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A Structural Matrix Autoregressive Model for the Joint Dynamics of Volume, Volatility, and Returns
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Keywords: Matrix-valued time series; Volatility; Trading volume decomposition; Structural model; MDH
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Recent global history has shown how extreme events cause financial markets to jitter, which experience periods of high volatility and extremely variable trading volumes Longin2001, Diebold2009, Bollerslev2018. In this context, it is crucial for market participants, institutions, and policymakers to understand how volatility and liquidity spillovers interact Acharya2005, Brunnermeier2009. Indeed, while from a microeconomic perspective, understanding the dynamics and correlations of asset prices leads investors and asset managers to portfolio optimization, asset management, and risk hedging, from a macroeconomic perspective, it should lead policymakers towards strategies aimed at economic and financial stability.
Although traditional financial theories have primarily focused on the dynamics between asset prices and their volatility, the analysis of trading volumes has received relatively less attention, despite being an indicator of market sentiment Darolles2015. For example, price increases during periods of low volume may only produce transitory effects, while similar price movements during periods of high volume may have long-lasting implications. As further discussed in Bollerslev2018, trading volume and return volatility tend to co-vary Chiang2010, especially after news shocks. This result underlies the common practice of using trading volume as an exogenous regressor in time varying volatility models.
Furthermore, the mixed distribution hypothesis (MDH), initially formulated by Tauchen1983, posits that the observed joint behavior between asset volatility and trading volumes is due to latent information flows that simultaneously drive price fluctuations and trading activity. In parallel, Darolles2015 emphasize the methodological challenges in jointly modeling price dynamics and trading volumes, particularly when one wants to separate the dynamics related to information shocks from those attributable to liquidity constraints. These insights point to the necessity of extending existing approaches to better capture the complex dynamics underlying trading activity. In this direction, He2014 develop a multi-asset mixture distribution hypothesis model to investigate commonality in stock returns and trading volumes based on factor structure for returns and trading volumes.
Although existing methodologies reliably assess the liquidity-volatility relationship for individual stocks Chordia2000, Pastor2003, they overlook different components of liquidity dynamics. To fill this gap in the literature, we propose here to model the cross-asset dynamics of volatility, volumes and prices through the Structural Matrix-valued AutoRegressive (SMAR) model. This method extends the structural vector autoregressive (SVAR) model of Sims1980 to multivariate settings in which data form matrices. As in the MAR model introduced by Chen2021, we collect $m$ assets in the rows and $n$ financial metrics in the columns, thus forming a data matrix for each time $t=1,2,\ldots,T$. In this way, SMAR jointly takes into account the spillover effects across assets and the dynamic interconnectedness of some financial variables that share a common latent structure in the MDH. The SMAR offers a parsimonious alternative to the traditional architectures, typically the Structural VAR (SVAR) or Panel VAR Canova2013 models, in which matrix data are stacked into a vector. In fact, excluding any deterministic component, the vectorization in these models foresees the estimation of $m^2n^2$ parameters per lag, while the number of parameters in the SMAR reduces to $m^2+n^2$. Hence, the vectorization results in a loss of efficiency since the relationships between rows and columns can no longer be maintained, and makes the estimation computationally cumbersome. In addition, while identifying structural innovations in panel SVAR models typically requires computationally intensive data-pooling or the assumption of statistical independence on a massive $mn \times mn$ covariance matrix $\bm{\Sigma}$ Herwartz2024, the SMAR framework leverages a separable covariance structure $\bm{\Sigma} = \bm{\Sigma}_{\text{r}} \otimes \bm{\Sigma}_{\text{c}}$, where $\bm{\Sigma}_{\text{c}}$ and $\bm{\Sigma}_{\text{r}}$ are the column- and row-wise error covariance matrices. This allows one to perform simultaneous structural identification exclusively within the lower-dimensional $n \times n$ variable space $\bm{\Sigma}_{\text{c}}$, leaving cross-sectional dependencies to be absorbed by $\bm{\Sigma}_{\text{r}}$ without requiring restrictive entity-level orderings. Furthermore, we can exploit the algebraic properties of our framework to solve analytically the identification issues and compute both single-asset-single-variable impulse response functions (IRFs) and averaged IRFs across either assets or financial variables. In addition, since our approach relies on the explicit structural impulse response functions, they represent a valid alternative to those defined by Chen2021 in the MAR context.
We apply the SMAR model to daily data for $m=30$ assets composing the Dow Jones index, covering the period from January 2021 to February 2025. The empirical analysis delivers several findings. First, volatility is a key determinant of trading activity. Second, a forecast error variance decomposition suggests that trading volume is primarily driven by its own shocks over the short run, while cross-asset spillovers account for more than half of the total volume variance over a 20-day time horizon. Moreover, an event study around the announcements of the Federal Open Market Committee (FOMC) shows a statistically significant spike in the information share of volume on announcement days Lucca2015, followed by a rapid mean reversion consistent with efficient price discovery Andersen2003.
The remainder of the paper is organized as follows. In Section (ref), we introduce the SMAR model, discuss identification issues, and define the impulse response functions. Section (ref) proposes a specification of the column-wise structural matrix based on some results from the finance literature and provides some financial considerations. Section (ref) is devoted to the empirical analysis, and finally Section (ref) concludes.
Let $\mathbf{Y}_t$ be a time series matrix $m \times n$, with $t=1,2,\ldots,T$, containing data for $m$ assets and $n$ financial measures. Since we are interested in a structural model that incorporates simultaneous relationships between all row and column variables of $\mathbf{Y}_t$, these simultaneous interactions are represented in a linear setup by the $m \times m$ matrix $\mathbf{R}_0$ and the $(n \times n)$ matrix $\mathbf{C}_0$, both of full rank. Therefore, the structural MAR($p$) model (hereafter SMAR) is defined as
where $p$ is the lag order, $\mathbf{R}_1, \ldots, \mathbf{R}_p$ are $m \times m$ row-wise effect parameter matrices whose Frobenius norm must be equal to 1 Chen2021, $\mathbf{C}_1, \ldots, \mathbf{C}_p$ are $n \times n$ matrices containing the parameters for column-wise relationships. The deterministic components have been suppressed for notational convenience. \(\mathbf{U}_t\) denotes an $m \times n$ matrix of serially uncorrelated structural shocks with zero mean and unit variances kilian2017. It follows that the $mn \times mn$ covariance matrix of $\text{vec}(\mathbf{U}_t)$ (where the vec operator transforms a matrix into a vector by stacking the columns of the matrix one underneath the other) is the identity matrix
where $\mathbf{I}_{m}$ and $\mathbf{I}_{n}$ represent the row-wise and the column-wise covariance matrices, respectively. Therefore, there are as many structural shocks as variables in the model, and these shocks are mutually uncorrelated with unit variance kilian2017.
The reduced-form representation of the structural MAR($p$) model, i.e., where $\mathbf{Y}_t$ is a function of the lags of $\mathbf{Y}_t$ only, is obtained by post-multiplying both sides of Equation (ref) by \(\mathbf{C}_0^{-\top}\) and pre-multiplying by \(\mathbf{R}_0^{-1}\). This leads to \[ \mathbf{R}_0^{-1}\mathbf{R}_0\mathbf{Y}_t\mathbf{C}_0^\top\mathbf{C}_0^{-\top} = \mathbf{R}_0^{-1}\mathbf{R}_1 \mathbf{Y}_{t-1} \mathbf{C}_1^\top \mathbf{C}_0^{-\top} + \dots + \mathbf{R}_0^{-1}\mathbf{R}_p \mathbf{Y}_{t-p}\mathbf{C}_p^\top \mathbf{C}_0^{-\top} + \mathbf{R}_0^{-1}\mathbf{U}_t\mathbf{C}_0^{-\top}, \] that corresponds to the reduced-form MAR($p$) by Chen2021 given by
where $\mathbf{A}_i=\mathbf{R}_0^{-1}\mathbf{R}_i$ are $m\times m$ matrices, and $\mathbf{B}_i=\mathbf{C}_0^{-1}\mathbf{C}_i$ are $n\times n$ matrices for $\forall\,i = 1,\ldots,p$. Consequently, the reduced-form innovation matrix at time $t$ is given by
To compute the covariance of $\mathbf{E}_t$, we can use the vectorized form of Equation (ref), given by
Since $\mathbf{R}_0$ and $\mathbf{C}_0$ are nonsingular, pre-multiplying by $\left(\mathbf{C}_0\otimes \mathbf{R}_0\right)^{-1}$ implies that
Further, we can obtain \[ \text{vec}(\mathbf{E}_t) = \text{vec}\left(\mathbf{R}_0^{-1}\mathbf{U}_t \mathbf{C}_0^{-\top}\right) = \left[\mathbf{C}_0^{-1} \otimes \mathbf{R}_0^{-1}\right]\text{vec}(\mathbf{U}_t). \] Considering the covariance matrix of structural shocks defined by the equation (ref), the covariance matrix of \(\text{vec}(\mathbf{E}_t)\) turns out to be
Assuming that the covariance of $\mathbf{E}_t$ in a MAR model Chen2021 is an $mn \times mn$ matrix expressed as \[ \bm{\Sigma}_E = \bm{\Sigma}_{\text{c}} \otimes \bm{\Sigma}_{\text{r}}, \] it follows that the term $\bm{\Sigma}_{\text{c}} = \left(\mathbf{C}_0^\top\mathbf{C}_0\right)^{-1}$ represents the column-wise covariance of $\mathbf{E}_t$, while $\bm{\Sigma}_{\text{r}} = \left(\mathbf{R}_0^\top\mathbf{R}_0\right)^{-1}$ denotes the row-wise covariance of $\mathbf{E}_t$. In this context, the column-wise and row-wise covariance matrices of the reduced-form error, $\bm{\Sigma}_{\text{c}}$ and $\bm{\Sigma}_{\text{r}}$, can be considered known kilian2017.
For the identification of $\mathbf{C}_0^{-\top}$, it can be noticed that the number of free parameters in $\bm{\Sigma}_{\text{c}}$ is $n(n +1)/2$ which is the upper limit for the number of parameters in $\mathbf{C}_0^{-\top}$ that can be uniquely identified kilian2017. This means that we have to impose at least $n(n-1)/2$ normalizing restrictions.
Since $\bm{\Sigma}_{\text{c}}^{-1} = \mathbf{C}_0^\top \mathbf{C}_0$, we can assume that $\mathbf{C}_0^\top$ is a lower triangular matrix with positive diagonal elements, as in the following section, i.e., the Cholesky factor of $\bm{\Sigma}_{\text{c}}^{-1}$.
Analogously, given that $\bm{\Sigma}_{\text{r}}$ has $m(m+1)/2$ unique elements, we must impose $m(m-1)/2$ conditions to achieve identification of the parameters of $\mathbf{R}_0$. This means that, in our model, identifying contemporaneous effects across assets can be challenging, as the number of required constraints increases quadratically with the number of assets ($m$). Defining such a large number of constraints from economic theory is often very difficult, as the simultaneous interactions between different assets could be extremely complex.
While we recognize that constraints can be imposed in the Equation (ref) by dividing assets into clusters using appropriate techniques (e.g., cluster analysis or principal components analysis), or by adopting some criterion to separate the elements (e.g., sector or return on assets), we assume that contemporaneous changes are due exclusively to structural shocks to financial measures, rather than being specific to individual assets; in practice, we impose the structure $\mathbf{R}_0 = \mathbf{I}_m$.
In this section, we present the log-likelihood of the SMAR(1) model. We start by defining the set of parameters contained in the vector $\bm{\psi} = \left[\text{vec}(\mathbf{C}_0)^\top, \text{vec}(\mathbf{C}_1)^\top, \text{vec}(\mathbf{R}_0)^\top, \text{vec}(\mathbf{R}_1)^\top\right]^\top$. The Gaussian log-likelihood Wu2026 is
where
Given that $\mathbf{U}_t = \mathbf{R}_0\mathbf{Y}_t\mathbf{C}_0^\top - \mathbf{R}_1 \mathbf{Y}_{t-1} \mathbf{C}_1^\top$, the first-order derivatives with respect to $\mathbf{R}_0, \mathbf{R}_1, \mathbf{C}_0$ and $\mathbf{C}_1$ are given by
Since the first-order conditions for $\mathbf{C}_0$ and $\mathbf{R}_0$ do not have an analytic solution, the parameters can be estimated by numerical optimization. However, this is hampered by the dimension of $\mathbf{C}_0$ and $\mathbf{R}_0$ which makes this approach unfeasible, especially for high-dimensional problems. Alternatively, one can rely on the iterative estimation Chen2021 of $\mathbf{A}_1, \mathbf{B}_1, \bm{\Sigma}_{\text{c}}, \bm{\Sigma}_{\text{r}}$, often referred to as the “flip-flop” algorithm Dutilleul1999. In this case, only one matrix is updated, while keeping fixed the others using the following iterations
where $\mathbf{A}_1$ and $\bm{\Sigma}_{\text{r}}$ have a unit Frobenius norm Chen2021,Bucci2026. Since we impose $\mathbf{R}_0 = \mathbf{I}_m$ in Section (ref), here we focus only on $\mathbf{C}_0$. We can recover $\hat{\mathbf{C}}_0^\top$ as the Cholesky factor of the estimated inverse of $\bm{\Sigma}_{\text{c}}$, i.e.,
For the purpose of interpretability, we normalize the effect of the structural shocks on the errors of the reduced-form model as follows:
where $\langle\hat{\mathbf{C}}_0\rangle = \text{diag}(\hat{\mathbf{C}}_0)$.
Using the following total score for $\bm{\beta} = \left[\text{vec}(\mathbf{A}_1)^\top, \text{vec}(\mathbf{B}_1)^\top, \text{vech}(\bm{\Sigma}_{\text{r}})^\top, \text{vech}(\bm{\Sigma}_{\text{c}})^\top\right]^\top$, $s_T(\bm{\beta}) = \sum_{t=2}^{T}s_t(\bm{\beta})$ (where $\operatorname{vech}$ is the operator that stacks the lower triangular elements of a matrix in a vector), derived in Appendix (ref)
where $\mathbf{D}_n$ is the duplication matrix that projects from the $n$-dimensional space onto the $n(n+1)/2$-dimensional space (the same is for $\mathbf{D}_m$), i.e., $\text{vec}(\mathbf{X})=\mathbf{D}_n \text{vech}(\mathbf{X})$. We can compute the asymptotic covariance matrix for the reduced-form parameters with the Fisher's Information matrix \[ \bm{\mathcal{I}}(\bm{\beta}) = \mathbb{E}\left[\mathbf{s}_t(\bm{\beta})\mathbf{s}_t(\bm{\beta})^\top\right] \] which, under regularity conditions, asymptotically converges to $-\mathbb{E}[\bm{\mathcal{H}}(\bm{\beta})]$, where $\bm{\mathcal{H}}(\bm{\beta})$ is the Hessian matrix of the log-likelihood function. Therefore, evaluating the score at the optimum, we can estimate the asymptotic covariance matrix using the outer-product-of-gradients (OPG) estimator $\widehat{\text{Var}}(\hat{\bm{\beta}}) = \left[ \sum_{t=2}^T \mathbf{s}_t(\hat{\bm{\beta}})\mathbf{s}_t(\hat{\bm{\beta}})^\top \right]^{-1}$.
In our case, we need to find the standard errors of the structural impact parameters. Since the structural matrix ${\mathbf{Q}}^{-\top}$ is a non-linear transformation of the reduced-form parameters, the standard errors of $\hat{\mathbf{Q}}^{-\top}$ are computed via the delta method\footnote{As a robust alternative to ensure validity in finite samples, standard errors can also be obtained through a residual-based bootstrap procedure, which accounts for the potential non-normality of the innovations $\mathbf{E}_t$.}. To this end, we define the $n(n-1)/2$-dimensional vector containing the sub-diagonal elements of $\hat{\mathbf{Q}}^{-\top}$, $\operatorname{vecl}(\hat{\mathbf{Q}}^{-\top})$. Proposition (ref) provides the Jacobian of $\text{vecl}(\hat{\mathbf{Q}}^{-\top})$ with respect to $\operatorname{vech}(\hat{\bm{\Sigma}}_c)$.
Applying the delta method, the asymptotic distribution of the structural parameters is given by
where $\mathbf{V}_{\bm{\Sigma}_{\text{c}}}$ is the asymptotic covariance matrix of the MLE estimator of the $n(n+1)/2$ singular elements of $\bm{\bm{\Sigma}_{\text{c}}}$.
Our approach computes structural impulse response functions (IRFs) using an explicit structural decomposition, in contrast to Chen2021. In fact, in our SMAR model the reduced-form errors ($\mathbf{E}_t$) are generated by a set of underlying uncorrelated structural shocks ($\mathbf{U}_t$) and the link is provided by the structural error in Equation (ref). Once we identify $\mathbf{C}_0$, we obtain the orthogonal shocks and thus can trace their causal impact through the IRFs. The orthogonalization is thus performed on the column-wise relationships, as captured by the matrix $\mathbf{C}_0$.
To compute the IRFs, for the sake of simplicity, we consider the vectorized form of a stable MAR(1) model \[ \mathbf{y}_t = \bm\Phi_1 \mathbf{y}_{t-1} + \mathbf{e}_t, \] where $\mathbf{y}_t = \text{vec}(\mathbf{Y}_t)$, $\bm \Phi_1 = (\mathbf{B}_1 \otimes \mathbf{A}_1)$ are the $mn \times mn$ coefficient matrices, $\mathbf{e}_t = \text{vec}(\mathbf{E}_t)$ is the $mn \times 1$ vector of reduced-form errors. The Wold representation of this model is given by
where each $mn \times mn$ matrix $\bm\Psi_h$ represent the $h$-th impulse response of $\mathbf{y}_t$ to the reduced-form shocks $\mathbf{e}_t$. These matrices can be computed recursively Lutkepohl1990, with $\bm\Psi_0 = \mathbf{I}_{mn}$, and \[ \bm\Psi_h = \bm\Phi_1^h = (\mathbf{B}_1 \otimes \mathbf{A}_1)^h = (\mathbf{B}_1^h \otimes \mathbf{A}_1^h) \qquad \text{for } h \geq 1. \] In the SMAR model, we have that $\mathbf{E}_t =\mathbf{R}_0^{-1}\mathbf{U}_t \mathbf{C}_0^{-\top}$, which in vectorized form becomes \[ \mathbf{e}_t = \text{vec}(\mathbf{R}_0^{-1}\mathbf{U}_t \mathbf{C}_0^{-\top}) = \left[\mathbf{C}_0^{-1} \otimes \mathbf{R}_0^{-1}\right] \mathbf{u}_t, \] where $\mathbf{u}_t = \text{vec}(\mathbf{U}_t)$. Let $\mathbf{S} = \left[\mathbf{C}_0^{-1} \otimes \mathbf{R}_0^{-1}\right]$ be the matrix that maps the structural shocks to the reduced-form shocks. Substituting it in Equation (ref), yields \[ \mathbf{y}_t = \sum_{h=0}^\infty \bm\Psi_h(\mathbf{S}\mathbf{u}_{t-h}) = \sum_{h=0}^\infty (\bm\Psi_h \mathbf{S})\mathbf{u}_{t-h}. \] The structural impulse response coefficient denotes the response of $y_{i,t+h}$ to a unit shock in $u_{j,t}$ which, from the MA version of the vectorized MAR in Equation (ref), is equal to \[ \frac{\partial y_{i,t+h}}{\partial u_{j,t}} = \bm{\eta}_i^\top \left(\bm{\Psi}_h \mathbf{S}\right) \bm{\eta}_j, \] where $\bm{\eta}_i$ and $\bm{\eta}_j$ represent the $i$-th and $j$-th columns of the identity matrix $\mathbf{I}_{mn}$. Since we imposed $\mathbf{R}_0 = \mathbf{I}_m$, in our model we have $\mathbf{S}=\mathbf{C}_0^{-1} \otimes \mathbf{I}_m.$
To compute the confidence intervals of the structural IRFs, we employ a residual bootstrap procedure Lutkepohl1990, Kilian1998. Let $\hat{\mathbf{E}}_t = \mathbf{Y}_t - \hat{\mathbf{A}}_1 \mathbf{Y}_{t-1} \hat{\mathbf{B}}_1^{\top}$ denote the matrix of reduced-form residuals obtained from the estimated SMAR model, and let $\hat{\mathbf{C}}_0$ be the corresponding structural matrix, we obtain a bootstrap sequence of innovations $\{\mathbf{E}_t^*\}_{t=1}^T$ by resampling from $\{\hat{\mathbf{E}}_t\}$ with replacement. Then, a bootstrap time series $\{\mathbf{Y}_t^*\}_{t=1}^T$ is generated recursively using the estimated matrices of parameters as follows:
initializing at $\mathbf{Y}_0^* = \mathbf{Y}_0$. The SMAR model is then re-estimated on $\{\mathbf{Y}_t^*\}_{t=1}^{T}$ to obtain bootstrap estimates $\hat{\mathbf{A}}_{1,i}^*$, $\hat{\mathbf{B}}_{1,i}^*$, and $\hat{\mathbf{C}}_0^*$, from which the corresponding structural IRFs, $\bm{\Theta}_h^* = \bm{\Psi}_h^* \mathbf{S}^*$, are computed as in Equation (ref). Repeating this procedure $V$ times yields the bootstrap distribution $\{\bm{\Theta}_h^{*(\nu)}\}_{\nu=1}^V$, from which pointwise $(1-\alpha)$ confidence bands at horizon $h$ are constructed as the empirical $\alpha/2$ and $1-\alpha/2$ quantiles Hall1992.
Since each asset $i$ enters the Kronecker structure symmetrically and we do not impose heterogeneous restrictions across assets in $\mathbf{R}_0 = \mathbf{I}_m$, we report the cross-sectional average of the structural IRF for each variable pair $(j \to k)$:
where the index $(k-1)m+i$ selects asset $i$ of variable $k$ in the $\operatorname{vec}(\mathbf{Y}_t)$ ordering. Averaging across assets exploits the panel dimension of the data and yields a parsimonious summary of the common propagation mechanism shared by the $m$ assets.
In this section, we present the structural form, i.e., the constraints on the $\mathbf{C}_0$ matrix in our model. Although the relationship between trading volumes, volatility, and returns is often complex and context-dependent, the structure of $\mathbf{C}_0$ is imposed by combining financial theory and empirical evidence. First, we refer to the Efficient Market Hypothesis Fama1970, which postulates that asset prices rapidly incorporate all available information, thus making financial returns unpredictable. Second, we consider the MDH, which establishes that both trading volume and volatility are influenced by a latent factor that typically corresponds to the flow of new information arriving in the market. Therefore, when new information becomes available, market participants tend to increase their trading activity, which often results in greater price fluctuations. However, return shocks have an immediate impact on both volatility and volume. Conversely, simultaneous changes in volatility or trading volume are very unlikely to influence returns within the same period Whitelaw1994.
Despite trading volume and volatility generally exhibit a positive correlation, the direction of causality is likely mixed. Some studies suggest that increased trading activity increases volatility, demonstrating that volume acts as a proxy for information inflow harris87,Lamoureux1990,lebaron92. In reality, the relationship is likely bidirectional, with feedback effects reinforcing both variables. The MDH Tauchen1983 explains that a new flow of information can simultaneously trigger a shock to price dispersion and trading activity. Empirical evidence consistently supports this mechanism across different markets and sample periods Karpoff1987,Gallant1992, Andersen1996. Based on this hypothesis, we choose to treat volume as the most endogenous variable, since it simultaneously responds to return shocks through leverage Black1976, Christie1982 and reacts to volatility shocks through the MDH channel. This shock structure (from asset returns to volatility and then to trading volume) is consistent with the idea that information shocks first influence prices, then generate a reassessment of risk reflected in volatility, and finally influence trading activity Lamoureux1990, Bessembinder1993.
To exemplify the structural representation, we discuss a two-asset case in which the matrix of time series is
where $\textsc{vol}_{i,t}$ is the trading volume of asset $i$ at time $t$. $\textsc{rbp}_{i,t}$ is the bipower variation of asset $i$ at time $t$: \[ \textsc{rbp}_{i,t} = \frac{\pi}{2} \sum_{\tau= 1}^{N_t-1} |r_{i,\tau}|\cdot |r_{i,\tau+1}| \] where $r_{i,\tau}$ being the $\tau$-th intraday return of asset $i$, and $N_t$ is the number of intraday observations on the $t$-th day, that we use as a proxy for volatility. In addition, daily returns are standardized through a GARCH(1,1) model as follows: \[ \textsc{ret}_{i,t} = \frac{\Delta \log p_{i,t}}{\hat{\sigma}_{i,t}} \] where $p_{i,t}$ is the $t$-th price of asset $i$ and $\hat{\sigma}_{i,t}$ is the conditional volatility estimated with the GARCH(1,1) model.
Given that the matrix $\mathbf{C}_0$ is an upper triangular kilian2017, we normalize the effect of the structural shocks on the errors of the reduced-form model by introducing the column-normalized lower triangular matrix \[ \mathbf{Q}^{-\top} = \mathbf{C}_0^{-\top}\langle\mathbf{C}_0\rangle =
, \] and the row-normalized triangular matrix \[ \mathbf{W}^{-1} = \langle\mathbf{R}_0\rangle\mathbf{R}_0^{-1} =
, \] where $\langle\hat{\mathbf{R}}_0\rangle = \operatorname{diag}(\mathbf{R}_0)$. It follows that the error of the model in Equation (ref) can be computed as \[ \mathbf{E}_t = \mathbf{R}_0^{-1}\mathbf{U}_t\mathbf{C}_0^{-\top} = \langle\mathbf{R}_0\rangle^{-1}\mathbf{W}^{-1} \mathbf{U}_t \mathbf{Q}^{-\top}\langle\mathbf{C}_0\rangle^{-1}, \] that is \[
=\langle\mathbf{R}_0\rangle^{-1}
\langle\mathbf{C}_0\rangle^{-1}. \] Defining $\tilde{\mathbf{E}}_t = \mathbf{W}^{-1} \mathbf{U}_t \mathbf{Q}^{-\top}$, which are the standardized errors of the reduced-form model, we get \[
\] for the first row of $\tilde{\mathbf{E}}_t$, and
for the second row of $\tilde{\mathbf{E}}_t$.
This decomposition highlights the explanatory power of the model. Indeed, the error equations for the first asset (the $n$-dimensional vector $\tilde{\mathbf{e}}_{1,t}$) show how structural shocks to a financial variable simultaneously impact other variables within the same asset. The parameters $c_{ij}$ quantify the magnitude of these effects across variables within the same asset.
The error equations for the second asset ($\tilde{\mathbf{e}}_{2,t}$) concern contagion between assets. The parameter $\rho_{21}$ governs the extent to which structural shocks originating in asset 1 are transmitted to asset 2. For example, the term $\rho_{21} u_{1,t}^{\textsc{ret}}+u_{2,t}^{\textsc{ret}}$ represents the total return shock affecting asset 2, which is a combination of its shock and the portion spilling over from asset 1. As already stated at the end of Section (ref), we set $\mathbf{R}_0 = \mathbf{I}_m$ to avoid any contagion effect between all assets. Analytically, the system (ref) reduces to
Our sample consists of the daily prices and volumes of the assets comprising the Dow Jones Industrial Average (as of April 2025), collected between January 2021 and February 2025 ($T=1059$ observations). The data are sourced from Bloomberg. The variables used in the analysis are computed as shown in Section (ref). Moreover, to avoid estimating the constant-value matrix\footnote{Through a preliminary analysis (available upon request from the authors) of the time series, we noted that the inclusion of a constant in the SMAR model can generate collinearity problems for a large number of assets, and can often lead to the estimation of singular matrices. This also affects the estimation time, which becomes increasingly longer as $m$ increases.}, all series are standardized. Based on the results of the Bayesian Information Criterion (BIC), we specify a SMAR(1).
As further detailed in Section (ref), the reduced-form MAR(1) defined as in Equation (ref) is estimated via maximum likelihood. The model is stationary since the product of the spectral radii is $\rho(\hat{\mathbf{A}}_1)\,\rho(\hat{\mathbf{B}}_1)\approx 0.989$ Chen2021. Equation (ref) reports the estimated $\hat{\mathbf{B}}_1$, with corresponding standard errors provided in parentheses and bold values indicating a 5% significance:
For instance, the first column shows the impact on financial variables from past volumes. Consistently with the literature on volatility clustering and long-memory properties in market activity, both trading volume and volatility exhibit a high degree of persistence. We also observe a significant cross-variable feedback between volume and volatility. Specifically, past trading volume has a positive and significant impact on current volatility, supporting the idea that intense activity anticipates higher price dispersion. Conversely, past volatility has a significant effect on current volume, suggesting a potential “cooling off” effect in trading activity following periods of high uncertainty. Finally, the coefficients related to lagged returns in the equations for volume and volatility are relatively small, although the negative impact on volume is statistically significant. This somehow provides evidence of asymmetric dynamics, where past price movements slightly influence market participation. In line with the efficient market hypothesis discussed in Section (ref), asset returns appear to be largely independent of the lagged values of all variables.
Then, using the estimated covariance matrix $\hat{\bm{\Sigma}}_{\text{c}}$, we derive the inverse of the contemporaneous column-wise matrix with normalized diagonal, $\hat{\mathbf{Q}}^{-\top}$, which takes the following values (with standard errors reported in parentheses and bold values indicating a 5% significance):
The interpretation of the estimated inverse of $\hat{\mathbf{Q}}^\top$ in Equation (ref) provides a clear and economically significant causal ordering. The highest coefficient is 0.5075, which indicates that a one-standard-deviation structural shock to market volatility (RBP) leads to a strong, significant, and immediate positive response in trading volumes. This finding suggests that unexpected changes in market uncertainty are a primary driver of contemporaneous trading activity, as market participants rapidly adjust their positions in response.
Furthermore, we find that a structural shock to returns has a negative impact on both trading volume (-0.0460) and volatility (-0.0473). This is consistent with the leverage effect Black1976: negative return shocks are associated with an immediate increase in perceived risk. The nearly equal magnitude of the two coefficients suggests that the information content of the return shocks is transmitted symmetrically to the other variables, rather than being absorbed first by one and then by the other. Notably, both effects are small relative to the volatility-volume coefficient ($0.5075$), reinforcing the view that uncertainty about future prices, rather than the return itself, is the dominant contemporaneous driver of trading activity.
Then, we analyze the average inverse response functions in Figure (ref). Looking at the diagonal (i.e., self-responses to a shock), it can be noticed that the self-response to a shock in returns tends to zero immediately after one lag. This is in large part coherent with the theory of market efficiency by Fama1970. Conversely, both the self-responses of volume and RBP decay slowly, consistent with the volatility clustering theory Engle1982, Andersen2001b and long-memory of the trading volume Bessembinder1993, Lobato2000.
Interestingly, a shock in the realized bipower leads to an immediate and persistent increase in the volume. This finding confirms what is stated in the MDH: news arrives and simultaneously moves prices and volumes. When we analyze the opposite direction (i.e., the effect of a shock in the volumes on $\textsc{rbp}_{}$), it emerges that the increase in the volatility is slightly lagged. Therefore, the intense trading activity precedes or feeds the dispersion in prices in the following periods. It should be further noticed that the magnitude of the response of trading volume to a shock in $\textsc{rbp}_{}$ is markedly different from the opposite. This implies market participants massively react increasing their trading activity when prices start fluctuating. In other words, volatility generates uncertainty, thus it forces buyers and sellers to immediately adjust their investment strategies. In addition, we find that the effect of a shock in the volumes on volatility is much more persistent than the opposite. This asymmetry in persistence constitutes, to our knowledge, a novel finding in the structural analysis of the volume--volatility nexus.
The existing literature has primarily documented the contemporaneous and short-run relationship between volumes and volatility. Gallant1992, using a semi-nonparametric bivariate framework, provide impulse response profiles for stock prices and volume and document that volume reacts strongly to price innovations, but do not examine the dynamic propagation between trading volume and volatility, nor distinguish the persistence of shocks across these two variables. Tauchen1983 and Andersen1996 focus on the contemporaneous co-movement implied by the latent information flow under the MDH, without characterizing the differential decay of cross-variable impulse responses at medium horizons. Bollerslev1999, examining the common fractional integration and long-run dependence between volume and volatility, find that both variables share a common fractionally integrated component driven by information arrival, but their framework abstracts from the short-to-medium-run asymmetry in impulse propagation that we document here.
The key contribution of our structural identification is that, by orthogonalizing shocks via $\hat{\mathbf C}_0$, we can isolate the dynamic effect of structural volume innovations on volatility. The resulting persistence is consistent with market microstructure models in which information is incorporated gradually through order flow Kyle1985, implying that the impact of trading activity on price variability unfolds over several trading days rather than instantaneously.
One of the aims of this paper is to disentangle the component of trading volume driven by informational shocks from that attributable to liquidity dynamics. The structural framework introduced in Section (ref) makes this decomposition possible through the forecast error variance decomposition (FEVD) of the model.
Since the SMAR operates on an $m \times n$ data matrix time series, the forecast error variance of trading volume for any given asset $i$ receives contributions from two distinct sources: shocks originating within the same asset (column-wise effects, governed by $\mathbf{C}_0$) and shocks originating in other assets (row-wise effects, transmitted through the lagged dynamics of $\hat{\mathbf{A}}_1$). We refer to these as the within-asset and cross-asset components, respectively.
Formally, we derive the FEVD from the structural moving average representation established in Section (ref). Recall that the vectorized SMAR admits the representation
where $\mathbf{S} = \left[\mathbf{C}_0^{-1} \otimes \mathbf{I}_m\right]$ (using $\mathbf{R}_0=\mathbf{I}_m$), $\mathbf{u}_t=\mathrm{vec}(\mathbf{U}_t)$ is the $mn\times 1$ vector of structural shocks with $\bm{\Omega} = \mathrm{Var}(\mathbf{u}_t)=\mathbf{I}_{mn}$ (see Equation (ref)), and the $mn\times mn$ matrices $\boldsymbol{\Psi}_l$ are the reduced-form impulse responses computed recursively as in Section (ref).
The $h$-step-ahead forecast error lutkepohl2005, conditional on information at time $t$, is
Let $\iota_{ik}\equiv(k-1)m+i$ denote the index of variable $k$ of asset $i$ in $\mathbf{y}_t$, and let $\bm{\eta}_s$ denote the $s$-th column of $\mathbf{I}_{mn}$, so that $\bm{\eta}_{\iota_{ik}}^\top\mathbf{y}_t$ selects the element corresponding to variable $k$ of asset $i$. Since the structural shocks are mutually uncorrelated with unit variance, the $h$-step forecast error variance of variable $k$ for asset $i$ is
Since $\bm{\eta}_{\iota_{ik}}^\top\boldsymbol{\Theta}_l\boldsymbol{\Theta}_l^\top\bm{\eta}_{\iota_{ik}} = \left\|\bm{\eta}_{\iota_{ik}}^\top\boldsymbol{\Theta}_l\right\|^2 = \sum_{s=1}^{mn}\!\left[\bm{\eta}_{\iota_{ik}}^\top\boldsymbol{\Theta}_l\,\bm{\eta}_s\right]^2$, Equation (ref) can be written as
where each term $\left[\bm{\eta}_{\iota_{ik}}^\top\boldsymbol{\Theta}_l\,\bm{\eta}_s\right]^2$ represents the squared structural impulse response at horizon $l$. Because the $mn$ structural shocks are orthogonal (i.e., $\bm{\Omega}=\mathbf{I}_{mn}$), the total variance decomposes additively lutkepohl2005, kilian2017. Denoting the contribution of the $s$-th structural shock as
it follows immediately from (ref) that \[ \sum_{s=1}^{mn}\mathcal{V}_{ik}^{(s)}(h) = \mathcal{V}_{ik}(h). \] In $\mathbf{u}_t$, the structural shocks originating from asset $i$ occupy the $n$ positions $\left\{(j-1)m+i\colon j=1,\ldots,n\right\}$, corresponding to the three financial variables ($\textsc{vol}_{},\textsc{rbp}_{},\textsc{ret}_{}$) of that asset. Summing over these positions gives the within-asset component
which we denote $\mathcal{V}_{ik}^{j}(h)\equiv\mathcal{V}_{ik}^{((j-1)m+i)}(h)$ for $j\in\{1,\ldots,n\}$. The cross-asset component collects all remaining shock contributions and is obtained by difference
Normalising by the total variance and averaging across the $m$ assets yields
and analogously for the cross-asset share
By construction,$\overline{\mathrm{FEVD}}_{\textsc{vol}_{\,},k}(h) + \overline{\mathrm{FEVD}}_{\textsc{rbp}_{\,},k}(h) + \overline{\mathrm{FEVD}}_{\textsc{ret}_{\,},k}(h) + \overline{\mathrm{FEVD}}_{\mathrm{cross},k}(h) = 1$ at every horizon $h$. At $h=1$, since $\mathbf{R}_0 = \mathbf{I}_m$ rules out any contemporaneous cross-asset transmission, $\overline{\mathrm{FEVD}}_{\mathrm{cross},k}(1) = 0$ exactly and the three within-asset components account for the full variance. As the horizon grows, cross-asset contributions accumulate through the lagged dynamics encoded in $\hat{\mathbf{A}}_1$.
We label $\overline{\mathrm{FEVD}}_{\textsc{rbp}_{},k}$ as the informative component of trading volume, since it captures the fraction of trading activity driven by the latent information flow that simultaneously moves prices and volumes under the MDH Tauchen1983. Conversely, $\overline{\mathrm{FEVD}}_{\textsc{vol}_{},k}$ constitutes the liquidity component, reflecting trading activity unrelated to contemporaneous changes in price dispersion.
Table (ref) reports the four components for trading volume ($k = \textsc{vol}_{}$) at horizons $h \in \{1, 5, 10, 20\}$. At the impact horizon ($h=1$), approximately 20.4% of volume variance is attributable to the RBP shock and 79.4% to the volume's own shock, confirming that same-day trading activity is dominated by asset-specific liquidity dynamics with a non-trivial informational contribution. The return shock accounts for a negligible 0.2%, reflecting the near-zero contemporaneous effect of returns on volume visible in Figure (ref).
As the horizon expands, both within-asset components decline in favour of cross-asset spillovers. At $h=20$, cross-asset dynamics explain 56.9% of total volume variance, while the informative and liquidity components decline to 7.2% and 35.8%, respectively. This decrease reflects the high integration of the Dow Jones constituents: after the initial within-asset response, volatility and liquidity shocks in one stock propagate to others through the estimated $\hat{\mathbf{A}}_1$ matrix, generating system-wide co-movement in trading activity that dwarfs the asset-level components at medium and long horizons. The stacked area chart in Figure (ref) visualizes this transition continuously over the 20-day horizon.
To further validate our structural identification, we analyze the time series dynamics of the informative component around exogenous information shocks. We define the daily informative share at time $t$ as
where $\tilde{\hat{e}}_{i,t}^{\textsc{vol}}$ is the standardized reduced-form residual for the volume of asset $i$ at day $t$, $\hat{u}_{i,t}^{\textsc{rbp}} = \sum_{j} [\hat{\mathbf{Q}}^\top]_{j,\textsc{rbp}}\, \tilde{\hat{e}}_{i,j,t}$ is the recovered structural RBP shock obtained by inverting $\tilde{\mathbf{E}}_t = \mathbf{U}_t\mathbf{Q}^{-\top}$, and $\hat{c}_{21}$ is the estimated element of $\mathbf{Q}^{-\top}$ governing the contemporaneous impact of volatility on volume. This sample analog of the FEVD informative component varies daily with the realized residuals and provides a time-varying measure of the information content of trading activity.
We then conduct an event study centered on Federal Open Market Committee (FOMC) announcements from 2021 to 2025 Lucca2015. As shown in Figure (ref), the informative share of trading volume remains stable and close to the unconditional mean (dotted line) during the pre-announcement window, followed by a sharp and statistically significant spike exactly on the day of the event ($t=0$). This immediate reaction confirms that our SMAR-based decomposition effectively isolates volume surges triggered by systematic information arrival. Furthermore, the rapid reversion toward the mean in the subsequent days ($t+1, t+2$) suggests that the Dow Jones constituents incorporate macroeconomic news efficiently, validating the RBP-volume channel as a reliable proxy for the latent information flow.
This paper introduces the Structural Matrix Autoregressive (SMAR) model, a flexible framework that jointly accounts for contemporaneous and dynamic interdependencies in high-dimensional financial time series. By extending the MAR model of Chen2021, the SMAR explicitly incorporates structural relationships between assets and financial variables via two contemporaneous interaction matrices, $\mathbf{C}_0$ and $\mathbf{R}_0$, thereby providing a tractable identification strategy for orthogonal structural shocks and a natural decomposition of forecast error variance even in large cross-sections.
The identification of the structural matrix $\mathbf{C}_0$ is based on a theory-driven recursive ordering, i.e., returns, realized bipower variance, and trading volume, consistent with the efficient market hypothesis and the mixture-of-distributions hypothesis. The assumption $\mathbf{R}_0 = \mathbf{I}_m$, while tractable, abstracts from contemporaneous cross-asset spillovers and represents a natural departure point for future extensions.
Applied to daily data for the constituents of the Dow Jones Industrial Average from January 2021 to February 2025, the SMAR delivers four main findings. First, the estimated contemporaneous structure confirms that unexpected changes in realized bipower variance are the primary driver of contemporaneous trading activity: a one-standard-deviation volatility shock results in an immediate and economically large increase in volumes, consistent with the MDH. Second, the structural impulse responses reveal a pronounced asymmetry. In fact, while volatility shocks generate a large and rapid volume response, the effect of volume shocks on volatility is comparatively modest at short horizons but markedly more persistent at longer horizons, suggesting that liquidity-driven trading gradually feeds back into price dispersion rather than producing an immediate reassessment of risk. Third, the forecast error variance decomposition shows that at the one-day horizon trading volume is mainly driven by own-liquidity dynamics (79.4%) with a non-trivial informational component (20.4%); as the horizon extends to twenty days, cross-asset spillovers account for over half of total volume variance (56.9%), highlighting the strong integration of Dow Jones constituents. Fourth, an event study around Federal Open Market Committee announcements confirms that the $\textsc{rbp}_{}$-based informative component spikes sharply and significantly on announcement days and reverts rapidly thereafter.
Despite the promising results of the SMAR, several challenges remain. For instance, the assumption of no contemporaneous cross-asset effects is an oversimplification that could be relaxed. Future work could explore more complex identification strategies for $\mathbf{R}_0$, e.g., using network theory or factor models. Additionally, the framework could be extended to incorporate time-varying parameters, allowing the model to capture evolving market structures and changing responses to different economic regimes.