2.1. Basic Framework

Suppose we are given some n-dimensional observable, weakly stationary vector process (y_t). The process (y_t) can be decomposed into the sum of two components, the conditional mean μ_t and some noise component ε_t,

where

and

is the sigma field generated by the past information of (y_t). We assume that the noise process (ε_t) is characterized by the relations

where H_t is some

-measurable (n × n)–dimensional, p.d., symmetric matrix and (z_t) is an n-dimensional vector process independently and identically distributed with mean zero and unit variance, z_t ∼ iid(0,I_n). Given all these assumptions, the variance matrix of y_t conditional on

equals the variance matrix of ε_t conditional on

,

In the following discussion, we concentrate on the modeling of the conditional variance H_t. The conditional mean μ_t may, e.g., be modeled by some autoregressive moving average system (ARMA).

For the modeling of the conditional variance matrices here we use VECH and BEKK models, which are well-known multivariate generalizations of the univariate ARCH (autoregressive conditional heteroskedastic) model proposed by Engle (1982) and the univariate GARCH model suggested by Bollerslev (1986).

2.2. The VECH Model

The VECH(p,q) model was proposed in Bollerslev et al. (1988) and is given by

where c is an (n(n + 1)/2)–dimensional vector and the A_i's are square matrices of order n(n + 1)/2. This is a very general formulation of the conditional second moments of (ε_t), because each component of H_t can be formed by a linear combination of its own past and the past of all the other elements in H_t and ε_tε_t′.

However, at least two drawbacks of this model are apparent. First, the high number of parameters to estimate, which increases rapidly with O(n⁴), implies that estimation of the model is infeasible for large n. Yet a close look at the theory of this very general model class is worthwhile, not only for understanding the model features but also for the construction or definition of reasonable parsimonious subclasses. Second, H_t has to be p.d. for all ε_t−i and all H_t−j in the sample space. This is ensured if we assume that the constant is strictly positive,

and that the contribution of each ARCH and GARCH term resp. is nonnegative; i.e.,

holds for i = 1,…, p + q. Here we have used the fact that any p.s.d. matrix H ≥ 0 may be factorized as

and thus the condition

is equivalent to (2.5). A VECH model that satisfies conditions (2.4) and (2.5) will be called admissible. However, note that admissibility is hard to check for given parameters c,A₁,…, A_p+q, and thus hard to impose during estimation.

To further analyze the admissibility constraint let us concentrate on one of the preceding terms, A vech(ee′), say. Any matrix

will be called VECH term in the following. A VECH term A is admissible iff

Thus A is admissible iff the optimization problem

has an optimal value,

, say, that is nonnegative. Note that (2.7) is a nonconvex optimization problem that in general is hard to solve.

The function f (ν,e) has the following interpretation: Suppose that A corresponds to the lag one ARCH term; then f (ν,e) is the contribution of this term to the conditional variance of the linear combination (ν′ε_t) for a given ε_t−1 = e. For fixed ν, f (ν,e) is a quadratic form in e and for fixed e, f (ν,e) is a quadratic form in ν.

By rearrangement of terms, f (·,·) may also be written as

where the symmetric matrix

is a solution of the following relations:

Here we have partitioned

into n × n subblocks of size (n × n) each and used the indexing as described in (1.1) and (1.2). Note that the first set of equations relates to the symmetry of Q and that because of the last set of equations the matrix Q is not uniquely determined for a given matrix A. It follows immediately that the set of Q matrices that satisfy (2.9) for given matrix A is given by

where Q₀ is an arbitrary matrix for which, given this matrix A, equations (2.9) hold; see also Section 1.1. Note that

is an affine subset of

of dimension (n(n − 1)/2)². These sets will be called

-sets in what follows.

One can of course always select a unique matrix Q out of

, if further normalizing conditions are imposed. For instance,

is the only element in

with symmetric subblocks:

However, as we will see later on, it is advantageous to consider other normalizing conditions.

Note that for a given symmetric

the relations (2.9) uniquely define a VECH term

. As an example the matrices Q(ω₁),

, for the simplest case, n = 2, are given here:

In this paper we will mainly deal with the admissibility constraints, i.e., conditions that ensure the positive definiteness of the conditional variance matrices. For conditions that ensure stationarity of the process (ε_t) we refer, e.g., to Engle and Kroner (1995, Prop. 2.7): The VECH model has a (unique) weakly stationary solution (ε_t) if and only if

holds. This is a highly nonlinear restriction, which is hard to impose on a parametrization. In particular it imposes a “cross restriction,” which jointly refers to all terms A_i. In special cases however, the stationarity condition simplifies significantly; see the diagonal VECH (DVECH) model and the F-GARCH model later on.

2.3. The BEKK Model

Baba et al. (1991) proposed the BEKK(p,q,K) model,

where C is a nonsingular n × n matrix, B_i,k are n × n parameter matrices, and K in a certain sense determines the generality of the process.

Of course admissibility is not an issue here, because the preceding recursions always give p.d. matrices H_t > 0 by construction. For completeness we state the corresponding stationarity condition; see Engle and Kroner (1995): The BEKK model has a weakly stationary solution (ε_t) if and only if

holds.

Because p.s.d. matrices H ≥ 0 may be factorized as H = [sum ]e_i e_i′, it follows that GARCH terms,

, may again be treated completely analogously to ARCH terms,

. Now consider one of these terms, [sum ]B_k′ee′B_k, say, and note that

This implies that any BEKK term has an equivalent representation as a VECH term, A, say, where A is computed from

and (2.9). Furthermore this VECH term is admissible, and the corresponding

-set

contains at least one p.s.d. element. On the other hand, if the

-set corresponding to a VECH term, A, say, contains a p.s.d. element, Q ≥ 0, say, then this A is admissible and has an equivalent BEKK representation. To see this, just note that Q ≥ 0 implies that Q may be factorized as

, where

and K ≤ n² is the rank of Q, and thus B_k = mat(U_k) is the desired BEKK representation.

As has been noted already it is hard to check whether a given VECH term, A, say, is admissible. However, it is easy to check whether A admits a BEKK representation.

To this end, let some VECH term A be given and consider the following semidefinite program (s.d.p.) for

and

:

where Q₀ is an arbitrary element of

.

One can easily prove that this s.d.p. always has an optimizer, (λ*,ω*), say. Furthermore, note that the optimal λ* is the minimal eigenvalue of Q(ω*) = Q₀ + [sum ]ω_k*Δ_k. In other words, the preceding s.d.p. maximizes λ_min(Q) over the set

, where λ_min(Q) denotes the minimal eigenvalue of Q. Thus,

contains a p.s.d. matrix Q iff the optimal λ is nonnegative. For a detailed description of s.d.p.'s and their properties we refer to Vandenberghe and Boyd (1996). Note that (2.17) is a convex relaxation of problem (2.7).

It is well known that any BEKK model may be cast as a VECH model. However to the best of our knowledge the previously given check whether a VECH model has an equivalent BEKK representation is new. By the preceding considerations one may embed the set of BEKK models/terms into the set of VECH models/terms. Thus we will, for simplicity of notation, from now on call a VECH term that allows for a BEKK representation simply a BEKK term.

The theorem that follows summarizes some basic facts on VECH and BEKK models. For simplicity of notation we restrict ourselves to single terms; however the generalization to complete models is straightforward. We define the following sets.

Let

denote the set of all admissible VECH terms A. Note that

holds; see (2.7). Let

denote the set of all VECH terms A that have a BEKK representation; i.e., using the previous convention

is the set of all BEKK terms. Let

denote the optimum value attained for problem (2.17) and note that

iff

contains a p.s.d. element Q ≥ 0, i.e., iff

holds. Finally, let

denote the set of all BEKK terms A where

contains a p.d. element Q > 0, i.e.,

.

THEOREM 2.1.

(i)

contains an open subset of

.

(ii)

contains an open subset of

.

(iii) The set

is open in

and dense in

.

(iv)

holds for n = 2, whereas for n > 2 the set

contains an open subset of

.

Theorem 2.1(i) makes precise that a parametrization of the set of admissible VECH terms

—and of any general subset—needs (n(n + 1)/2)² = O(n⁴) free parameters. Here “general” means that this subclass contains an open set of VECH terms.

Theorem 2.1(ii) implies that this applies in particular to the set of all BEKK terms

. In the next section we will consider the subclass

of BEKK terms and give a parametrization for this class. From Theorem 2.1(iii) it follows that this subset covers “almost all” BEKK terms. Finally, the preceding theorem shows that in the bivariate case, n = 2, BEKK models are as general as (admissible) VECH models. Thus in this case there is no loss of generality if one considers BEKK models only. However for n > 2 there is a “thick” set of admissible VECH models that have no equivalent BEKK representation.

3.1. A Parametrization for

Here we apply the preceding scheme to derive a parametrization to the generic set

. By definition the

-sets for all

contain at least one p.d. element Q > 0. We then choose a representative Q > 0 that is, in a certain sense to be defined subsequently, the center of all positive elements in

. In particular this implies that K = n².

Let a

-set

be given where

is nonvoid. We then define a function φ(·) on

by

Note that the set

is a compact, convex subset of

and that, by assumption, the interior of this set, i.e., the set

, is nonvoid. The function φ(·), when restricted to

, is convex and analytic and thus has a unique minimizer, ω, say. The corresponding Q = Q(ω) = Q₀ + [sum ]_k ω_k Δ_k is called the analytic center of

; see Vandenberghe and Boyd (1996). The gradient, [dtri ]φ(ω), of φ for

is given by

Hence, in the analytic center, Q = Q(ω), we have trace(Q⁻¹Δ_k) = 0 for all k = 1,…, n. This is guaranteed if and only if each subblock matrix of Q⁻¹ is symmetric; see Section 1.1.

Next, let Q⁻¹ = VV′ be the Cholesky decomposition of Q⁻¹; i.e.,

is a lower triangular matrix where without loss of generality all diagonal elements are assumed to be positive. The parameter vector θ is defined as

i.e., we stack the lower triangular entries of all lower triangular blocks of V. Note that we have put all the diagonal elements of V to the first n² positions and thus the positivity constraint implies that

Because of the block symmetric structure of Q⁻¹ these entries are sufficient to reconstruct Q⁻¹.

Let us recapitulate the previous procedure. Suppose we have given a VECH term,

, say. Then a unique set of BEKK parameter matrices (B₁,…, B_n²) corresponding to this VECH term may be constructed as follows. First, compute the analytic center Q of

as described before. Next, compute a Cholesky factor V of Q⁻¹ with strictly positive diagonal elements. A parameter vector θ is given by (3.3), and the BEKK parameter matrices are given by B_k = mat(U_k), where U_k is the kth column of U = V^−T.

For the reverse direction, suppose we have given a parameter vector θ ∈ Θ⁺; then we have to construct the BEKK matrices B₁,…, B_n² and the corresponding VECH parameter matrix A. First, note that the matrix V is a lower triangular matrix, i.e., v_ij,kl = 0 for i < j and for i = j and k < l. The lower triangular entries v_ij,kl, i ≥ j, and k ≥ l are stored in the parameter vector θ; see (3.3). Thus, it remains to construct the upper triangular elements v_ij,kl, k < l for i > j from the condition that the product P = Q⁻¹ = VV′ has symmetric subblocks P_ij = P_ij′. For i ≥ j such a subblock is given by P_ij = V_i1V_j1′ + ··· + V_ijV_jj′. Because V_jj is a lower triangular matrix the (k,l)th entry of subblock P_ij is of the form

From the symmetry condition p_ij,kl = p_ij,lk and by the constraint v_jj,ll > 0 we get

For i > j and k < l this expression only depends on V_jj, V_ir, and V_jr with r < j, and on v_ij,kr, v_ij,lr with r < l. Thus, V may be recursively reconstructed from θ. The lower diagonal blocks of V are reconstructed in the sequence V₂₁,…,V_n1,V₃₂,…,V_n,n−1, and in each block V_ij the upper diagonal elements are computed in the sequence v_ij,12,…,v_ij,1n,v_ij,23,…,v_ij,(n−1),n. Finally, we get Q = UU′ from U = V^−T, the BEKK parameter matrices from B_k = mat(U_k), and the VECH matrix A from the relations (2.9).

As an illustration the structure of V for the simplest case n = 2 is given here:

THEOREM 3.1. The previously defined mapping

is a diffeomorphism, and the previously defined mapping

is differentiable and injective.

We want to emphasize that this parametrization covers a generic set of BEKK models; i.e., the corresponding set is an open and dense subset of all BEKK models. We here considered only the case of one term, i.e., the case of a BEKK(0,1,K) model, but the generalization to the case of BEKK(p,q,K) models is straightforward.

3.2. BEKK Parametrization of Engle and Kroner

In Proposition 2.3 of Engle and Kroner (1995) a parametrization for BEKK models is presented. The authors claim that this parametrization is fully general and identifiable. Here “fully general” means that the parametrization covers as many VECH models as possible and “identifiable” means that two different parameter vectors do not represent the same BEKK model. However, both statements are not correct, as we will show in this section.

In their representation K = n² and the parameter matrices B_k are chosen such that the corresponding matrix U = [vec(B₁),…,vec(B_K)] is a lower block triangular matrix where all (n × n) subblocks are again lower triangular; i.e., U is a Cholesky factor of Q = UU′ with the additional zero restrictions given by u_ij,kl = 0 for i > j and k < l. By these additional zero restrictions implicitly a unique representative,

, say, in the

-set

is chosen. In addition, the entries of the last row of U are assumed to be positive, u_nj,nl > 0, for all j,l to get rid of the nonuniqueness of the Cholesky factorization of

.

Using the relations between VECH models, BEKK models, and

-sets, respectively, it is easy to see that two problems occur with this parametrization:

1. The positivity constraints do not guarantee uniqueness in all cases.
2. There is a “thick” set of BEKK models that is not covered by this parametrization, because the corresponding
   
   -sets do not contain a p.s.d. representative
   
   of this structure.

To see this consider the case n = 2. The Engle–Kroner parametrization leads to a U-matrix of the form

where θ₅ > 0, θ₆ > 0, θ₈ > 0, θ₉ > 0 and thus

is given by

Now, consider a BEKK term—in Engle–Kroner form—given by

In this case we lose identifiability, because θ₇ = 0, and thus only the sum θ₈² + θ₉² is identified from Q. (In other words, the positivity constraint, in this case, is not sufficient to select a unique Cholesky factor.) However, this problem may be circumvented by replacing the restrictions u_nj,nl > 0 by the restriction that the diagonal elements of U are positive, i.e., u_jj,ll > 0 for all j,l.

Next, consider a BEKK term of the form

The Engle–Kroner parameters θ_k and thus

may be computed from Q as follows:

Note that θ₅ is computed from the sum q_21,21 + q_21,12 because only the sum is uniquely defined. However, this procedure breaks down for “large” x, because 4 − (1 + x)² − 1 − (1 + x)²/(1 + x²) < 0 for sufficiently large x. In this case the BEKK model defined as before has no representation in the form suggested by Engle and Kroner because the set

does not contain a p.s.d. element

conforming to the restrictions imposed by the Engle–Kroner parametrization. It is clear that this problem occurs on a “thick” set of BEKK terms, i.e., on a set that contains an open subset of all BEKK terms. In this sense, using the Engle–Kroner parametrization we lose a nonnegligible set of BEKK models.

The preceding examples were given for the case n = 2; however, the reasoning may be easily extended to the case n > 2.

4.1. The DVECH Model

The DVECH model has been proposed by Bollerslev et al. (1988). This model class is very popular in practice because of its parsimony and the fact that the admissibility and stationarity conditions are very simple. Here we give a short review of this model using the previously derived methodology.

A VECH model is called a DVECH model if all parameter matrices A_i in (2.3) are diagonal matrices. Thus, the elements of H_t depend only on their own past and the respective elements of ε_t−iε_t−i′. The number of parameters reduces to (p + q + 1)n(n + 1)/2 = O(n²).

In the following discussion we will again concentrate on one (diagonal) VECH term, A, say. Because

is diagonal we may write A = diag(vech(Ω)) for some matrix

. The term A is admissible if f (ν,e) = ν′ math(A vech(ee′))ν ≥ 0 holds for all vectors

. By some easy algebra it follows that

where

is a selection matrix that satisfies S(ν [otimes ] e) = (ν [odot ] e) for all

. From (4.1) it follows that A = diag(vech(Ω)) is admissible if and only if Ω ≥ 0 holds. Furthermore (4.2) implies Q ≥ 0, and thus any admissible DVECH term A has a BEKK representation. Note that Q is the only p.s.d. element in the corresponding

-set. To see this just note that the diagonal blocks of Q only have one nonzero diagonal element: q_ii,kk = 0 for all k ≠ i. Thus Q = Q + Δ ≥ 0,

implies Δ = 0.

To construct the BEKK parameter matrices consider a factorization of Ω ≥ 0 as

where L_k are vectors in

and K = rank Ω ≤ n. This gives a factorization of Q, and thus B_k = mat(SL_k) = diag(L_k). Therefore the corresponding BEKK matrices are diagonal too, and we obtain a diagonal BEKK model. Note that the number of components K is in any case less than or equal to n.

To get a unique BEKK representation for A it suffices to define a unique factorization of Ω. For example, a Cholesky decomposition leads to diagonal matrices B_k, for which the kth diagonal element is positive and the first k − 1 diagonal elements are zero. (Strictly speaking this holds true for “generic” Ω, which have a regular K × K left upper block.)

It is obvious that a diagonal BEKK model gives a diagonal (and admissible) VECH model. Therefore there is a one to one correspondence of admissible DVECH models and diagonal BEKK models.

The stationarity conditions in this case are especially simple. Because of the diagonal structure (2.13) simplifies to

where a_i,jj is the jth diagonal element of the DVECH term A_i.

4.2. The Factor GARCH Model

This model class has been proposed in Engle, Ng, and Rothschild (1990). An F-GARCH model is a BEKK(p,q,K) model (see equation (2.14)), where the following restrictions are imposed:

1. K ≤ n.
2. The BEKK matrices B_i,k are rank one matrices of the form
   

1. Let
   
   and
   
   . Then
   

Let us first consider the case 0 < K < n. By (4.4) there exist two matrices Γ,

such that

Therefore (ε_t) has a factor model representation of the form

where (f_k,t = Γ_k′ε_t) are the factors and where v_t = ΛΓ′ε_t is the “noise.” The following relations give the conditional variances and covariances, respectively:

Thus, the factors (f_k,t) are univariate GARCH processes. The conditional variance matrix of the noise (v_t) and the conditional covariance matrices are constant. For K = n there is no noise term (v_t), and hence ε_t is a linear combination of scalar GARCH processes only.

Note that the preceding model imposes strong cross-restrictions on the parameters for the respective ARCH and GARCH terms. The number of parameters reduces to n(n + 1)/2 + (p + q)K + 2nK − K − K² = O(n²).

The stationarity condition (2.15) simplifies to the condition that the K univariate factor processes (f_k,t) are stationary:

4.3. A Generalization of F-GARCH Models

One of the main difficulties in defining a suitable parametrization of BEKK models is to define a rule for how to pick a unique p.s.d. element

, Q ≥ 0. A natural candidate would be

, which is the only element with symmetric subblocks; see (2.11). However, this element does not have to be p.s.d.; see, e.g., the DVECH model. Therefore here we simply assume that

holds; i.e., we only consider the corresponding subset of BEKK terms. In addition we demand that the rank of

is equal to K, for some K ≤ n, to reduce the number of parameters needed.

Let

denote the set of BEKK terms where the block symmetric representative

is a p.d. rank K matrix. To avoid some degenerated cases we also impose the restriction that there exists a transformation of the form (S [otimes ] T) with full rank matrices

such that the transformed matrix

has a full rank K × K left upper block.

The following theorem states that this subclass of BEKK terms is closely related to the F-GARCH model.

THEOREM 4.1. For K ≤ n the set

is the set of all BEKK terms that have a representation of the form

where ∥Λ_k∥ = ∥Γ_k∥ = 1 and μ_k > 0. Here Λ and Γ are two n × K matrices where Γ has full rank K and where Λ_k and Γ_k denote the respective kth columns. This representation is unique up to sign changes and permutations if and only if Λ_k ≠ Λ_s holds for all k ≠ s.

Hence, the preceding restrictions essentially state that the BEKK matrices B_k are rank one matrices as in the F-GARCH model; see (4.3). In this sense we get a generalization of F-GARCH models for which the following restrictions hold:

1. K ≤ n.
2. The BEKK matrices B_i,k are rank one matrices:
   

Of course one has to impose some more technical conditions to get rid of trivial nonidentifiabilities, such as sign changes and permutations. (The same holds true for the F-GARCH model.)

This model is more flexible than the F-GARCH model, of course at the cost of the number of parameters, which is now equal to n(n + 1)/2 + (p + q)(2nK − K). We think this might be a valuable alternative in cases where the F-GARCH model is too restrictive. However, we have to admit that the stationarity condition here is much more involved than for the F-GARCH model. In addition, the “factor interpretation” gets lost, because the parameters Γ_i,k, Λ_i,k may depend on the lag i and because the condition (4.4) is not imposed.

Let us conclude this section with some final remarks: A natural question is whether the rank constraint K ≤ n is sufficient to define unique BEKK parameter matrices. For K = 1 the question has been answered by Engle and Kroner (1995): If a VECH term A has a BEKK representation with K = 1 components, then the corresponding BEKK matrix B₁ is unique up to sign changes.

The case 1 < K ≤ n has been discussed in Ribarits (2006). The main findings are as follows: Let

denote the set of BEKK terms that have a BEKK representation with K components. There is a generic (i.e., open and dense) subset of

for which

is a singleton. This means that for BEKK terms within this set the BEKK parameter matrices (B₁,B₂,…, B_K) are unique up to orthogonal transformations. Using, e.g., a Cholesky decomposition one may define unique BEKK matrices.

However there are also “nasty” BEKK terms in

for which

is not a singleton and thus the BEKK matrices are not uniquely defined without adding further structure.