Cross-Asset Feature Engineering for Correlated Time Series

Welcome to the next stage of our journey into advanced feature engineering. In our last lesson, we developed a powerful "microscope" for time series analysis—the wavelet transform—allowing us to dissect a single series into its time-frequency components. We learned to see not just what was happening, but when it was happening. Now, we will zoom out to a "macro" perspective. Instead of looking inside one signal, we will look at the intricate dance between multiple signals.

In finance, as in many complex systems, nothing exists in a vacuum. The price of an airline stock is affected by the price of oil. The value of the Australian dollar is related to that of the New Zealand dollar. The stock prices of Google and Microsoft, while competitors, often move in response to broader tech sector trends. These interconnections are not noise; they are rich sources of predictive information.

This lesson is dedicated to engineering features that capture these cross-asset dynamics. We will learn how to quantify and model correlations, identify stable long-term relationships, and detect subtle lead-lag effects where one asset's movement precedes another's. You will learn to build features that represent not just an asset's own history, but its context within the wider market ecosystem.

Capturing Lead-Lag Relationships with Cross-Correlation

One of the most fundamental relationships between two time series is that of a leader and a follower. Does a change in asset A consistently happen a few days before a similar change in asset B? If so, asset A's recent behavior is a powerful feature for predicting asset B's future. The classic tool for measuring this is cross-correlation.

Cross-correlation systematically calculates the correlation between two series at different time offsets or "lags." It involves sliding one series forward and backward in time relative to the other and computing the correlation at each step. The resulting plot of correlation versus lag reveals the strength of the relationship at different time horizons.

To get a precise definition and see a practical example, the following article provides a concise walkthrough.

As the article demonstrates, the output of a cross-correlation analysis gives you direct, interpretable features:

• The lag with the maximum correlation: This tells you the typical lead/lag time between the two series. For instance, a peak at lag = +2 suggests that series Y's movements at time $t$ are most correlated with series X's movements at time $t-2$, meaning X leads Y by two time steps. This lag value itself can be a feature.
• The value of the maximum correlation: The height of this peak quantifies the strength of the lead-lag relationship.

Engineering Spreads and Relative Value Features

While cross-correlation is excellent for finding a time-shifted relationship, another powerful class of features arises from modeling the simultaneous relationship between assets. Often, two related assets have a "normal" price relationship that they tend to maintain. Deviations from this normal relationship can be temporary and predictive of a reversion to the mean. Features that capture these deviations are central to strategies like pairs trading.

The simplest way to model this is by creating a spread feature, which is often just the difference between the two asset prices or, more robustly, their log returns. A model can then learn how the behavior of this spread predicts the future movement of one or both assets.

The following video provides an excellent, code-oriented explanation of how to engineer features based on both an asset's own past (auto-regression) and another asset's past (cross-regression), culminating in the spread feature.

The video demonstrates a clear hierarchy: you can use asset Y's returns to predict asset X (cross-regression), or you can create a more sophisticated feature—the spread—which represents the relative value between them. An increasing spread might signal that asset X is becoming overvalued relative to Y, and a reversion is likely.

A More Robust Foundation: Cointegration

Basing a spread feature on a simple difference of prices or returns is intuitive, but it rests on a shaky assumption: that the relationship between the assets is stable. Stock prices are typically non-stationary (they have trends, and their statistical properties change over time). Simply calculating the correlation or difference between two non-stationary series can lead to spurious, unreliable results.

A much more statistically sound approach is to test for cointegration. Two or more non-stationary time series are said to be cointegrated if a specific linear combination of them is stationary. In essence, this means that even though each series may wander off on its own, they are bound by a long-term equilibrium relationship that they tend to revert to.

The spread created from a cointegrated pair is, by definition, stationary (mean-reverting), making it a far more reliable feature for a predictive model.

This next video explains the concept of cointegration and why it is superior to simple correlation for pairs trading strategies.

Interview Alert: You are tasked with building a model to trade the relationship between West Texas Intermediate (WTI) crude oil and United Airlines (UAL) stock. How would you engineer a feature to capture their relative value?

A strong answer would be: "My primary approach would be to test for cointegration between the WTI and UAL price series. Both are likely non-stationary, so a simple correlation of their prices would be misleading. A cointegration test, like the Engle-Granger or Johansen test, would determine if a stable, long-term equilibrium relationship exists between them.

If they are cointegrated, the test will provide a cointegrating vector, [1, -β]. I would then engineer a 'spread' feature defined as Spread = log(UAL_price) - β * log(WTI_price). This resulting spread series would be stationary by definition.

The key advantages of this feature are twofold: First, its stationarity means it has a stable mean and variance, making it a reliable input for a machine learning model. Second, its value represents the deviation from the long-term equilibrium. A large positive or negative value for the spread would be a strong signal that a price correction is likely, providing significant predictive power for short-term price movements in either UAL or WTI."

Advanced Technique: Cross-Asset Embeddings

So far, we've focused on pairwise relationships. But what if we want to capture an asset's relationship to the entire market or a large sector simultaneously? Inspired by techniques from NLP, where a word's meaning is captured by its context (the "company it keeps"), we can create a dense vector representation—an embedding—for each asset based on its similarity to other assets.

The process is as follows:

1. Define a set of "base" assets. These could be major market indices (S&P 500, NASDAQ), key commodities (Oil, Gold), or representative stocks from various sectors.
2. Create a similarity matrix. For each asset in our universe, we calculate a vector of similarity scores between it and every asset in our base set. The similarity metric can be Pearson correlation, but more advanced, non-linear measures like Dynamic Time Warping (DTW) can capture more complex relationships.
3. Create dense embeddings. This similarity matrix will be wide and sparse. We can use a dimensionality reduction technique like Principal Component Analysis (PCA) on this matrix to create a low-dimensional, dense embedding for each asset.

Each asset is now represented by a vector (e.g., of length 10) that encodes its relationship to the broader market. Assets that behave similarly will have embeddings that are close to each other in this new vector space. This embedding can then be fed as a powerful contextual feature into a deep learning model like an LSTM.

The following research paper details exactly such a system. It's a comprehensive resource, so we will focus on the conceptual and implementation sections.

This embedding approach is a perfect example of the "out-of-the-box" thinking you're interested in. It transforms the problem from simple pairwise analysis to a holistic, market-aware feature representation. As a final thought, this idea of representing assets and their relationships can be taken even further by modeling the entire market as a graph, where assets are nodes and their relationships are edges. You could then apply Graph Neural Networks (GNNs) to learn features directly on this structure, as shown in the conceptual diagram below.

Conclusion

In this lesson, we have significantly expanded our feature engineering toolkit for time series. We've moved beyond analyzing single series in isolation and learned to harness the rich information encoded in the relationships between them.

Key Takeaways:

• Lead-Lag Detection: Cross-correlation is a fundamental tool for identifying which assets lead and which follow, allowing you to use one asset's past to predict another's future.
• Relative Value & Spreads: The deviation of two assets from their normal relationship is a powerful predictive signal. While simple spreads are useful, features based on cointegration are more statistically robust as they model long-term equilibrium.
• Scalable Context with Embeddings: For complex, many-asset systems, you can create dense vector embeddings for each asset. These embeddings capture an asset's "identity" within the broader market by encoding its similarity to a set of reference assets.

So far in this module, we have engineered features from individual series dynamics, their volatility, their multi-scale properties, and now, their relationships to each other. In our final lesson for this module, we will drill down to an even finer level of granularity and explore feature engineering for high-frequency trading by looking at order book data.