Cross-Impact of Order Flow Imbalance
in Equity Markets

We first measure the model performance via in-sample adjusted-$R^{2}$, denoted as the in-sample $R^{2}$ or IS $R^{2}$. From Table 3, we first observe that $\textbf{PI}^{[1]}$ can explain 71.16% of the in-sample variation of a stock’s contemporaneous returns, consistent with the findings of Cont et al. [17]. Meanwhile, $\textbf{PI}^{I}$ displays higher and more consistent explanation power, with an average adjusted $R^{2}$ value of 87.14% and a standard deviation of 9.16%, indicating the effectiveness of our integrated OFIs.⁷⁷7We also investigate price impact with multi-level OFIs in Appendix B. The results demonstrate that the price impact model using integrated OFIs outperforms those using multi-level OFIs in out-of-sample tests.

Table 3 also shows that the in-sample $R^{2}$ values increase as cross-asset OFIs are included as additional features, which is not surprising given that $\textbf{PI}^{[1]}$ (respectively, $\textbf{PI}^{I}$) is a nested model of $\textbf{CI}^{[1]}$ (respectively, $\textbf{CI}^{I}$). However, the increments of the in-sample $R^{2}$ are smaller when using integrated OFIs (87.85%-87.14%=0.71%), compared to the counterpart using best-level OFIs (73.87%-71.16%=2.71%). This indicates that cross-asset multi-level OFIs may not provide additional information on the variance in returns compared to the price impact model with integrated OFIs.

Next, we take a closer look at the cross-impact coefficients based on either the best-level or integrated OFIs, i.e. $\beta_{i,j}^{[1]}$ and $\beta_{i,j}^{I}\,\,(i,j=1,\dots,N)$. Table 4 reveals the frequency of self-impact and cross-impact variables selected by LASSO, i.e. the frequency of $\beta_{i,j}^{[1]}\neq 0$ (respectively, $\beta_{i,j}^{I}\neq 0$). We observe that self-impact variables are consistently chosen in both $\textbf{CI}^{[1]}$ and $\textbf{CI}^{I}$, as found in Cont et al. [17]. However, another interesting observation is that the frequency of a cross-asset integrated OFI variable selected by $\textbf{CI}^{I}$ is around 1/2 of its counterpart in $\textbf{CI}^{[1]}$. When we turn to the size of the average regression coefficients as shown in Table 4, we obtain reasonably consistent results. The self-impact is much higher than the cross-impact in both the $\textbf{CI}^{[1]}$ and $\textbf{CI}^{I}$ models, while the cross-impact coefficients in $\textbf{CI}^{I}$ are about 1/3 in scale of their counterparts in $\textbf{CI}^{[1]}$. This difference in scale may suggest that the cross-impact terms are less important in the $\textbf{CI}^{I}$ model, however, it is worth noting that even small cross-term coefficients can have a non-negligeable effect when aggregated at portfolio level.

In addition, cross-impact being large/small is a statement about a matrix, more related with its singular values and relative magnitudes, rather than the individual value of the coefficients. Figure 4 shows a comparison of the top 20 singular values of the coefficient matrices given by the best-level and integrated OFIs.⁸⁸8Here we only use the off-diagonal elements, i.e. $\big{[}\beta_{i,j}^{[1]}\big{]}_{i\neq j}$ and $\big{[}\beta_{i,j}^{I}\big{]}_{i\neq j}$. The relatively large singular values of the best-level OFI matrix are a consequence of the higher edge density, and thus average degree, of the network. Note that both networks exhibit a large top singular value of the adjacency matrix (akin to the usual market mode in Laloux et al. [38]), and the integrated OFI network has a faster decay of the spectrum, thus revealing its low-rank structure.

We visualize a network for each coefficient matrix, which only preserves the edges larger than a given threshold (following Kenett et al. [34], Curme et al. [20]), as shown in Figure 5. We color stocks according to the GICS sector division, and sort them by their market capitalization within each sector.⁹⁹9The Global Industry Classification Standard (GICS) is an industry taxonomy developed in 1999 by MSCI and Standard & Poor’s (S&P) for use by the global financial community. As one can see from Figure 5(a), the cross-impact coefficient matrix $\left[\beta_{i,j}^{[1]}\right]_{j\neq i}$ displays a sectorial structure, in accordance with previous studies (e.g. Benzaquen et al. [5]). This behavior could be fueled by index arbitrage strategies, where traders may, for example, trade an entire basket of stocks coming from the same sector against an index.

Figure 5(b) presents the network of cross-impact coefficients based on integrated OFIs, i.e. $\left[\beta_{i,j}^{I}\right]_{j\neq i}$. Compared with Figure 5(a), the connections in Figure 5(b) are much weaker, implying that the cross-impact from stocks can be potentially explained by a stock’s own multi-level OFIs, to a large extent. Note that there is only one connection from GOOGL to GOOG, as pointed out at the top of Figure 5(b). This stems from the fact that both stock ticker symbols pertain to Alphabet (Google). Our study also reveals that OFIs of GOOGL have more influence on the returns of GOOG, not the other way around. The main reason might be that GOOGL shares have voting rights, while GOOG shares do not.

In Figures 5(c) and 5(d), we set lower threshold values (75-th, respectively, 25-th percentile of coefficients) in order to promote more edges in the networks based on integrated OFIs. Interestingly, we observe only four connections in Figure 5(c). Except from bidirectional links between GOOGL and GOOG, there exists a one-way link from Cigna (CI) to Anthem (ANTM), and another one-way link from Duke Energy (DUK) to NextEra Energy (NEE). Anthem announced to acquire Cigna in 2015. After a prolonged breakup, this merger finally failed in 2020. Therefore, it is unsurprising that the OFIs of Cigna can affect the price movements of Anthem. Conversely, Anthem’s OFIs also have an impact on the price movements of Cigna, but to a lesser extent. Further research should be undertaken to investigate this phenomenon. In terms of the second pair, Duke Energy rebuffed NextEra’s acquisition interest in 2020. Note that 2020 is not in our sample period. This finding hints that certain market participants may have noticed the special relationship between Duke Energy and NextEra Energy before this mega-merger was proposed.

Although the in-sample estimation yields interesting findings, practitioners are eventually concerned about the out-of-sample estimation. Therefore, we propose to perform the following out-of-sample tests. We use the above fitted models to estimate returns on the following 30-minute data and compute the corresponding $R^{2}$, denoted as out-of-sample $R^{2}$ or OS $R^{2}$.¹⁰¹⁰10Previous studies either investigated the in-sample $R^{2}$ (including Cont et al. [17], Capponi and Cont [8]), or adopted a cross-validation method (e.g. Xu et al. [53]). However, these works failed to consider the generalization error of their models or damaged the chronological order of the time-series data. In contrast, we obey the temporal ordering in our study. These matters are vital to practitioners, as only the historical data are accessible for the model fit in practice.

Table 5 reports the average values and their standard deviations of out-of-sample $R^{2}$ of $\textbf{PI}^{[1]}$, $\textbf{CI}^{[1]}$, $\textbf{PI}^{I}$, and $\textbf{CI}^{I}$. We first focus on the models using best-level OFIs. It appears $\textbf{CI}^{[1]}$ has a slight advantage compared with $\textbf{PI}^{[1]}$ for out-of-sample tests with an improvement of 1.39% (=66.03%-64.64%). However, when involving multi-level or integrated OFIs, the performance of $\textbf{CI}^{I}$ is slightly worse than $\textbf{PI}^{I}$, indicating that the cross-impact model with integrated OFIs cannot provide extra explanatory power to the price impact model with integrated OFIs. Overall, we observe that the models using integrated OFIs unveil significant and consistent improvements over those using only best-level OFIs.

In general, we observe strong evidence implying $\textbf{CI}^{[1]}$ provides a better out-of-sample estimate than $\textbf{PI}^{[1]}$, while for $\textbf{PI}^{I}$ and $\textbf{CI}^{I}$, the evidence is opposite. However, it is important to note that these conclusions are based on a point estimate and do not necessarily indicate statistical significance. Therefore, we perform the following hypothesis test for each stock on the out-of-sample data to assess statistical significance,

We employ the approach from Giacomini and White [24] and Chinco et al. [12] to assess statistical significance through a Wald-type test (see Ward and Ahlquist [52]). Theorem 1 in Giacomini and White [24] implies that we can use a standard $t$-test to evaluate the statistical significance of changes in $R^{2}$. A $p$-value less than a given significance level $\alpha$ rejects the null hypothesis in favor of the alternative at the $1-\alpha$ confidence level, implying $\textbf{CI}^{[1]}$ has significantly better estimation than $\textbf{PI}^{[1]}$. We also implement this test for the comparison between $\textbf{PI}^{I}$ and $\textbf{CI}^{I}$.

Figure 6 illustrates the main results from the above hypothesis tests. When using only the best-level OFIs, the cross-impact model is superior to the price impact model for 91.0% (94.4%) of stocks, at the 1% (5%) confidence level. However, when examining the models using integrated OFIs, we reject the null hypothesis (i.e., in favor of the cross-impact model) only for 28.1% (33.7%) of stocks at the 1% (5%) confidence level. As expected, cross-impact terms can significantly improve the explanatory power of the price impact model for GOOG and GOOGL.

Dynamics of limit order book may depend on the tick-to-price ratio, or alternatively, the fraction of time that the bid-ask spread is equal to one tick for a given stock (Curato and Lillo [19]). We examine whether this dependence also extends to cross-asset OFIs.¹¹¹¹11We would like to thank an anonymous reviewer for suggesting this analysis. Our findings, presented in Table 6, suggest that cross-asset OFIs can better explain the price dynamics of stocks with a larger tick-to-price ratio.

In summary, our previous results mainly show that when considering only the best-level OFI of a single stock, the addition of the best-level OFI from other stocks slightly increases the explanatory power. On the other hand, when the information from multiple levels is integrated into the OFI, the improvement is negligible. In the meantime, it is unsurprising that taking into account more levels in the LOB ($\textbf{PI}^{I}$) could better explain price changes, compared to only considering best-level orders ($\textbf{PI}^{[1]}$).

After observing these results, several natural questions may arise: How can the above facts be reconciled?¹²¹²12One possible explanation for those facts is that the duration of the cross-impact terms might be shorter than the current time interval (30 minutes) used in our experiments, rendering the cross-impact terms vanish in out-of-sample tests. To verify this assertion, we implement additional experiments where models are updated more frequently. The results (deferred to Appendix D) reveal that even under higher-frequency updates (1-min) of the models, there is no benefit from introducing cross-impact terms to the price impact model with integrated OFIs. How do the cross-asset best-level OFIs interact with the multi-level OFIs, when modeling contemporaneous returns?

To address these questions, we consider the following scenario, also depicted in Figure 7. For simplicity, we denote the order from trading strategy $A$ on stock $i$ (respectively, $j$) as $A_{i}$ (respectively, $A_{j}$). Analogously, we define orders from strategy $B$ and $S$. Let us next consider the orders of stock $i$. There are three orders from different portfolios, given by $A_{i}$, $B_{i}$ and $S_{i}$. $A_{i}$ is at the third bid level of stock $i$ and linked to an order at the best ask level of stock $j$, i.e. $A_{j}$. Also, $B_{i}$ is at the best ask level of stock $i$ and linked to an order at the best bid level of stock $j$, i.e. $B_{j}$. Finally, $S_{i}$ is an individual bid order at the best level of stock $i$.

We observe that the best-level limit orders from stock $j$ may be linked to price movements of stock $i$ through paths $B_{j}\to B_{i}\to\text{ofi}_{i}^{1}\to r_{i}$ and $A_{j}\to A_{i}\to\text{ofi}_{i}^{3}\to r_{i}$. Thus the price impact model for stock $i$ which only utilizes its own best-level orders will ignore the information of $A_{i}$, while the cross-impact model can partially collect it along the path $A_{j}\to A_{i}$. This might illustrate why the best-level OFIs of multiple assets can provide slightly additional explanatory power to the price impact model using only the best-level OFIs.

Nonetheless, if we can integrate multi-level OFIs in an efficient way (in our example, aggregate order imbalances caused by orders $A_{i}$, $B_{i}$ and $S_{i}$), then there is no need to consider OFIs from other stocks for modeling price dynamics. For example, information hidden in the path $A_{j}\to A_{i}\to\text{ofi}_{i}^{3}\to r_{i}$ can be leveraged as long as $A_{i}$ is well absorbed into new integrated OFIs. In this sense, for stock $i$, cross-asset best-level OFIs (including $A_{j}$) are surrogates of its own OFIs at different levels (here $A_{i}$), to a certain extent. The likelihood of this relationship is attributed to massive portfolio trades that submit or cancel limit orders across a variety of assets at different levels.¹³¹³13Cao et al. [7], Hautsch and Huang [30], Sirignano [45], Chakrabarty et al. [11] showed that the depth of some deeper levels (such 2-3) is higher than the best level depth. We put forward this mechanism which potentially explains why the cross-impact model with integrated OFIs cannot provide additional explanatory power compared to the price impact model with integrated OFIs.¹⁴¹⁴14It would be a very interesting research direction to derive testable predictions for this mechanism in future work. However, so far it hinges on the availability of client-ID based LOB data, i.e. knowing that the same market participant is behind the orders for two related instruments, released at approximately the same time, such as $(A_{i},A_{j})$, $(B_{i},B_{j})$.

A related question is about the aggregation of cross-impact at portfolio level. Let us consider the OFI of portfolio $p$ as $\mathrm{ofi}_{p,t}^{1,h}:=\sum_{i=1}^{N}w_{i}\mathrm{ofi}_{i,t}^{1,h}$, where $w_{i}$ is the weight of asset $i$ in a portfolio. Then the price impact for portfolio $p$ is

where $a_{p}^{[1]}$ is the intercept and $e_{p,t}^{[1]}$ is the noise term.

On the other hand, the cross-impact for portfolio $p$ is

where $\alpha_{p}^{[1]}=\sum_{i=1}^{N}w_{i}\alpha_{i}^{[1]}$ is the intercept.

Comparing (10) and (11) shows that cross-impact at portfolio level depends on the angles of $\vec{\beta}=(\beta_{1}^{[1]},\dots,\beta_{N}^{[1]})$ and $\vec{w}=(w_{1},\dots,w_{N})$. On one hand, if individual assets exhibit a universal pattern of price dynamics, i.e. $\beta_{i}^{[1]}\approx\beta_{j}^{[1]},\forall i\neq j$, we may conclude that the portfolio return is driven by its own order flows, to a large extent. On the other hand, if the portfolio places most of the weight on a specific stock or a set of stocks with similar patterns of price dynamics, then the portfolio returns are also driven by its own order flows, rather than by cross-impact. In other scenarios, it is necessary to consider the cross-impact. One can perform an analogous analysis for models with integrated OFIs.

In fact, the mechanism depicted in Figure 7 aligns with this analysis. For example, orders placed on stock $i$ represented by $A_{i}$ and $S_{i}$ have the potential to affect the price movements of that stock, which subsequently affect the returns of portfolio $B$, even if $A_{i}$ and $S_{i}$ have no direct association with $B$. Additionally, $A_{j}$ may have a different influence on the performance of $B$ because of the (potentially) different price dynamics of stock $j$. Therefore, it may be necessary to take into account cross-asset OFIs when developing models for portfolio returns.

To examine the potential of cross-asset OFIs in explaining portfolio returns, we choose two widely-used portfolio construction methods: the equal-weighted portfolio (EW), and the eigenportfolio¹⁵¹⁵15See Avellaneda and Lee [3]. using the 1st principal component (Eig1). Table 7 summarizes the out-of-sample $R^{2}$ of various models on these two portfolios. As Table 7 shows, there is a significant difference between $\textbf{PI}^{[1]}$ and $\textbf{CI}^{[1]}$, $\textbf{PI}^{I}$ and $\textbf{CI}^{I}$, indicating that cross-impact is a crucial factor when modeling portfolio returns. Again, the models using the integrated OFIs outperform their counterparts using the best-level OFIs.

Table 8 summarizes the out-of-sample $R^{2}$ values of the aforementioned predictive models when predicting the subsequent 1-minute returns, i.e. $f=1$. It appears the cross-impact models $\textbf{FCI}^{[1]}$ (respectively, $\textbf{FCI}^{I}$, CAR) achieve higher out-of-sample $R^{2}$ statistics compared to the price impact models $\textbf{FPI}^{[1]}$ (respectively, $\textbf{FPI}^{I}$, AR). We also implement the same hypothesis test described in Section 3 to investigate the statistical significance (unreported) of these results. We observe that the cross-impact models exhibit significantly superior performance than the price impact models across all stocks, at the 1% confidence level.

Most of the empirical literature in return prediction focuses its evaluations on out-of-sample $R^{2}$. However, we remark that negative $R^{2}$ values do not imply that the forecasts are economically meaningless (see more discussions in Kelly et al. [33], Choi et al. [13]).¹⁷¹⁷17A simple example can be framed as follows. Consider a model with one predictor and suppose that the estimated predictive coefficient is a significantly large multiple of the actual value. In this case, the $R^{2}$ will become negative. However, the predictions will be perfectly correlated with the true expected return, resulting in a positive expected return for our strategy. Proposition 4 in Kelly et al. [33] further proposed that we can worry less about the positivity of out-of-sample $R^{2}$ from a prediction model and focus more on the out-of-sample performance of specific trading strategies based on predicted returns. To emphasize this point, we will incorporate these return forecasts into a forecast-based trading strategy, and showcase their profitability in the following subsection.

Considering the different magnitudes of the OFIs and returns, we first normalize the coefficient matrix of each model by dividing by the average of the absolute coefficients. Figure E.1 (deferred to Appendix E) shows the average coefficient matrices of $\textbf{FCI}^{[1]}$, $\textbf{FCI}^{I}$, and CAR. For example, as revealed in Figure E.1(a) ($\textbf{FCI}^{[1]}$), for a specific stock, the main influence comes from its own OFI, i.e. the absolute values of diagonal elements are significantly larger than the off-diagonal ones. We observe that cross-impact is often negative, consistent with Pasquariello and Vega [41]. Except for the self-impact, most stocks are also influenced by stocks in Communication Services, Consumer Discretionary and Information Technology.

To better illustrate the interactions between different stocks, we construct a network for each normalized coefficient matrix and only preserve the cross-asset edges (i.e. off-diagonal elements) larger than the 95-th percentile of coefficients. Figure 8 illustrates some of the main characteristics of the coefficient networks for $\textbf{FCI}^{[1]}$, $\textbf{FCI}^{I}$, and CAR. For example, we again observe that there are more edges from Communication Services, Consumer Discretionary and Information Technology, indicating they may contain more predictive power for others.

To gain a better understanding of the structural properties of the resulting network, we aggregate node centrality measures (see Everett and Borgatti [22]) at the sector level, and also perform a spectral analysis of the adjacency matrix. From Table 9, we observe that the out-degree centrality of Communication Services, Consumer Discretionary and Information Technology is significantly larger than that of others, consistent with previous findings. Figure 8(c) also shows that the network based on returns contains more inner-sector connections than the other two counterparts, thus implying a sectorial structure. Table 10 presents the top five most significant stocks in terms of out-degree centrality in each network, which exhibit more impact on the prices of other stocks.

Figure 9 shows a barplot with the average value for the top 20 largest singular values of the network adjacency matrix, for best-level OFIs, integrated OFIs, and returns, where the average is performed over all constructed networks. For ease of visualization and comparison, we first normalize the adjacency matrix before computing the top singular values, which exhibit a fast decay. In addition to the significantly large top singular value revealing that the networks have a strong rank-1 structure, the next 6-8 singular values are likely to correspond to the more prominent industry sectors.

On the basis of return forecasts, we employ a portfolio construction method, proposed by Chinco et al. [12], to evaluate the economic gains of the aforementioned predictive models.