Stochastic volatility (SV)¶

Overview¶

Stochastic volatility (SV) allows the variance of VAR residuals to change over time. This is important in macro/financial data where volatility can shift markedly across regimes.

This toolkit implements SV variants commonly used in the Bayesian VAR literature:

log-volatility dynamics: random walk (SVRW) or AR(1)
residual covariance: diagonal covariance (independent shocks) or a triangular factorization with time-invariant correlations (CCC-style)

Model sketch¶

Let \(h_{t,j}\) be the log-variance state for series \(j\) at time \(t\).

Random walk (SVRW)¶

For each series \(j\) the log-variance evolves as:

\[ h_{t,j} = h_{t-1,j} + \eta_{t,j}, \qquad \eta_{t,j} \sim \mathcal{N}(0, \sigma_{\eta,j}^2). \]

AR(1)¶

Optionally, the log-variance follows an AR(1):

\[ h_{t,j} = \gamma_{0,j} + \phi_j h_{t-1,j} + \eta_{t,j}. \]

Residual covariance structure¶

By default, conditional residual covariance is diagonal:

\[ \Sigma_t = \mathrm{diag}(\exp(h_t)). \]

With the triangular factorization, the model uses:

\[ \Sigma_t = Q^{-1}\,\mathrm{diag}(\exp(h_t))\,(Q^{-1})', \]

where \(Q\) is upper-triangular with ones on the diagonal. This yields time-varying variances with a time-invariant correlation structure.

Inference approach (KSC mixture)¶

The toolkit uses a standard auxiliary-mixture method (Kim, Shephard and Chib) to sample log-volatilities efficiently by approximating the log-\(\chi^2\) distribution with a discrete mixture.

Implementation notes:

the log-volatility state is sampled with a banded precision representation;
when covariance="triangular", the triangular factor is updated via a Gaussian prior on off-diagonal elements.

MCMC overview