Zero-inflated Poisson leads four rivals on empty 10-second NYSE bins

Forecasting an empty ten seconds on the NYSE is the result a trader could use from Mussayeva and Moussa. Whether that forecast saves a cent in execution costs goes untested. The authors acknowledge that their in-sample winner falls behind out of sample. For JPM, it finishes last among their four model variants.

Inside the count forecast

The paper proposes forecasts of 10-second trade counts for VWAP and other execution schedules. Its starting point is a Markov-switching multifractal (MSM) intensity, which lets activity change at several speeds. Each 10-second bin has an expected trade rate equal to a base rate multiplied by seven two-state chains. Each chain switches at its own frequency. The slow chains capture the day's regime; the fast ones capture bursts. A Fourier term supplies the intraday U-shape. With seven binary chains and 128 joint states, an exact Hamilton-style filter computes the likelihood without simulation.

The authors' contribution changes how that intensity produces observed counts. In the basic model, counts follow a Poisson conditional on intensity. ZIP, the zero-inflated Poisson, adds a structural-zero probability pi0 alongside the zeros the Poisson already generates. GP, the Consul-Jain generalized Poisson, uses a shape parameter alpha to allow conditional over- or underdispersion. All fitted alpha values are positive, with a mean of 0.498. ZIGP brings both additions together.

The sample contains TAQ prints from ten NYSE-listed stocks during February 2023. It covers 19 days from 10:00 to 16:00 and includes only NYSE executions. Each stock supplies 41040 bins, divided into 30000 for training and 11040 for testing. For six of the ten stocks, the median bin contains zero trades. Average activity ranges from 0.512 trades per bin for AIZ to 3.785 for JPM, whose peak is 368. The benchmarks are ACP (Heinen's autoregressive conditional Poisson), Croston's method and a random walk.

Does the in-sample winner travel?

GP achieves the lowest in-sample BIC for eight stocks; ZIGP wins the remaining two. Its JPM BIC is 186636, compared with 188474 for the basic model and 235169 for ACP.

The variance decomposition helps explain those fits. The authors use psi for the fraction of count variance attributable to the conditional distribution rather than latent intensity. For the basic model, psi ranges from 0.546 (F) to 0.785 (AEE). GP raises that range to 0.882 through 0.993. Among the four liquid names, the fastest chain's renewal probability drops from 0.999 to a mean of 0.018. The authors write that GP "can account for part of the volatility generated by the intensity process."

I would go further for the liquid stocks. GP's psi reaches 0.900 or more for all four liquid names, while the fastest chain renews with probability 0.018. Most of the fast regime machinery is left idle.

The out-of-sample ordering reverses, as the abstract acknowledges. ZIP leads on average predictive log likelihood for six stocks. GP wins two (AIZ, KO), and ZIGP wins two (F, MLM). The BIC winner's failures are more telling than that tally. GP ranks last among the four MSM variants on JPM, scoring -2.725. Its AEE score of -1.514 also falls short of the basic model's -1.394.

The authors assess calibration through probability integral transform (PIT) plots, which check the forecast distribution. They themselves flag GP's in-sample PIT miscalibration for F and JPM. JPM's out-of-sample likelihood supports that concern. F goes the other way: GP outperforms the basic model out of sample (-1.861 vs -1.884).

For the zero-event log score, ZIP wins nine stocks. F is the exception, with ZIGP at -0.653 against -0.660. Zero inflation contributes most among the illiquid names. The six low-activity stocks have an average pi0 estimate of 0.415, compared with 0.076 for the four high-activity stocks.

Squared error favors ACP

Despite posting the worst BIC for every stock, ACP delivers the lowest one-step RMSE for six: AEE, AIZ, F, FMC, MLM and WRB. The paper describes ACP as typically best except for KO, where GP records 3.983 against 4.029. Table 9 also contains three ZIP wins over ACP omitted from the text: JPM (5.853 vs 5.864), MMC (2.763 vs 2.781) and WMT (4.553 vs 4.556). Those margins range from 0.003 to 0.018. At 1-minute bins, ACP wins six stocks and the basic MSM-Poisson wins the other four. The extensions win none.

The choice of score follows the trading decision. Sizing participation slices against expected prints calls for the conditional mean, where ACP leads for six of ten stocks. Choosing whether to rest passively or wait calls for P(zero), where ZIP leads for nine of ten. The abstract captures the distinction honestly: "for point forecasting, other models are preferred". A desk would need to run two models.

One venue, one split

Following Barndorff-Nielsen et al., the authors retain only NYSE executions to reduce reporting delays. That cleaning decision is defensible. It leaves a precise interpretation of a zero bin: no NYSE prints. We did not find the proportion of each stock's consolidated trades represented by those prints in the paper. The average pi0 of 0.415 for the six illiquid names therefore measures venue selection in part. A router choosing whether to rest on the NYSE may want exactly that quantity. A desk timing a parent order across the market needs a different one.

All out-of-sample evidence comes from a single 30000/11040 split. We found no forecast-comparison test. Its absence matters for a gap such as F, where ZIP trails the basic model by -1.885 to -1.884. According to the appendix, out-of-sample fits begin at the in-sample estimates. Those estimates were fitted to all 41040 bins, test period included.

The starting point matters. An alternative initialization reduces the log likelihood by as much as 745 (GP on F). The authors consider these differences minor beside log likelihoods that "all exceed 28000 in absolute terms." Their chosen start produces an equal or higher log likelihood in every cell except Poisson on WRB (-39), so the reported results use the better optima. Model gaps are the more relevant comparison. For F, the move from Poisson to GP adds 1292 points (-83373 vs -82081); 745 exceeds half that gap. F's ranking still holds: even the worse start would leave GP on F about 547 points above the reported Poisson likelihood. The narrower concern is that local optima can differ by amounts comparable to the gaps between models.

There are two further discrepancies. The 10-second discussion identifies FMC and MLM as ZIGP's BIC wins. Table 7 instead gives ZIGP the lower BIC for AEE (115010 vs 115031) and MLM, with GP winning FMC (69481 vs 69485). The 1-minute discussion credits GP with the lowest BIC for eight stocks. Table 14 gives it seven; the basic model wins WMT (46587 vs 46706).

We could not test any of this ourselves. Rebuilding NYSE-only 10-second counts requires transaction-level prints, which we do not have.

Across 19 days, the zero-event forecast wins nine of ten stocks, with its clearest gains among the six low-activity names. I would change my view if someone used ZIP's zero probability to time passive child orders and reported fill rates and costs over more than one month. Until then, the paper offers a better count model and no execution evidence.