Engle-Kroner (1995) Multivariate Simultaneous Generalized ARCH

Summary

Engle and Kroner give the theoretical formulation and estimation of multivariate GARCH within simultaneous-equations systems. Their central contribution is a new parameterization — the BEKK form (Baba-Engle-Kraft-Kroner) — that guarantees the conditional covariance matrix $H_t$ is positive definite at every date without imposing diagonal restrictions. They establish equivalence relations among the various multivariate ARCH parameterizations (notably the general VEC form), derive constraints sufficient for positive definiteness, give necessary and sufficient conditions for covariance stationarity, and treat identification and maximum-likelihood estimation.

Key Claims

• BEKK parameterization. The quadratic (outer-product) form $H_t = C'C + \sum_k A_k'\varepsilon_{t-1}\varepsilon_{t-1}'A_k + \sum_k G_k'H_{t-1}G_k$ is positive definite by construction (a constant term $C'C$ plus quadratic forms), so it needs no ad-hoc constraints to keep $H_t \succ 0$.
• Relation to VEC. BEKK is a restricted case of the general vech (VEC) multivariate GARCH; the paper works out equivalence relations and when different parameterizations coincide, clarifying identification (BEKK parameters are identified only up to sign/orthogonal transformations).
• Stationarity and estimation. Necessary and sufficient conditions for covariance stationarity are given; the parameters are estimated by maximum likelihood, including within a simultaneous-equations context.
• Spillovers without exploding parameters. BEKK allows volatility and shock spillovers across equations (off-diagonal $A_k, G_k$) while remaining far more parsimonious and tractable than the full VEC form.

Concepts Introduced or Extended

• GARCH and BEKK-GARCH

Entities Mentioned

• Robert Engle
• Kenneth F. Kroner

Quotes

"A new parameterization of the multivariate ARCH process is proposed, and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed."

My Take

BEKK is the multivariate GARCH parameterization that solved the field's original headache — how to let a whole covariance matrix evolve over time while staying positive definite — and it did so with an elegant trick (write $H_t$ as a sum of quadratic forms, which are automatically PSD). It remains the reference specification for genuinely multivariate volatility with cross-equation spillovers, and is the natural foil to the later DCC approach (Engle 2002): BEKK is more flexible but its parameter count grows as $O(N^2)$, so it is used for small systems while DCC scales to large ones. On the wiki it is the second half of the GARCH and BEKK-GARCH page's story, and the one place where positive-definiteness — not fit — is the binding design constraint.