# VAR with Minnesota Prior (`Minnesota`)

## Description
VAR estimation with Minnesota (Litterman) prior that shrinks coefficients toward a random walk specification. This is the most widely used Bayesian VAR prior in macroeconomic applications.

## Technical Details
- Applies Minnesota/Litterman prior beliefs:
  - Own lags: coefficients shrunk toward random walk (prior mean = 1 for first lag, 0 for higher lags)
  - Cross-variable lags: coefficients shrunk toward zero
  - Exogenous variables: coefficients shrunk toward zero
- Tightness decreases with lag length (geometric decay)
- Different covariance matrix specifications available

## User Settings (Hyperparameters)

### Minnesota-Specific Settings

#### `Sigma` (string)
- **Description**: Method for calculating priors on error covariance matrix
- **Default**: `"ar"`
- **Options**: 
  - `"ar"` (prior = 11): Use residual variances from univariate AR models
  - `"diag"` (prior = 12): Use diagonal elements of OLS VAR covariance matrix  
  - `"full"` (prior = 13): Use full OLS VAR covariance matrix
- **Details**: Determines how the "true" error covariance matrix σ is specified for prior construction

### 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 Minnesota Hyperparameters

#### `Lambda1` (double)
- **Description**: Overall tightness of the Minnesota prior
- **Default**: `0.1`
- **Details**: Controls the overall shrinkage intensity. Smaller values = tighter prior (more shrinkage toward prior beliefs). Standard range: 0.01-1.0

#### `Lambda2` (double)
- **Description**: Cross-variable weighting parameter
- **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. Standard range: 0.1-1.0

#### `Lambda3` (double)
- **Description**: Lag decay parameter
- **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). Standard range: 1.0-2.0

#### `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). Can be specified as scalar (applied to all) or matrix (variable-specific)

#### `Lambda5` (double)
- **Description**: Block exogeneity shrinkage parameter
- **Default**: `0.001`
- **Details**: Additional shrinkage factor applied when block exogeneity restrictions are imposed. Smaller values = stronger enforcement of restrictions

#### `Autoregression` (double vector)
- **Description**: Prior mean for first-order autoregressive coefficients
- **Default**: `0.8`
- **Details**: Prior mean for coefficients on own first lags. Standard setting is between 0.8-1.0 (near random walk). Can be variable-specific vector

### 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. Matrix structure allows variable-equation specific control

#### `BlockExogenous` (logical)
- **Description**: Block exogeneity restriction flag
- **Default**: `false`
- **Details**: Imposes block exogeneity restrictions where specified variables do not affect others (used with lambda5 shrinkage)

## Implementation Notes
- Prior mean β₀ is mostly zeros except for first own-lag coefficients (set to `Autoregression` values)
- Prior covariance Ω₀ follows Minnesota formula with hyperparameter combinations:
  - Own lags: (λ₁/j^λ₃)² where j is lag length
  - Cross lags: (σᵢᵢ/σⱼⱼ)((λ₁λ₂)/j^λ₃)² where σᵢᵢ, σⱼⱼ are residual variances
  - Exogenous: σᵢᵢ(λ₁λ₄)²
- When block exogeneity is enabled, additional λ₅² factor applied to restricted coefficients

## Usage Recommendations
- **Standard macro applications**: λ₁=0.2, λ₂=0.5, λ₃=1.0
- **High-frequency data**: Consider lower λ₁ (more shrinkage)
- **Large VARs**: Consider lower λ₁ and λ₂ (more shrinkage)
- **Covariance specification**: "ar" is most common; "full" provides more flexibility but less shrinkage