A Structural Matrix Autoregressive Model for the Joint Dynamics of Volume, Volatility, and Returns

Abstract: This paper proposes a Structural Matrix Autoregressive (SMAR) model for the joint analysis of asset returns, realized volatility, and trading volume in a large-dimensional setting. This framework simultaneously captures dynamic spillovers across financial variables and cross-sectional dependence across assets while preserving a parsimonious parameterization relative to conventional vector autoregressive models. The model is estimated on daily data for the constituents of the Dow Jones Industrial Average over the period 2021-2025 and is structurally identified through restrictions consistent with the Mixture of Distributions Hypothesis and efficient market theory. The empirical findings indicate that volatility is the primary driver of trading activity, suggesting that informational shocks are predominantly incorporated into markets through price variability. Forecast error variance decompositions further reveal that, although internal shocks dominate short-term volume dynamics, cross-asset spillovers account for more than 50% of trading volume variation at longer horizons. Finally, an event-study analysis around FOMC announcements supports the proposed decomposition by identifying significant increases in the informative component of trading activity on announcement days followed by rapid mean reversion.

• The paper introduces the SMAR model, a parsimonious framework that reduces parameter complexity from O(m²n²) to O(m²+n²) per lag for high-dimensional data.
• It demonstrates that volatility shocks significantly drive trading volume with a coefficient of 0.5075, while return shocks have modest negative impacts.
• Forecast error variance decompositions reveal strong cross-asset spillovers and rapid mean reversion, highlighting asymmetric and efficient market behavior.

Structural Matrix Autoregressive Model for Joint Dynamics of Volume, Volatility, and Returns

Model Formulation and Identification

The paper introduces the Structural Matrix Autoregressive (SMAR) model, designed for simultaneous modeling of asset returns, realized volatility, and trading volume in large cross-sectional settings. The SMAR extends the SVAR methodology of Sims (1980) to multivariate matrix-valued contexts, preserving both dynamic spillovers between financial variables and cross-sectional dependence between assets while offering substantial parsimony. Specifically, instead of vectorizing the data matrix and incurring $O(m^2n^2)$ parameters per lag—as in conventional SVAR or PVAR—the SMAR requires only $O(m^2 + n^2)$ parameters per lag. This efficient parameterization enables high-dimensional dynamic modeling without sacrificing tractability.

The SMAR employs two contemporaneous interaction matrices: $\mathbf{R}_0$ for inter-asset effects and $\mathbf{C}_0$ for inter-variable effects. Identification is achieved via theory-driven recursive ordering, consistent with the Efficient Market Hypothesis and the Mixture of Distributions Hypothesis (MDH). In the empirical application, contemporaneous cross-asset shocks are ruled out by setting $\mathbf{R}_0 = \mathbf{I}_m$, focusing identification on the structural matrix $\mathbf{C}_0$.

Structural shock orthogonalization is performed on financial variables, with $\mathbf{C}_0$ recursively identified as lower triangular. The estimation strategy leverages iterative “flip-flop” algorithms and analytical properties of separable covariance structure to resolve structural identification, impulse responses, and forecast error variance decompositions.

Empirical Application and Results

The SMAR model is estimated on daily data for the Dow Jones Industrial Average constituents (2021-2025) using realized bipower variation as volatility measure, standardized returns, and log volumes. Stationarity is confirmed by spectral radius, and the choice of lag length (MAR(1)) is validated by BIC.

Empirical findings highlight economically significant causal relationships:

• Volatility is the dominant driver of trading volume: The contemporaneous structure yields a statistically and economically large coefficient (0.5075) for the effect of volatility shocks on trading volume.
• Return shocks have immediate but modest negative impacts on both volume and volatility, consistent with leverage effects; these coefficients (-0.0460 for volume, -0.0473 for volatility) are significant but small relative to volatility-volume linkage.
• Feedback effects are strong and asymmetric: Past trading volume positively influences current volatility, while past volatility exerts a “cooling off” effect on trading activity. Lagged returns exert negligible effect on other variables, supporting efficient market theory.

Structural impulse response functions (IRFs) show rapid mean reversion for return self-responses, slow decay for volume and volatility self-responses indicative of volatility clustering and persistent market activity, and pronounced asymmetry in cross-variable dynamics. Volatility shocks instantaneously increase volume, reflecting immediate information incorporation, whereas volume-driven volatility increases are delayed and more persistent.

Forecast Error Variance Decomposition and Information Dynamics

A central contribution is the decomposition of trading volume forecast error variance into within-asset informative (volatility-driven), liquidity (own-volume shock), and cross-asset spillover components. At the one-day horizon, 79.4% of trading volume variance is due to liquidity (own-volume), 20.4% to volatility (information shock), and only 0.2% to returns. At the 20-day horizon, cross-asset spillovers dominate, accounting for 56.9% of total variance, highlighting the high degree of interconnectedness among Dow constituents.

The decomposition isolates the informative component driven by volatility, validated by an event study around FOMC announcements. On announcement days, the informative share of trading volume surges sharply, followed by rapid mean reversion, consistent with efficient price discovery and market microstructure theory.

Theoretical and Practical Implications

The SMAR model provides a theoretically sound and empirically effective framework for disentangling informational and liquidity shocks in high-dimensional financial systems. Its parsimony and analytical tractability enable structural inference and impulse response analysis even in large asset panels, overcoming limitations of vectorized SVAR and Panel VAR approaches. The findings underscore the central role of volatility in driving trading activity, validate asymmetric cross-variable dynamics, and quantify the rapid integration and contagion among assets in response to shocks.

Potential future directions include relaxing the assumption of $\mathbf{R}_0 = \mathbf{I}_m$ to allow contemporaneous cross-asset effects, incorporating more sophisticated identification strategies via network or factor models, and enabling time-varying parameterizations for regime-sensitive modeling. The SMAR framework is suitable for applications in risk management, liquidity monitoring, contagion analysis, and market microstructure studies.

Conclusion

The SMAR model advances structural matrix-valued time series analysis for financial markets, offering parsimonious estimation, rigorous identification, and interpretable decompositions for joint dynamics of returns, volatility, and trading volume. Empirical results demonstrate strong volatility-volume links, asymmetric impulse responses, and persistent cross-asset propagation consistent with latent information flow theories. The framework provides a robust basis for further exploration of high-dimensional market interdependencies and their implications for market stability and information aggregation (2606.08141).