10.2.1 Diagonal Model

A popular specification, known as the diagonal representation, is obtained by assuming that each element h _{kℓ, t} of the covariance matrix H _t is formulated in terms only of the product of the prior k and ℓ returns. Specifically,

with ω _kℓ = ω _ℓk , , for all (k, ℓ). For m = 1, this model coincides with the usual univariate formulation. When m > 1, the model obviously has a large number of parameters and will not in general produce positive definite covariance matrices H _t . We have

where ⊙ denotes the Hadamard product, that is, the element by element product. ² Thus, in the ARCH case ( p = 0), sufficient positivity conditions are that Ω is positive definite and the A ⁽ⁱ⁾ are positive semi‐definite, but these constraints do not easily generalise to the GARCH case. We shall give further positivity conditions obtained by expressing the model in a different way, viewing it as a particular case of a more general class.

It is easy to see that the model is not stable by aggregation: for instance, the conditional variance of ε_{1, t}  + ε_{2, t} can in general be expressed as a function of the and , but not of the (ε_{1, t − i}  + ε_{2, t − i} )² . A final drawback of this model is that there is no interaction between the different components of the conditional covariance, which appears unrealistic for applications to financial series.

In what follows, we present the main specifications introduced in the literature, before turning to the existence of solutions. Let η denote a probability distribution on ℝ ^m , with zero mean and unit covariance matrix.

10.2.2 Vector GARCH Model

The vector GARCH (VEC‐GARCH) model is the most direct generalisation of univariate GARCH: every conditional covariance is a function of lagged conditional variances as well as lagged cross‐products of all components. In some sense, everything is explained by everything, which makes this model not only very general but also not very parsimonious.

Denote by vech(·) the operator that stacks the columns of the lower triangular part of its argument square matrix (if A = (a _ij ), then vech(A) = (a ₁₁, a ₂₁,…, a _m1, a ₂₂,…, a _m2,…, a _mm )^′ ). The next definition is a natural extension of the standard GARCH( p, q ) specification.

Positivity Conditions

To ensure the positive semi-definiteness of H_t , the initial values and Ω (where ω = vech(Ω)) have to be positive semi‐definite, and the matrices A⁽ⁱ⁾ and B^(j) need to map the vectorized positive semi‐definite matrices into themselves. To be more specific, a generic element of

is denoted by h _{kℓ, t} ( k ≥ ℓ), and we will denote by ( ) the entry of A ⁽ⁱ⁾ ( B ^(j) ) located on the same row as h _{kℓ, t} and belonging to the same column as the element of . We thus have an expression of the form

Denoting by the m × m symmetric matrix with (k ^′, ℓ^′)th entry , for k ^′ ≠ ℓ^′ , and the elements on the diagonal, the preceding equality is written as

10.12

In order to obtain a more compact form for the last part of this expression, let us introduce the spectral decomposition of the symmetric matrices H _t , assumed to be positive semi‐definite. We have , where is an orthogonal matrix of eigenvectors associated with the (positive) eigenvalues of H _t . Defining the matrices by analogy with the , we get

10.13

Finally, consider the m ² × m ² matrix admitting the block form , and let . The preceding expressions are equivalent to

10.14

where Ω is the symmetric matrix such that vech(Ω) = ω .

In this form, it is evident that the assumption

10.15

ensures that if the H _t − j are almost surely positive definite, then so is H _t .

10.2.3 Constant Conditional Correlations Models

Suppose that, for a MGARCH process of the form ( 10.6), all the past information on ε _kt , involving all the variables ε_{ℓ, t − i} , is summarised in the variable h _{kk, t} , with . Then, letting , we define for all k a sequence of iid variables with zero mean and unit variance. The variables are generally correlated, so let , where . The conditional variance of

is then written as

10.16

By construction, the conditional correlations between the components of ε _t are time‐invariant:

To complete the specification, the dynamics of the conditional variances h _{kk, t} has to be defined. The simplest constant conditional correlations (CCC) model relies on the following univariate GARCH specifications:

10.17

where ω _k  > 0, a _{k, i}  ≥ 0, b _{k, j}  ≥ 0, −1 ≤ ρ _kℓ ≤ 1, ρ _kk  = 1, and R is symmetric and positive semi‐definite. Observe that the conditional variances are specified as in the diagonal model. The conditional covariances clearly are not linear in the squares and cross products of the returns.

In a multivariate framework, it seems natural to extend the specification (10.17) by allowing h _{kk, t} to depend not only on its own past, but also on the past of all the variables ε_{ℓ, t} . Set

We have where is a centred vector with covariance matrix R . The components of ε _t thus have the usual expression, , but the conditional variance h _{kk, t} depends on the past of all the components of ε _t .

Note that the conditional covariances are generally non‐linear functions of the components of and of past values of the components of H _t . Model (10.18) is thus not a VEC‐GARCH model, defined by ( 10.9), except when R is the identity matrix.

One advantage of this specification is that a simple condition ensuring the positive definiteness of H _t is obtained through the positive coefficients for the matrices A _i and B _j and the choice of a positive definite matrix for R . Conrad and Karanasos (2010) showed that less restrictive assumptions ensuring the positive definiteness of H_t can be found. Moreover there exists a representation of the CCC model in which the matrices B_j are diagonal. Another advantage of the CCC specification is that the study of the stationarity is remarkably simple.

Two limitations of the CCC model are, however, (i) its non‐stability by aggregation and (ii) the arbitrary nature of the assumption of constant conditional correlations.

10.2.4 Dynamic Conditional Correlations Models

Dynamic conditional correlations GARCH (DCC‐GARCH) models are an extension of CCC‐GARCH, obtained by introducing a dynamic for the conditional correlation. Hence, the constant matrix R in Definition 10.4 is replaced by a matrix R _t which is measurable with respect to the past variables {ε _u , u < t}. For reasons of parsimony, it seems reasonable to choose diagonal matrices A _i and B _i in ( 10.18), corresponding to univariate GARCH models for each component as in ( 10.17). Different DCC models are obtained depending on the specification of R _t . A simple formulation is

10.19

where the θ _i are positive weights summing to 1, R is a constant correlation matrix, and Ψ _t − 1 is the empirical correlation matrix of ε _t − 1,…, ε _t − M . The matrix R _t is thus a correlation matrix (see Exercise 10.9). Equation (10.19) is reminiscent of the GARCH(1, 1) specification, θ ₁ R playing the role of the parameter ω , θ ₂ that of α , and θ ₃ that of β .

Another way of specifying the dynamics of R _t is by setting

where diag Q _t is the diagonal matrix constructed with the diagonal elements of Q _t , and Q _t is a sequence of covariance matrices which is measurable with respect to σ(ε _u , u < t). In the original DCC model of Engle (2002), the dynamics of is given by

where denotes the vector of standardized returns, , and is a positive‐definite matrix.

Matrix turns out to be difficult to interpret, and Aielli (2013) pointed out that in general. Thus, the commonly used estimator of defined as the sample second moment of the standardized returns is not consistent in this formulation.

In the so‐called corrected DCC (cDCC) model of Aielli (2013), the dynamics of is reformulated as

under the same constraints on the coefficients. In this model, under stationarity conditions, .

Multiplying the left‐hand side and the right‐hand side of by an arbitrary positive‐definite matrix yields the same conditional correlation matrix . It is thus necessary to introduce an identifiability condition, as for instance imposing that be a correlation matrix.

10.2.5 BEKK‐GARCH Model

The BEKK acronym refers to a specific parameterisation of the MGARCH model developed by Baba, Engle, Kraft, and Kroner, in a preliminary version of Engle and Kroner (1995).

The specification obviously ensures that if the matrices, H _t − i , i = 1,…, p , are almost surely positive definite, then so is H _t .

To compare this model with the representation ( 10.9), let us derive the vector form of the equation for H _t . Using the relations (10.10) and (10.11), we get

The model can thus be written in the VEC‐GARCH(p,q) form ( 10.9), with

10.21

for i = 1,…, q and j = 1,…, p . In particular, it can be seen that the number of coefficients of a matrix A ⁽ⁱ⁾ in ( 10.9) is [m(m + 1)/2]² , whereas it is Km ² in this particular case. However, the converse is not true. Stelzer (2008) showed that, for m ≥ 3, there exist VEC-GARCH models that cannot be represented in the BEKK form.

The BEKK class contains (Exercise 10.13) the diagonal models obtained by choosing diagonal matrices A _ik and B _jk . The following theorem establishes a converse to this property.

It is interesting to consider the stability by aggregation of the BEKK class.

As in the univariate case, the ‘square’ of the (ε _t ) process is the solution of an ARMA model. Indeed, define the innovation of the process :

10.22

Applying the vec operator, and substituting the variables vec(H _t − j ) in the model of Definition 10.5 by , we get the representation

10.23

where r = max(p, q), with the convention A _ik  = 0 ( B _jk  = 0) if i > q ( j > p ). This representation cannot be used to obtain stationarity conditions because the process (ν _t ) is not iid in general. However, it can be used to derive the second‐order moment, when it exists, of the process ε _t as

that is,

provided that the matrix in braces is non‐singular.

10.2.6 Factor GARCH Models

In these models, it is assumed that a non‐singular linear combination f _t of the m components of ε _t , or an exogenous variable summarising the co‐movements of the components, has a GARCH structure.

Factor Models with Idiosyncratic Noise

A very popular factor model links individual returns ε _it to the market return f _t through a regression model

10.24

The parameter β _i can be interpreted as a sensitivity to the factor, and the noise η _it as a specific risk (often called idiosyncratic risk) which is conditionally uncorrelated with f _t . It follows that H _t  = Ω + λ _t β β ^′ , where β = (β₁,…, β _m )^′ is the vector of sensitivities, λ _t is the conditional variance of f _t , and Ω is the covariance matrix of the idiosyncratic terms. More generally, assuming the existence of r conditionally uncorrelated factors, we obtain the decomposition

10.25

It is not restrictive to assume that the factors are linear combinations of the components of ε _t (Exercise 10.10). If, in addition, the conditional variances λ _jt are specified as univariate GARCH, the model remains parsimonious in terms of unknown parameters and (10.25) reduces to a particular BEKK model (Exercise 10.11). If Ω is chosen to be positive definite and if the univariate series (λ _jt ) _t , j = 1,…, r are independent, strictly and second‐order stationary, then it is clear that ( 10.25) defines a sequence of positive definite matrices (H _t ) that are strictly and second‐order stationary.

Principal Components GARCH Model

The concept of factor is central to principal components analysis (PCA) and to other methods of exploratory data analysis. PCA relies on decomposing the covariance matrix V of m quantitative variables as V = PΛP ^′ , where Λ is a diagonal matrix whose elements are the eigenvalues λ ₁ ≥ λ ₂ ≥ ⋯ ≥ λ _m of V , and where P is the orthonormal matrix of the corresponding eigenvectors. The first principal component is the linear combination of the m variables, with weights given by the first column of P, which, in some sense, is the factor which best summarises the set of m variables (Exercise 10.12). There exist m principal components, which are uncorrelated and whose variances λ ₁,…, λ _m (and hence whose explanatory powers) are in decreasing order. It is natural to consider this method for extracting the key factors of the volatilities of the m components of ε _t .

We obtain a principal component GARCH (PC‐GARCH) or orthogonal GARCH (O‐GARCH) model by assuming that

10.26

where P is an orthogonal matrix ( P ^′ = P ⁻¹ ) and Λ _t  = diag(λ _1t ,…, λ _mt ), where the λ _it are the volatilities, which can be obtained from univariate GARCH‐type models. This is equivalent to assuming

10.27

where f _t  = P ^′ε _t is the principal component vector, whose components are orthogonal factors. If univariate GARCH(1, 1) models are used for the factors , then

10.28

10.2.7 Cholesky GARCH

Suppose that the conditional covariance matrix H _t of ε _t is positive‐definite, i.e. that the components of ε _t are not multicolinear. Given the information ℱ _t − 1 generated by the past values of ε _t , let ℓ_{21, t} be the conditional beta in the regression of ε_2t on v _1t  ≔ ε_1t . One can write

with β _{21, t}  = ℓ_{21, t}  ∈ ℱ _t − 1 , and υ _2t is orthogonal to ε_1t conditionally on ℱ _t − 1 . More generally, we have

10.31

where υ _it is uncorrelated with υ _1t ,…, υ _{i − 1, t} , and thus uncorrelated with ε_1t ,…, ε _{i − 1, t} , conditionally on ℱ _t − 1 . In matrix form, Eq. (10.31) is written as

where L _t and are lower unitriangular (i.e. triangular with 1 on the diagonal) matrices, with ℓ _{ij, t} (respectively −β _{ij, t} ) at row i and column j of L _t (respectively B _t ) for i > j . For m = 3, we have

The vector υ _t of the error terms in the linear regressions ( 10.31) can be interpreted as a vector of orthogonal factors, whose covariance matrix is G _t  = diag(g _1t ,…, g _mt ) with g _it  > 0 for i = 1,…, m . We then obtain the so‐called Cholesky decomposition of the covariance matrix of ε _t :

10.32

Note that the Cholesky decomposition also extends to positive semi‐definite matrices.

Taking , we obtain . A simple parametric form of the volatility of the i th factor υ _it could be of the GARCH(1,1)‐type

10.33

In view of the results of the Chapter 2, a necessary and sufficient condition for the existence of a strictly stationary process (υ _t ) in this parameterisation is

10.34

Of course, Eq. (10.33) could also include explanatory variables with j ≠ i , or other variables belonging .

To obtain a complete parametric specification for H _t , it now suffices to specify a time series model for the conditional betas, i.e. for the elements of L _t (or alternatively for those of B _t ). Note that this model can be quite general because, to ensure the positive‐definiteness of H _t , the conditional betas are not a priori subject to any constraint.

For instance, one can assume a dynamics of the form

10.35

where f _ij is a real‐valued function, depending on υ _t − 1 and on some parameter θ , and c _ij is a real coefficient. Under the conditions ∣c _ij  ∣  < 1, and the condition (10.34) ensuring the existence of the stationary sequence (υ _t ), there exists a stationary solution to Eq. (10.35). The Cholesky decomposition (10.32) then provides a model for the conditional covariance matrix H _t .

The absence of strong constraints on the coefficients constitutes an attractive feature of the Cholesky decomposition ( 10.32), compared in particular to the DCC decomposition for which the unexplicit constraints of positive definiteness of R _t render the determination of stationarity conditions challenging. Another obvious interest of the Cholesky approach is to provide a direct way to predict the conditional betas, which appear to be of primary interest for particular financial applications (see Engle 2016).

Now, let us briefly explore the relationships between the Cholesky decomposition and the factor models. Suppose that the first k < m columns of the matrix L _t are constant so that L _t  = [P : P _t ] with P a non‐random matrix of size m × k and P _t of size m × (m − r). The Cholesky GARCH can then be interpreted as a factor model:

where f _t  = (υ _1t ,…, υ _kt )^′ is a vector of k orthogonal factors, P is called the loading matrix and u _t  = P _t (υ _{k + 1, t} ,…, υ _mt )^′ is a so‐called idiosyncratic disturbance (independent of f _t ). If ε _t is a vector of excess returns (i.e. each component is the return of a risky asset minus the return of a risk free asset), and if the first component is the excess return of the market portfolio, in the one‐factor case ( k = 1) the previous factor model corresponds to the Capital Asset Pricing Model (CAPM) of Sharpe (1964), Lintner (1965), and Merton (1973). More precisely, we have P = (1, β ^′)^′ , where β represents the ‘sensitivity of returns to market returns’, and . Denoting by r _t  = (ε_2t ,…, ε _mt )^′ the vector of excess returns, the last (m − 1) lines of the one‐factor model gives the CAPM equation

where υ _1t is the market excess return.

10.3.1 Stationarity of VEC and BEKK Models

It is not possible to provide stationary solutions, in explicit form, for the general VEC model ( 10.9). To illustrate the difficulty, recall that a univariate ARCH(1) model admits a solution ε _t  = σ _t η _t with σ _t explicitly given as a function of {η _t − u , u > 0} as the square root of

provided that the series converges almost surely. Now consider a bivariate model of the form ( 10.6) with where α is assumed, for the sake of simplicity, to be scalar and positive. Also choose to be a lower triangular so as to have Eq. (10.7). Then

It can be seen that given η _t − 1 , the relationship between h _{11, t} and h _{11, t − 1} is linear, and can be iterated to yield

under the constraint . In contrast, the relationships between h _{12, t} , or h _{22, t} , and the components of H _t − 1 are not linear, which makes it impossible to express h _{12, t} and h _{22, t} as a simple function of α , {η _t − 1, η _t − 2,…, η _t − k } and H _t − k for k ≥ 1. This constitutes a major obstacle for determining sufficient stationarity conditions.

In the particular case of the BEKK model of Definition 10.5, condition (iii) takes the form

The proof of Theorem 10.5 relies on sophisticated algebraic tools. Assumption (ii) is a standard technical condition for showing the β ‐mixing property (but is of no use for stationarity). Note that condition (iii), written as in the univariate case, is generally not necessary for the strict stationarity.

This theorem does not provide explicit stationary solutions, that is, a relationship between ε _t and the η _t − i . However, it is possible to construct an algorithm which, when it converges, allows a stationary solution to the vector GARCH model ( 10.9) to be defined.

Construction of a Stationary Solution

For any t, k ∈ ℤ, we define

and, recursively on k ≥ 0,

10.36

with .

Observe that, for k ≥ 1,

where f _k is a measurable function and H ^(k) is a square matrix. The processes and are thus stationary with components in the Banach space L ² of the (equivalence classes of) square integral random variables. It is then clear that Eq. ( 10.9) admits a strictly stationary solution, which is non‐anticipative and ergodic, if, for all t ,

10.37

Indeed, letting and , and taking the limit of each side of (10.36), we note that ( 10.9) is satisfied. Moreover, (ε _t ) constitutes a strictly stationary and non‐anticipative solution, because ε _t is a measurable function of {η _u , u ≤ t}. In view of Theorem A.1, such a process is also ergodic. Note also that if H _t exists, it is symmetric and positive definite because the matrices are symmetric and satisfy

This solution (ε _t ) is also second‐order stationary if

10.38

Let

From Exercise 10.8 and its proof, we obtain (10.37), and hence the existence of strictly stationary solution to the vector GARCH ( 10.9), if there exists ρ ∈ (0, 1) such that almost surely as k → ∞, which is equivalent to

10.39

Similarly, we obtain (10.38) if . The criterion in (10.39) is not very explicit but the left‐hand side of the inequality can be evaluated by simulation, just as for a Lyapunov coefficient.

10.3.2 Stationarity of the CCC Model

In model ( 10.18), letting , we get

Multiplying by ϒ _t the equation for , we thus have

which can be written

10.40

where

and

10.41

is a (p + q)m × (p + q)m matrix.

We obtain a vector representation, analogous to (2.16) obtained in the univariate case. This allows us to state the following result.

The following result provides a necessary strict stationarity condition which is simple to check.

10.3.3 Stationarity of DCC models

Stationarity conditions for the DCC model, with the specification (10.19) for R _t , have been established by Fermanian and Malongo (2016). However, except in very specific cases, such conditions are non explicit. The corrected DCC model allows for more tractable conditions. Sufficient stationarity conditions – based on the results by Boussama, Fuchs and Stelzer (2011) – are provided for the cDDD model by Aielli (2013).

10.4.1 Asymptotic Properties of the QMLE

Let ,

Recall that ρ denotes a generic constant belonging to [0, 1), and K denotes a positive constant or a positive random variable measurable with respect to {ε _u , u < 0} (and thus which does not depend on n ). The regularity conditions (which do not depend on the choice of the norms) are the following.

1. A1: a.s.
2. A2: a.s.
3. A3: E‖ε _t ^s ‖ < ∞ and E‖H _t (θ ₀)‖ ^s  < ∞ for some s > 0.
4. A4: For θ ∈ Θ, H _t (θ) = H _t (θ ₀) a.s. implies θ = θ ₀ .
5. A5: For any sequence x ₁, x ₂,… of vectors of ℝ ^m , the function θ ↦ H(x ₁, x ₂,…; θ) is continuous on Θ.
6. A6: θ ₀ belongs to the interior of Θ.
7. A7: For any sequence x ₁, x ₂,… of vectors of ℝ ^m , the function θ ↦ H(x ₁, x ₂,…; θ) admits continuous second‐order derivatives.
8. A8: For some neighbourhood V(θ ₀) of θ ₀ ,
   
9. A9: For some neighbourhood V(θ ₀) of θ ₀ , for all i, j ∈ {1,…, m} and p > 1, q > 2 and r > 2 such that 2q ⁻¹ + 2r ⁻¹ = 1 and p ⁻¹ + 2r ⁻¹ = 1, we have ⁵
   
10. A10: E ∥ η _t ∥⁴ < ∞.
11. A11: The matrices {∂ H _t (θ ₀)/∂θ _i , i = 1,…, d} are linearly independent with non‐zero probability.

In the case m = 1, Assumption A1 requires that the volatility be bounded away from zero uniformly on Θ. For a standard GARCH satisfying ω > 0 and Θ compact, the assumption is satisfied. Assumption A2 is related to the invertibility and entails that H _t (θ) is well estimated by the statistic when t is large. For some models it will be useful to replace A2 by the weaker, but more complicated, assumption

1. A2': where ρ _t is a random variable satisfying , for some p > 1 and s ∈ (0, 1) satisfying A3.

This assumption, as well as A8, will be used to show that the choice of the initial values is asymptotically unimportant. Corollary 10.2 shows that, for some particular models, the strict stationarity implies the existence of marginal moments, as required in A3. Assumption A4 is an identifiability condition. As shown in Section 8.2, A6 is necessary to obtain the asymptotic normality (AN) of the QMLE. Assumptions A9 and A10 are used to show the existence of the information matrices I and J involved in the sandwich form J ⁻¹ IJ ⁻¹ of the asymptotic variance of the QMLE. Assumption A11 is used to show the invertibility of J .

10.5.1 Identifiability Conditions

Let By convention, if q = 0 and ℬ _θ (z) = I _m if p = 0.

If ℬ _θ (z) is non‐singular, that is, if the roots of det(ℬ _θ (z)) = 0 are outside the unit disk, we deduce from the representation

10.51

In the vector case, assuming that the polynomials and have no common root is insufficient to ensure that there exists no other pair , with the same degrees (p, q), such that

10.52

This condition is equivalent to the existence of an operator U(B) such that

10.53

this common factor vanishing in ℬ _θ (B)⁻¹ 풜 _θ (B) (Exercise 10.2).

The polynomial U(B) is called unimodular if det{U(B)} is a non‐zero constant. When the only common factors of the polynomials P(B) and Q(B) are unimodular, that is, when

then P(B) and Q(B) are called left coprime.

The following example shows that, in the vector case, assuming that and are left coprime is insufficient to ensure that condition (10.52) has no solution θ ≠ θ ₀ (in the univariate case this is sufficient because the condition imposes U(B) = U(0) = 1).

10.5.2 Asymptotic Properties of the QMLE of the CCC‐GARCH model

For particular classes of MGARCH models, the assumptions of the previous section can be made more explicit. Let us consider the CCC‐GARCH model. Let (ε₁,…, ε _n ) be an observation of length n of the unique non‐anticipative and strictly stationary solution (ε _t ) of model (10.50). Conditionally on non‐negative initial values , the Gaussian quasi‐likelihood is written as

where the are recursively defined, for t ≥ 1, by

A QMLE of θ is defined as in ( 10.47) by:

10.56

To establish the AN we require the following additional assumptions:

1. CC6: , where is the interior of Θ .
2. CC7:

10.6.1 Variance Targeting Estimation

Variance targeting (VT) estimation is a two‐step procedure in which the unconditional variance–covariance matrix of the returns vector process is estimated by a moment estimator in a first step. VT is based on a reparameterisation of the MGARCH model, in which the matrix of intercepts in the volatility equation is replaced by the unconditional covariance matrix.

To be more specific, consider the CCC‐GARCH( p, q ) model

10.58

where and R ₀ is a correlation matrix, A _0i and B _0j are m × m matrices with positive coefficients. Model (10.58) admits a strict and second‐order non‐anticipative stationary solution (ε _t ) when

A: the spectral radius of is strictly less than 1.

Moreover, under this assumption, we have that

10.59

It follows that the last equation in model ( 10.58) can be equivalently written

10.60

With this new formulation, the generic parameter value consists of the coefficients of the vector h and the matrices A _i and B _j (corresponding to the true values h ₀, A _0i and B _0j , respectively), and the coefficients of the lower triangular part (excluding the diagonal) of the correlation matrix R  = (ρ _ij ). One advantage of this parameterisation over the initial one is that the parameter h ₀ can be straightforwardly estimated by the empirical mean

In the VT estimation method, the components of h are thus estimated empirically in a first step, and the other parameters are estimated in a second step, via a QML optimisation. Under appropriate assumptions, the resulting estimator is shown to be consistent and asymptotically normal (see Francq, Horváth, and Zakoian 2016).

One interest of this procedure is computational. Table 10.1 shows the reduction of computational time compared to the full quasi‐maximum likelihood (FQML), when both the dimension m of the series and the sample size n vary. For both methods, the computation time increases rapidly with m , but the relative time‐computation gain does not depend much on m , nor on n .

One issue is whether the time‐computation gain is paid in terms of accuracy of the estimator. In the univariate case, m = 1 (that is, for a standard GARCH( p, q ) model), it can be shown that the VTE is never asymptotically more efficient than the QMLE, regardless of the values of the GARCH parameters and the distribution of the iid process (see Francq, Horváth, and Zakoian 2011). In the multivariate setting, the asymptotic distributions of the two estimators are difficult to compare but, on simulation experiments, the accuracy loss entailed by the two‐step VT procedure is often barely visible. In finite sample, the VT estimator may even perform much better than the QML estimator.

A nice feature of the VT estimator is that it ensures robust estimation of the marginal variance, provided that it exists. Indeed, the variance of a model estimated by VT converges to the theoretical variance, even if the model is mis‐specified. For the convergence to hold true, it suffices that the observed process be stationary and ergodic with a finite second‐order moment. This is generally not the case when the misspecified model is estimated by QML. For some specific purposes such as long‐horizon prediction or long‐term value‐at‐risk evaluation, the essential point is to well estimate the marginal distribution, in particular the marginal moments. The fact that the VTE guarantees a consistent estimation of the marginal variance may then be a crucial advantage over the QMLE.

10.6.2 Equation‐by‐Equation Estimation

Another approach for alleviating the dimensionality curse in the estimation of MGARCH models consists in estimating the individual volatilities in a first step – ‘equation‐by‐equation’ (EbE) – and to estimate the remaining parameters in a second step.

More precisely, any ℝ ^m ‐valued process (ε _t ) with zero‐conditional mean and positive‐definite conditional variance matrix H _t can be represented as

10.61

where is the σ ‐field generated by {ε _u , u < t}, D _t  = {diag( H _t )}^1/2 and Let the k th diagonal element of H _t , that is the variance of ε _kt conditional on . Assuming that is parameterised by some parameter , belonging to a compact parameter space Θ _k , we can write

10.62

where σ _k is a positive function. The vector satisfies and Because and , can be called the vector of EbE innovations of (ε _t ).

It is important to note that model (10.62), satisfied by the components of (ε _t ), is not a standard univariate GARCH. First, because the volatility σ _kt depends on the past of all components of ε _t – not only the past of ε _kt . And second, because the innovation sequence is not iid (except when the conditional correlation matrix is constant, R _t  =  R ). For instance, a parametric specification of σ _kt mimicking the classical GARCH(1,1) is given by

10.63

Of course, other formulations – for instance including leverage effects – could be considered as well.

Having parameterised the individual conditional variances of (ε _t ), the model can be completed by specifying the dynamics of the conditional correlation matrix R _t . For instance, in the DCC model of Engle (2002a), the conditional correlation matrix is modelled as a function of the past standardized returns, as follows

where α, β ≥ 0, α + β < 1, S is a positive‐definite matrix, and is the diagonal matrix with the same diagonal elements as Q _t .

More generally, suppose that matrix R _t is parameterised by some parameter ρ ₀ ∈ ℝ ^r , together with the volatility parameter θ ₀ , as

Given observations ε₁,…, ε _n , and arbitrary initial values for i ≤ 0, we define for k = 1,…, m . A two‐step estimation procedure can be developed as follows.

1. First step: EbE estimation of the volatility parameters , by
   

   and extraction of the vectors of residuals where ;

2. Second step: QML estimation of the conditional correlation matrix ρ ₀ , as a solution of
   10.64

   where Λ ⊂ ℝ ^r is a compact set, , and the 's are initial values.

Asymptotic properties of the procedure can be established for particular models (BEKK, CCC, etc.).

Numerical experiments confirm that the gains in computation time can be huge compared with the FQML estimator in which all the parameters are estimated in one step. Table 10.2 compares the effective computation times required by the two estimators as a function of the dimension m , for the CCC‐GARCH(1, 1) model with A ₁ = 0.05 I _m , B ₁ = 0.9 I _m , R  =  I _m , and . The m(m − 1)/2 sub‐diagonal terms of R were estimated, together with the 3m other parameters of the model. The two estimators were fitted on simulations of length n = 2000. The comparison of the CPUs is clearly in favour of the EbEE, the FQML being in failure for m ≥ 10. When m increases, the computation time of the FQML estimator becomes prohibitive, and more importantly, the optimisation fails to provide a reasonable value for . Table  10.2 also compares the relative efficiencies (RE) of the two approaches. To this aim, we first computed the approximated information matrix . The quadratic form is used as a measure of accuracy of an estimator (the Euclidean distance, obtained by replacing J _n by the identity matrix, has the drawback of being scale dependent). The relative efficiency displayed in Table  10.2 is defined by

where and denote, respectively, the EbE and FQML estimators. The computation time of the FQML estimator being huge when m is large, the RE and CPU times are only computed on 1 simulation, but they are representative of what is generally observed. When m ≤ 9, the accuracies are very similar, with a slight advantage for the FQML (which corresponds here to the ML). When the number of parameters becomes too large, the optimisation fails to give a reasonable value of , and the RE clearly indicates the superiority of the EbEE over the QMLE for m ≥ 10.

10.7.1 Proof of the CAN in Theorem 10.7

We shall use the multiplicative norm (see Exercises 10.5 and 10.6) defined by

10.67

where A is a d ₁ × d ₂ matrix, ∥x∥ is the Euclidean norm of vector , and ρ(·) denotes the spectral radius. This norm satisfies, for any d ₂ × d ₁ matrix B ,

10.68

10.69

When A is a square d × d matrix, the last inequality of (10.68) yields

10.70

We also often use the elementary relation

10.71

when A is a d ₁ × d ₂ matrix and B is d ₂ × d ₁ matrix.

To show the AN, we will use the following elementary results on the differentiation of expressions involving matrices. If f(A) is a real‐valued function of a matrix A whose entries a _ij are functions of some variable x , the chain rule for differentiation of compositions of functions states that

10.72

Moreover, for A invertible we have

10.73

Proof of the Consistency

We shall establish the intermediate results (a), (c), and (d) which are stated as in the univariate case (see the proof of Theorem 7.1 in Section 7.4), the result (b) being satisfied by Assumption A4.

Proof of (a): initial values are forgotten asymptotically. We have

10.79

Using (10.71), (10.69), A1–A2, and omitting the subscript ‘( θ )’, the first sum of (10.79) satisfies

as n → ∞. Indeed is finite a.s. since, for some s < 1,

by A3. If A2 is replaced by A2^′, the previous result follows from the Hölder inequality. Now, consider the second term of the right‐hand side of ( 10.79). The relation (10.70), the Minkowski inequality, the elementary inequality log(x) ≤ x + 1 and the multiplicativity of the spectral matrix norm imply

and, by symmetry,

Using again A1, A2, or A2^′, we thus have shown that

10.80

Proof of (c): the limit criterion is minimised at the true value. As in the univariate case, we first show that Eℓ _t ( θ ) is well defined in ℝ ∪ {+∞} for all θ , and in ℝ for θ  =  θ ₀ . By A1 and ( 10.70) we have and thus

At θ ₀ , Jensen's inequality and A3 entail

Now we show that Eℓ _t ( θ ) is minimised at θ ₀ . Without loss of generality, assume that . Let λ _{1, t} ,…, λ _{m, t} be the eigenvalues of , which are positive (see Exercise 10.15). We have

where the inequality is strict unless if λ _it  = 1 a.s. for all i , that is iff H _t ( θ ) =  H _t ( θ ₀) a.s., which is equivalent to θ  =  θ ₀ under A4.

Proof of (d). The previous results and the ergodic theorem then entail that

Similarly, (10.80) and the ergodic theorem applied to the stationary process (X _t ) with show that

where V _m ( θ ) denotes the ball of centre θ and radius 1/m . By A1 we have . Subtracting the constant K ₀ to ℓ _t ( θ ^*) if necessary, one can always assume that is positive. If E ∣ ℓ _t ( θ ) ∣  < ∞, by Fatou's lemma and A5, for any ε > 0 there exists m sufficiently large such that

If , then the left‐hand side of the previous inequality can be made arbitrarily large. The consistency follows.

Proof of the Asymptotic Normality (AN)

The matrix derivative rules (10.72), (10.77), and (10.75) yield

10.81

We thus have

10.82

Under A10, the matrix K is well defined, and the second moment condition in A9 entails E ∥  C _{i, t} ∥² < ∞. The existence of the matrix I is thus guaranteed by A9 and A10. It is then clear from (10.82) that is a square integrable stationary and ergodic martingale difference. The central limit theorem in Corollary A.1 and the Cramér–Wold device then entail that

10.83

Differentiating (10.81), we also have

with

Under A9, noting that Ec ₁( θ ₀) =  − Ec ₄( θ ₀) and Ec ₃( θ ₀) =  − Ec ₅( θ ₀), we thus have

using elementary properties of the vec and Kronecker operators. By the consistency and A6, we have , and thus almost surely for n large enough. Taylor expansions and A7–A8 thus show that almost surely

where the 's are between and θ ₀ component‐wise. To show that the previous matrix into brackets converges almost surely to J , it suffices to use the ergodic theorem, the continuity of the derivatives, and to show that

for some neighbourhood V( θ ₀) of θ ₀ , which follows from A7 and A9.

If J was singular, there would exist some non‐zero vector λ  ∈ ℝ ^d such that λ ^′ Jλ  = 0. Since is almost surely positive definite, this entails that Δ _t λ  = 0 with probability one, which is excluded by A10. The AN, as well as the Bahadur linearisation (10.49), easily follow from (10.83).

10.7.2 Proof of the CAN in Theorems 10.8 and 10.9

The proof consists in showing that the regularity conditions of Theorem 10.7 are satisfied.

Proof of Theorem 10.8

Note that CC2 implies that the invertibility condition ( 10.46) holds for all θ  ∈ Θ , and that H _t  =  H _t ( θ ₀) is a measurable function of { η _u , u < t}. Corollary 10.2 shows that, under the stationarity condition CC2, the moment conditions A3 are satisfied. The smoothness conditions A5 and A7 are obviously satisfied for the CCC model.

Proof that A1 and A2^′ are satisfied: Rewrite Eq. (10.57) in matrix form as

10.84

where is defined in Corollary 10.1 and

10.85

In view of assumption CC2 and Corollary 10.1, we have By the compactness of Θ , we even have

10.86

Corollary 10.2 and (10.86) thus entail that

10.87

for some s > 0. Using Eq. (10.84) iteratively, as in the univariate case, we deduce that almost surely

where denotes the vector obtained by replacing the variables by in H _t . Observe that K is a random variable that depends on the past values {ε _t , t ≤ 0}. Since K does not depend on n , it can be considered as a constant, like ρ . We deduce that

10.88

where the second inequality is obtained by arguing that the elements of are strictly positive uniformly in Θ . We then have

10.89

with ρ _t  = ρ ^t sup _{θ  ∈ Θ}  ∥  D _t ∥. Therefore, the second moment condition of (10.87) shows that A2^′ holds true, for any p and for s sufficiently small.

Noting that ∥ R ⁻¹∥ is the inverse of the eigenvalue of smallest modulus of R , and that , we have

10.90

using CC5, the compactness of Θ and the strict positivity of the components of . Similarly, we have

10.91

We thus have shown that A1 and A2^′ hold true, meaning that the initial values are asymptotically irrelevant, in the sense of ( 10.80).

Proof of A4: Suppose that, for some θ  ≠  θ ₀ ,

Then it readily follows that ρ = ρ ₀ and, using the invertibility of the polynomial ℬ _θ (B) under assumption CC2, by ( 10.87),

that is,

Let . Noting that and isolating the terms that are functions of η _t − 1 ,

where Z _t − 2 belongs to the σ ‐field generated by {η _t − 2, η _t − 3,…}. Since η _t − 1 is independent of this σ ‐field, Exercise 10.3 shows that the latter equality contradicts CC3 unless when p _ij h _{jj, t}  = 0 almost surely, where the p _ij are the entries of for i, j = 1,…, m . Because h _{jj, t}  > 0 for all j , we thus have . Similarly, we show that by successively considering the past values of η _t − 1 . Therefore, in view of CC4 (or CC4 ^′ ), we have α = α ₀ and β = β ₀ by arguments already used. It readily follows that . Hence θ  =  θ ₀ . We have thus established the identifiability condition A4, and the proof of Theorem 10.8 follows from Theorem 10.7.□

Proof of Theorem 10.9

It remains to show that A8, A9 and A11 hold.

Proof of A9: Denoting by the i ₁ th component of ,

where c ₀ is a strictly positive constant and, by the usual convention, the index 0 corresponds to quantities evaluated at θ  =  θ ₀ . Under CC6, for a sufficiently small neighbourhood of θ ₀ , we have

for all i ₁, j ₁, j ₂ ∈ {1,…, m} and all δ > 0. Moreover, in , the coefficient of is bounded below by a constant c > 0 uniformly on . We thus have

for some ρ ∈ [0, 1), all δ > 0 and all s ∈ [0, 1]. Corollary 10.2 then implies that, for all r ₀ ≥ 0, there exists such that

Since

where R ₀ denotes the true value of R , the last moment condition of A9 is satisfied (for all r ).

By ( 10.84)–( 10.86),

and, setting s ₂ = m + qm ² ,

Setting s ₁ = m + (p + q)m ² , we have

where is a matrix whose entries are all 0, apart from a 1 located at the same place as θ _i in . By abuse of notation, we denote by H _t (i ₁) and the i ₁ th components of H _t and . Note that, by the arguments used to show ( 10.87), the previous expressions show that

10.92

for some s > 0. With arguments similar to those used in the univariate case, that is, the inequality x/(1 + x) ≤ x ^s for all x ≥ 0 and s ∈ [0, 1], and the inequalities

and, setting ,

we obtain

where the constants (which also depend on i ₁ , s and r ₀ ) belong to the interval [0, 1). Noting that these inequalities are uniform on a neighbourhood of , that they can be extended to higher‐order derivatives, and that Corollary 10.2 implies that , we can show, as in the univariate case, that for all i ₁ = 1,…, m , all i, j, k = 1,…, s ₁ and all r ₀ ≥ 0, there exists a neighbourhood of θ ₀ such that

10.93

and

10.94

Omitting ‘( θ )’ to lighten the notation, note that

for i = 1,…, s ₁ , and

for i = s ₁ + 1,…, s ₀ . It follows that (10.93) entails the second moment condition of A9 (for any q ). Similarly, (10.94) entails the first moment condition of A9 (for any p ).

Proof of A8: First note that by

and (10.88)–(10.91), we have

10.95

From Eq. ( 10.84), we have

where r = max {p, q} and the tilde means that initial values are taken into account. Since for all t > r , we have and

Thus condition ( 10.86) entails that

10.96

Because

the second inequality of ( 10.88) and (10.96) imply that

10.97

Note that ( 10.81) continues to hold when ℓ _t ( θ ) and H _t ( θ ) are replaced by and . Therefore,

where

Using conditions ( 10.87), (10.90), ( 10.91), (10.92), (10.95), and (10.97), it can be shown that Tr( C ₁ +  C ₂) ≤ Kρ ^t u _t , where u _t is a random variable such that sup _t E|u _t | ^s  < ∞ for some small s ∈ (0, 1). Arguing that is almost surely finite because its moment of order s is finite, the conclusion follows.

Proof of A11: Recall that we applied Theorem 10.7 with d = s ₀ . We thus have to show that it cannot exist as a non‐zero vector , such that with probability 1. Decompose c into with and , where s ₃ = s ₀ − s ₁ = m(m − 1)/2. Rows 1, m + 1,…, m ² of the equations

10.98

give

10.99

Differentiating Eq. ( 10.57) yields

where

Because (10.99) is satisfied for all t , we have

where quantities evaluated at θ  =  θ ₀ are indexed by 0. This entails that

and finally, introducing a vector θ ₁ whose s ₁ first components are

we have

by choosing c ₁ small enough so that θ ₁ ∈ Θ . If c ₁  ≠ 0 then θ ₁ ≠  θ ₀ . This is in contradiction to the identifiability of the parameter (see the proof of A4), hence c ₁  = 0. Equations (10.98) thus become

Therefore,

Because the vectors, ∂vec  R /∂θ _i , i = s ₁ + 1,…, s ₀ , are linearly independent, the vector is null, and thus c = 0, which completes the proof.□

10.9 Exercises

1. 10.2 (Identifiability of a matrix rational fraction) Let 풜 _θ (z), ℬ _θ (z), , and denote square matrices of polynomials. Show that
   10.100

   for all z such that if and only if there exists an operator U(z) such that

   10.101
2. 10.3 (Two independent non‐degenerate random variables cannot be equal) Let X and Y be two independent real random variables such that Y = X almost surely. We aim to prove that X and Y are almost surely constant.
   1. Suppose that Var(X) exists. Compute Var(X) and show the stated result in this case.
   2. Suppose that X is discrete and P(X = x ₁)P(X = x ₂) ≠ 0. Show that necessarily x ₁ = x ₂ and show the result in this case.
   3. Prove the result in the general case.
3. 10.4 (Duplication and elimination) Consider the duplication matrix D _m and the elimination matrix defined by
   

   where A is any symmetric m × m matrix. Show that

   
4. 10.8 (Norm and spectral radius) Show that
   
5. 10.6 (Elementary results on matrix norms) Show the equalities and inequalities of ( 10.68) and ( 10.69).
6. 10.7 (Scalar GARCH) The scalar GARCH model has a volatility of the form
   

   where the α _i and β _j are positive numbers. Give the positivity and second‐order stationarity conditions.

7. 10.8 (Condition for the L ^p and almost sure convergence) Let p ∈ [1, ∞ ) and let (u _n ) be a sequence of real random variables of L ^p such that
   

   for some positive constant C , and some constant ρ in (0, 1). Prove that

   

   to some random variable u of L ^p .

8. 10.9 (An average of correlation matrices is a correlation matrix) Let R and Q be two correlation matrices of the same size and let p ∈ [0, 1]. Show that pR + (1 − p)Q is a correlation matrix.
9. 10.10 (Factors as linear combinations of individual returns) Consider the factor model
   

   where the β _j are linearly independent. Show there exist vectors α _j such that

   

   where the are conditional variances of the portfolios . Compute the conditional covariance between these factors.

10. 10.11 (BEKK representation of factor models) Consider the factor model
    

    where the β _j are linearly independent, ω _j  > 0, a _j  ≥ 0, and 0 ≤ b _j  < 1 for j = 1,…, r . Show that a BEKK representation holds, of the form

    
11. 10.12 (PCA of a covariance matrix) Let X be a random vector of ℝ ^m with variance matrix Σ.
    1. Find the (or a) first principal component of X , that is a random variable of maximal variance, where . Is C ¹ unique?
    2. Find the second principal component, that is, a random variable of maximal variance, where and Cov(C ¹, C ²) = 0.
    3. Find all the principal components.
12. 10.13 (BEKK‐GARCH models with a diagonal representation) Show that the matrices A ⁽ⁱ⁾ and B ^(j) defined in ( 10.21) are diagonal when the matrices A _ik and B _jk are diagonal.
13. 10.14 (Determinant of a block companion matrix) If A and D are square matrices, with D invertible, we have
    

    Use this property to show that matrix in Corollary 10.1 satisfies

    
14. 10.15 (Eigenvalues of a product of positive definite matrices) Let A and B denote symmetric positive definite matrices of the same size. Show that AB is diagonalizable and that its eigenvalues are positive.
15. 10.16 (Positive definiteness of a sum of positive semi‐definite matrices) Consider two matrices of the same size, symmetric, and positive semi‐definite, of the form
    

    where A ₁₁ and B ₁₁ are also square matrices of the same size. Show that if A ₂₂ and B ₁₁ are positive definite, then so is A + B .

16. 10.17 (Positive definite matrix and almost surely positive definite matrix) Let A by a symmetric random matrix such that for all real vectors c ≠ 0,
    

    Show that this does not entail that A is almost surely positive definite.