Weighted GARCH Estimation for GOGARCH Models

Implements weighted maximum likelihood estimation for Generalized Orthogonal GARCH (GOGARCH) models in the context of Markov-Switching frameworks.

GOGARCH differs fundamentally from DCC and Copula GARCH in how it models dynamic correlations:

DCC/CGARCH: Estimate explicit correlation dynamics parameters (alpha, beta) that govern how correlations evolve over time.
GOGARCH: Assume observations arise from independent latent factors. Time-varying correlations emerge from the time-varying volatilities of these factors, transformed through a fixed mixing matrix.

Usage

estimate_garch_weighted_gogarch(
  residuals,
  weights,
  spec,
  diagnostics = NULL,
  verbose = FALSE
)

Arguments

residuals

Numeric matrix of residuals with dimensions T x k, where T is the number of observations and k is the number of series.

weights

Numeric vector of state probabilities/weights with length T. These weights come from the E-step of the EM algorithm in the MS-VARMA-GARCH framework.

spec

List containing the model specification with the following elements:

garch_spec_args

List with:

model: GARCH model type (default: "garch"). Valid choices are “garch” for vanilla GARCH, “gjrgarch” for asymmetric GARCH, “egarch” for exponential GARCH, “aparch” for asymmetric power ARCH, “csGARCH” for the component GARCH, “igarch” for the integrated GARCH.
order: GARCH order as c(p, q) (default: c(1, 1))
ica: ICA algorithm. Currently only "radical" is supported (via tsmarch v1.0.0)
components: Number of ICA components to extract

distribution

Component distribution: "norm", "nig", or "gh"

start_pars

List with:

garch_pars: List of starting GARCH parameters for each component (e.g., list(list(omega=0.1, alpha1=0.1, beta1=0.8), ...))
dist_pars: Distribution parameters (e.g., list(shape=1, skew=0) for NIG)

diagnostics

Optional diagnostics collector object for logging estimation progress and issues.

verbose

Logical; if TRUE, print progress information during ICA decomposition and GARCH estimation. Default is FALSE.

Value

A list with the following components:

coefficients

List containing:

garch_pars: List of estimated GARCH parameters for each ICA component
ica_info: List with ICA results:
- A: Mixing matrix (k x n_components)
- W: Unmixing matrix (n_components x k)
- K: Pre-whitening matrix (for Jacobian)
- S: Independent components matrix (T x n_components)
- method: ICA algorithm used
- n_components: Number of components extracted
dist_pars: Distribution parameters
correlation_type: Always "gogarch"

warnings

List of warning messages generated during estimation

diagnostics

Updated diagnostics object

Details

The estimation proceeds in two stages:

Stage 1: ICA Decomposition

Independent Component Analysis extracts statistically independent components from the multivariate residuals: $$S = Y \cdot W$$ where $Y$ is the T x k residual matrix, $W$ is the unmixing matrix, and $S$ contains the independent components.

The RADICAL algorithm (Learned-Miller, 2003) is used for ICA decomposition, which is robust to outliers. For improved convergence with multiple restarts and quality diagnostics, see improved_ica_decomposition.

Stage 2: Component GARCH Estimation

Univariate GARCH models are fitted to each independent component using weighted MLE. The weights come from the state probabilities in the Markov-Switching framework.

Covariance Reconstruction

The time-varying covariance matrix is reconstructed as: $$H_t = A \cdot D_t \cdot A'$$ where $A$ is the mixing matrix from ICA and $D_t = \text{diag}(\sigma^2_{1,t}, \ldots, \sigma^2_{k,t})$ contains the component GARCH variances.

Log-Likelihood

The GOGARCH log-likelihood includes a Jacobian adjustment for the ICA transformation: $$LL = \sum_i LL_{component,i} + \log|det(K)|$$ where $K$ is the pre-whitening matrix. See compute_gogarch_loglik_ms.

Comparison with DCC and CGARCH

Aspect	DCC	CGARCH	GOGARCH
Correlation source	alpha, beta params	DCC on copula	ICA mixing matrix
Dynamics	DCC recursion	DCC/ADCC	Component volatilities
Marginal treatment	Assumed Normal	Flexible (PIT)	ICA components
Key strength	Interpretable	Tail dependence	Non-Gaussian dependence

When to Use GOGARCH

GOGARCH is particularly suitable when:

The dependence structure arises from independent underlying factors
Heavy-tailed distributions (NIG, GH) are needed for the components
Dimension reduction is desired (fewer components than series)
Non-linear dependence structures are suspected

ICA Quality Assessment

The quality of the ICA decomposition is critical for GOGARCH reliability. Key diagnostics include:

Independence score: Measures how uncorrelated the extracted components are (target: > 0.8)
Reconstruction error: How well Y = S * A' holds (target: < 1\
Negentropy: Non-Gaussianity of components (higher = better separation

If ICA convergence is problematic, consider using improved_ica_decomposition which provides multiple restarts and automatic quality assessment. For post-estimation diagnostics, see gogarch_diagnostics.

References

van der Weide, R. (2002). GO-GARCH: A multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics, 17(5), 549-564.

Learned-Miller, E. G. (2003). ICA using spacings estimates of entropy. Journal of Machine Learning Research, 4, 1271-1295.