Multi-Level Order Flow Imbalance (MLOFI)

• Multi-Level Order Flow Imbalance (MLOFI) is a vector metric that measures net buy-minus-sell pressure across several limit order book levels instead of only the best quotes.
• It employs statistical modeling with ridge regression to address multicollinearity among depth-wise imbalances, significantly enhancing the accuracy of mid-price change predictions.
• Dimensionality reduction via PCA integrates detailed level information into a single factor, supporting more robust algorithmic trading strategies and improved market microstructure analysis.

Multi-Level Order Flow Imbalance (MLOFI) is a vector metric designed to capture net buy-minus-sell pressure across multiple depths of the limit order book (LOB). It extends best-level Order Flow Imbalance (OFI) by systematically tracking imbalances not only at the best quotes but at multiple price levels, thereby providing a more nuanced view of liquidity and latent supply/demand dynamics within financial markets (Xu et al., 2019, Zhang et al., 2020, Cont et al., 2021).

1. Formal Definition and Computation

Let the LOB for an asset have $M$ discrete price levels on both bid and ask sides, with updates indexed by event time $\tau_n$ (where each event may be an order addition, cancellation, or trade execution). For each level $m = 1,\ldots,M$ at event $n$, let:

• $b^m(\tau_n)$, $r^m(\tau_n)$: the $m$-th best bid price and its associated volume,
• $a^m(\tau_n)$, $q^m(\tau_n)$: the $m$-th best ask price and its associated volume.

The per-event changes at each LOB level are decomposed as follows:

$$\Delta W^m(\tau_n) = \begin{cases} \phantom{-}r^m(\tau_n) & \text{if } b^m(\tau_n) > b^m(\tau_{n-1}) \ \phantom{-}r^m(\tau_n) - r^m(\tau_{n-1}) & \text{if } b^m(\tau_n) = b^m(\tau_{n-1}) \ -r^m(\tau_{n-1}) & \text{if } b^m(\tau_n) < b^m(\tau_{n-1}) \end{cases}$$

$$\Delta V^m(\tau_n) = \begin{cases} -q^m(\tau_{n-1}) & \text{if } a^m(\tau_n) > a^m(\tau_{n-1}) \ q^m(\tau_n) - q^m(\tau_{n-1}) & \text{if } a^m(\tau_n) = a^m(\tau_{n-1}) \ \phantom{-}q^m(\tau_n) & \text{if } a^m(\tau_n) < a^m(\tau_{n-1}) \end{cases}$$

The net signed flow at level $m$ is then $$e^m(\tau_n) = \Delta W^m(\tau_n) - \Delta V^m(\tau_n)$$.

Over a time interval $(t_{k-1}, t_k]$, the $m$-th component of the MLOFI vector is:

$$\text{MLOFI}^m(t_{k-1}, t_k) = \sum_{n : t_{k-1} < \tau_n \leq t_k} e^m(\tau_n)$$

For $M=1$, this construction recovers the standard best-level OFI; for $M>1$, MLOFI aggregates level-by-level order imbalances up to depth $M$ (Xu et al., 2019, Zhang et al., 2020, Cont et al., 2021).

2. Statistical Modeling and Estimation

The contemporaneous impact of MLOFI on mid-price changes is commonly modeled linearly:

$$\Delta P_k = \alpha + \sum_{m=1}^{M} \beta^m\,\text{MLOFI}^m(t_{k-1}, t_k) + \epsilon_k$$

where $\Delta P_k$ is the mid-price change, $\beta^m$ quantifies the price impact of imbalance at level $m$, and $\epsilon_k$ captures residual variation (Xu et al., 2019).

The strong collinearity among MLOFI components—pairwise correlations frequently exceeding 0.7 in large-tick stocks and 0.5 in small-tick stocks—is a key empirical feature. This renders ordinary least squares (OLS) regression unstable, with large variances and diminished statistical significance for coefficients beyond the best levels. Ridge regression, which introduces a penalty $\lambda \|\beta\|_2^2$, effectively regularizes these estimates, reducing standard errors and yielding nearly all $\beta^m$ coefficients significantly nonzero at conventional confidence levels (Xu et al., 2019).

Empirical evaluation uses out-of-sample root-mean-squared error (RMSE) and adjusted $R^2$ via 5-fold cross-validation. Ridge-MLOFI with $M=10$ achieves RMSE reductions of 15–30% (small-tick stocks) and 65–75% (large-tick stocks) relative to single-level OFI, with $R^2$ approaching unity in large-tick stocks (Xu et al., 2019).

3. Dimensionality Reduction: Integrated MLOFI via PCA

Due to strong cross-correlations among depthwise OFIs, principal component analysis (PCA) is employed to derive a single integrated MLOFI factor. Denote the vector of depth-scaled order-flow imbalances as $$\mathbf{ofi}_{i,t}^{(h)} = (\mathrm{ofi}_{i,t}^{1,h}, \ldots, \mathrm{ofi}_{i,t}^{M,h})^T$$ for stock $i$ and interval $h$ (Cont et al., 2021). The first principal component $\mathbf w_1$ of historical $\mathbf{ofi}_{i,t}^{(h)}$ typically explains over 89% of total variance.

The scalar integrated MLOFI is computed as:

$$\mathrm{ofi}_{i,t}^{I,h} = \frac{\mathbf w_1^T \mathbf{ofi}_{i,t}^{(h)}}{\|\mathbf w_1\|_1}$$

This integrated OFI subsumes the information in individual levels and allows parsimonious univariate modeling that preserves most explanatory power while mitigating multicollinearity and overfitting (Cont et al., 2021).

4. Empirical Performance and Economic Interpretation

In large-scale studies (e.g., S&P 100, 2017–2019), integrated MLOFI consistently outperforms best-level OFI in explanatory and predictive regressions for short-horizon mid-price returns. In-sample and out-of-sample adjusted $R^2$ for integrated MLOFI models reach approximately 87% and 83%, respectively—substantially higher than best-level versions. The addition of cross-asset MLOFI terms yields only marginal improvement for contemporaneous returns, indicating that integrated MLOFI internalizes most of the relevant information (Cont et al., 2021).

For 1-minute-ahead prediction, cross-asset lagged MLOFI terms provide statistically significant improvement (out-of-sample adjusted $R^2$ up to 1.34%, annualized PnL up to 15.5%), but this cross-impact decays rapidly beyond a few minutes. The PCA-based depth aggregation in MLOFI efficiently compresses both depthwise and cross-asset information (Cont et al., 2021).

Mechanistically, deeper-level imbalances can foreshadow larger price moves, especially when best-level liquidity is depleted. Informationally, activity at deeper LOB levels may reflect private signals or strategic behavior, suggesting that monitoring multi-level imbalances provides forward-looking insight into price discovery (Xu et al., 2019).

5. Implementation in Algorithmic Trading Systems

MLOFI has been incorporated into several algorithmic trading agent frameworks, substantially improving sensitivity to market impact. In contrast to imbalances computed solely from top-of-book prices/volumes (e.g., $\Delta m = P_\text{micro} - P_\text{mid}$), MLOFI-based decision rules robustly detect block orders or latent imbalance at deeper levels (Zhang et al., 2020).

Pseudocode implementations maintain per-level updates and may use geometric decay factors ($\alpha \in [0.7,0.9]$) to weight deeper levels less heavily. Parameters such as depth $M$, recalculation interval, and scaling factor are selected via empirical tuning, with $M=5$ often balancing signal and noise (Zhang et al., 2020).

Agent modifications (e.g., ISHV $\rightarrow$ ZZISHV, AA $\rightarrow$ ZZIAA, ZIP $\rightarrow$ ZZIZIP) apply MLOFI-derived offsets to quoting rules, often via an additive factor scaled by average depths and weighted by decay, yielding statistically significant profit improvements under block-order pressure (Zhang et al., 2020).

6. Practical Considerations and Limitations

MLOFI computation is $O(M)$ per LOB event and remains tractable for $M\leq10$ at high-frequency data rates. Empirical recommendations include:

• $M=3$–$7$ to capture relevant block-order signals,
• Decay parameter $\alpha=0.8$ as a default,
• Scaling factor $c=5$, matching prior studies,
• Periodic recomputation every 10–50 events or per fixed time bucket.

In agent-based simulations, MLOFI-sensitive agents demonstrate performance parity with their original counterparts absent block imbalance, but significantly outpace them when latent block pressure is present. This suggests robust detection and response to hidden liquidity or information (Zhang et al., 2020).

7. Implications for Price-Formation and Market Microstructure Research

MLOFI establishes a rigorous and empirically validated link between multi-layered order flow activity and near-term price moves. It demonstrates that liquidity and imbalance beyond the proximate quotes play a nontrivial role in price formation, challenging approaches that only monitor the best bid and ask. These findings carry direct implications for high-frequency trading, market-making, and regulatory analysis of market impact.

The empirical superiority of integrated MLOFI over best-level approaches, both for single-asset and cross-asset modeling, underscores the importance of depth-wise order flow—and its summarization via principled statistical techniques—in both academic research and practical market applications. The rapid decay of cross-impact beyond a few minutes is consistent with transient lead–lag dependencies among assets and suggests a short time horizon for exploiting multi-asset LOB signals (Cont et al., 2021).