H-SGDLM: Heterogeneous Graphical Dynamic Linear Model

• H-SGDLM is a fully Bayesian multivariate time series model that couples individual dynamic linear models through a dynamically learned, sparse precision matrix.
• It integrates heterogeneous autoregressive components to capture endogenous temporal signals and exogenous cross-series influences, enhancing model interpretability.
• The model supports scalable, GPU-accelerated inference for high-dimensional financial modeling, enabling efficient portfolio optimization and variance decomposition.

The Heterogeneous Simultaneous Graphical Dynamic Linear Model (H-SGDLM) is a fully Bayesian, multivariate time series model which contemporaneously couples a universe of Dynamic Linear Models (DLMs) via a dynamic, sparsely parameterized precision matrix. H-SGDLM extends standard simultaneous graphical DLMs by incorporating heterogeneous autoregressive components, designed to capture both endogenous temporal dependencies (e.g., autoregressive signals) and exogenous cross-series influences (through a dynamically learned, sparse network structure). The approach supports efficient GPU-based inference and compositional variance decomposition, and underpins scalable procedures for high-dimensional, interpretable financial modeling, notably sparse mean-reverting portfolio construction via quasi-convex optimization and cyclical coordinate descent.

1. State-Space Formulation and Model Specification

H-SGDLM observes an $n$-dimensional time series, typically log-prices $S_t = (s_{1,t}, \ldots, s_{n,t})^T$. For each asset $j = 1,\ldots,n$, the model posits a univariate DLM of the form

$$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$

with observation noise $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$ and state noise $\omega_{j,t} \sim N(0, W_{j,t})$.

The regressor vector $F_{j,t}$ is partitioned into:

• Endogenous terms $e_{j,t}$ (lags of returns, moving averages, leverage terms, and derived signals).
• Exogenous terms $r_{sp_t(j),t}$ given by the real-time values of assets in a dynamically selected parent set $sp_t(j)$.

The state vector $S_t = (s_{1,t}, \ldots, s_{n,t})^T$0 encapsulates regression coefficients on endogenous ($S_t = (s_{1,t}, \ldots, s_{n,t})^T$1) and exogenous ($S_t = (s_{1,t}, \ldots, s_{n,t})^T$2) predictors, with $S_t = (s_{1,t}, \ldots, s_{n,t})^T$3 conforming to conjugate Inverse Gamma prior structure.

To capture cross-asset dependencies, the DLMs are coupled via a sparse, time-varying exogenous coefficient matrix $S_t = (s_{1,t}, \ldots, s_{n,t})^T$4, forming a dynamic, non-symmetric graphical structure. The joint law at time $S_t = (s_{1,t}, \ldots, s_{n,t})^T$5 is then

$S_t = (s_{1,t}, \ldots, s_{n,t})^T$6

with implied precision and covariance

$S_t = (s_{1,t}, \ldots, s_{n,t})^T$7

where $S_t = (s_{1,t}, \ldots, s_{n,t})^T$8.

2. Bayesian Filtering, Priors, and Sequential Inference

For each series $S_t = (s_{1,t}, \ldots, s_{n,t})^T$9, H-SGDLM maintains Normal–Gamma conjugate priors: $j = 1,\ldots,n$0 Forecast and filtering through Kalman-type recursions yield: $j = 1,\ldots,n$1

$j = 1,\ldots,n$2

$j = 1,\ldots,n$3

$j = 1,\ldots,n$4

$j = 1,\ldots,n$5

with all hyperparameter updates following standard Normal–Gamma algebra.

3. Sparse Graphical Structure and Parent Selection

The cross-sectional (graph) structure is built through data-driven selection and shrinkage:

• For each $j = 1,\ldots,n$6, an empirical covariance or sparse Wishart filter identifies high-magnitude conditional dependencies, producing candidate parent sets $j = 1,\ldots,n$7.
• Candidate parents are dynamically promoted or demoted between core $j = 1,\ldots,n$8 and down-set via their estimated signal-to-noise ratios.
• This yields a sparse adjacency matrix $j = 1,\ldots,n$9 encoding current conditional relationships among assets, which is generally both directed and time-varying.
• Sparsity is induced by limiting parent-set sizes ($$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$0) and is dynamically adapted at each time step.

4. Computational Implementation and Scalability

The model is architected for efficient high-dimensional inference:

• All univariate DLM updates are independent conditional on parent-sets, enabling perfect parallelism—well-matched to GPU tensor operations.
• The only cross-series computation is the "recoupling" step, involving the assembly of $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$1, $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$2 and determinant evaluations, typically handled with $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$3 few-hundred Monte Carlo samples.
• GPU implementations with TensorFlow and sparse-pattern storage for $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$4 achieve real-time updates for dimensions $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$5, $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$6, $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$7.
• Per-update complexity: parent selection $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$8 (with thresholding), individual DLM update $$s_{j,t} = F_{j,t}^\top \theta_{j,t} + v_{j,t}, \qquad \theta_{j,t} = G_{j,t} \theta_{j,t-1} + \omega_{j,t},$$9 ($v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$0), and inversion $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$1 at $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$2 by exploiting sparsity.

5. Mean-Reverting Portfolio Construction

A primary application is the construction of sparse, mean-reverting portfolios:

• At time $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$3, given H-SGDLM estimates $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$4 (candidate predictive covariance, e.g., $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$5) and empirical covariance $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$6, the following quasi-convex optimization is solved: $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$7 interpreted as the “predictability difference” $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$8 (Box), with sparsity arising from restricting $v_{j,t} \sim N(0, \lambda_{j,t}^{-1})$9 to nonzero entries corresponding to a graph block of assets.
• The block is defined by the current H-SGDLM parent graph; further sparsity can be imposed by referencing nonzero entries in $\omega_{j,t} \sim N(0, W_{j,t})$0.

The optimization is efficiently solved via cyclical coordinate descent (CCD):

• For $\omega_{j,t} \sim N(0, W_{j,t})$1, the per-coordinate update for $\omega_{j,t} \sim N(0, W_{j,t})$2 is: $\omega_{j,t} \sim N(0, W_{j,t})$3 (normed after each sweep to enforce $\omega_{j,t} \sim N(0, W_{j,t})$4).
• An $\omega_{j,t} \sim N(0, W_{j,t})$5-penalty can be incorporated for further sparsification via soft-thresholding.
• CCD exhibits geometric convergence; the objective is differentiable and quasi-convex with convergence guarantee by Tseng (2001, Thm 5.1).

CCD Pseudocode (as specified in (Griveau-Billion et al., 2019))

$F_{j,t}$4

6. Variance Decomposition and Interpretability

H-SGDLM enables granular variance decomposition for each forecast:

• The predictive mean for asset $\omega_{j,t} \sim N(0, W_{j,t})$6 decomposes into endogenous and exogenous contributions: $\omega_{j,t} \sim N(0, W_{j,t})$7
• Summing absolute values of coefficient vectors $\omega_{j,t} \sim N(0, W_{j,t})$8 vs.\ $\omega_{j,t} \sim N(0, W_{j,t})$9 enables a time series of “endogenous vs.\ exogenous signal strength,” empirically found to be predictive of subsequent variance moves.
• Change-point statistics on these signals facilitate interpretable financial forecasting, including the construction of “directional” portfolios responsive to endogenous/exogenous regime shifts.

7. Empirical Performance and Applications

Empirical studies benchmark H-SGDLM and its variants on large universes of equities and derivatives:

• On 487 European stocks (2001–2019), one-step median absolute deviation in $F_{j,t}$0-realized variance is 0.013 for H-SGDLM, outperforming classical HAR-RV (0.012) and SGDLM (0.015).
• H-SGDLM captures large one-day-ahead moves with 63.89% coverage inside an 18.06%-wide forecast interval, compared to 53.24% (HAR-RV) and 34.69% (SGDLM). For positive jumps $F_{j,t}$1, H-SGDLM achieves 63.95% correctness.
• On S&P 500 data, out-of-sample accuracy for large jumps is 65.93% (vs. 54.21% for HAR-RV).
• Variance-decomposition-derived signals, when used for change-point-timed equal-weight portfolios, yield steadily increasing cumulative returns over multi-year periods including stress episodes such as the 2008 crisis.
• Thresholded signals deliver >60% directional forecasting accuracy for the STLFSI Financial Stress Index when applied to S&P 500 subsets.
• GPU acceleration yields real-time inference for up to 500 assets with $F_{j,t}$2 parent connections and several hundred Monte Carlo recoupling steps per update.

8. Context, Significance, and Research Directions

H-SGDLM provides a unified framework for:

• Multi-horizon volatility modeling (via HAR-RV),
• Dynamic graphical network discovery (via time-varying sparse $F_{j,t}$3),
• Fully Bayesian inference in high dimensions (via Normal–Gamma and Variational Bayes decoupling), and
• Interpretable variance attribution across endogenous/exogenous factors.

The approach overcomes the scalability, flexibility, and interpretability limitations of classical co-integration and factor models, facilitating real-time high-dimensional inference and portfolio construction for financial and econometric applications (Griveau-Billion et al., 2019, Griveau-Billion et al., 2019).

The method's capacity for interpretable decomposition, efficient implementation, and robust empirical performance underpins its adoption for risk management, mean-reverting portfolio discovery, and higher-order market structure analysis. Future extensions may involve alternative network learning strategies, alternative Bayesian shrinkage priors, or applications to non-financial multivariate time series.