cogley_sargent_sv

Cogley-Sargent Stochastic-Volatility VAR (CogleySargentSV)

Description

Standard stochastic volatility VAR model following Cogley and Sargent's specification. This model allows for time-varying error covariances while keeping VAR coefficients constant, capturing changing volatility patterns in macroeconomic time series.

Technical Details

• Constant VAR coefficients with time-varying error covariance matrix
• Stochastic volatility modeled through time-varying Cholesky factorization
• Log-volatilities follow AR(1) processes
• Off-diagonal elements of Cholesky factor can vary over time
• Standard model corresponds to stvol = 1 in BEAR5

User Settings (Hyperparameters)

Inherited Base Settings

Burnin (double)

• Description: Number of burn-in draws for posterior sampling
• Default: 0
• Details: Number of initial MCMC draws to discard before collecting posterior samples

StabilityThreshold (double)

• Description: Threshold for maximum eigenvalue magnitude to ensure VAR stability
• Default: Inf (no stability check)
• Details: Maximum allowed eigenvalue magnitude; draws with eigenvalues exceeding this are rejected

MaxNumUnstableAttempts (double)

• Description: Maximum number of unstable sampling attempts before giving up
• Default: 1000
• Details: If stability checking is enabled, this limits the number of consecutive unstable draws before termination

Core VAR Hyperparameters (Minnesota-style priors for coefficients)

Lambda1 (double)

• Description: Overall tightness of the Minnesota prior on VAR coefficients
• Default: 0.1
• Details: Controls the overall shrinkage intensity for VAR coefficients. Smaller values = tighter prior (more shrinkage toward prior beliefs)

Lambda2 (double)

• Description: Cross-variable weighting parameter for VAR coefficients
• Default: 0.5
• Details: Controls the relative tightness of priors on cross-variable coefficients vs. own-lag coefficients. Smaller values = more shrinkage of cross-variable effects

Lambda3 (double)

• Description: Lag decay parameter for VAR coefficients
• Default: 1.0
• Details: Controls the rate of decay of prior variance with lag length. Higher values = faster decay (more aggressive shrinkage of higher-order lags)

Lambda4 (double matrix)

• Description: Exogenous variable tightness parameter
• Default: 100
• Details: Controls the tightness of priors on exogenous variables and constants. Higher values = looser priors (less shrinkage)

Lambda5 (double)

• Description: Block exogeneity shrinkage parameter
• Default: 0.001
• Details: Additional shrinkage factor applied when block exogeneity restrictions are imposed

Autoregression (double vector)

• Description: Prior mean for first-order autoregressive coefficients
• Default: 0.8
• Details: Prior mean for coefficients on own first lags. Can be variable-specific vector

Stochastic Volatility Hyperparameters (from SVMixin)

HeteroskedasticityAutoRegression (double)

• Description: AR coefficient on log-volatility processes (γ parameter)
• Default: 1.0
• Details: Controls persistence in volatility. Values close to 1.0 indicate highly persistent volatility changes. Standard range: 0.8-1.0 ?

HeteroskedasticityShape (double)

• Description: Inverse-Gamma shape parameter for volatility innovations (α₀ parameter)
• Default: 1e-3 (0.001)
• Details: Shape parameter for prior on volatility innovation variances. Small values indicate informative prior with low variance ?

HeteroskedasticityScale (double)

• Description: Inverse-Gamma scale parameter for volatility innovations (δ₀ parameter)
• Default: 1e-3 (0.001)
• Details: Scale parameter for prior on volatility innovation variances. Together with shape, determines prior mean and variance ?

Block Structure Settings

Exogenous (logical matrix)

• Description: Flags for applying priors to exogenous variables
• Default: false
• Details: Controls whether to apply informative priors to exogenous variables

BlockExogenous (logical)

• Description: Block exogeneity restriction flag
• Default: false
• Details: Imposes block exogeneity restrictions where specified variables do not affect others

Implementation Details

• Time-varying covariance structure: Σₜ = F·Λₜ·F' where F is lower triangular and Λₜ is diagonal
• Log-volatility evolution: log(λᵢₜ) = γ·log(λᵢₜ₋₁) + ηᵢₜ, where ηᵢₜ ~ N(0, φᵢ)
• Prior specifications:
  • VAR coefficients: Minnesota-style priors with hyperparameters λ₁-λ₅
  • Cholesky elements: Diffuse priors with large variance (10,000) ?
  • Volatility innovations: IG(α₀, δ₀) priors on φᵢ parameters
  • Initial volatility: Set to ω = 10,000 ?

Usage Recommendations

• Standard applications: Use default stochastic volatility parameters (γ=1.0, α₀=δ₀=0.001)
• Less persistent volatility: Reduce HeteroskedasticityAutoRegression below 1.0 ?
• More informative volatility priors: Increase HeteroskedasticityShape and HeteroskedasticityScale ?
• Computational considerations: Requires MCMC sampling; computationally intensive compared to plain estimators
• Model comparison: Compare to constant volatility models to assess importance of time-varying volatility

Limitations

• Cannot be used with dummy observations (CanHaveDummies = false)
• Requires sufficient sample size for stable MCMC estimation
• Assumes log-volatilities follow AR(1) processes (may be restrictive for some applications)