Conditional Correlation Models

Definition

Conditional correlation models decompose the conditional covariance matrix $H_t$ of a multivariate return vector $y_t$ as $H_t = D_t R_t D_t,$ where $D_t = \mathrm{diag}(\sigma_{1t},\ldots,\sigma_{Nt})$ contains individual conditional standard deviations from univariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models and $R_t$ is the $N\times N$ conditional correlation matrix. The key virtue of this decomposition is parsimony: the $N$ univariate volatility models and the correlation dynamics are estimated separately, reducing the curse of dimensionality relative to full vech-GARCH (VEC) or Baba-Engle-Kraft-Kroner (BEKK) models.

Key Ideas

Positive definiteness is guaranteed by construction: $H_t = D_t R_t D_t$ is positive definite (PD) whenever $R_t$ is a valid correlation matrix ( $R_t \succ 0$ , unit diagonal).
Constant Conditional Correlation (CCC, Bollerslev 1990): $R_t = R$ for all $t$ . Simple and consistent to estimate, but empirically rejected in most financial data. $[N(N+5)]/2$ parameters total. NOT invariant to linear transformations of $\varepsilon_t$ .
Dynamic Conditional Correlation (DCC-E, Engle 2002): $R_t$ is time-varying via the auxiliary process $Q_t = (1-\alpha-\beta)\bar{Q} + \alpha u_{t-1}u_{t-1}' + \beta Q_{t-1}, \qquad R_t = (\mathrm{diag}\,Q_t)^{-1/2}\,Q_t\,(\mathrm{diag}\,Q_t)^{-1/2},$ where $u_t = D_t^{-1}\varepsilon_t$ are standardized residuals and $\bar{Q} = E[u_t u_t']$ . Scalar $(\alpha, \beta)$ : $[(N+1)(N+4)]/2$ parameters total. Mean-reverting when $\alpha + \beta < 1$ .
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 sample correlation of $u_t$ ; requires $M \geq N$ .
Generalized Dynamic Covariance (GDC, Kroner-Ng 1998): $H_t = D_t R_t D_t + \Phi \odot \Theta_t$ , nesting DCC, CCC, diagonal VEC, BEKK, and factor GARCH. The Asymmetric Dynamic Covariance (ADC) variant adds leverage via $b_i' v_{t-1}v_{t-1}' b_j$ where $v_t = \max(0, -\varepsilon_t)$ .
Copula-GARCH: By Sklar's (1959) theorem, the joint distribution factors as marginals plus copula. Fit $N$ univariate GARCH models, then estimate copula parameters on standardized residuals. Allows non-Gaussian joint distributions (Student- $t$ , Clayton, Gumbel) while retaining flexible marginals.

How It Works

Two-Step DCC Estimation

DCC-E is estimated by decomposing the log-likelihood $L_T = QL_1 + QL_2$ :

$QL_1 = \sum_{i=1}^N \ell_i(\theta_i), \qquad QL_2 = -\frac{1}{2}\sum_{t=1}^T \bigl[\log|R_t| + u_t' R_t^{-1} u_t\bigr].$

Step 1 maximizes $QL_1$ — the sum of $N$ independent univariate GARCH log-likelihoods — yielding $\hat{D}_t$ and $\hat{u}_t$ . Step 2 maximizes $QL_2$ over $(\alpha,\beta)$ given $\hat{u}_t$ . The estimator is consistent but not fully efficient; a single Newton-Raphson step at the two-step estimates achieves asymptotic efficiency.

Variance targeting replaces the free parameter $\bar{Q}$ by its sample analogue $T^{-1}\sum_t \hat{u}_t \hat{u}_t'$ , further reducing the parameter count.

Invariance and Aggregation

General VEC and BEKK are invariant to linear transformations of $\varepsilon_t$ ; diagonal VEC, diagonal BEKK, CCC, and DCC are not invariant. A bivariate VEC(1,1) implies each marginal is at most weak GARCH(3,3); DCC marginalization properties are unknown. Weak multivariate GARCH (MGARCH) is closed under temporal aggregation (Hafner 2003).

Quasi-Maximum Likelihood (QML) Theory

The Gaussian quasi-log-likelihood $L_T(\theta) = -\frac{1}{2}\sum_{t=1}^T \log|H_t(\theta)| - \frac{1}{2}\sum_{t=1}^T (y_t - \mu_t)' H_t(\theta)^{-1}(y_t - \mu_t)$ yields consistent estimates if the first two conditional moments are correctly specified (Jeantheau 1998). Asymptotic normality has been established only for BEKK (Comte-Lieberman 2003); the asymptotic theory for CCC/DCC relies on more informal arguments. Robust sandwich standard errors handle non-normality.

CCC-MGARCH Application: Korean Currency Crisis (Kim-Tsurumi 2000)

Kim and Tsurumi (2000) apply CCC-MGARCH to four daily Korean financial returns (Korea Composite Stock Price Index (KOSPI) spot/futures and Won/Dollar spot/non-deliverable forward (NDF)) over October 1996 – April 1998. Marginal GARCH(1,1) estimates are integrated GARCH (IGARCH, $\hat\alpha_i + \hat\beta_i \approx 1$ ) for all four series. Full-sample estimate: $\hat\rho_{KF,KS} = 0.8597$ ; pooled log-likelihood = −2100.77.

An unknown structural break date $t^*$ is estimated via Laplace-approximation Bayesian posterior: the log-posterior is proportional to split-sample log-likelihoods minus half the log-determinant of associated Hessians. Grid search over 5 July – 10 December 1997 yields a multivariate break mode at 20 October 1997, before the 8 November official Won devaluation. Derivative markets (futures: 11 August; NDF: 14 August 1997) broke earlier than spot markets (spot: 30 September; WS [Won/Dollar spot]: 22 October 1997), consistent with informed trading anticipating the crisis. See Kim-Tsurumi (2000).

Why It Matters

Conditional correlations are central to portfolio construction (minimum-variance weights, risk parity), dynamic hedging ratios, and Value-at-Risk aggregation across assets.
DCC-E offers a practical balance between flexibility and parsimony: $O(N)$ correlation parameters vs. $O(N^2)$ for full BEKK, making it feasible for $N$ up to 50–100.
Time-varying correlations spike during crises (equity-bond correlation flips negative; within-equity correlations surge), making CCC empirically inadequate and DCC/GDC necessary.

Open Questions

Marginalization properties of CCC/DCC: do they imply valid univariate GARCH for each component?
Consistent estimation of large- $N$ DCC ( $N > 100$ ): the $O(N^2)$ sample correlation $\bar{Q}$ is noisy; regularization (shrinkage, factor structure) is active research.
Asymptotic theory for CCC/DCC QML estimators: full proofs analogous to Comte-Lieberman (2003) for BEKK are not yet available.
Identification of copula dynamics: time-varying copula parameters are hard to distinguish from marginal misspecification.