Multivariate GARCH: BEKK and Dynamic Correlations

Why Model Covariances Jointly?

When several financial or economic series are observed together, their conditional variances may change over time, and their conditional covariances may change too. A covariance matrix is more than a collection of separate volatility estimates: its off-diagonal entries describe how series move together, conditional on information available at a given time. Ignoring those entries can misrepresent joint risk even when each series’ volatility is modeled well.

Multivariate GARCH models address this by specifying a time-varying conditional covariance matrix. Two influential approaches are BEKK, which builds the covariance recursion from matrix products, and dynamic conditional correlation (DCC), which combines univariate volatility models with a separate dynamic correlation process. Both are designed to keep the resulting covariance matrix mathematically valid.

Definition: Let \(u_t\) be a \(k\)-dimensional innovation from a conditional-mean model, and let \(\mathcal{F}_{t-1}\) denote information available before observing time \(t\). Its conditional covariance matrix is \(H_t=E(u_tu_t'\mid\mathcal{F}_{t-1})\), assuming the conditional mean of \(u_t\) is zero. The diagonal entries are conditional variances; the off-diagonal entries are conditional covariances. A valid covariance matrix must be positive semidefinite. These models commonly aim for positive definiteness, which also makes the matrix invertible.

BEKK: A Matrix Recursion for Covariance

A first-order BEKK specification is

H_t=CC'+A u_{t-1}u_{t-1}'A'+B H_{t-1}B'.

Here \(C\), \(A\), and \(B\) are \(k\times k\) parameter matrices. The constant matrix \(C\) is typically restricted to be triangular for convenient parameterization, with nonzero diagonal entries. The term involving \(A\) responds to the most recent innovation, while the term involving \(B\) carries forward information in the previous conditional covariance matrix. The model is conditional on the specified mean equation and the information set \(\mathcal{F}_{t-1}\).

The matrix products are important. For any vector \(v\), \(vv'\) is positive semidefinite, since \(x'vv'x=(v'x)^2\geq 0\) for every vector \(x\). Therefore, \(A u_{t-1}u_{t-1}'A'\) is positive semidefinite. Likewise, if \(H_{t-1}\) is positive semidefinite, then \(B H_{t-1}B'\) is positive semidefinite. If \(C\) is nonsingular, \(CC'\) is positive definite. Adding that positive-definite matrix to the two positive-semidefinite terms gives a positive-definite \(H_t\).

This establishes the key preservation property: a nonsingular \(C\) ensures positive definiteness at every recursion step, provided the starting \(H\) is positive semidefinite. In practice, the triangular restriction and nonzero diagonal entries of \(C\) ensure it is nonsingular. The argument concerns validity of the covariance matrix; it does not by itself guarantee that the process is stationary or that a fitted model is empirically adequate.

Worked Example: One BEKK Covariance Update

Suppose a hypothetical two-series model uses \(C=\begin{pmatrix}1&0\\0.2&0.8\end{pmatrix}\), \(A=\begin{pmatrix}0.2&0.1\\0.3&0.2\end{pmatrix}\), and \(B=\begin{pmatrix}0.7&0\\0&0.6\end{pmatrix}\). Let the previous innovation be \(u_{t-1}=(2,-1)'\), and let \(H_{t-1}=\begin{pmatrix}1&0.2\\0.2&0.8\end{pmatrix}\). Treat all entries as hypothetical and expressed in the appropriate series units.

First, the constant component is \(CC'=\begin{pmatrix}1&0.2\\0.2&0.68\end{pmatrix}\). For the shock component, \(Au_{t-1}=(0.3,0.4)'\), so

A u_{t-1}u_{t-1}'A' =(Au_{t-1})(Au_{t-1})' =\begin{pmatrix}0.09&0.12\\0.12&0.16\end{pmatrix}.

For the persistence component, \(BH_{t-1}B'=\begin{pmatrix}0.49&0.084\\0.084&0.288\end{pmatrix}\). Adding the three terms gives

H_t= \begin{pmatrix}1.58&0.404\\0.404&1.128\end{pmatrix}.

The conditional variances are \(1.58\) and \(1.128\) in squared units, and the conditional covariance is \(0.404\) in the product of the two series’ units. The implied conditional correlation is \(0.404/\sqrt{1.58(1.128)}\approx 0.3026\). It is positive: conditional on the model’s information set, the two innovations have positive covariance at this time.

Interpreting BEKK Spillovers

The BEKK matrices permit a shock in one series to affect the conditional variance of another. Under the specification above, the shock contribution is formed from \(Au_{t-1}\). Its \(i\)th component is a weighted combination of all previous innovations, and the shock contribution to the \(i\)th conditional variance is \((Au_{t-1})_i^2\). Thus, an innovation in series \(j\) can affect series \(i\)’s next conditional variance when the corresponding element \(A_{ij}\) is nonzero. The effect depends on the realized innovation vector, not just on one coefficient in isolation.

The \(B\) term propagates the previous covariance matrix. Its off-diagonal coefficients can transmit elements of past covariance into multiple entries of the next matrix. Consequently, a nonzero cross-series coefficient is evidence of a modeled dynamic channel, not by itself proof of a causal effect. Also, a covariance contribution can be negative: the off-diagonal entry of \(vv'\) is \(v_i v_j\), which is negative when the two components have opposite signs. That is compatible with a positive-semidefinite matrix; it is not evidence that the covariance matrix is invalid.

BEKK’s matrix structure is useful when interactions among variances and covariances are substantively important. Its flexibility has a cost: even a first-order specification can have many parameters as \(k\) grows. Interpretation should focus on the complete matrix products and their implied covariance dynamics rather than treating each coefficient as a standalone spillover measure.

DCC: Separate Volatility and Correlation Dynamics

DCC models represent the conditional covariance matrix as

$$ H_t=D_t R_t D_t, $$

where \(D_t\) is diagonal with conditional standard deviations on its diagonal, and \(R_t\) is a conditional correlation matrix. Each diagonal variance can be modeled with a univariate GARCH specification. Define standardized innovations \(z_t=D_t^{-1}u_t\). A common first-order DCC recursion is

Q_t=(1-a-b)\bar{Q}+a z_{t-1}z_{t-1}'+bQ_{t-1}, \qquad R_t=\operatorname{diag}(Q_t)^{-1/2}Q_t\operatorname{diag}(Q_t)^{-1/2}.

Here \(Q_t\) is a positive-definite auxiliary matrix, \(\bar{Q}\) is a positive-definite target matrix, and \(a\) and \(b\) are nonnegative scalar parameters satisfying \(a+b<1\). The notation \(\operatorname{diag}(Q_t)^{-1/2}\) means a diagonal matrix whose entries are the inverse square roots of the diagonal entries of \(Q_t\). Normalization converts \(Q_t\) into a correlation matrix with ones on the diagonal.

The positive-definiteness argument mirrors the logic of BEKK but uses a weighted sum. If \(Q_{t-1}\) and \(\bar{Q}\) are positive definite, and \(a,b\geq 0\) with \(a+b<1\), then \(1-a-b>0\). The outer product \(z_{t-1}z_{t-1}'\) is positive semidefinite. For any nonzero \(x\),

x'Q_tx=(1-a-b)x'\bar{Q}x +a(x'z_{t-1})^2+b\,x'Q_{t-1}x>0.

The strict inequality follows from the first term, which is positive for nonzero \(x\), while the other terms are nonnegative. Thus \(Q_t\) is positive definite. Scaling by a diagonal matrix with positive entries preserves positive definiteness, so \(R_t\) is positive definite as well. Since \(D_t\) has positive standard deviations, \(H_t=D_tR_tD_t\) is positive definite.

The parameter \(a\) controls how strongly recent standardized co-movements affect \(Q_t\); \(b\) controls persistence in the auxiliary matrix. The condition \(a+b<1\) makes the target’s weight positive and is the usual scalar-DCC restriction. A large \(a+b\) often corresponds to persistent correlation dynamics, but it is not a substitute for checking whether the fitted model is appropriate.

Worked Example: Updating a DCC Matrix

Consider a hypothetical two-series DCC model with \(a=0.1\), \(b=0.8\), \(\bar{Q}=\begin{pmatrix}1&0.3\\0.3&1\end{pmatrix}\), and \(Q_{t-1}=\begin{pmatrix}1.2&0.5\\0.5&0.8\end{pmatrix}\). Suppose \(z_{t-1}=(1.5,-0.5)'\). The parameter restrictions hold because \(a,b\geq0\) and \(a+b=0.9<1\). Both supplied matrices are positive definite: their determinants are \(0.91\) and \(0.71\), respectively, and their diagonal entries are positive.

The standardized innovation outer product is \(z_{t-1}z_{t-1}'=\begin{pmatrix}2.25&-0.75\\-0.75&0.25\end{pmatrix}\). Since \(1-a-b=0.1\), the update is

Q_t= 0.1\begin{pmatrix}1&0.3\\0.3&1\end{pmatrix} +0.1\begin{pmatrix}2.25&-0.75\\-0.75&0.25\end{pmatrix} +0.8\begin{pmatrix}1.2&0.5\\0.5&0.8\end{pmatrix} = \begin{pmatrix}1.285&0.355\\0.355&0.765\end{pmatrix}.

The corresponding correlation is \(R_{12,t}=0.355/\sqrt{1.285(0.765)}\approx0.3581\). The first standardized innovation is positive and the second negative, so their outer product contributes a negative off-diagonal term. The resulting conditional correlation is nevertheless positive because the full update also includes the target and persistence terms.

Worked Example: From DCC Correlation to Covariance

Continue with the DCC update above. Suppose, hypothetically, that the two conditional standard deviations at time \(t\) are \(2\) units and \(3\) units. Then \(D_t=\operatorname{diag}(2,3)\), and the correlation matrix has diagonal entries one and off-diagonal entries approximately \(0.3581\). The conditional covariance matrix is

H_t=D_tR_tD_t = \begin{pmatrix}2&0\\0&3\end{pmatrix} \begin{pmatrix}1&0.3581\\0.3581&1\end{pmatrix} \begin{pmatrix}2&0\\0&3\end{pmatrix} \approx \begin{pmatrix}4&2.149\\2.149&9\end{pmatrix}.

The conditional variances are \(4\) and \(9\) squared units, and the conditional covariance is about \(2.149\) in the product of the series’ units. The correlation remains \(0.3581\), which is dimensionless. This illustrates why a correlation matrix alone does not give the scale of joint risk: the diagonal volatility models are also needed to recover \(H_t\).

Choosing What to Interpret

BEKK and DCC both model a full conditional covariance matrix, but they organize its dynamics differently. BEKK uses matrix interactions in the shock and persistence terms. DCC first models each series’ conditional scale, then updates dependence among standardized innovations. DCC is often more parsimonious when the primary goal is to let correlations vary without estimating a large collection of unrestricted covariance-dynamics parameters. BEKK offers a more direct matrix representation of interactions, while potentially requiring more parameters.

In either model, the conditional correlation at time \(t\) is a model-based measure of co-movement given \(\mathcal{F}_{t-1}\). It can change over time even if a series’ conditional standard deviation is stable, and covariance can change when correlation is fixed because the standard deviations change. Neither a conditional correlation nor a volatility spillover coefficient establishes causation. Such interpretations require a design and assumptions beyond covariance modeling.

Common Pitfalls

Confusing covariance and correlation. Covariance has units and depends on both series’ scales; correlation is dimensionless and normalized. Use \(H_t=D_tR_tD_t\) to keep the distinction clear.
Assuming any covariance recursion stays valid. Arbitrary updates to matrix entries can produce a non-positive-definite matrix. BEKK’s matrix products or DCC’s positive-weight construction provide a structural safeguard under their stated conditions.
Reading one parameter as a complete spillover effect. In BEKK, the shock impact depends on \(Au_{t-1}\), and persistence depends on \(BH_{t-1}B'\). Interpret the resulting matrix contributions, not a coefficient in isolation.
Calling a negative outer-product entry invalid. The off-diagonal entry can be negative while the whole outer-product matrix remains positive semidefinite. Check the matrix property, not the sign of one entry.
Equating co-movement with causal transmission. Conditional covariance dynamics describe statistical dependence given the model information set; they do not alone identify causal effects.

Key takeaway: BEKK preserves covariance validity by adding transformed positive-semidefinite shock and persistence terms to a positive-definite constant. DCC combines conditional scales with a normalized, dynamically updated correlation matrix. In both, interpret the resulting conditional covariance and correlation in context, rather than treating model coefficients as causal effects.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

In a BEKK model, what role does the nonsingular matrix \(C\) play in preserving positive definiteness?
Why can an off-diagonal entry of \(z_{t-1}z_{t-1}'\) be negative without invalidating the DCC update?
If the DCC correlation between two series is unchanged but one conditional standard deviation doubles, can their conditional covariance change? Explain using \(H_t=D_tR_tD_t\).
What do the \(A\) and \(B\) terms in BEKK contribute to the conditional covariance recursion?
Why is a nonzero BEKK cross-series coefficient not, by itself, evidence of a causal volatility spillover?