Bayesian VAR with Stochastic Volatility-in-Mean

• The paper develops a framework for Bayesian VAR with stochastic volatility-in-mean that directly links latent volatility to the conditional mean for robust macroeconomic forecasting.
• The model integrates factor stochastic volatility and global-local shrinkage priors, reducing dimensionality and enhancing computational efficiency in high-dimensional datasets.
• Empirical applications demonstrate improved forecast densities and risk-premium inference, validating the model's effectiveness in capturing dynamic uncertainty and tail risk.

Bayesian Vector Autoregression (BVAR) with Stochastic Volatility-in-Mean refers to a class of state-space time series models wherein the conditional mean of the endogenous variables is influenced by latent volatility processes, and the error covariance structure features stochastic volatility that evolves contemporaneously with the observed data. Recent research has expanded the modeling framework to include high-dimensional systems, flexible shrinkage priors, mixed-frequency and tensor representations, order-invariant specifications, and direct modeling of volatility (and, more recently, skewness) in the conditional mean for structural and forecasting purposes.

1. Mathematical Formulation of BVAR with Stochastic Volatility-in-Mean

A BVAR with SV-in-Mean is typically formulated as follows: $$y_t = A_1 y_{t-1} + \cdots + A_p y_{t-p} + b \cdot h_t + \varepsilon_t, \qquad \varepsilon_t \sim \mathcal{N}(0, \Sigma_t)$$ where

• $y_t$ is an $m$-dimensional vector of endogenous variables,
• $A_1, \ldots, A_p$ are lag coefficient matrices,
• $b$ is a vector (or possibly a matrix) of coefficients linking the latent volatility $h_t$ (typically the log-variance or its function) directly to the conditional mean (“volatility-in-mean”),
• $\Sigma_t$ is a time-varying covariance matrix; most modern approaches build $\Sigma_t$ using factor stochastic volatility (FSV) or other multivariate SV structures: $$\Sigma_t = \Lambda V_t \Lambda' + \Sigma_t^\text{id}$$ with $\Lambda$ factor loadings, $V_t$ diagonal factor volatilities, and $\Sigma_t^\text{id}$ diagonal idiosyncratic volatilities.

The evolution of the latent volatility $h_t$ follows a process such as

$$h_t = \mu + \phi(h_{t-1} - \mu) + \sigma \eta_t, \qquad \eta_t \sim \mathcal{N}(0,1)$$

and, in full multivariate settings, each error variance or factor volatility evolves via its own AR(1) dynamics.

The SV-in-Mean specification allows $h_t$ (or its exponential, or multiple lags thereof) to impact the conditional mean and thus macroeconomic variables are directly affected by contemporaneous and lagged uncertainty.

2. Stochastic Volatility Structures and High-Dimensional Covariance Modeling

The factor stochastic volatility structure is pivotal in modern large-scale BVARs as it facilitates dimension reduction and conditional equation-by-equation estimation. The error term is modeled: $\varepsilon_t = \Lambda f_t + \eta_t$ with $f_t \sim \mathcal{N}_q(0, V_t)$ and $\eta_t \sim \mathcal{N}_m(0, \Sigma_t^\text{id})$, so the full covariance structure is

$$\Omega_t = \Lambda V_t \Lambda' + \Sigma_t^\text{id}$$

This approach reduces the parameter count from $m(m+1)/2$ per $t$ to $mq + m$, which is essential when $m$ is large (e.g., $m=215$ in macroeconomic datasets) (Kastner et al., 2017).

Conditional on sampled latent factors and loadings, each equation is estimated separately: $z_t = y_t - \Lambda f_t = X_t \beta + \eta_t$ with known idiosyncratic variance. This “decoupling” is critical for computational tractability.

3. Shrinkage Priors and Posterior Sampling in High Dimension

To control the curse of dimensionality, global-local shrinkage priors are standard. The Dirichlet–Laplace prior has proven effective: $$b_j \mid \psi_j, \theta_j, \zeta \sim \mathcal{N}(0, \psi_j \theta_j^2 \zeta^2), \qquad \psi_j \sim \exp(\tfrac{1}{2}), \qquad (\theta_1, \ldots, \theta_K) \sim \text{Dirichlet}(a, \ldots, a), \qquad \zeta \sim \text{Gamma}(Ka, 1/2)$$ Such priors concentrate most mass near zero but retain heavy tails, allowing for signal-adaptive sparsification while avoiding excessive shrinkage (Kastner et al., 2017).

Posterior sampling leverages algorithms avoiding $K \times K$ inversion, e.g., the Bhattacharya et al. (2015) data augmentation scheme:

• Draw $u \sim \mathcal{N}(0, \Phi)$ and $\delta \sim \mathcal{N}(0, I_T)$
• Form $v = \tilde{X}u + \delta$, solve $$(\tilde{X}\Phi \tilde{X}' + I_T)w = (\tilde{z} - v)$$
• Set $\hat{b} = u + \Phi \tilde{X}'w$

This allows fully Bayesian MCMC inference for models with hundreds of thousands of coefficients when the number of predictors $k$ far exceeds $T$.

4. SV-in-Mean Effect, Structural Identification, and Extensions

Direct modeling of volatility in the mean equation enables economic uncertainty to affect macroeconomic outcomes (e.g., GDP, inflation, interest rates). This can be implemented as: $$y_{t,i} = \mu_e + \gamma \exp(h_{i,t}/2) + X_{t,i}\beta_i + \varepsilon_{t,i}, \qquad h_{i,t} \sim \text{AR(1)}$$ Both volatility and, in recent extensions, time-varying skewness can enter the mean, e.g.,

$$Y_t = c + \sum_j \beta_j Y_{t-j} + \sum_\ell b_\ell \tilde{h}_{t-\ell} + \sum_\ell a_\ell \tilde{d}_{t-\ell} + V_t$$

where $\tilde{h}_{t-\ell}$ are latent log-volatilities and $\tilde{d}_{t-\ell}$ are dynamic skewness states, with $V_t$ constructed from skew-normal mixtures (Ferreira et al., 9 Oct 2025).

Statistical identification of multivariate SV models frequently leverages order-invariant designs, as in factor SV approaches (Chan et al., 2021, Chan et al., 2022), where parameterizations and likelihood construction do not rely on variable ordering and sign restrictions on factor loadings enable point-identification in structural analysis.

5. Practical Estimation Strategies and Computational Techniques

Scalable estimation in high-dimensional settings is achieved via:

• Equation-by-equation (or block) MCMC, exploiting conditional independence in FSV models (Kastner et al., 2017, Ankargren et al., 2019, Chan, 2023).
• Integration with specialized SV samplers, e.g., stochvol with ASIS and AWOL algorithms for latent volatility (Kastner, 2019).
• Use of variational Bayes for approximate inference and real-time forecasting in high dimensions, with both batch and sequential updating (Gunawan et al., 2020, Chan et al., 2022).
• Marginal likelihood estimation via conditional Monte Carlo and adaptive importance sampling to compare SV specifications (Chan, 2022).

Tensor VAR representations and low-rank CP decompositions further reduce the parameter space, enabling multi-way shrinkage and facilitating estimation and forecasting with hundreds of variables (Chan et al., 24 Sep 2024).

6. Empirical Applications and Model Comparisons

Empirical studies demonstrate the effectiveness of BVAR–SV-in-Mean models:

• Large-scale applications (e.g., US macroeconomic data with $m=215$, $T=200$) show the latent factor volatility tracks business cycle uncertainty, with the VAR structure improving forecast densities over FSV-only models (Kastner et al., 2017).
• Exchange rate and financial time series benefit from SV-in-Mean modeling, leading to improved predictive densities and risk-premium inference (Kastner, 2019, Hiraki et al., 22 Apr 2024).
• Models incorporating dynamic skewness, volatility-in-mean and flexible structural innovations yield sharper tail risk measures and outperform standard SV models in weighted density scoring and CRPS, particularly during crisis periods (Ferreira et al., 9 Oct 2025).
• Comparative studies underline the importance of parsimonious SV structures and cross-variable shrinkage for density and point forecast accuracy; factor SV and order-invariant models are robust to ordering and preferable in large systems (Chan, 2022, Chan et al., 2021, Chan et al., 2022).

7. Outlook: Modeling Extensions and Future Directions

Ongoing developments focus on:

• Efficient high-dimensional MCMC and approximate inference for nonlinear and mixed-frequency BVAR–SV-in-Mean models (Gunawan et al., 2020, Chan et al., 2022, Ankargren et al., 2019).
• Enhanced state-space models combining factor SV, stochastic volatility-in-mean, dynamic skewness, leverage effects, and structural identification via sign restrictions or other economic constraints (Ferreira et al., 9 Oct 2025, Chan et al., 2022).
• Parsimonious time-varying coefficient structures and the integration of tensor and nonparametric techniques for further dimensionality reduction and regularization in large macro-financial datasets (Chan, 2023, Chan et al., 24 Sep 2024).

Modeling stochastic volatility-in-mean, especially with order invariance and dynamic higher-order moments, is a central route for empirical macroeconomic research seeking to capture and forecast time-varying uncertainty, tail risk, and nonlinear policy transmission mechanisms in high-dimensional environments.