IQR of 2.0-4.2. The ratio exceeds one in every one of the 112 stock-months. Raw n(1 s) is 0.63, including 0.46 in the first millisecond. Clean n(1 s) is 0.20.

That first millisecond accounts for the multiple. Measure both fits over a common window starting at the threshold, and raw mass is 0.71 of clean. Start at 5 ms and it is 0.48. There, the raw fit sees real excitation diluted by the doubled event count.

The observed mid cannot support the raw excitation

Cleaning removes the sub-millisecond spike partly by construction: E1m leaves no sub-millisecond gaps to fit. The argument that the raw spike is mechanical therefore rests on the states' properties, stability across thresholds and clocks, and the simulation. A check in price units supplies further evidence.

Lee simulates each fitted model and compares its implied post-event mid change with the observed path. The mid continues by +0.09 to +0.21 ticks after a raw event, completing the second half of the move. The raw model predicts +0.29 to +0.59 ticks. Within 100 ms and 1 s, its prediction is 2.7 to 3.1 times the observed path. Across all four stocks tested, the clean model follows the observed path to within 0.021 ticks out to one second.

A synthetic-feed experiment provides a second line of evidence. Lee generates moves from the clean fitted model, then adds only the measured brief states. Refitting produces raw/clean n(1 s) ratios of 5.3, 2.6, 6.4 and 1.6 for INTC, MDLZ, BAC and TGT. In the real data, those ratios are 5.4, 2.8, 8.0 and 1.8. The simulation permits one brief state per move. It consequently undershoots raw n(1 s), reaching 86-97% of the real value.

Most of this disappears inside the total branching ratio. Raw is just 1.33 times clean, or 0.72 against 0.54, because two effects offset each other. Brief states contribute roughly s ≈ 0.45-0.50 to the raw total. Double-counting multiplies real excitation by Nm/N0, whose median is 0.46. This relation predicts raw totals with a median absolute error of 0.035. The raw fits also compress differences across stocks: clean n(1 s) ranges from 0.10 in BAC to 0.45 in TGT, whereas raw spans only 0.57 to 0.70.

A threshold with costs

Lee treats one millisecond as a measurement choice and tests alternatives. Lowering it to h = 0.25 ms brings the fast component back in 14 of 16 stock-months. Between 1 and 5 ms, kernel mass measured from 5 ms changes little at the median: at most 3% for one second and 4% for ten seconds. Locks receive a separate 5 ms allowance because 97% last less than 5 ms. Otherwise, BAC's 1-5 ms locks would escape merging and appear as fast excitation.

Rebuilding the NBBO in exchange time leaves the problem in place. Locks decline by 20-48%, but many become brief widenings. The share of sub-millisecond gaps remains 0.47-0.55. Across clocks, E1m counts change by at most 2.3%. The clean fitted models also agree across clocks except in BAC, where exchange time lowers n(1 s) by 18-28%. The paper recommends a 2 ms threshold as the safer choice there.

Two costs remain. Genuine sub-millisecond reactions cannot be recovered, a limit Lee acknowledges. In simulation, a real 0.3 ms component with mass 0.10 is lost under both definitions once brief states are included. E1m is also retrospective: persistence becomes known only 1 ms after a state starts. The paper describes the series as a reconstruction of the price path and rules out treating it as a real-time event stream. A signal built from it must restore that delay.

Baseline choice changes the total branching ratio

The abstract already acknowledges the trade-off between the total branching ratio and the baseline. The body goes further: this measure cannot settle whether the market is near criticality. For INTC in May 2017, the clean total branching ratio ranges from 0.18 with 5-minute baseline bins to 0.74 with one level per day. Over that range, n(1 s) stays within 0.100-0.103.

AIC selects 30-minute bins in all 16 stock-months tested. BIC chooses daily levels in 14 of the 16, comparing 30-minute, 60-minute and daily baselines. These information criteria cannot settle a near-critical interpretation.

Horizon masses are the quantities worth reporting, with honest error bars. For a month's n(1 s), the day-block bootstrap gives a median standard error of 0.015. The parametric bootstrap gives 0.002-0.003 in May 2017, when block SEs ranged from 0.003-0.05. Excitation changes across days, with earnings days running at 2.03 times ordinary days. The clean model still fails 11 of 16 calibrated KS tests for May 2017 and all 16 for November. Lee attributes the misfit to stronger excitation at the open, bursts lasting less than five minutes, and more reversals than a linear symmetric kernel can generate.

We could not test any of this ourselves. E1m requires microsecond-stamped NBBO states to establish whether each level persisted for 1 ms; our data stop at one-minute bars.

The paper reports neither costs nor a trading test. Its application comes upstream of a strategy: a measurement rule for calibrating execution or short-horizon risk models from event clustering. Anyone fitting a Hawkes execution or market-making model of the kind we examined in our note on state-dependent sizing to raw quote events should first check the share of gaps under 1 ms. If that share is large, the fit is measuring the feed.