minnesota

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