Uhlig (1997) Bayesian Vector Autoregressions with Stochastic Volatility

Summary

Extends the standard Bayesian Vector Autoregression (VAR) to allow a time-varying $m \times m$ error precision matrix $H_t$, where $H_t$ evolves according to a multiplicative random walk driven by multivariate Beta variates. The key result is that despite time-varying $H_t$, the joint posterior of $(B, H_t)$ at each time step remains in the matrix-normal inverse-Wishart (MNIW) family and is computed by exact closed-form recursions analogous to a Kalman filter — no Markov Chain Monte Carlo (MCMC) is required for the per-step update. The marginal posterior of $B$ (integrating over all $H_t$ paths) is evaluated via importance sampling and shown to work well for 4,000 draws on a 4-variable U.S. macro VAR.

Key Claims

Model Specification

The VAR(p) with time-varying precision:

$$Y_t = R_t B' + H_t^{-1/2} e_t, \qquad e_t \sim \mathcal{N}(0, I_m)$$

where $R_t = [1, Y_{t-1}', \ldots, Y_{t-p}']$ is the $1 \times k$ regressor row ($k = mp+1$), $B$ is $k \times m$, and $H_t$ is the $m \times m$ time-varying precision matrix. The stochastic volatility law of motion:

$$H_{t+1} = A_{t+1}^{1/2} H_t A_{t+1}^{1/2}$$

where $A_{t+1}$ is drawn from the multivariate Beta distribution (defined as a ratio of Wishart variates; see Uhlig 1994b). The scalar parameter $n > 0$ controls persistence: large $n$ implies $H_t$ is nearly constant; small $n$ allows large swings. In the univariate case this reduces to a scalar multiplicative random walk $\lambda_{t+1} = A_t \lambda_t$ with $A_t \sim \text{Beta}(n/2, 1/2)$.

The figure in the paper shows the density of $\lambda/\theta$ for $n = 10$ and $n = 2$: for small $n$ the distribution has a long right tail (allowing large precision jumps) and a spike near zero (allowing near-zero precision), while large $n$ concentrates around 1.

Conjugate Prior and Posterior

The prior on $(B, H_0)$ is matrix-normal inverse-Wishart (MNIW): $B | H_0 \sim \mathcal{MN}(\bar{B}, \ldots)$ and $H_0 \sim \mathcal{W}(\Phi_0, \nu_0)$. The prior mean $\bar{B}$ is centered on Litterman's random walk prior (1's on the first own-lag diagonal, 0's elsewhere).

Exact conjugate updating. After accounting for the multiplicative evolution of $H_t$ and observing $Y_t$, the posterior of $(B, H_t)$ remains MNIW with parameters $(N_t, S_t, s_t, \nu_t)$ updated via closed-form recursions. At each step:

1. Age the precision (discount step): reduce the effective degrees of freedom by $\nu_t^{\mathrm{prior}} = n \cdot \nu_{t-1}^{\mathrm{post}} / (n+1)$ — this "forgets" old information at rate $\delta = n/(n+1)$.
2. Update with data (standard MNIW conjugate update): compute the new sufficient statistics from the observation $Y_t$.

This structure is analogous to the Kalman filter for linear Gaussian models. The discount factor $\delta = n/(n+1)$ plays the same role as in West-Harrison (1989) dynamic linear models.

Importance Sampling for the Marginal Posterior of B

The full joint posterior $p(B, H_1, \ldots, H_T | Y_1, \ldots, Y_T)$ is not closed-form. Uhlig's importance sampling procedure:

1. Draw $H_t$ paths from their conditional posteriors (obtained from the forward recursion).
2. Compute importance weights for each draw.
3. Form importance-weighted estimates of any functional of $B$.

Figure 2 (scatter of log-weight vs. log-marginal-posterior for 4,000 draws) shows a near-diagonal relationship: the importance weights are well-matched to the posterior mass, confirming the sampler does not collapse. The scatter is well-concentrated around the main ridge.

Sensitivity note: the method can produce extreme weights when the precision $\lambda$ is very large (near-zero error variance), creating a spike in the importance distribution. Uhlig recommends monitoring the maximum weight as a diagnostic.

Contrast with ARCH/GARCH and Standard SV

• ARCH/GARCH: AutoRegressive Conditional Heteroskedasticity (ARCH)/Generalized ARCH (GARCH) — $H_t$ is a deterministic function of past residuals, fully observable once residuals are known. Uhlig's model has $H_t$ as a genuine latent stochastic process.
• Standard univariate Stochastic Volatility (SV) (Jacquier-Polson-Rossi (JPR) 1994, Kim-Shephard-Chib (KSC) 1998): also latent, but MCMC is required because the log-normal/normal likelihood is not conjugate. Uhlig's multiplicative Wishart specification restores conjugacy for the per-step update.
• The connection to ARCH is noted: "ARCH captures the 'average' level of volatility; the stochastic volatility approach here additionally models randomness in the volatility process itself."

Empirical Application

Applied to a 4-variable U.S. macro VAR with $p = 5$ lags (from Uhlig 1996 draft). Key findings:

• There is "modest evidence of a genuine change" in macroeconomic volatility over the sample — consistent with early documentation of the Great Moderation.
• Results are not highly sensitive to the discount factor: $\delta = 3/5$ and $\delta = 2/3$ yield similar posteriors.
• The effective sample size from 4,000 importance draws is adequate: the scatter of log-weight vs. log-posterior (Figure 2) confirms a non-degenerate sampler.

Concepts Introduced or Extended

• Stochastic Volatility — multivariate BVAR extension via multiplicative Wishart random walk; conjugate updating contrast with MCMC-based SV; importance sampling for joint posterior of B; Great Moderation evidence; discount-factor interpretation
• Vector Autoregression — time-varying precision matrix; MNIW conjugate prior with B centered on Litterman prior; importance sampling posterior
• Minnesota Prior — Litterman random-walk prior used as $\bar{B}$ in the empirical application

Entities Mentioned

• Harald Uhlig

Quotes

"The key for providing the validity of those updating formulas is the conjugacy between the beta and the Wishart distribution, established in Theorem 2 in Appendix B."

My Take

The paper is elegant in its mathematical economy: the multivariate Beta (Wishart-ratio) distribution is exactly the right object to maintain conjugacy for a multiplicative random walk in the precision matrix, and Uhlig proves this with minimal apparatus. The per-step closed-form updating is a genuine advance over MCMC-dependent approaches to multivariate SV. The limitation is the restrictive multiplicative structure — $H_t = A_t^{1/2} H_{t-1} A_t^{1/2}$ does not allow the kind of asymmetric GARCH-like dynamics (differential response to positive vs. negative shocks) that Baba-Engle-Kraft-Kroner (BEKK)-GARCH captures. The importance sampling approach for the marginal posterior of B is reasonable for 4 variables and short samples but would face a curse-of-dimensionality challenge for larger systems. The empirical application is deliberately thin (one reference to a draft paper), which limits independent verification. The paper's lasting influence is primarily methodological: it established the template for conjugate Bayesian SV treatment of multivariate systems, later extended by Cogley-Sargent (2005) and Primiceri (2005) in the Time-Varying Parameter VAR (TVP-VAR) with SV literature.