# Two-Step FAVAR with Minnesota Prior (`MinnesotaFAVARTwostep`)

## Description
Factor-Augmented Vector Autoregression (FAVAR) using two-step estimation with Minnesota prior on VAR coefficients. Factors are first extracted via principal components, then used in a standard Minnesota VAR. Suitable for incorporating information from large datasets into VAR analysis.

## Technical Details
- **Two-step procedure**:
  1. Extract factors from large dataset using principal component analysis
  2. Estimate VAR using factors and observed variables with Minnesota prior
- Factors summarize information from many variables (informational variables)
- Observed variables are directly included in the VAR
- Standard Minnesota prior applied to the factor-augmented VAR

## User Settings (Hyperparameters)

### FAVAR-Specific Settings
*Note: FAVAR-specific settings (number of factors, factor datasets, etc.) are configured through the Meta and DataHolder components, not through estimator settings.*

### Minnesota Prior Settings (Inherited from Minnesota)

#### `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**: Applied to the factor-augmented VAR system (factors + observed variables)

### 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

### Minnesota Hyperparameters (Applied to Factor-Augmented System)

#### `Lambda1` (double)
- **Description**: Overall tightness of the Minnesota prior
- **Default**: `0.1`
- **Details**: Controls overall shrinkage intensity for all variables in the factor-augmented system (factors + observed variables). May need adjustment for FAVAR systems ???

#### `Lambda2` (double)
- **Description**: Cross-variable weighting parameter
- **Default**: `0.5`
- **Details**: Controls relative tightness of priors on cross-variable coefficients. Applied to interactions between factors, between observed variables, and between factors and observed variables

#### `Lambda3` (double)
- **Description**: Lag decay parameter
- **Default**: `1.0`
- **Details**: Controls rate of decay of prior variance with lag length. Applied uniformly across the factor-augmented system

#### `Lambda4` (double matrix)
- **Description**: Exogenous variable tightness parameter
- **Default**: `100`
- **Details**: Controls tightness of priors on exogenous variables and constants in the factor-augmented VAR

#### `Lambda5` (double)
- **Description**: Block exogeneity shrinkage parameter
- **Default**: `0.001`
- **Details**: Additional shrinkage factor for block exogeneity restrictions in the factor-augmented system ???

#### `Autoregression` (double vector)
- **Description**: Prior mean for first-order autoregressive coefficients
- **Default**: `0.8`
- **Details**: Prior mean for coefficients on own first lags. Applied to both factors and observed variables

### 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 in the factor-augmented system

#### `BlockExogenous` (logical)
- **Description**: Block exogeneity restriction flag
- **Default**: `false`
- **Details**: Imposes block exogeneity restrictions in the factor-augmented VAR (e.g., factors don't respond to observed variables within period) ???

## FAVAR Implementation Notes
- **Factor extraction**: Factors are extracted from a large panel of informational variables using principal components
- **Factor selection**: Number of factors typically chosen using information criteria or cross-validation ?
- **Identification**: Factors are only identified up to rotation; economic interpretation requires careful analysis
- **Minnesota prior application**: Standard Minnesota shrinkage applied to the augmented system treating factors as additional variables

## Usage Recommendations
### Prior Tuning for FAVAR
- **Lambda1**: May need to be smaller than standard VAR applications due to larger system size ???
- **Lambda2**: Standard values (0.3-0.7) usually work well for factor-variable interactions
- **Factor treatment**: Factors often have similar prior treatment to observed variables, but may warrant different λ₂ weighting ???

### FAVAR-Specific Considerations
- **Sample size**: Requires sufficient observations for both factor extraction and VAR estimation
- **Factor interpretation**: Factors may represent common economic forces but lack direct economic meaning
- **Two-step uncertainty**: Standard errors don't account for factor extraction uncertainty ???

## Comparison to Other FAVAR Estimators
- **vs. One-step FAVAR**: Simpler computation but ignores factor extraction uncertainty
- **vs. Other two-step priors**: Minnesota provides good shrinkage for macro applications
- **vs. Time-varying FAVAR**: Constant parameters vs. time-varying relationships

## Limitations
- Factor extraction uncertainty not accounted for in final inference ???
- Assumes factors are adequately represented by principal components
- Two-step procedure may be less efficient than joint estimation
- Requires specification of number of factors (model selection issue)

## Technical Validation
- Check factor stability over time
- Verify adequate factor representation of informational variables
- Compare results with benchmark VAR using only observed variables
- Assess sensitivity to number of factors chosen