# 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)