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Integrated OFI

Definition

Integrated OFI is a single scalar signal that aggregates order-flow imbalance across the top \(M\) levels of the limit order book. It is constructed as the first principal component of the multi-level OFI vector, normalised so the weights sum to 1. Introduced by Cont, Cucuringu & Zhang (2023) as the base feature for contemporaneous and predictive impact models.

Formally, for stock \(i\) and interval \((t-h, t]\):

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

where \(\mathbf{w}_1\) is the first principal vector of the depth-scaled multi-level OFI vector \(\mathbf{of}^{(h)}_{i,t} = (\text{ofi}^1_{i,t,h}, \ldots, \text{ofi}^M_{i,t,h})^T\), computed from historical minute-level data.

Construction algorithm

  1. Compute per-level OFIs. For each level \(m = 1, \ldots, M\):

$\(\text{OF}^{m,b}_{i,n} = \begin{cases} q^{m,b}_{i,n} & \text{if } P^{m,b}_{i,n} > P^{m,b}_{i,n-1} \\ q^{m,b}_{i,n} - q^{m,b}_{i,n-1} & \text{if } P^{m,b}_{i,n} = P^{m,b}_{i,n-1} \\ -q^{m,b}_{i,n-1} & \text{if } P^{m,b}_{i,n} < P^{m,b}_{i,n-1} \end{cases}\)$

Symmetrically for the ask side. Sum over events \(n\) in the interval, subtract ask from bid, and scale by the average top-\(M\) depth \(Q^{M,h}_{i,t}\) to neutralise intraday depth patterns.

  1. Stack into vector. \(\mathbf{of}^{(h)}_{i,t} = (\text{ofi}^1_{i,t,h}, \ldots, \text{ofi}^M_{i,t,h})^T \in \mathbb{R}^M\).

  2. Fit PCA on historical data. Using minute-level observations across a chosen calibration window, compute the first principal component \(\mathbf{w}_1 \in \mathbb{R}^M\) of the empirical covariance of \(\mathbf{of}^{(h)}\).

  3. Normalise. \(\tilde{\mathbf{w}}_1 = \mathbf{w}_1 / \|\mathbf{w}_1\|_1\) so the scalar retains OFI-like units and the weights sum to 1.

  4. Project. For any future \((t, h)\): \(\text{ofi}^I_{i,t,h} = \tilde{\mathbf{w}}_1^T \mathbf{of}^{(h)}_{i,t}\).

In the original paper \(M = 10\) and the calibration is refreshed on a rolling basis (implicitly through the 30-minute re-estimation windows of the downstream regressions).

Why PCA-normalised aggregation?

The problem. Multi-level OFIs are highly collinear — pairwise correlations across levels exceed 75% in the Nasdaq-100 sample. A plain OLS with 10 OFI features is numerically unstable and most coefficients are statistically indistinguishable from zero.

The resolution. The first PC captures >89% of total variance (std 6%) in the 10-OFI covariance matrix (Cont-Cucuringu-Zhang, Table 2), while the remaining components contribute <5% each. Collapsing onto the first PC retains almost all signal with \(10\times\) dimensionality reduction.

The \(\ell_1\) normalisation \(\|\mathbf{w}_1\|_1 = 1\) is a normalisation choice, not a constraint on the fit — it ensures the resulting scalar has interpretable units (a convex combination of per-level OFIs) and is directly comparable with the per-level OFIs at the same scale.

Empirical weight structure

The first principal component weights show:

  • Lowest weight on best-level OFI, highest standard deviation across stocks — the top-of-book is the most idiosyncratic level.
  • Deeper levels (2–3) receive most weight on average — consistent with evidence that depth at levels 2–3 often exceeds depth at level 1 (Hautsch-Huang, Chakrabarty et al.).
  • Stock-characteristic dependence:
  • High-volume stocks: deeper-level OFIs dominate the PC weights.
  • Low-volatility stocks: same — deeper OFIs win.
  • Low-volume / large-spread stocks: best-level OFI weight rises.

This suggests that for actively traded liquid names, the useful signal is not where the order book starts but where institutional flow parks itself.

Performance (from Cont, Cucuringu, Zhang 2023)

Feature Model IS \(R^2\) OOS \(R^2\)
Best-level OFI PI¹ 71.16% 64.64%
Integrated OFI PIᴵ 87.14% 83.83%

A +16–20 point \(R^2\) jump from a single aggregation step.

When to use / when not to

Use integrated OFI when: - You want a single feature that captures most multi-level information for minute-resolution impact modelling. - You're running a linear / penalised linear regression and want to avoid collinearity headaches from raw multi-level OFIs. - Interpretability matters (weights are a convex combination of per-level OFIs).

Don't use when: - You need horizon-specific level information. The PC collapse loses which level is "doing the work" — for questions about strategic order placement, use per-level OFIs directly. - Your data window is too short to estimate a stable first PC (need at least several hundred minute bars per stock). - You're training a deep model that can absorb the raw multi-level vector — DNNs may benefit from the extra dimensions (Kolm-Turiel-Westray).

Variants

Aggregation Source Notes
Integrated OFI (PCA, \(\ell_1\)-norm) cross-impact-ofi-equity-markets Default recommendation; variance-weighted across levels
Multi-level OFI vector (no aggregation) mlofi-xu-gould-howison Preserves level info; used as input to deep models
log-GOFI price-impact-generalized-ofi Log-transform for non-minimum-tick stocks; generalised to R² ~84% on CSI 500
Equal-weighted multi-level OFI Baseline (not recommended) Ignores that best-level is idiosyncratic

Implementation notes

  • Scale before PCA: divide each \(\text{OF}^{m,\cdot}_{i,n}\) by \(Q^{M,h}_{i,t}\) (average depth across the top \(M\) levels in the interval) before forming the vector. Not doing this bakes intraday depth patterns into the PC weights.
  • Cross-sectional or stock-specific PC? The original paper fits the PC cross-sectionally per stock, not globally. Globally-fitted PCs are more stable but may lose stock-idiosyncratic level structure.
  • Refresh frequency: the paper rolls the calibration with the 30-min regression window. A much longer window (e.g. one day) would bias against recent regime changes; much shorter is noisy.

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