Bauwens-Laurent-Rombouts (2006) Multivariate GARCH Models: A Survey

Summary

Bauwens, Laurent, and Rombouts (2006) provide a comprehensive survey of multivariate GARCH (MGARCH) models, organizing the field into three broad families: (i) direct generalizations of univariate GARCH (VEC [vech-form], BEKK [Baba-Engle-Kraft-Kroner], Factor GARCH); (ii) linear combinations of univariate GARCH models (orthogonal GARCH [O-GARCH], generalized orthogonal GARCH [GO-GARCH], latent factor); and (iii) nonlinear combinations (constant conditional correlation [CCC], dynamic conditional correlation [DCC], generalized dynamic covariance [GDC], Copula-GARCH). The paper covers parameter counts, identification conditions, positive-definiteness (PD) guarantees, and algebraic relationships between models. Estimation is treated via quasi-maximum likelihood (QML) (scalar and two-step DCC variants), analytical score formulas, and variance targeting. A final section catalogs diagnostic tests and lists ten open research questions.

Key Claims

VEC(1,1) — $h_t = c + A\eta_{t-1} + Gh_{t-1}$ where $h_t = \mathrm{vech}(H_t)$ and $\eta_t = \mathrm{vech}(\varepsilon_t\varepsilon_t')$ . Full VEC has $[N(N+1)(N(N+1)+1)]/2$ parameters (78 for $N=3$ ); diagonal VEC has $[N(N+5)]/2$ parameters.
RiskMetrics is a scalar VEC with $\lambda=0.94$ (daily) or $\lambda=0.97$ (monthly); parameters are pre-set, not estimated.
BEKK(1,1,K) — $H_t = C^{*\prime}C^* + \sum_k A_k^{*\prime}\varepsilon_{t-1}\varepsilon_{t-1}' A_k^* + \sum_k G_k^{*\prime}H_{t-1}G_k^*$ . PD by construction. For $K=1$ : $[N(5N+1)]/2$ parameters. Covariance-stationary iff spectral radius of $A\otimes A + G\otimes G < 1$ . Special case of VEC (Engle-Kroner 1995).
Factor GARCH — BEKK with rank-1 matrices; each factor $h_{kt}$ is univariate GARCH(1,1). Implies common persistence. $[N(N+5)]/2$ parameters for $K=1$ factor.
FF-GARCH (full-factor GARCH; Vrontos-Dellaportas-Politis 2003) — $H_t = W\Sigma_t W'$ where $W$ is triangular with ones on diagonal; always PD.
O-GARCH — eigenvectors of sample correlation matrix; $m \leq N$ factors from principal component analysis (PCA). Simple but relies on orthogonality assumption.
GO-GARCH (van der Weide 2002) — $\Lambda = PL^{1/2}U$ where $U$ is an orthogonal matrix parametrized via $N(N-1)/2$ rotation angles; two-step estimation.
CCC (Bollerslev 1990) — $H_t = D_t R D_t$ ; constant correlation $R$ ; $[N(N+5)]/2$ parameters. NOT invariant to linear transformations of $\varepsilon_t$ .
DCC-T (Tse-Tsui 2002) — $R_t = (1-\theta_1-\theta_2)R + \theta_1\Psi_{t-1} + \theta_2 R_{t-1}$ where $\Psi_{t-1}$ is the $M$ -period rolling correlation of standardized residuals; requires $M \geq N$ .
DCC-E (Engle 2002) — $Q_t = (1-\alpha-\beta)\bar{Q} + \alpha u_{t-1}u_{t-1}' + \beta Q_{t-1}$ ; $R_t = (\mathrm{diag}Q_t)^{-1/2}Q_t(\mathrm{diag}Q_t)^{-1/2}$ . Scalar $\alpha,\beta$ : $[(N+1)(N+4)]/2$ parameters. Extended DCC uses $N\times N$ matrices $A$ , $B$ .
GDC (Kroner-Ng 1998) — $H_t = D_t R_t D_t + \Phi \odot \Theta_t$ ; nests DCC, CCC, DVEC, BEKK, F-GARCH; $[N(7N-1)+4]/2$ parameters. ADC (asymmetric dynamic covariance) adds leverage via $b_i' v_{t-1}v_{t-1}'b_j$ where $v_t = \max(0,-\varepsilon_t)$ .
Copula-GARCH — Sklar (1959): joint distribution = marginals + copula. Time-varying copula parameters. Two-step maximum likelihood estimation (MLE): fit univariate GARCH for each margin, then fit copula to standardized residuals (Patton 2000, 2002; Jondeau-Rockinger 2001).
Invariance — General VEC and BEKK are invariant to linear transformations; diagonal VEC/BEKK and CCC are NOT invariant.
Marginalization — A bivariate VEC(1,1) implies each margin is at most weak GARCH(3,3); DVEC(1,1) implies strong GARCH. CCC/DCC/GDC marginalization properties are unknown.
Temporal aggregation — Weak multivariate GARCH is closed under temporal aggregation (Hafner 2003).
QML — $L_T(\theta) = -\frac{1}{2}\sum_t \log|H_t| - \frac{1}{2}\sum_t (y_t-\mu_t)'H_t^{-1}(y_t-\mu_t)$ . Consistent if the first two conditional moments are correctly specified (Jeantheau 1998). Asymptotic normality proved only for BEKK (Comte-Lieberman 2003).
DCC two-step — $QL_1 = \sum_{i=1}^N \ell_i(\theta_i)$ (eq. 54); $QL_2 = -\frac{1}{2}\sum_t[\log|R_t| + u_t'R_t^{-1}u_t]$ (eq. 55). Consistent but not efficient; one Newton-Raphson step achieves asymptotic efficiency.
Diagnostics — Hosking (1980) multivariate Ljung-Box $HM(M)\sim\chi^2(K^2M)$ ; Ling-Li (1997) $LL(M)\sim\chi^2(M)$ distribution-free; Tse (2002) residual-based $RB(M)_1,RB(M)_2\sim\chi^2(M)$ ; Lagrange multiplier (LM) test for CCC (Tse 2000) $\sim\chi^2(N(N-1)/2)$ ; DCC test (Engle-Sheppard 2001).

How It Works

Three-Family Taxonomy

Family 1 — Direct generalizations. VEC and BEKK directly generalize the univariate GARCH recursion to the conditional covariance matrix. BEKK guarantees PD at every $t$ (Engle-Kroner 1995). Full BEKK is a special case of full VEC but without the need to impose PD constraints. Factor GARCH restricts BEKK to $K$ common volatility factors.

Family 2 — Linear combinations. O-GARCH extracts $m$ principal components from the sample correlation matrix and fits univariate GARCH to each. GO-GARCH (van der Weide 2002) generalizes by allowing an orthogonal rotation $U$ parametrized via Givens angles, estimated by two-step ML. FF-GARCH (Vrontos-Dellaportas-Politis 2003) uses a lower-triangular $W$ that makes $H_t = W\Sigma_t W'$ always PD.

Family 3 — Nonlinear combinations. CCC and DCC decompose $H_t = D_tR_tD_t$ where $D_t$ contains individual GARCH standard deviations and $R_t$ is the conditional correlation matrix (constant in CCC, dynamic in DCC). GDC (Kroner-Ng 1998) nests all parsimonious MGARCH models and adds cross-product leverage. Copula-GARCH (Patton 2000; Jondeau-Rockinger 2001) separates marginal dynamics from dependence structure via Sklar's theorem.

Estimation

QML maximizes the Gaussian log-likelihood: $L_T(\theta) = -\frac{1}{2}\sum_{t=1}^T \log|H_t(\theta)| - \frac{1}{2}\sum_{t=1}^T (y_t-\mu_t)'H_t(\theta)^{-1}(y_t-\mu_t)$ Consistency requires only correctly specified first two conditional moments (Jeantheau 1998). Asymptotic normality has been proved only for BEKK (Comte-Lieberman 2003). Robust sandwich standard errors handle non-normality.

DCC two-step decomposes $L_T = QL_1 + QL_2$ where $QL_1 = \sum_{i=1}^N \ell_i(\theta_i)$ is the sum of $N$ univariate GARCH log-likelihoods and $QL_2 = -\frac{1}{2}\sum_t[\log|R_t| + u_t'R_t^{-1}u_t]$ involves the correlation part alone. The estimator is consistent but not efficient; a single Newton-Raphson iteration at the two-step estimates achieves asymptotic efficiency.

Analytical score (eq. 56) avoids numerical differentiation. Variance targeting (Engle-Mezrich 1996) replaces $\bar{Q}$ (or $C^*'C^*$ ) by the sample unconditional covariance, reducing the free-parameter count.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The multivariate GARCH models can be broadly classified into three categories: (i) direct generalizations of the univariate GARCH models; (ii) linear combinations of univariate GARCH models; and (iii) nonlinear combinations." (§1)

"The DCC model of Engle (2002) is probably the most promising as it offers a good compromise between flexibility and parsimony." (§5)

My Take

This is the canonical survey of MGARCH as of the mid-2000s and remains an essential reference for the field. The three-family taxonomy is clarifying and has become standard in textbook treatments. The paper is particularly strong on parameter counts, invariance properties, and the nesting relationships between models. Weaknesses: (1) the QML asymptotic theory is incomplete — normality results are available only for BEKK, leaving CCC/DCC on shakier theoretical footing; (2) estimation of large ( $N > 5$ ) MGARCH systems is discussed but not fully solved; (3) the 10 open questions listed in Section 5 remain largely open as of 2006, suggesting the field had reached a plateau of theoretical tools relative to empirical needs. The DCC endorsement has been borne out in subsequent applied work.