# Hierarchical Panel (`HierarchicalPanel`)

## Description
Hierarchical Bayesian panel VAR that allows for heterogeneity across units while sharing information through a hierarchical prior structure. Each unit has its own VAR parameters, but these are drawn from a common distribution whose hyperparameters are themselves estimated.

## Technical Details
- Each unit i has its own coefficient vector βᵢ and error covariance matrix Σᵢ
- Unit-specific coefficients are drawn from a common distribution: βᵢ ~ N(b, λ₁Ω_b)
- Hyperparameters (b, λ₁) are estimated from the data
- Information sharing occurs through the hierarchical prior structure
- Suitable for panels where units are heterogeneous but share common characteristics

## User Settings (Hyperparameters)

### Hierarchical Panel-Specific Settings

#### `S0` (double)
- **Description**: Inverse-Gamma shape parameter for overall tightness hyperparameter λ₁
- **Default**: `0.001`
- **Details**: Shape parameter for the IG prior on λ₁. Small values provide diffuse prior on the common variance across units ?

#### `V0` (double)  
- **Description**: Inverse-Gamma scale parameter for overall tightness hyperparameter λ₁
- **Default**: `0.001`
- **Details**: Scale parameter for the IG prior on λ₁. Together with S0, controls the prior distribution of cross-unit heterogeneity ?

### 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 ?

### Panel VAR Hyperparameters

#### `Lambda2` (double)
- **Description**: Cross-variable weighting parameter for unit-specific priors
- **Default**: `0.5`
- **Details**: Controls the relative tightness of priors on cross-variable coefficients vs. own-lag coefficients within each unit

#### `Lambda3` (double)
- **Description**: Lag decay parameter for unit-specific priors  
- **Default**: `1.0`
- **Details**: Controls the rate of decay of prior variance with lag length within each unit. Higher values = faster decay

#### `Lambda4` (double matrix)
- **Description**: Exogenous variable tightness parameter for unit-specific priors
- **Default**: `100`
- **Details**: Controls the tightness of priors on exogenous variables and constants within each unit. Higher values = looser priors

## Implementation Details
- **Three-level hierarchy**:
  1. Data level: yᵢₜ = Xᵢₜβᵢ + εᵢₜ, εᵢₜ ~ N(0, Σᵢ)
  2. Unit level: βᵢ ~ N(b, λ₁Ω_b), Σᵢ ~ IW(S̃ᵢ, T)  
  3. Hyperparameter level: b ~ N(β̄, (1/N)λ₁Ω_b), λ₁ ~ IG(s₀, v₀)
- **Sampling sequence**:
  1. Draw common mean b from its conditional posterior
  2. Draw variance hyperparameter λ₁ from its conditional posterior  
  3. Draw unit-specific parameters (βᵢ, Σᵢ) for each unit
- **Prior structure**: Unit-specific priors Ω_b follow Minnesota-style structure with λ₂, λ₃, λ₄ parameters

## Usage Recommendations
- **Cross-unit heterogeneity**: Increase S0 and V0 for more diffuse priors on λ₁, allowing more heterogeneity ?
- **Strong pooling**: Use smaller values of S0 and V0 to encourage more similar parameters across units ?
- **Standard panel applications**: Use default hierarchical hyperparameters (S0=V0=0.001)
- **Unit-specific priors**: Adjust λ₂, λ₃, λ₄ similar to Minnesota VAR guidelines
- **Sample size considerations**: Requires sufficient observations per unit for stable estimation

## Limitations
- Cannot be used with dummy observations (CanHaveDummies = false)
- Requires balanced panel data with sufficient time series length per unit ?
- Computational intensity scales with number of units
- Assumes common prior structure across all units (may be restrictive)

## Comparison to Other Panel Estimators
- **vs. MeanOLSPanel**: More flexible, allows for unit-specific error covariances
- **vs. NormalWishartPanel**: Hierarchical structure provides automatic determination of pooling degree
- **vs. Cross-unit panels**: No cross-unit spillovers, units are separable