Skip to content

Cross-Impact of Order Flow Imbalance in Equity Markets

Authors: Rama Cont, Mihai Cucuringu, Chao Zhang Institution: University of Oxford (Mathematical Institute, Department of Statistics, Oxford-Man Institute), The Alan Turing Institute Year: 2021 (first draft); published Quantitative Finance 23(10), 2023 arXiv: 2112.13213 Categories: q-fin.TR, q-fin.CP, q-fin.ST

Abstract (plain-language)

When a large trader moves AAPL's order book, does it also shift GOOG's price? This paper tests that question — cross-impact — on Nasdaq-100 stocks and arrives at a subtle answer. Using top-10 LOB levels instead of just the best-level, they build a single aggregate signal (integrated OFI) by taking the first principal component of the per-level OFI vector. That signal alone explains 87% of contemporaneous minute-return variance, up from 71% for best-level OFI. Crucially, once you use integrated OFI, contemporaneous cross-impact from other stocks adds nothing — the multi-level aggregation has already absorbed whatever information cross-asset OFIs carried. But lagged cross-asset OFI does help forecast the next minute's return and translates into higher PnL in a forecast-implied trading strategy. Cross-impact is a short-horizon phenomenon that decays within minutes.

Key contributions

  1. Integrated OFI — First systematic procedure for aggregating top-\(M\) multi-level OFIs into a single signal. PCA on the 10-level OFI vector captures >89% of variance in the first component; re-normalising by \(\ell_1\) norm gives weights that sum to 1. See integrated-ofi.
  2. Contemporaneous-impact result — Best-level single-asset OFI model (PI¹): adjusted \(R^2 = 71\%\) IS, \(65\%\) OOS. Integrated-OFI single-asset model (PIᴵ): \(R^2 = 87\%\) IS, \(84\%\) OOS. Adding cross-asset OFIs (LASSO-selected) to the best-level model buys \(\sim\!1.4\%\) OOS; adding them to the integrated-OFI model buys essentially nothing (\(-0.2\%\)). Conclusion: integrated OFI subsumes contemporaneous cross-asset information.
  3. Predictive cross-impact — Lagged cross-asset OFIs (up to 30 min of lags) improve 1-minute-ahead return forecasts in both best-level and integrated variants. At the 1% confidence level, the cross-impact predictive model (FCI) beats the own-OFI predictive model (FPI) for all tested stocks.
  4. Economic gains — In a Chinco-et-al.-style forecast-implied trading strategy (trade only when $|\hat{r}| > $ spread, weight by signal/volatility), FCI doubles the annualised PnL versus FPI (0.43 vs. 0.21 for best-level; 0.39 vs. 0.23 for integrated). Predictability decays rapidly with horizon — by 30 min cross-impact provides no advantage.
  5. Portfolio-level cross-impact is real — Even when individual-stock cross-impact is absent (integrated OFI case), the portfolio-level projection depends on the angle between \(\vec{\beta}\) and the portfolio weight vector \(\vec{w}\). For eigenportfolios and equal-weighted portfolios, cross-impact adds \(\sim\!3\%\) OOS \(R^2\).
  6. Mechanism — Proposes that integrated OFI captures "multi-asset portfolio trades" (a trader simultaneously placing correlated orders across assets) as paths like \(A_j \to A_i \to \text{ofi}^3_i \to r_i\), whereas best-level OFI misses them and has to borrow information via the cross-impact terms.

Method summary

Integrated OFI construction

For stock \(i\), interval \((t-h, t]\):

\[ \text{ofi}^m_{i,t,h} \;=\; \frac{1}{Q^{M,h}_{i,t}} \sum_{n=N(t-h)+1}^{N(t)} \text{OF}^{m,b}_{i,n} - \text{OF}^{m,a}_{i,n} \]

where \(\text{OF}^{m,b}_{i,n}\) is the signed event contribution at bid level \(m\) (positive for arrivals / upticks, negative for cancels / downticks); \(Q^{M,h}_{i,t}\) is the average book depth across the top \(M\) levels, used to scale for intraday depth patterns.

Stacking across \(m = 1, \ldots, 10\) gives the multi-level OFI vector \(\mathbf{of}^{(h)}_{i,t}\). The integrated OFI is the first principal component normalised so its weights sum to 1:

\[ \text{ofi}^I_{i,t,h} \;=\; \langle \tilde{\mathbf{w}}_1, \mathbf{of}^{(h)}_{i,t}\rangle, \quad \tilde{\mathbf{w}}_1 = \frac{\mathbf{w}_1}{\|\mathbf{w}_1\|_1} \]

where \(\mathbf{w}_1\) is the first principal vector from historical data. Full algorithm on integrated-ofi.

Four regression models

Let \(r^{(h)}_{i,t}\) be the \(h\)-minute log return of stock \(i\). The paper defines four contemporaneous regressions:

Model Equation Fit
PI¹ \(r = \alpha + \beta \cdot \text{ofi}^1 + \epsilon\) OLS
PIᴵ \(r = \alpha + \beta \cdot \text{ofi}^I + \epsilon\) OLS
CI¹ \(r_i = \alpha + \beta_{i,i} \cdot \text{ofi}^1_i + \sum_{j \ne i} \beta_{i,j} \cdot \text{ofi}^1_j + \epsilon\) LASSO
CIᴵ \(r_i = \alpha + \beta_{i,i} \cdot \text{ofi}^I_i + \sum_{j \ne i} \beta_{i,j} \cdot \text{ofi}^I_j + \epsilon\) LASSO

LASSO is essential — with ~100 stocks and 30-minute estimation windows at 1-minute resolution, OLS is ill-posed and multicollinearity (cross-asset OFI correlations up to 0.6) makes unpenalised estimation hopeless.

Forward-looking variants (FPI¹, FPIᴵ, FCI¹, FCIᴵ) use lagged OFIs in \(L = \{1, 2, 3, 5, 10, 20, 30\}\) to predict \(r^{(f)}_{i,t+f}\) for forecasting horizons \(f \in \{1, 2, 3, 5, 10, 20, 30\}\) minutes.

Data

  • Source: Nasdaq ITCH via LOBSTER.
  • Universe: top 100 S&P 500 constituents by market cap as of 2019-12-31.
  • Period: 2017-01-01 to 2019-12-31.
  • Frequency: minute-level OFIs + returns.
  • Windowing: 30-minute rolling estimation windows, excluding first and last 30 minutes of the trading day.

Main results

Contemporaneous (\(R^2\), OOS, %)

Best-level OFIs Integrated OFIs
PI¹ / PIᴵ 64.64 (21.82) 83.83 (16.90)
CI¹ / CIᴵ 66.03 (19.51) 83.62 (14.53)

Integrated OFI gives a +19-point OOS jump over best-level OFI. Adding cross-impact to integrated OFI is worse OOS (overfitting signal).

Tick-size dependency (OOS \(R^2\) by tick-to-price quartile)

Larger tick-to-price ratio → cross-asset OFIs explain more. For lowest-quartile (small tick) stocks, PIᴵ = 68%, CIᴵ = 72%. For highest-quartile (large tick), PIᴵ = 90%, CIᴵ = 91%. Cross-impact mattering more for discrete-price stocks is consistent with tick-size regime literature.

Predictive (\(R^2\), 1-min-ahead, OOS, %)

Best-level Integrated Returns
FPI / FPIᴵ / AR −0.37 −0.36 −0.36
FCI / FCIᴵ / CAR −0.10 −0.10 −0.10

All models have negative OOS \(R^2\) (signal-to-noise is low at 1 min), but cross-impact variants consistently less negative. Per Kelly et al. (2022), negative \(R^2\) does not rule out positive economic performance — confirmed below.

Economic performance (annualised PnL)

Best-level Integrated Returns
FPI / FPIᴵ / AR 0.21 0.23 0.23
FCI / FCIᴵ / CAR 0.43 0.39 0.40

Cross-impact roughly doubles the annualised PnL of the forecast-implied strategy. Ignores transaction costs — authors note this is not the focus.

Network structure

Coefficient matrices exhibit low-rank structure dominated by a "market mode" (top singular value). Out-degree centrality is concentrated in Communication Services, Consumer Discretionary, and Information Technology — these sectors lead the others. Highest-centrality individual stocks: AMZN, NFLX, NVDA, GOOG/GOOGL.

Limitations

  • Horizon is minute-level. The mesoscopic analysis deliberately ignores sub-minute microstructure effects. Cross-impact findings might not transfer to sub-second trading.
  • Transaction costs ignored in the PnL comparison.
  • Integrated OFI drops level information. A limitation acknowledged in Section 4.4 — depth-specific strategic order placement is collapsed into a single number. A multi-level cross-impact model with explicit level awareness is left as future work.
  • US equities only. No evidence it generalises to futures, FX, crypto, or markets with different tick-size regimes at the universe level.
  • Static factor structure assumption. LASSO cross-impact coefficients are re-estimated every 30 minutes; the paper does not model how the cross-impact network itself evolves.
  • In-sample PC fitted globally. Principal vector \(\mathbf{w}_1\) is computed from historical data across all stocks, not stock-specific. This is a feature for stability, but may lose stock-level structure.

Connections to other wiki pages

  • Extends: price-impact-order-book-events (single-asset best-level OFI) and mlofi-xu-gould-howison (multi-level OFI as a vector) — this paper unifies them with an aggregation step.
  • Relates to: price-impact-generalized-ofi — both address the multi-level-OFI question, but with different aggregations (log-GOFI vs. PCA-normalised integrated OFI).
  • Uses: integrated-ofi (new, introduced here) as primary feature, LASSO for sparse cross-impact selection.
  • Companion result: Kolm, Turiel, Westray (Mathematical Finance, 2023) — "Deep Order Flow Imbalance: Extracting Alpha at Multiple Horizons" — shows DNNs on multi-level OFIs beat LOB-direct inputs. This paper is the interpretable-linear-model counterpart.
  • Contrasts with: Benzaquen, Mastromatteo, Eisler, Bouchaud (2017) — which argued for meaningful cross-impact — by showing that cross-impact disappears once within-asset multi-level information is integrated.
  • Concept page: cross-impact (new).

Direct signal-design takeaways

For anyone building an LOB-snapshot or L3 signal:

  1. Use multiple levels. Level-1 OFI leaves \(\sim\!20\) points of \(R^2\) on the table vs. integrated OFI.
  2. PCA-aggregate across levels. The first PC is stable across stocks and captures most of the signal; re-normalising by \(\ell_1\) keeps weights interpretable.
  3. Don't bother with contemporaneous cross-asset features if you already aggregate levels properly — it's overfitting.
  4. Do use cross-asset OFI for short-horizon forecasting (\(\leq 3\) min). LASSO keeps it tractable.
  5. Scale OFIs by intraday depth (\(Q^{M,h}_{i,t}\)), not just by the previous day's average.