Reduced-order autoregressive dynamics of a complex financial system: a PCA-based approach

TL;DR

This work tackles the problem of deciphering the joint dynamics of the NASDAQ, crude oil, gold, and the US dollar by combining time-delay embedding with PCA to build a reduced-order representation, followed by autoregression in the PCA subspace. The framework focuses on cross-asset interactions through correlation and lagged regression, and evaluates predictive performance using $R^2$ on held-out data, across varying numbers of retained principal components. The key finding is that a small set of principal components suffices to capture the dominant dynamics, with the exact dimensionality showing asset-dependent differences and a saturation of predictive gain beyond a critical dimension. This approach offers a transparent, interpretable reduced-order modeling pathway for multi-asset financial dynamics and suggests extensions to nonlinear or regime-dependent modeling for further improvements.

Abstract

This study analyzes the dynamic interactions among the NASDAQ index, crude oil, gold, and the US dollar using a reduced-order modeling approach. Time-delay embedding and principal component analysis are employed to encode high-dimensional financial dynamics, followed by linear regression in the reduced space. Correlation and lagged regression analyses reveal heterogeneous cross-asset dependencies. Model performance, evaluated using the coefficient of determination ($R^2$), demonstrates that a limited number of principal components is sufficient to capture the dominant dynamics of each asset, with varying complexity across markets.

Figures (6)

• Figure 1: Pairwise Pearson correlation matrix among NASDAQ, crude oil (Oil), gold (Gold), and the US dollar index (USD). Color intensity represents the strength of linear correlation, while numerical values indicate the corresponding coefficients.
• Figure 2: Estimated regression coefficients for the lagged multivariate linear models with self-lag terms excluded. Each panel shows the conditional influence of lagged explanatory variables at time $t-1$ on the corresponding target asset at time $t$. The sign and magnitude of each coefficient indicate the direction and strength of the cross-asset effect.
• Figure 3: $R^2$ performance of the PCA--regression model for Nasdaq. The saturation of test performance indicates that most predictive information is captured by a limited number of dominant components.
• Figure 4: $R^2$ performance of the PCA--regression model for Crude Oil. Test accuracy improves steadily with increasing dimensionality before reaching a plateau, suggesting diminishing returns beyond the optimal subspace.
• Figure 5: $R^2$ performance of the PCA--regression model for Gold. The close agreement between training and test curves indicates robust generalization and limited overfitting across a wide range of components.