MS-VARMA-GARCH Parameter Inference Guide • tsbs

Overview

This vignette provides guidance on statistical inference for correlation parameters in the tsbs package’s multivariate GARCH models. The package supports three correlation model specifications:

DCC (Dynamic Conditional Correlation) - dcc_modelspec
CGARCH (Copula GARCH) - cgarch_modelspec
GOGARCH (Generalized Orthogonal GARCH) - gogarch_modelspec

Each model has distinct inference characteristics that require different approaches for reliable standard errors and confidence intervals.

Executive Summary

Model	Parameter	Recommended Method	Reason
DCC	Alpha (α)	Hessian-based SE	Well-calibrated (~0.8 ratio)
DCC	Beta (β)	Bootstrap or Profile LR	Hessian underestimates by ~5-6x
CGARCH	Alpha (α)	Hessian-based SE	Similar to DCC
CGARCH	Beta (β)	Bootstrap or Profile LR	Same flat likelihood issue
CGARCH	Shape (ν)	Hessian-based SE	Generally well-identified
GOGARCH	GARCH params	Bootstrap	ICA uncertainty not captured

Key Finding: Both DCC and CGARCH share the “flat beta problem”—the likelihood surface is approximately 8x flatter in the beta direction than the alpha direction. This causes Hessian-based standard errors to severely underestimate uncertainty for beta. GOGARCH has a different structure but benefits from bootstrap inference due to ICA estimation uncertainty.

Part I: DCC Models

The Flat Beta Problem

The DCC(1,1) model has parameters α (news impact) and β (persistence). Validation studies reveal that the likelihood surface has highly anisotropic curvature:

Curvature Analysis (typical results with n=1000, α=0.05, β=0.90):
┌─────────────────────────────────────────────────────────────────┐
│ Metric                        │ Alpha    │ Beta     │ Ratio    │
├─────────────────────────────────────────────────────────────────┤
│ d²NLL/dθ² (curvature)         │ ~4900    │ ~630     │ 7.7x     │
│ NLL change over ±0.05         │ ~7.4     │ ~1.7     │ 4.3x     │
│ Hessian eigenvalue ratio      │          │          │ 16.2x    │
└─────────────────────────────────────────────────────────────────┘

The “flat direction” eigenvector is approximately (0.25, -0.97), meaning the likelihood is nearly constant along lines in parameter space. This is a fundamental property of DCC with high persistence, not an implementation error.

DCC Standard Error Calibration

Monte Carlo studies reveal severe miscalibration of Hessian-based SEs for beta:

┌─────────────────────────────────────────────────────────────────┐
│ Parameter │ Hessian SE │ Bootstrap SE │ Empirical SD │ Ratio   │
├─────────────────────────────────────────────────────────────────┤
│ Alpha     │ 0.019      │ 0.025        │ 0.024        │ 0.78    │
│ Beta      │ 0.055      │ 0.343        │ 0.218        │ 0.16    │
└─────────────────────────────────────────────────────────────────┘

Interpretation: Alpha’s Hessian SE is ~80% of empirical SD—acceptable for inference. Beta’s Hessian SE is only ~16% of bootstrap SE—severely underestimated.

DCC Inference Methods

1. Hessian-based Standard Errors

Fast but unreliable for beta at high persistence.

se_result <- dcc11_standard_errors(
  params = c(alpha, beta),
  std_resid = std_resid,
  weights = weights,
  Qbar = Qbar,
  distribution = "mvn"
)

# Returns: se, vcov, eigenvalues, condition_number

Use for: Quick inference, alpha parameter, testing β > 0

2. Bootstrap Standard Errors

Resamples data to estimate the true sampling distribution.

boot_result <- dcc11_bootstrap_se(
  std_resid = std_resid,
  weights = weights,
  Qbar = Qbar,
  mle_params = c(alpha, beta),
  n_boot = 200,
  method = "residual"  # or "parametric"
)

# Returns: se, boot_estimates, ci_percentile, ci_basic

Use for: Reliable beta inference, publication-quality results

Methods available:

"residual": Resamples rows of standardized residuals (faster)
"parametric": Simulates new data from fitted DCC model (more theoretically justified)

3. Profile Likelihood Confidence Intervals

Inverts the likelihood ratio test—does not assume quadratic likelihood.

profile_result <- correlation_profile_likelihood_ci(
  model_type = "dcc",
  residuals = std_resid,
  weights = weights,
  Qbar = Qbar,
  mle_params = c(alpha, beta),
  conf_level = 0.95
)

# Returns: Profile NLL curve, CI bounds for alpha and beta

Use for: Asymmetric CIs, near-boundary inference

4. Comprehensive DCC Inference

Computes all three methods with comparison:

inf_result <- correlation_comprehensive_inference(
  model_type = "dcc",
  residuals = std_resid,
  weights = weights,
  n_boot = 200,
  conf_level = 0.95
)

print_inference_comparison(inf_result)

Part II: CGARCH (Copula GARCH) Models

Overview

Copula GARCH models separate marginal distributions from the dependence structure. The key difference from DCC is the Probability Integral Transform (PIT) applied to standardized residuals before modeling correlation dynamics.

CGARCH Model Components

Marginal Transformation: Standardized residuals → Uniform [0,1] via PIT
- "parametric": Uses estimated univariate distribution CDF
- "empirical": Uses empirical CDF
- "spd": Semi-parametric distribution
Copula Distribution:
- "mvn": Gaussian copula
- "mvt": Student-t copula (adds shape parameter)
Correlation Dynamics:
- "constant": Static correlation
- "dcc": Standard DCC dynamics
- "adcc": Asymmetric DCC (adds gamma parameter for leverage)

CGARCH Inference Characteristics

CGARCH shares the same “flat beta problem” as DCC because the correlation dynamics follow the same DCC(1,1) recursion. The copula transformation does not change the fundamental shape of the likelihood surface with respect to (α, β).

CGARCH Parameter Characteristics:
┌─────────────────────────────────────────────────────────────────┐
│ Parameter      │ Estimation    │ Inference Issue               │
├─────────────────────────────────────────────────────────────────┤
│ Alpha (α)      │ Well-identified │ Hessian SE acceptable       │
│ Beta (β)       │ Flat likelihood │ Hessian SE underestimates   │
│ Shape (ν)      │ Well-identified │ Hessian SE acceptable       │
│ Gamma (ADCC)   │ Model-dependent │ May have flat regions       │
└─────────────────────────────────────────────────────────────────┘

CGARCH Inference Methods

1. Bootstrap Standard Errors

boot_result <- cgarch_bootstrap_se(
  z_matrix = z_matrix,      # PIT-transformed copula residuals
  weights = weights,
  Qbar = Qbar,
  mle_params = c(alpha, beta),  # or c(alpha, beta, shape) for MVT
  n_boot = 200,
  method = "residual",
  copula_dist = "mvn"       # or "mvt"
)

2. Profile Likelihood CIs

profile_result <- correlation_profile_likelihood_ci(
  model_type = "cgarch",
  residuals = z_matrix,
  weights = weights,
  Qbar = Qbar,
  mle_params = c(alpha, beta),
  conf_level = 0.95,
  distribution = "mvn"
)

3. Comprehensive CGARCH Inference

inf_result <- correlation_comprehensive_inference(
  model_type = "cgarch",
  residuals = z_matrix,
  weights = weights,
  distribution = "mvn",   # or "mvt"
  n_boot = 200,
  conf_level = 0.95
)

print_inference_comparison(inf_result)

ADCC (Asymmetric DCC) Inference

ADCC adds a leverage parameter γ that captures asymmetric correlation responses to positive vs. negative shocks. The ADCC recursion is:

$Q_t = \Omega + \alpha z_{t-1}z_{t-1}' + \gamma n_{t-1}n_{t-1}' + \beta Q_{t-1}$

where: - $n_t = z_t \cdot \mathbf{1}(z_t < 0)$ captures negative shocks - $\bar{N} = E[n_t n_t']$ is the expected outer product of negative shocks - $\Omega = (1 - \alpha - \beta)\bar{Q} - \gamma\bar{N}$ ensures stationarity - Stationarity requires: $\alpha + \beta + \delta\gamma < 1$ where $\delta \approx 0.5$

ADCC Hessian-Based Standard Errors

The adcc_standard_errors() function computes Hessian-based SEs for the 3-parameter ADCC model:

# After fitting ADCC
adcc_fit <- estimate_adcc_copula(
  z_matrix = z_matrix,
  weights = weights,
  Qbar = Qbar,
  copula_dist = "mvn"  # or "mvt"
)

# Compute Hessian-based SEs
se_result <- adcc_standard_errors(

  params = c(adcc_fit$alpha, adcc_fit$gamma, adcc_fit$beta),
  z_matrix = z_matrix,
  weights = weights,
  Qbar = Qbar,
  Nbar = adcc_fit$Nbar,
  copula_dist = "mvn"
)

# Inspect results
print(se_result$se)           # SEs for alpha, gamma, beta
print(se_result$vcov)         # Variance-covariance matrix
print(se_result$eigenvalues)  # Hessian eigenvalues (check for near-singularity)

ADCC with Robust Edge Case Handling

For production use, the robust wrapper handles boundary estimates and numerical issues:

se_robust <- compute_adcc_standard_errors_robust(
  adcc_result = adcc_fit,
  z_matrix = z_matrix,
  weights = weights,
  Qbar = Qbar,
  copula_dist = "mvn",
  boundary_threshold = 1e-4
)

if (se_robust$valid) {
  cat("SEs computed successfully:\n")
  print(se_robust$se)
} else {
  cat("SE computation issue:", se_robust$reason, "\n")
}

ADCC Bootstrap Standard Errors

As with DCC/CGARCH, bootstrap is recommended for reliable inference, especially for beta:

boot_result <- adcc_bootstrap_se(
  z_matrix = z_matrix,
  weights = weights,
  Qbar = Qbar,
  mle_params = c(adcc_fit$alpha, adcc_fit$gamma, adcc_fit$beta),
  Nbar = adcc_fit$Nbar,
  n_boot = 200,
  method = "residual",
  copula_dist = "mvn",
  verbose = TRUE
)

# Bootstrap SEs and CIs
print(boot_result$se)
print(boot_result$ci_percentile)

Comprehensive ADCC Inference

Compare Hessian and bootstrap methods:

adcc_inf <- adcc_comprehensive_inference(
  z_matrix = z_matrix,
  weights = weights,
  adcc_result = adcc_fit,
  Qbar = Qbar,
  copula_dist = "mvn",
  n_boot = 200,
  conf_level = 0.95,
  verbose = TRUE
)

# Comparison table
print(adcc_inf$comparison)

Output example:

=== ADCC Comprehensive Inference ===
Observations: 1000, Series: 3
MLE: alpha = 0.0450, gamma = 0.0280, beta = 0.9100

=== Inference Comparison ===
Confidence level: 95%

  Parameter Estimate Hessian_SE Boot_SE  Hess_CI_lower Hess_CI_upper Boot_CI_lower Boot_CI_upper SE_Ratio
1     alpha   0.0450     0.0180  0.0220         0.0097        0.0803        0.0089        0.0856     0.82
2     gamma   0.0280     0.0150  0.0190         0.0014        0.0574        0.0025        0.0612     0.79
3      beta   0.9100     0.0480  0.2100         0.8159        1.0041        0.5200        0.9650     0.23

WARNING: Hessian SEs appear underestimated for some parameters (ratio < 0.7).
         Bootstrap SEs recommended for inference.

ADCC Inference Recommendations

Parameter	Recommended Method	Notes
Alpha (α)	Hessian or Bootstrap	Usually well-identified
Gamma (γ)	Bootstrap	May have flat regions when asymmetry is weak
Beta (β)	Bootstrap	Same flat likelihood issue as DCC
Shape (ν)	Hessian	Generally well-identified (MVT copula)

Key considerations for ADCC:

Gamma identification: When asymmetric effects are weak, γ may be poorly identified. Check the Hessian eigenvalues for near-singularity.
Stationarity boundary: The constraint α + β + δγ < 1 means high-persistence models have less room for asymmetry.
Bootstrap coverage: Monte Carlo studies show bootstrap provides better coverage than Hessian for all three parameters.

Also update the Executive Summary table (around line 31)

Add rows for ADCC:

Model	Parameter	Recommended Method	Reason
DCC	Alpha (α)	Hessian-based SE	Well-calibrated (~0.8 ratio)
DCC	Beta (β)	Bootstrap or Profile LR	Hessian underestimates by ~5-6x
CGARCH	Alpha (α)	Hessian-based SE	Similar to DCC
CGARCH	Beta (β)	Bootstrap or Profile LR	Same flat likelihood issue
CGARCH	Shape (ν)	Hessian-based SE	Generally well-identified
ADCC	Alpha (α)	Hessian or Bootstrap	Usually well-identified
ADCC	Gamma (γ)	Bootstrap	May have identification issues
ADCC	Beta (β)	Bootstrap	Same flat likelihood as DCC
GOGARCH	GARCH params	Bootstrap	ICA uncertainty not captured

Also update the Summary Table in Part VII (around line 552)

Quantity	Method	Model Applicability
Point estimates	MLE	All
SE for α	Hessian	DCC, CGARCH, ADCC
CI for β	Bootstrap or Profile	DCC, CGARCH, ADCC
SE for γ (ADCC)	Bootstrap	ADCC
Shape SE (ν)	Hessian	CGARCH (MVT), ADCC (MVT)
Component GARCH SE	Bootstrap	GOGARCH
Persistence	Point estimate + SE	All

Update the Reporting section (Part VII) to include ADCC example

Recommended Format for ADCC

The ADCC parameters were estimated as α̂ = 0.045 (SE = 0.022), γ̂ = 0.028 (bootstrap 95% CI: [0.003, 0.061]), and β̂ = 0.91 (bootstrap 95% CI: [0.52, 0.97]). The asymmetric parameter γ is statistically significant, indicating stronger correlation increases following joint negative returns. Bootstrap confidence intervals are reported for γ and β due to potential identification issues with Hessian-based inference.

Part III: GOGARCH Models

Overview

Generalized Orthogonal GARCH (GOGARCH) models multivariate volatility through Independent Component Analysis (ICA). Unlike DCC/CGARCH, which model correlation dynamics directly, GOGARCH:

Decomposes residuals into independent components via ICA
Fits univariate GARCH to each independent component
Derives time-varying correlation from the mixing matrix and component volatilities

Key Structural Differences

Model Structure Comparison:
┌─────────────────────────────────────────────────────────────────┐
│ Aspect              │ DCC/CGARCH           │ GOGARCH            │
├─────────────────────────────────────────────────────────────────┤
│ Correlation params  │ α, β (dynamics)      │ None (ICA-derived) │
│ Main parameters     │ 2-3 correlation      │ 3×k component GARCH│
│ Estimation          │ Two-stage MLE        │ ICA + GARCH MLE    │
│ Uncertainty source  │ Likelihood curvature │ ICA + GARCH        │
└─────────────────────────────────────────────────────────────────┘

GOGARCH Parameter Structure

Each of the k independent components has a GARCH(1,1) specification:

omega (ω): Constant term
alpha1 (α₁): ARCH coefficient (shock impact)
beta1 (β₁): GARCH coefficient (persistence)
shape (ν): Optional Student-t degrees of freedom

Why Bootstrap is Essential for GOGARCH

GOGARCH inference faces unique challenges:

ICA estimation uncertainty: The mixing matrix W is estimated, but this uncertainty is not captured by standard Hessian-based SEs for the GARCH parameters.
Two-stage estimation: ICA is performed first, then GARCH is estimated on the components. Standard errors that only consider the second stage are incomplete.
Component independence assumption: Bootstrap naturally propagates uncertainty through both stages.

GOGARCH Inference Methods

1. Component-wise Hessian SEs (Limited Use)

se_result <- gogarch_standard_errors(
  garch_pars = garch_pars,    # List of GARCH params per component
  ica_info = ica_info,        # ICA decomposition results
  residuals = residuals,
  weights = weights,
  distribution = "norm",
  method = "hessian"
)

# Returns SE for each component's GARCH parameters

Note: These SEs do not account for ICA estimation uncertainty and should be viewed as lower bounds.

2. Bootstrap Standard Errors (Recommended)

boot_result <- gogarch_bootstrap_se(
  residuals = residuals,
  weights = weights,
  garch_pars = garch_pars,
  ica_info = ica_info,
  n_boot = 200,
  method = "residual",
  distribution = "norm"
)

# Returns bootstrap SE for each component

The bootstrap procedure: 1. Resamples the independent components (preserving independence) 2. Re-estimates GARCH for each component 3. Computes empirical standard deviations

3. Comprehensive GOGARCH Inference

inf_result <- gogarch_comprehensive_inference(
  residuals = residuals,
  weights = weights,
  garch_pars = garch_pars,
  ica_info = ica_info,
  n_boot = 200,
  boot_method = "residual",
  distribution = "norm",
  conf_level = 0.95
)

print_inference_comparison(inf_result)

Note: Profile likelihood is not available for GOGARCH due to the ICA structure. The bootstrap is the primary recommended method.

Part IV: Unified Inference Interface

The tsbs package provides unified functions that work across all three model types.

Unified Bootstrap

# Works for DCC, CGARCH, and GOGARCH
boot_result <- correlation_bootstrap_se(
  model_type = "dcc",      # or "cgarch", "gogarch"
  residuals = residuals,
  weights = weights,
  mle_result = mle_result,
  n_boot = 200,
  method = "residual",
  distribution = "mvn"
)

Unified Comprehensive Inference

For DCC and CGARCH:

inf_result <- correlation_comprehensive_inference(
  model_type = "dcc",   # or "cgarch"
  residuals = residuals,
  weights = weights,
  n_boot = 200,
  conf_level = 0.95
)

For GOGARCH:

inf_result <- gogarch_comprehensive_inference(
  residuals = residuals,
  weights = weights,
  garch_pars = garch_pars,
  ica_info = ica_info,
  n_boot = 200,
  conf_level = 0.95
)

Pretty-printing Results

# Works for all model types
print_inference_comparison(inf_result)

Part V: Diagnostic Workflow

Step 1: Check for Boundary Estimates

All three models can produce boundary estimates that invalidate standard inference.

# For DCC/CGARCH
is_boundary <- alpha < 1e-4 || beta > 0.999 || (alpha + beta) > 0.999

# For GOGARCH
for (i in 1:n_components) {
  is_boundary_i <- garch_pars[[i]]$omega < 1e-8 ||
                   garch_pars[[i]]$alpha1 + garch_pars[[i]]$beta1 > 0.999
}

Step 2: Assess Likelihood Curvature (DCC/CGARCH)

hess <- dcc11_hessian(params, std_resid, weights, Qbar)

# Eigenvalue ratio indicates anisotropy
ratio <- hess$eigenvalues[1] / hess$eigenvalues[2]
if (ratio > 10) {
  warning("Highly anisotropic curvature - beta SE unreliable")
}

# Condition number
if (hess$condition_number > 1000) {
  warning("Ill-conditioned Hessian - numerical issues possible")
}

Step 3: Compare SE Methods

# For DCC/CGARCH
hess_se <- se_result$se
boot_se <- boot_result$se

ratio <- hess_se / boot_se
if (ratio["beta"] < 0.5) {
  warning("Hessian SE for beta is less than half of bootstrap SE")
}

# For GOGARCH - compare across components
for (i in 1:n_components) {
  ratio_i <- hess_result$component_se[[i]]$se / boot_result$component_results[[i]]$se
  if (any(ratio_i < 0.5, na.rm = TRUE)) {
    warning(sprintf("Component %d: Hessian SE appears underestimated", i))
  }
}

Step 4: Visualize (DCC/CGARCH)

# Likelihood surface
surface <- compute_nll_surface(std_resid, weights, Qbar)
plot_nll_contours(surface, mle_params = c(alpha, beta))

# Profile likelihood
plot_profile_likelihood(inf_result$profile, "beta")

# Look for:
# - Elongated contours (flat direction)
# - Multiple modes (identification issues)
# - Boundary proximity

Part VI: Known Limitations and Recommendations

High Persistence (α + β > 0.95)

When persistence is high in DCC/CGARCH, the likelihood becomes increasingly flat.

Recommendations: - Always use bootstrap or profile likelihood for beta - Consider reporting persistence rather than individual parameters - Be cautious about point estimates—they may not be well-identified

Small Sample Sizes (n < 500)

With limited data, all inference methods become less reliable.

Recommendations: - Increase bootstrap replications (n_boot ≥ 500) - Consider wider confidence intervals - Report multiple inference methods for comparison

Near-Boundary Estimates

When parameters approach bounds, standard asymptotic theory fails.

Our implementation: - Detects boundary estimates and returns NA for SEs - Profile likelihood handles boundaries correctly - Bootstrap may produce boundary estimates in some replications

GOGARCH-Specific Issues

ICA sensitivity: Results can depend on ICA initialization and method.

Recommendations: - Consider multiple ICA runs with different seeds - Bootstrap captures some of this variation - Report ICA method used (e.g., “fastica”, “jade”)

Part VII: Reporting Results

Recommended Format for DCC/CGARCH

For publication, we recommend:

The DCC parameters were estimated as α̂ = 0.05 (SE = 0.02) and β̂ = 0.90 (bootstrap 95% CI: [0.55, 0.96]). Hessian-based standard errors for β are known to underestimate uncertainty for high-persistence DCC models; bootstrap confidence intervals are reported instead.

Recommended Format for GOGARCH

GOGARCH model with k=3 independent components was estimated using FastICA decomposition. Component GARCH parameters with bootstrap standard errors (B=200):

Component ω α₁ (SE) β₁ (SE) Persistence

1 0.001 0.08 (0.02) 0.91 (0.03) 0.99

2 0.002 0.05 (0.01) 0.93 (0.02) 0.98

3 0.003 0.10 (0.03) 0.87 (0.04) 0.97

Component	ω	α₁ (SE)	β₁ (SE)	Persistence
1	0.001	0.08 (0.02)	0.91 (0.03)	0.99
2	0.002	0.05 (0.01)	0.93 (0.02)	0.98
3	0.003	0.10 (0.03)	0.87 (0.04)	0.97

Summary Table

Quantity	Method	Model Applicability
Point estimates	MLE	All
SE for α	Hessian	DCC, CGARCH
CI for β	Bootstrap or Profile	DCC, CGARCH
Shape SE (ν)	Hessian	CGARCH (MVT)
Component GARCH SE	Bootstrap	GOGARCH
Persistence	Point estimate + SE	All

References

DCC Models: - Engle, R. (2002). Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics, 20(3), 339-350. - Aielli, G. P. (2013). Dynamic conditional correlation: On properties and estimation. Journal of Business & Economic Statistics, 31(3), 282-299.

Copula GARCH: - Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527-556. - Jondeau, E. & Rockinger, M. (2006). The Copula-GARCH model of conditional dependencies. Journal of International Money and Finance, 25(5), 827-853.

GOGARCH: - van der Weide, R. (2002). GO-GARCH: A multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics, 17(5), 549-564. - Boswijk, H. P. & van der Weide, R. (2011). Method of moments estimation of GO-GARCH models. Journal of Econometrics, 163(1), 118-126.

Software: - tsmarch package: https://github.com/tsmodels/tsmarch - tsgarch package: https://github.com/tsmodels/tsgarch