carriero_sv

Carriero Stochastic Volatility VAR (CarrieroSV)

Overview

The Carriero Stochastic Volatility VAR implements the stochastic volatility framework developed by Carriero, Kapetanios, and Marcellino. This estimator combines constant VAR coefficients with time-varying error volatilities, providing a computationally efficient approach to modeling changing uncertainty while maintaining stable structural relationships.

Methodology

The CarrieroSV specification features:

• Constant VAR coefficients with Minnesota-type priors
• Time-varying error volatilities following stochastic processes
• Efficient posterior simulation algorithms
• Focuses on volatility dynamics rather than parameter changes
• Balances flexibility with computational tractability

This approach is particularly useful when structural relationships are believed to be stable but error volatilities exhibit time-variation.

Parameters

Minnesota-Style Prior Parameters

• Autoregression: Prior on first-order autoregression

  • Range: [0, 1]
  • Default: 0.8
  • Controls the belief in random walk behavior for own lags

• Lambda1: Overall tightness of priors

  • Range: (0, ∞)
  • Default: 0.1
  • Smaller values impose tighter priors (more shrinkage)

• Lambda2: Variable weighting

  • Range: (0, ∞)
  • Default: 0.5
  • Controls relative weights between own-variable and cross-variable lags

• Lambda3: Lag decay

  • Range: (0, ∞)
  • Default: 1
  • Controls how quickly prior importance decays with lag length

• Lambda4: Exogenous variable tightness

  • Range: (0, ∞)
  • Default: 100
  • Controls prior tightness on exogenous variable coefficients

• Lambda5: Block exogeneity shrinkage

  • Range: (0, ∞)
  • Default: 0.001
  • Controls shrinkage when block exogeneity is imposed

Stochastic Volatility Parameters

• HeteroskedasticityAutoRegression: AR coefficient on residual variance

  • Range: [0, 1)
  • Default: 1
  • Controls persistence in stochastic volatility process

• HeteroskedasticityShape: IG shape parameter on residual variance

  • Range: (0, ∞)
  • Default: 0.001
  • Controls prior beliefs about volatility distribution shape

• HeteroskedasticityScale: IG scale parameter on residual variance

  • Range: (0, ∞)
  • Default: 0.001
  • Controls prior beliefs about volatility level

Advanced Options

• Exogenous: Priors on exogenous variables flag

  • Default: false
  • Enables specialized priors for exogenous variables

• BlockExogenous: Block exogeneity flag

  • Default: false
  • Imposes block exogeneity restrictions

MCMC Settings

• Burnin: Number of burn-in draws

  • Range: [0, ∞)
  • Default: 0
  • Recommended: Several thousand for proper convergence

• StabilityThreshold: Threshold for maximum eigenvalue magnitude

  • Range: (0, ∞)
  • Default: null (no stability check)
  • Ensures VAR stability by rejecting draws with eigenvalues above threshold

• MaxNumUnstableAttempts: Maximum number of unstable sampling attempts

  • Range: [1, ∞)
  • Default: 1000
  • Maximum attempts to find stable draw before stopping

Usage Guidelines

When to Use

• Time-varying volatility: When error variances change over time but relationships are stable
• Financial applications: Modeling financial time series with volatility clustering
• Crisis periods: When volatility increases during economic stress
• Efficient SV estimation: When computational efficiency is important for stochastic volatility

When Not to Use

• Parameter instability: When coefficient relationships change over time
• Constant volatility: When error variances are stable over time
• Very short series: When insufficient data for volatility estimation
• Simple applications: When constant-parameter models are adequate

Practical Considerations

• Monitor convergence of both coefficients and volatility parameters
• Consider volatility forecasting performance
• Assess whether volatility changes are economically meaningful
• Compare with constant-volatility alternatives for model selection

Model Specification

Observation Equation

y_t = X_t β + A^{-1} Σ_t^{1/2} ε_t

where β are constant coefficients and Σ_t varies over time.

Volatility Process

Log volatilities follow stochastic processes:

log(σ²_{i,t}) = γᵢ log(σ²_{i,t-1}) + η_{i,t}

Computational Characteristics

Performance

• Speed: Moderate - slower than constant volatility, faster than full time-variation
• Memory: Moderate memory requirements for volatility storage
• Convergence: Generally good convergence properties
• Efficiency: More efficient than full time-varying specifications

Implementation Features

• Efficient posterior simulation algorithms
• Specialized treatment of stochastic volatility
• Good numerical stability
• Reasonable computational requirements

Comparison with Related Models

vs. Constant Volatility VARs

• Flexibility: Captures volatility changes
• Complexity: More complex estimation
• Performance: Better fit during volatile periods
• Interpretation: Provides volatility evolution insights

vs. Time-Varying Coefficient Models

• Focus: Volatility vs. parameter changes
• Efficiency: More efficient when parameters are stable
• Applicability: Better when structural relationships are constant
• Computation: Less computationally demanding

Diagnostic Considerations

Volatility Assessment

• Plot estimated volatilities over time
• Compare with known economic events
• Assess volatility persistence and clustering
• Validate volatility forecasts out-of-sample

Model Validation

• Compare with constant volatility models using information criteria
• Assess forecasting performance during volatile periods
• Check residual properties after volatility adjustment
• Evaluate economic significance of volatility changes

References

• Carriero, A., G. Kapetanios, and M. Marcellino (2009). "Forecasting Exchange Rates with a Large Bayesian VAR"
• Carriero, A., T.E. Clark, and M. Marcellino (2019). "Large Vector Autoregressions with Stochastic Volatility and Non-Conjugate Priors"
• Primiceri, G.E. (2005). "Time Varying Structural Vector Autoregressions and Monetary Policy"