Multivariate GARCH¶

All models in mfe.multivariate are missing from the arch package.

Model summary¶

Model	Params	PSD guaranteed	Notes
CCC	3K (GARCH)	Yes	Constant correlation
DCC	3K + 2	Yes	Dynamic correlation
BEKK scalar	K(K+1)/2 + 2	Yes	Covariance targeting
BEKK diagonal	K(K+1)/2 + 2K	Yes	Per-asset persistence
OGARCH	3K (GARCH)	Yes	PCA factors
GOGARCH	3K + K(K-1)/2	Yes	Independent factors
RCC	3K (GARCH) + 2	Yes	Rotation + targeting

DCC¶

from mfe.multivariate import DCC

dcc = DCC(variant="dcc").fit(returns)  # (T, K)
sigma_t = dcc.conditional_covariances   # (T, K, K)
print(f"a={dcc.diagnostics['a']:.4f}  b={dcc.diagnostics['b']:.4f}")

RCC¶

from mfe.multivariate import RCC

rcc = RCC(rotation="symmetric").fit(returns)
# G_t recursion in rotated space: G_t = (1-a-b)I + a u_{t-1}u_{t-1}' + b G_{t-1}
# u_t = P^{-1/2} r_t;  Sigma_t = P^{1/2} G_t P^{1/2}
print(f"a={rcc.a:.4f}  b={rcc.b:.4f}")
corr_t = rcc.conditional_correlations()  # (T, K, K) — diagonal = 1

BEKK¶

from mfe.multivariate import BEKK

bekk = BEKK("scalar").fit(returns)
# H_t = C'C + a² eps_{t-1}eps_{t-1}' + b² H_{t-1}
print(bekk.diagnostics["a"], bekk.diagnostics["b"])

GO-GARCH¶

from mfe.multivariate import GOGARCH

gg = GOGARCH(rotation="ica", n_components=None).fit(returns)
# W = W_pca @ U.T;  Sigma_t = W diag(h_{k,t}) W'
print(f"Rotation U orthogonality: {np.max(np.abs(gg.rotation_matrix @ gg.rotation_matrix.T - np.eye(K))):.2e}")