A trade in E-mini S&P moves the price less when it hits a thick queue, and execution models should account for that. Hafsi, Lillo and Szymanski make the case convincingly. Their estimate of how much posting liquidity moves the price is far less settled: by their own account, one calibration choice changes it by a factor of fifty.
The book changes the price response
In the standard propagator, each market order delivers the same price kick, κ. The kick decays through a kernel determined by how strongly order flow excites itself, modelled here as a Hawkes process. The authors extend the passive-impact model of Ouazzani Chahdi, Rosenbaum and Szymanski by making κ depend on volume at the best quote consumed by the trade. Price then has two components: the usual propagator evaluated at average sensitivity, and a correction for the queue state.
For the affine sensitivity κ(q) = κ0 + κ1·q, the correction contains an observable signal and an unobserved anticipation term. The observable signal, H, weights each trade's signed flow by the queue it consumed. Limit orders and cancels affect price only through the queues that subsequent market orders encounter. In the authors' words, "Passive impact is thus not a separate signed-flow term."
The sample consists of CME ES order-book messages from 2025, restricted to the front month and 10:00 to 16:00. After cleaning, it contains 263 sessions, each with roughly 330,000 trades; 97% of trades face a one-tick spread. The regressions explain weighted-mid changes over 212-second windows using the propagator signal and H. Each fit uses 16 days of coefficients to test the next day, yielding 193 test days.
The estimated queue slope is κ1 = −8.1×10⁻⁶, with a Fama-MacBeth s.e. of 1.7×10⁻⁶. Every rolling 20-session window produces a negative slope, as do all 24 fifteen-minute bins. The weighted mid itself depends on the queues, a concern the authors acknowledge. Using the plain mid preserves the sign, with t = −7.7. When sessions are sorted by book thickness, a unit trade produces an immediate response of 9.3×10⁻³ index points in thin books and 4.0×10⁻³ in thick books, a difference of about 2.3 times.
The sign holds throughout; the magnitude halves over the year. In windows ending mid-May 2025, the rolling slope reaches −2.5×10⁻⁵. The authors attribute that to the spring volatility episode. The first-half estimate is −1.8×10⁻⁵, compared with −9.5×10⁻⁶ in the second half, with t = −2.6 for the difference.
How much R² belongs to the queue term?
The abstract credits "the passive-impact term" with substantial explanatory power beyond the classical propagator. The term derived in the theorem delivers less than that phrasing suggests. Adding the affine queue term raises unclipped out-of-sample R² from 0.422 to 0.437.
A much larger gain comes from a different specification: a dictionary of 102 book-state signals, with 105 parameters altogether, estimated by partial ridge. Its unclipped R² reaches 0.597. Plain OLS using that same dictionary manages 0.425, with two test days below −1. Shrinkage matters here. Across all sixteen cells, ridge beats the propagator; Newey-West t ranges from 7 to 13 for equities and from 7 to 12 for Treasuries. This result supports the claim that book state helps explain price moves.
Forecast clipping also earns some of the improvement. A 3σ forecast clip raises the propagator's R² from 0.422 to 0.458, and the affine model's to 0.469. The benchmark gains 0.036 from clipping alone, more than double the incremental contribution of the queue term. Across the sixteen instrument-years, the affine gain with the 3σ forecast clip ranges from 0.002 for ES 2023 to 0.015 for Russell 2000 futures, RTY, in 2023.
On that same clipped basis, ridge adds 0.06 to 0.19. Most of the improvement therefore comes from the dictionary, which is outside the theorem. The conclusion displays the figures together: 0.458 to 0.469 for the queue term, 0.473 for log sensitivity, and 0.603 for the full book state. In the abstract, the gain from the 105-parameter dictionary is attributed to the passive-impact term.
These fits are contemporaneous.
Signed volume and H, the main regressors, use realized order flow from the very window whose price change they explain. The coefficients come from earlier days, which makes the test out-of-sample, but the flow remains unknown before trading. An R² of 0.6 describes the share of the price move explained after that window's flow is observed.
Posting 122 ES contracts: six percent of a tick, give or take fiftyfold
The passive experiment posts 100 units of average event size (AES), where the mean best-quote insertion is 1.22 contracts. That amounts to about 122 contracts, submitted as unit sell limit orders over 50 seconds. With affine sensitivity, the weighted mid reaches a peak decline of 1.5×10⁻² index points, about 6% of a tick. It finishes within 1 to 4% of that peak. Extending the posting period to 100 seconds alters the peak by under 1%. The log-log volume slope is 0.98 for V ≥ 10 AES; at V = 50, trading the same volume aggressively moves price 22 times more.
Much of the near-linear scaling follows from the setup. The simulation producing 0.98 assumes affine sensitivity. Even log and sum-of-powers sensitivities give an exponent of 0.99 in the stationary-shift calculation. At full participation, the stationary queue shifts by only about 16 AES, and every fitted sensitivity is close to linear over that interval. Ouazzani Chahdi, Rosenbaum and Szymanski report 0.54 with a square-root sensitivity.
The level remains the problem. The authors describe calibration of the queue mean-reversion rate as "the dominant uncertainty". For V = 100, fitting queue drift over queues of 60 to 140 AES produces a peak of 1.5×10⁻². A drift fit over the entire state range gives a reversion rate of 1.6 per second and a peak of 1.0×10⁻². With the interquartile-filtered global fit, the rate is 77 per second and the peak falls to 3.0×10⁻⁴. As the authors explain, "a book that absorbs a posted unit in a hundredth of a second leaves almost no trace on the price." Their estimates leave unresolved which book describes ES.
The authors also state two modelling choices that constrain any reading of this estimate as a cost. The experiment measures posted volume without a fill or adverse-selection model. It also holds market-order intensity independent of queues, although the empirical projection λ̂_M(q) visibly changes with q.
A place in TCA
Queue conditioning is the usable result. A child market order's expected impact should decline as the consumed-side queue grows thicker. The 2.3-fold difference between thin and thick sessions is too large to average away. The passive estimate, however, depends on a reversion rate that remains unpinned. It should therefore stay a model output rather than serve as a cost of posting.
We could not test any of this. Our ES history consists of daily bars. The model requires event timestamps and trade signs, along with pre-trade queue sizes and cancellation messages; aggregation preserves none of them.
The authors propose queue-reactive market orders and fill-adjusted costs as extensions. A version incorporating both, with estimates that remain within a factor of two across the reversion fits, would change our view of the passive number.