2. THE EXTENDED CONSTANT CONDITIONAL CORRELATION GARCH MODEL

Following Jeantheau (1998), consider the following vector stochastic process:

where D_t = diag{h_1t,…,h_Mt} and h_it is the conditional standard deviation of ε_it, i = 1,…,M. Furthermore, the stochastic vector z_t = (z_1t,…,z_Mt)′ is independent and identically distributed (i.i.d.) with mean 0 and positive definite covariance matrix R = [ρ_ij] such that ρ_ii = 1 and ρ_ij ≠ 0, i,j = 1,…,M. The main diagonal elements of R are restricted to unity for identification reasons; compare this with the univariate case, in which customarily

. Furthermore, h_t = (h_1t,…,h_Mt)′ is an M × 1 vector of conditional standard deviations of ε_t. Let

where h_t⁽²⁾ = (h_1t²,…,h_Mt²)′ and Z_t = diag{z_1t,…,z_Mt}. Define the vector GARCH(p,q) process

where a₀ is an M × 1 vector with positive elements and A_i, i = 1,…,q, and B_j, j = 1,…,p, are M × M matrices such that each element of h_t⁽²⁾ is positive for every t. Note that (4) defines the diagonal elements of D_t. From (2) it follows that

where

is the σ-field generated by all the available information up through time t − 1.

Remark 1. A sufficient condition for h_t⁽²⁾ > 0 for all t is that all elements in a₀ be positive and all elements in A_i and B_j for each i and j be nonnegative. (Note that all vector and matrix inequality signs in this paper represent element-by-element inequality.) From Nelson and Cao (1992) we conjecture that this condition is not necessary, at least not if p > 1 or q > 1 or both. It follows from z_t ∼ iid(0,R) that

is positive definite.

Remark 2. The vector GARCH process defined by equations (2)–(6) is a multivariate GARCH model with constant conditional correlations. The CCC-GARCH model of Bollerslev (1990) is obtained by assuming that A_i, B_j, i = 1,…,q, and j = 1,…,p are diagonal matrices. In particular, setting B_j = 0, j = 1,…,p, yields the constant conditional correlation ARCH model introduced in Cecchetti et al. (1988).

3. THE FOURTH-MOMENT STRUCTURE OF THE SECOND-ORDER EXTENDED CCC-GARCH MODEL

In this section we consider the vector GARCH(2,2) model defined in (2)–(6) and set C_it = [c_i,jlt] = A_iZ_t² + B_i, i = 1,2. Note that {C_it} is a sequence of i.i.d. random matrices such that C_it is independent of h_t⁽²⁾. By (3) we may rewrite (4) as

Let

, where [otimes ] denotes the Kronecker product of two matrices. Let λ(Γ) denote the modulus of the largest eigenvalue of Γ. We can now state the following result.

THEOREM 1. The vector GARCH(2,2) process defined in (2) and (3) and (7) is weakly and strictly stationary if

Proof. Apply Proposition 3.1 of Jeantheau (1998) to {ε_t} defined in (2) and (3) and (7). █

Remark 3. Bollerslev and Engle (1993) and Engle and Kroner (1995) derived a necessary and sufficient condition for weak stationarity of a vector GARCH(p,q) model without the assumption of the constant conditional correlation. Condition (8) is a special case of their result. When it holds, the unconditional variances of the elements of ε_t in the vector GARCH(2,2) model (2) and (3) and (7) are

We are now ready to state our result concerning the fourth-order unconditional moment matrix of model (2) and (3) with (7). Let vec(A) be a vector in which the columns of the M × M matrix A are stacked one underneath the other. Then vec(A′) = K_MM vec(A), where K_MM is the M² × M² commutation matrix (see, e.g., Magnus, 1988, pp. 35–37). We have the following theorem.

THEOREM 2. Consider the vector GARCH(2,2) model (2) and (3) and (7). Assume that condition (8) holds and

exists. Then the fourth-order moment matrix

of {ε_t} exists if

where

Under (10),

Proof. See the Appendix.

Remark 4. Figure 1 helps to compare the largest absolute eigenvalues in condition (10) and the one of Ling and McAleer (2003) for the existence of the fourth-order unconditional moments in CCC-GARCH(2,2) models. The graphs are obtained by fixing values of all parameters of the model but b_2,11 and letting b_2,11 increase from 0.2. The moduli of the largest eigenvalues of matrix Γ in CCC-GARCH(2,2) models are monotonically increasing functions of the parameter b_2,11. The solid curve is for λ(Γ), where Γ is defined in (10), whereas the dashed-dotted one is the counterpart of the solid one with

defined in Theorem 2.2 of Ling and McAleer (2003). It is seen that these two curves have an intersection exactly at λ(Γ) = 1 (dashed line). Similar graphs can be obtained for any other parameter combination, and it appears that those two conditions, although they look different, always give the same answer. The advantage of condition (10) is that it can be used in deriving the fourth-moment matrix of ε_t.

Moduli of the largest eigenvalues of matrices Γ for a CCC-GARCH(2,2) model as a function of parameter b2,11 when (a) Γ is defined by condition (10) (solid curve) and (b) is defined in Theorem 2.2 of Ling and McAleer (2003) (dashed-dotted curve). The parameters of the model are a1,11 = 0.06, a1,12 = 0.08, a1,21 = 0.1, a1,22 = 0.12, a2,11 = 0.07, a2,12 = 0.1, a2,21 = 0.08, a2,22 = 0.12, b1,11 = 0.05, b1,12 = 0.08, b1,21 = 0.11, b1,22 = 0.2, b2,12 = 0.09, b2,21 = 0.1, and b2,22 = 0.12, and the nondiagonal element of the correlation matrix of the standard normal error process {zt} equals 0.2.

Remark 5. Setting M = 1 in (10) and (11) yields the condition for the existence of the fourth-order moment for the univariate GARCH(2,2) model

where

, and

, and the expression for the fourth unconditional moment of ε_1t:

Both are derived in He and Teräsvirta (1999a). The necessary and sufficient conditions for finite fourth-order moments of GARCH(2,1) and GARCH(1,2) models of Bollerslev (1986) are nested in (12). The result illustrated in Figure 1 also holds for the univariate GARCH(2,2) model: the left-hand side of (12) and the corresponding eigenvalue in Theorem (2.1) of Ling and McAleer (2002) intersect at λ(Γ) = 1.¹

This requires, however, that in Theorem 2.1 of Ling and McAleer (2002), λ(Γ) is redefined to be the largest eigenvalue of Γ instead of being the smallest eigenvalue.

The proof of the necessary condition for the existence of the fourth moment of the GARCH(p,q) model in He and Teräsvirta (1999a) is complete. It is important to note that the authors begin by representing the recursion formula for the squared conditional variance function h_t² such that it has a bilinear form (see pp. 835–836). The advantage of this form is obvious from formula (A.21). The authors show that under some conditions,

implies λ(Γ) < 1, where a = (a₁,…,a_p*)′, b = (b₁,…,b_p*)′ for a_i > 0, b_i ≥ 0 (i = 1,…,p*) and b ≠ 0, and Γ^s = [γ_ij^(s)] , which is a positive (p* × p*) matrix (γ_ij^(s) > 0 for any s ≥ 2), and Γ is well defined. To see that the necessary condition λ(Γ) < 1 holds for (A.21), set

. That is, for any given ε > 0, there exists an integer N > 0 such that

for any k > N. It follows from the positiveness of a and Γⁱ and nonnegativeness of b that for any i,j = 1,…,p* and ε and N given previously,

, which implies that

converges as k approaches infinity. This shows that the necessary condition λ(Γ) < 1 in Lemma 4 of He and Teräsvirta (1999a) holds. Ling and McAleer (2002, note 1) argue that this result is proved by concluding that

follows from

. This is not the case.

Next we consider the relationship between the distributions of ε_t and z_t through their fourth-moment structure. For this purpose, let

and define the multivariate “rescaled fourth-moment matrix” of {ε_t} in (2)–(4) as

The jth diagonal element of K is the kurtosis of ε_jt, j = 1,…,M.

COROLLARY 1. Assume that Γ_{Z[otimes ]Z} exists and (10) holds for the vector GARCH(2,2) model (2) and (3) and (7). Then

where the inequality sign refers to element-by-element inequality and K(z_t) is the “rescaled fourth-moment matrix” of {z_t}.

Inequality (13) follows from Jensen's inequality. To illustrate, if z_t has a multivariate normal distribution, the unconditional distribution of ε_t is leptokurtic in the sense of (13).

The cross-moment matrix Γ_{Ci[otimes ]Cj}, i ≠ j, is required in the considerations if the vector model is of higher order than one. Expressions in Theorem 2 simplify for the vector GARCH(1,1) model. This is seen from the following result.

COROLLARY 2. Let the assumptions of Theorem 2 hold and assume C_2t = 0 in (7). Then the fourth-order moment matrix

of {ε_t} for the vector GARCH(1,1) model (2) and (3) and (7) exists if

Under (14),

4. AUTOCORRELATION FUNCTION OF SQUARES FOR THE EXTENDED CCC-GARCH MODEL

In this section we shall derive the multidimensional correlation function of {ε_t⁽²⁾} for our vector GARCH(2,2) process. Let

as before and

. Furthermore, let D_M = diag{γ₁₁^1/2(0),…,γ_MM^1/2(0)}. To fix notation, write the nth-order autocorrelation matrix of {ε_t⁽²⁾} for the vector GARCH(2,2) process as

for n ≥ 1.

The ith diagonal element, r_ii(n), i = 1,…,M, of R_M(n) in (16), is the nth autocorrelation of the squared observations for the ith component {ε_it}, whereas r_ij(n), i,j = 1,…,M, i ≠ j, the off-diagonal elements of R_M(n), represent the cross-correlations between ε_it² and ε_jt−n².

To obtain R_M(n), we must find an expression for

. This can be done by applying (7) to ε_t⁽²⁾ recursively up to the nth step and further applying Theorem 2 to

. The final result appears in the following theorem.

THEOREM 3. Assume that Γ_{Z[otimes ]Z} and μ₂ in (9) exist and condition (10) holds for the vector GARCH(2,2) model (2) and (3) and (7). Then the nth-order autocorrelation matrix R_M(n) of {ε_t⁽²⁾} has the stacked form

where

such that for n = 1,2,

Furthermore, for n ≥ 3,

with

In (19),

and

.

Proof. See the Appendix.

If the assumptions of Theorem 3 are satisfied and the first two autocorrelation matrices R_M(1) and R_M(2) are known we can compute the nth autocorrelation matrix R_M(n) recursively through equations (17)–(23). The autocorrelation structure simplifies for the first-order model. We state this result in the following corollary.

COROLLARY 3. Assume that the assumptions of Theorem 3 hold and C_2t = 0 in (7). Then the vectors (19)–(21) in the definition of the autocorrelation matrix R_M(n) of {ε_t⁽²⁾} for the CCC-GARCH(1,1) model (2) and (3) and (7) are

with

A number of theoretical properties of the autocorrelation matrix {R_M(n)}, n = 1,2,…, for the first-order model, obtained through Corollary 3, are listed here:

1. R_M(n) → 0 as n → ∞.

It follows from Corollary 3 that we can write

Under condition (10),

. This implies

2. The autocorrelation matrices R_M(n) satisfy the Yule–Walker equations.

Suppose that

is known so that vec(R_M(1)) is known (see (17) and (18)). It follows from equation (25) that vec(R_M(n)), for any n ≥ 2, can be solved by R_M(1) and (I_M [otimes ] Γ_C1) through

In particular,

3. The first-order auto- and cross-correlations r_ij(1) for i,j = 1,…,M, in R_M(1) are positive if the positivity restrictions a_0i > 0, a_l,ij and b_l,ij ≥ 0, i,j = 1,…,M, l = 1,…,max{p,q}, mentioned in Section 2, are satisfied.

4. The decay rate of R_M(n) as a function of n depends on the eigenvalues of (I_M [otimes ] Γ_C1). When M = 2 the autocorrelations in R₂(n) will exhibit a mixture of decaying exponential decay, because (I_M [otimes ] Γ_C1) only has real roots. When M ≥ 3 the autocorrelations display a mixture of exponential decay if there exists a dominant real root in (I_M [otimes ] Γ_C1) and dampening sinusoidal behavior if the moduli of the complex conjugate pairs of eigenvalues are sufficiently large. An example of the latter case is depicted in Figure 2. When M = 1, the decay rate of the autocorrelation function of the squared observations for the univariate GARCH(1,1) model is exactly

. In this case the decay is exponential as r(n) = γ_c1ⁿ⁻¹r(1), n ≥ 2. This property also holds when M > 1 and A₁ and B₁ are diagonal matrices as in Bollerslev (1990). Thus our extension of the CCC-GARCH model allows a considerably richer autocorrelation structure than the original CCC-GARCH model.

Auto- and cross-correlations of squared observations of a three-dimensional extended CCC-GARCH(1,1) process, lags 1–8. Here ΓC has a pair of complex conjugate eigenvalues with modulus = 0.432 and a real root = 0.858. The parameter values are a01 = 0.1, a02 = 0.2, a03 = 0.25, a11 = 0.05, a12 = 0.25, a13 = 0.06, a21 = 0.07, a22 = 0.08, a23 = 0.25, a31 = 0.25, a32 = 0.08, a33 = 0.08, b11 = 0.05, b12 = 0.4, b13 = 0.05, b21 = 0.05, b22 = 0.07, b23 = 0.3, b31 = 0.25, b32 = 0.08, and b33 = 0.08. The correlation matrix of the standard normal error process {zt} has the following nondiagonal elements: ρ12 = 0.2, ρ13 = 0.3, and ρ23 = 0.23.

5. BIVARIATE GARCH(1,1) MODEL

To illustrate the general correlation results we consider the bivariate GARCH(1,1) process (2) and (3) and (7). The correlation matrix of {ε_t⁽²⁾} for this process is obtained as a special case of Corollary 3.

COROLLARY 4. Let M = 2, C_2t = 0, and c_1,12t = c_1,21t = 0 in (7). Furthermore, let Γ_C1 = [γ_ij] where

for i,j = 1,2, Γ_{C1[otimes ]C1} = [γ_ij,kl] where

for i,j,k,l = 1,2, and, finally, Γ_{Z[otimes ]C1} = [γ_{zi c1,jk}] where γ_{zi cjk} = E(z_it² c_1,jkt), for i,j,k = 1,2. Assume that z = (z_1t,z_2t)′ ∼ NID(0,R) and, furthermore, that condition (10) holds for the bivariate GARCH(1,1) model (2) and (3) with (7). Then R₂(n) = [r_ij(n)] in (17) for i,j = 1,2 has the simplified form

According to Corollary 4, the elements in R₂(n), n ≥ 1, decay exponentially. The autocorrelations r₁₁(n) and cross-correlations r₁₂(n), n ≥ 1, have the decay rate γ₁₁, whereas r₂₁(n) and r₂₂(n) also have a common decay rate γ₂₂. The exponential decay rates generalize to the case M > 2 and are a characteristic feature of the standard CCC-GARCH model.

6. AN EMPIRICAL EXAMPLE

The purpose of this section is to illustrate practical usefulness of our theoretical results. Bollerslev (1990) fitted two CCC-GARCH models with nonzero constant conditional correlations to a set of five weekly exchange rate return series. His purpose was to analyze the dynamic behavior of these returns before the introduction of the European monetary system (EMS) and thereafter. The pre-EMS period ran from July 1973 to the second week of March 1979 (299 observations) and the second, the EMS period, from the third week of March 1979 to the second week of August 1985 (333 observations). The estimated constant correlations from the pre-EMS and EMS models appear in Table 1. The parameter estimates are not reproduced and can be found in Tables 1 and 2 of Bollerslev (1990). The exchange rates that are rates against the U.S. dollar are indexed as follows: 1 = DM (the Deutschmark), 2 = FF (the French franc), 3 = IL (the Italian lira), 4 = SF (the Swiss franc), and 5 = BP (the British pound). Bollerslev concluded, among other things, that the conditional correlations are significantly higher for the EMS period than for the period preceding it.

The estimated constant correlations , i,j = DM, FF, IL, SF, BP, from the two estimated CCC-GARCH models (EMS period and pre-EMS period) of Bollerslev (1990)

Auto- and cross-correlations rij(n),i,j = DM, FF, IL, SF, BP, n = 1, 2, 5, 10, are computed from the two estimated CCC-GARCH models (EMS period and pre-EMS period) of Bollerslev (1990)

Using plug-in auto- and cross-correlation estimates based on the definitions of the true quantities derived in the previous sections we are able to complete Bollerslev's analysis. However, in the pre-EMS model,

, so that the IL process does not have a finite variance. As this would invalidate our small example, we shrink the estimate of a₃₃ to 0.287 to satisfy the fourth-moment existence condition for the pre-EMS system. Estimated auto- and cross-correlations of the squared observations from the two models after this adjustment can be found in Table 2. The individual autocorrelations are relatively high and persistent for FF and SF and, of course, very high and persistent for IL (the adjusted

remains just below unity). On the other hand, they are low for DM. It may also be noted that most cross-correlations are negligible.

This can be compared with the results from the model for the EMS period. The autocorrelation structure of DM is practically unaffected by EMS. But then, the three rates with previously large autocorrelations of squared returns are now much more weakly autocorrelated than before. The autocorrelations in the BP not included in the EMS increase slightly. Thus, although the conditional correlations between the returns of EMS currencies increase as a result of the EMS, the autocorrelations of squared returns decrease. The anchor currency of the monetary system, DM, constitutes an exception. On the other hand, cross-correlations appear where none were observed before the EMS: note the ones between DM and IL and FF and IL. It seems that in the EMS period, the changes in the volatility of DM and FF have dynamic effects on the volatility of IL. This may not be unexpected as these currencies belong to the EMS during the observation period. Finally, about the only nonnegligible pre-EMS cross-correlations, the ones between BP and DM, practically vanish in the EMS period. As BP has not been a part of the EMS during the observation period, this may not be an unexpected result either.

These results should be viewed with caution because one parameter estimate was adjusted to allow the estimated pre-EMS process to have finite unconditional fourth moments. Besides, the uncertainty in the parameter estimates is not accounted for in the discussion. We emphasize, however, that the main purpose of this example is to demonstrate practical uses of the theoretical results of the paper. The example shows that the fourth-moment results are useful already in the case of the standard first-order CCC-GARCH model, and the same can be said about more general situations also.