large_shock_sv

Stochastic Volatility VAR for Large Shocks (LargeShockSV)

Overview

The Large Shock Stochastic Volatility VAR is specifically designed to handle periods with unusually large innovations while maintaining stochastic volatility throughout the sample. This estimator combines Minnesota-style priors on VAR coefficients with sophisticated modeling of both regular volatility evolution and extreme shock periods, making it particularly suitable for samples that include crisis periods or structural breaks.

Methodology

The LargeShockSV specification features:

• Constant VAR coefficients with Minnesota-type priors
• Stochastic volatility with scaling factors for large shocks
• Pareto-distributed scaling factors to handle extreme events
• Time-varying persistence in volatility scaling
• Specialized turning point modeling for crisis periods

This approach explicitly models periods of heightened volatility while maintaining the overall stochastic volatility framework.

Parameters

Large Shock Parameters

• Mult0: Initial mean of scaling factors

  • Type: Vector of numbers
  • Default: "" (empty, estimated)
  • Controls initial beliefs about volatility scaling magnitude

• ScaleMult: Scale parameter for Pareto distribution of scaling factors

  • Type: Vector of numbers
  • Default: "" (empty, estimated)
  • Controls the scale of the Pareto distribution for volatility multipliers

• ShapeMult: Shape parameter for Pareto distribution of scaling factors

  • Type: Vector of numbers
  • Default: "" (empty, estimated)
  • Controls the tail behavior of large shock distribution

• PropStdMult: Proposal standard deviation for scaling factors

  • Type: Vector of numbers
  • Default: "" (empty, calibrated)
  • MCMC tuning parameter for scaling factor updates

Autoregressive Scaling Parameters

• MultAR0: Initial mean for scaling factor AR parameter

  • Range: [0, 1]
  • Default: 0.5
  • Controls initial persistence in volatility scaling

• AlphaMultAR: Alpha parameter for beta distribution of AR coefficient

  • Type: Number
  • Default: "" (empty, estimated)
  • Beta distribution shape parameter for AR coefficient

• BetaMultAR: Beta parameter for beta distribution of AR coefficient

  • Type: Number
  • Default: "" (empty, estimated)
  • Beta distribution shape parameter for AR coefficient

• PropStdAR: Proposal standard deviation for AR parameter

  • Type: Number
  • Default: "" (empty, calibrated)
  • MCMC tuning parameter for AR coefficient updates

Crisis Period Settings

• Turningpoint: Start date of high-volatility period
  • Type: Date
  • Default: "" (empty)
  • Specifies when the large shock period begins

Minnesota-Style Prior Parameters (Active)

• 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)

• Lambda3: Lag decay

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

MCMC Settings

• Burnin: Number of burn-in draws

  • Range: [0, ∞)
  • Default: 0
  • Recommended: Extensive burn-in due to complex volatility structure

• 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

• Crisis periods: When sample includes financial crises or extreme economic events
• Structural breaks: When large shocks coincide with volatility regime changes
• Heavy-tailed data: When residuals exhibit extreme observations
• Policy intervention periods: When extraordinary policy measures affect volatility

When Not to Use

• Normal periods: When extreme shocks are not a primary concern
• Short samples: Insufficient data to identify large shock parameters
• Computational constraints: Very computationally intensive
• Simple volatility patterns: When basic stochastic volatility is adequate

Practical Considerations

• Requires careful specification of turning point if known crisis dates exist
• Scaling factor parameters need careful tuning for convergence
• Monitor whether large shock identification aligns with economic events
• Consider computational resources - this is among the most complex estimators

Model Specification

Basic Structure

y_t = X_t β + A^{-1} Σ_t^{1/2} M_t^{1/2} ε_t

where M_t represents time-varying scaling factors for large shocks.

Scaling Factor Process

M_{i,t} ~ Pareto(scale_i, shape_i)
log(M_{i,t}) = ρ_i log(M_{i,t-1}) + ν_{i,t}

where ρ_i follows a Beta distribution.

Volatility Evolution

Standard stochastic volatility combined with large shock scaling:

log(σ²_{i,t}) = log(σ²_{i,t-1}) + η_{i,t}
Var(ε_{i,t}) = σ²_{i,t} × M_{i,t}

Computational Characteristics

Performance

• Speed: Very slow due to complex volatility structure
• Memory: High memory requirements for scaling factors and volatility
• Convergence: Challenging convergence due to parameter interactions
• Numerical stability: Requires careful tuning of proposal distributions

Implementation Challenges

• Scaling factor identification can be difficult
• Strong parameter interactions require joint sampling
• Sensitivity to prior specifications on Pareto parameters
• MCMC acceptance rates need careful monitoring

Comparison with Related Models

vs. Basic Stochastic Volatility

• Robustness: Much more robust to extreme observations
• Complexity: Significantly more complex parameter structure
• Identification: Requires longer time series for parameter identification
• Performance: Better handling of crisis periods vs. normal periods

vs. CCMM Outlier Models

• Approach: Scaling factors vs. outlier indicators
• Persistence: Allows for persistent large shock periods
• Flexibility: More flexible shock magnitude modeling
• Computation: Similar computational complexity

Implementation Strategy

Parameter Initialization

• Use simpler models for initial volatility estimates
• Carefully initialize scaling factors near unity
• Consider sequential estimation approach
• Monitor acceptance rates across all parameter blocks

Prior Specification

• Pareto parameters require domain expertise
• Consider data-driven prior calibration
• Monitor sensitivity to shape and scale parameters
• Balance identification with prior informativeness

Diagnostic Considerations

Model Validation

• Plot estimated scaling factors over time
• Compare large shock periods with known economic events
• Assess improvement in volatility forecasting during crises
• Validate parameter identification through simulation

Economic Interpretation

• Large scaling factors should align with known crisis periods
• Persistence parameters should reflect economic intuition
• Consider policy implications of identified large shock periods
• Compare with alternative crisis dating methodologies

References

• Carriero, A., T.E. Clark, and M. Marcellino (2019). "Large Vector Autoregressions with Stochastic Volatility and Non-Conjugate Priors"
• Carriero, A., G. Kapetanios, and M. Marcellino (2012). "Forecasting Large Datasets with Bayesian Reduced Rank Multivariate Models"
• Stock, J.H. and M.W. Watson (2012). "Disentangling the Channels of the 2007-09 Recession"