This section collects models that may be viewed as nonlinear combinations of univariate GARCH models. This allows for models where one can specify separately, on the one hand, the individual conditional variances, and on the other hand, the conditional correlation matrix or another measure of dependence between the individual series (like the copula of the conditional joint density). For models of this category, theoretical results on stationarity, ergodicity and moments may not be so straightforward to obtain as for models presented in the preceding sections. Nevertheless, they are less greedy in parameters than the models of the first category, and therefore they are more easily estimable.
Conditional Correlation Models
The conditional variance matrix for this class of models is specified in a hierarchical way. First, one chooses a GARCH-type model for each conditional variance. For example, some conditional variances may follow a conventional GARCH model while others may be described as an EGARCH model. Second, based on the conditional variances one models the conditional correlation matrix (imposing its positive definiteness ).
Bollerslev (1990) proposes a class of MGARCH models in which the conditional correlations are constant and thus the conditional covariances are proportional to the product of the corresponding conditional standard deviations. This restriction greatly reduces the number of unknown parameters and thus simplifies the estimation.
Definition 7 The CCC model is defined as:
hiitcan be defined as any univariate GARCH model, and
is a symmetric positive definite matrix with.
R is the matrix containing the constant conditional correlations ρij. The original CCC model has a GARCH(1, 1) specification for each conditional variance in Dt:
This CCC model contains N(N + 5)/2 parameters. Ht is positive definite if and only if all the N conditional variances are positive and R is positive definite. The unconditional variances are easily obtained, as in the univariate case, but the unconditional covariances are difficult to calculate because of the nonlinearity in (31). He and Teräsvirta (2002b) use a VEC-type formulation for (h11t, h22t, …, hNNt)′, to allow for interactions between the conditional variances. They call this the extended CCC model.
The assumption that the conditional correlations are constant may seem unrealistic in many empirical applications. Christodoulakis and Satchell (2002), Engle (2002) and Tse and Tsui (2002) propose a generalization of the CCC model by making the conditional correlation matrix time-dependent. The model is then called a dynamic conditional correlation (DCC) model. An additional difficulty is that the time-dependent conditional correlation matrix has to be positive definite . The DCC models guarantee this under simple conditions on the parameters.
Definition 8 The DCC model of Tse and Tsui (2002) orDCCT(M) is defined as:
whereDtis defined in(32), hiitcan be defined as any univariate GARCH model and
In(36), θ1and θ2are non-negative parameters satisfying θ1 + θ2 < 1, R is a symmetric N × N positive definite parameter matrix with ρii = 1 and Ψt−1 is the N × N correlation matrix of ϵτ for τ = t − M, t − M + 1, …, t − 1. Its i,jth element is given by:
where. The matrix Ψt−1 can be expressed as:
where Bt−1 is a N × N diagonal matrix with ith diagonal element given by and Lt−1 = (ut−1, …, ut−M) is a N × M matrix, with ut = (u1tu2t…uNt)′.
A necessary condition to ensure the positivity of Ψt−1, and therefore also of Rt, is that M ≥ N.9 Then Rt is itself a correlation matrix if Rt−1 is also a correlation matrix (notice that ).
Alternatively, Engle (2002) proposes a different DCC model (see also Engle and Sheppard, 2001).
Definition 9 The DCC model of Engle (2002) or DCCE(1, 1) is defined as in(35)with
where the N × N symmetric positive definite matrix Qt = (qij, t) is given by:
with ut as in Definition 8. Q̄ is the N × N unconditional variance matrix of ut, and α and β are non-negative scalar parameters satisfying α + β < 1.
The elements of Q̄ can be estimated or alternatively set to their empirical counterpart to render the estimation even simpler (see Section 3). To show more explicitly the difference between DCCT and DCCE, we write the expression of the correlation coefficient in the bivariate case: for the DCCT(M)
and for the DCCE(1, 1)
Unlike in the DCCT model, the DCCE model does not formulate the conditional correlation as a weighted sum of past correlations. Indeed, the matrix Qt is written like a GARCH equation, and then transformed to a correlation matrix. However, for both the DCCT and DCCE models, one can test θ1 = θ2 = 0 or α = β = 0, respectively to check whether imposing constant conditional correlations is empirically relevant.
A drawback of the DCC models is that θ1, θ2 in DCCT and α, β in DCCE are scalars, so that all the conditional correlations obey the same dynamics. This is necessary to ensure that Rt is positive definite through sufficient conditions on the parameters. If the conditional variances are specified as GARCH(1,1) models then the DCCT and DCCE models contain (N + 1)(N + 4)/2 parameters.
Interestingly, DCC models can be estimated consistently in two steps (see Section 3.2), which makes this approach feasible when N is high. Of course, when N is large, the restriction of common dynamics gets tighter, but for large N the problem of maintaining tractability also gets harder. In this respect, several variants of the DCC model are proposed in the literature. For example, Billio et al. (2003) argue that constraining the dynamics of the conditional correlation matrix to be the same for all the correlations is not desirable. To solve this problem, they propose a block-diagonal structure where the dynamics is constrained to be identical only within each block. The price to pay for this additional flexibility is that the block members have to be defined a priori, which may be cumbersome in some applications. Pelletier (2003) proposes a model where the conditional correlations follow a switching regime driven by an unobserved Markov chain so that the correlation matrix is constant in each regime but may vary across regimes. Another extension proposed by Engle (2002) consists of changing (40) into
where i is a vector of ones and A and B are N × N matrices of parameters. This increases the number of parameters considerably, but the matrices A and B could be defined to depend on a small number of parameters (e.g. A = aa′).
To conclude, DCC models open the door to using flexible GARCH specifications in the variance part. Indeed, as the conditional variances (together with the conditional means) can be estimated using N univariate models, one can easily extend the DCC-GARCH models to more complex GARCH-type structures (as mentioned at the beginning of Section 2.2). One can also extend the bivariate CCC FIGARCH model of Brunetti and Gilbert (2000) to a model of the DCC family.
General Dynamic Covariance Model
A model somewhat different from the previous ones but that nests several of them is the general dynamic covariance (GDC) model proposed by Kroner and Ng (1998). They illustrate that the choice of a multivariate volatility model can lead to substantially different conclusions in an application that involves forecasting dynamic variance matrices. We extend the definition of Kroner and Ng (1998) to cover models with dynamic conditional correlations.
Definition 10 The GDC model is defined as:
Elementwise we have:
where the θijt are given by the BEKK formulation in (45). The GDC model contains several MGARCH models as special cases. To show this we adapt a proposition from Kroner and Ng (1998). Consider the following set of conditions:
θ1 = θ2 = 0(DCCT) or α = β = 0(DCCE);
R = IN(DCCT) or Q̄ = IN(DCCE);
, where li
is the i
th column of an (N
) identity matrix, and αi
= 1, …, N
A = α(wλ′) and G = β(wλ′) where A = [a1, …, aN] and G = [g1, …, gN] are N × N matrices, w and λ are N × 1 vectors, and α and β are scalars.
The GDC model reduces to different multivariate GARCH models under different combinations of these conditions. Specifically, the GDC model becomes:
the DCCT or the DCCE(1, 1) model with GARCH(1,1) conditional variances under conditions (ii) and (iii);
the CCC model with GARCH(1,1) conditional variances under conditions (ia), (ii) and (iii);
a restricted DVEC(1,1) model under conditions (i) and (ii);
the BEKK(1,1,1) model under conditions (i) and (iv);
the F-GARCH(1,1,1) model under conditions (i), (iv) and (v).
Condition (ib) serves as an identification restriction for the VEC, BEKK and F-GARCH models. As we can see, the GDC model is an encompassing model. This requires a large number of parameters (i.e. [N(7N − 1) + 4]/2). For example, in the bivariate case there are 11 parameters in θt, 3 in Rt and 1 in Φ, which makes a total of 15. This is less than for an unrestricted VEC model (21 parameters), but more than for the BEKK model (11 parameters).