A square-root impact coefficient fitted to calendar-time prints can measure two different things for two different names. Angstmann and Gebbie explain the clock mechanics behind that mismatch. They leave its size unanswered, although that is the question facing anyone who has to cost a parent order.
The model under the clocks
The model begins with two liquidity densities on a log-price lattice, one for the bid side and one for the ask. Each updates once per order-book event. Between events, liquidity jumps to neighbouring price levels with weights r/2 left, 1-r stay and r/2 right. It decays at rate nu, source terms replenish it, and meta-order forcing pushes the front. Bid and ask liquidity annihilate when they meet, at rate kappa. The diffusive limit is D = lim (r/2) dx^2 / tau_m, where tau_m is the waiting time between event m-1 and event m.
Subtraction does the main work. The imbalance is phi = rho_B - rho_A. Because the reaction term kappa tau_m rho_A rho_B is the same on both sides, it vanishes from the difference and leaves a closed linear recursion for phi. The authors state the scope precisely. Cancellation holds conditional on the realised waiting-time sequence, regardless of the law governing those waits. Their mid-price proxy is the zero of phi, located by linear interpolation through the grid cell in which the sign changes.
The impact calculation then follows the latent-order-book treatment of Tóth and co-authors and Donier and co-authors. The model moves to a continuum operational clock u. Around its zero, the stationary imbalance profile is linearised. Its local slope, L_u, measures local liquidity depth. The background sources are frozen over the execution horizon, while a delta forcing on the moving front drives the perturbation. The resulting front position satisfies a Volterra equation.
For constant-rate execution m_e(s) = m0 on [0,U], the kernel reduces to Abel under weak cancellation and small displacement. The front then moves as y(u) - y_0 approximately (m0/L_u) sqrt(u / (pi D_u)).
The sign-memory argument puts Lillo, Mike and Farmer on the event clock. A uniformly sampled child order encounters the length-biased meta-order law l p_L(l)/E[L]. Its sign autocorrelation is exactly (1/E[L]) times the sum over l > tau of (l - tau) p_L(l). When meta-order lengths have power-law tails with exponent alpha+1, C_tau decays as tau^{-(alpha-1)}, giving gamma = alpha - 1.
Both observables are then composed with an inverse subordinator E_t. Finite-mean waiting times make E_t of order t/tau_bar, recovering the ordinary square-root law after a deterministic rescaling for activity. Under a stable clock with index beta in (0,1), the impact scale instead follows t^{beta/2}.
No market evidence accompanies the construction. The paper contains no data, no calibration and no numerical solution of the Volterra equation. Its single figure is explicitly schematic. As the caption says, the drawn front segments "are not intended as an exact solution for the concatenated sequence."
What can be implemented now?
We read three components as sufficiently specified to code without filling in missing choices. First comes the imbalance recursion, including the exact cancellation and the rule for interpolating the mid-price. Next is the renewal expression for C_tau, which requires only a distribution of meta-order lengths. The final component is C_t = sum over tau of P(N_t - N_0 = tau) C_tau. Once a counting process has been chosen, this becomes a convolution.
The continuum-limit results demand further judgement. The authors describe the limit, in their own word, as "formally" derived. We did not find either a convergence statement for the interpolant or an error bound. The passage from the Volterra equation to the Abel kernel is also asymptotic in weak cancellation and small displacement, with no threshold supplied. Anyone implementing it must choose when nu U is sufficiently small. The same judgement applies to dropping (y(u) - y(s))^2 / (4 D_u (u-s)) from the exponential.
The location of the concavity matters. At a fixed operational-time participation rate, the paper states proportionality to sqrt(Q_U), where Q_U = m0 U. Yet the formula is linear in m0, so our reading is that size enters linearly when duration is fixed. A pooled fit across different rates and durations tests Eq. (12) only when rate and duration are held fixed.
A desk still needs a schedule
Execution is scheduled here on the operational clock. The authors say directly that moving a calendar-time schedule into the framework would require subordinating both the forcing process and the front. They leave that step undone.
A trading desk faces precisely this setup. VWAP or TWAP instructions are expressed in wall-clock or volume time, while child orders arrive with an intensity partly caused by the execution itself. The conclusion acknowledges another restriction: the event clock is exogenous rather than state dependent. The authors defend both choices immediately. In their account, those restrictions separate statements that are exact for the event-indexed bid/ask reduction from results belonging to operational-time transport and effects introduced only by subordination. The bookkeeping is fair. It still cannot price a schedule written on the calendar clock, which is what the desk must cost.
For a liquid US large cap, the anomalous branch also appears poorly matched to the market. A stable index beta below one entails an infinite expected inter-event time. Our working assumption for liquid names is intraday variation in intensity around a finite mean. By the paper's own result, that assumption selects the t/tau_bar case: the standard square-root law with deterministic activity rescaling. The authors also discuss tempered clocks that cross over to finite-mean behaviour at long horizons, an explicit concession to the same issue.
We could not test any of this.
The required observables are event-indexed signed order flow and inter-event waiting times, together with bid and ask depth for constructing the imbalance field. Our highest-frequency data consists of one-minute OHLCV. It has no trade prints, no sign classification and no quotes. Minute-bar volume cannot reconstruct event ordering or hidden-order fragmentation without proxies that would alter the method under test.
Evidence that would change the verdict
The paper ends with a clean empirical claim. Square-root coefficients estimated across assets or venues combine liquidity response with clock projection unless the sampling clock is held fixed. Testing it is cheap. On the same tape, fit the coefficient in trade time, volume time and calendar time, then report the spread.
Agreement among the three exponents to within 0.02 would leave the clock hierarchy as bookkeeping with no empirical bite. A difference of 0.1 or more between trade-time and calendar-time exponents on the same fills would change how cross-venue impact calibrations should be interpreted.
We have written before about the Sato and Kanazawa line against which the paper positions itself. In that work, a Lévy-walk representation allows square-root impact to coexist with Brownian prices (our note). We have also examined how the order of composing the impact rate and decay kernel determines whether a propagator can be manipulated (here). Angstmann and Gebbie apply the same kind of argument to the clock, and the separation is useful. Yet the paper remains six pages of construction without a number drawn from a market.