Some of the results in this section, though not all, are available in the extant literature, but the interpretation of the models and their non-existent underlying stochastic processes, as well as the discussions and caveats in the following section, are not available. Much of the basic material relating to the univariate and multivariate specifications in

Section 2.1 and

Section 2.3 overlap with the presentation in

McAleer (

2019).

The first step in estimating Full BEKK is to estimate the standardized shocks from the univariate conditional mean returns shocks. The most widely used univariate conditional volatility model, namely the Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model, will be presented briefly, followed by Full BEKK. Consider the conditional mean of financial returns, as follows:

where the returns,

${y}_{t}=\mathsf{\Delta}\mathrm{log}{P}_{t}$, represents the log-difference in financial asset prices

$\left({P}_{t}\right),\text{}{I}_{t-1}$ is the information set at time

t − 1, and

${\epsilon}_{t}$ is a conditionally heteroskedastic returns shock that has the same unit of measurement as the returns. In order to derive conditional volatility specifications, it is necessary to specify, wherever possible, the stochastic processes underlying the returns shocks,

${\epsilon}_{t}$.

#### 2.2. Random Coefficient Autoregressive Process and GARCH

Consider the random coefficient autoregressive process of order one:

where

${\varphi}_{t}~iid\left(0,\alpha \right)$,

${\eta}_{t}~iid\left(0,\omega \right)$,

and ${\eta}_{t}={\epsilon}_{t}/\sqrt{{h}_{t}}$ is the standardized residual.

The standardized residual is unit-free of measurement, and is a financial fundamental as it represents a riskless asset.

Tsay (

1987) derived the ARCH(1) model of

Engle (

1982) from Equation (1) as:

where

${h}_{t}$ is conditional volatility, and

${I}_{t-1}$ is the information set available at time

t − 1. The mathematical regularity condition of invertibility is used to relate the conditional variance,

${h}_{t},$ in Equation (3) to the returns shocks,

${\epsilon}_{t}$, which has the same measurement as

${y}_{t}$ in Equation (1), thereby yielding a valid likelihood function of the parameters given the data.

The use of an infinite lag length for the random coefficient autoregressive process in Equation (2), with appropriate geometric restrictions (or stability conditions) on the random coefficients, leads to the GARCH model of

Bollerslev (

1986). From the specification of Equation (2), it is clear that both

$\omega $ and

$\alpha $ should be positive as they are the unconditional variances of two independent stochastic processes. The GARCH model is given as:

where

$\alpha $ is the short run ARCH effect, and

$\beta $, which lies in the range (−1, 1), is the GARCH contribution to the long run persistence of returns shocks.

From the specification of Equation (2), it is clear that both $\omega $ and $\alpha $ should be positive as they are the unconditional variances of two independent stochastic processes. It should be emphasized that the random coefficient autoregressive process is a sufficient condition to derive ARCH, but to date the ARCH specification has not been derived from any other underlying stochastic process.

#### 2.3. Multivariate Conditional Volatility Models

Multivariate conditional volatility GARCH models are often used to analyze the interaction between the second moments of returns shocks to a portfolio of assets, and can model and the possible risk transmission or spillovers among different assets.

In order to establish volatility spillovers in a multivariate framework, it is useful to define the multivariate extension of the relationship between the returns shocks and the standardized residuals, that is,

${\eta}_{t}={\epsilon}_{t}/\sqrt{{h}_{t}}$. The multivariate extension of Equation (1), namely:

can remain unchanged by assuming that the three components in the above equation are now

$m\times 1$ vectors, where

$m$ is the number of financial assets.

The multivariate definition of the relationship between

${\epsilon}_{t}$ and

${\eta}_{t}$ is given as:

where

${D}_{t}=diag\left({h}_{1t},{h}_{2t},\dots ,{h}_{mt}\right)$ is a diagonal matrix comprising the univariate conditional volatilities. Define the conditional covariance matrix of

${\epsilon}_{t}$ as

${Q}_{t}$. As the

$m\times 1$ vector,

${\eta}_{t}$, is assumed to be

iid for all

$m$ elements, the conditional correlation matrix of

${\epsilon}_{t}$, which is equivalent to the conditional correlation matrix of

${\eta}_{t}$, is given by

${\Gamma}_{t}$.

Therefore, the conditional expectation of the process in Equation (4) is defined as:

Equivalently, the conditional correlation matrix,

${\Gamma}_{t}$, can be defined as:

Equation (5) is useful if a model of ${\Gamma}_{t}$ is available for purposes of estimating the conditional covariance matrix, ${Q}_{t}$, whereas Equation (6) is useful if a model of ${Q}_{t}$ is available for purposes of estimating the conditional correlation matrix, ${\Gamma}_{\mathrm{t}}$.

Both Equations (5) and (6) are instructive for a discussion of asymptotic properties. As the elements of ${D}_{t}$ are consistent and asymptotically normal, the consistency of ${Q}_{t}$ in Equation (5) depends on consistent estimation of ${\Gamma}_{t}$, whereas the consistency of ${\Gamma}_{t}$ in Equation (6) depends on consistent estimation of ${H}_{t}$. As both ${Q}_{t}$ and ${\Gamma}_{t}$ are products of matrices, and the inverse of the matrix D is not asymptotically normal, even when D is asymptotically normal, neither the QMLE of ${Q}_{t}$ nor ${\Gamma}_{t}$ will be asymptotically normal, especially based on the definitions that relate the conditional covariances and conditional correlations given in Equations (5) and (6).

#### 2.4. Full BEKK and Diagonal BEKK Models

The vector random coefficient autoregressive process of order one is the multivariate extension of Equation (2), and is given as:

where

${\epsilon}_{t}$ and ${\eta}_{t}$ are $m\times 1$ vectors,

${\Phi}_{t}$ is an $m\times m$ matrix of random coefficients,

${\Phi}_{t}~iid\left(0,A\right)$,

${\eta}_{t}~iid\left(0,Q{Q}^{\prime}\right)$

Technically, a vectorization of a full (that is, non-diagonal) matrix A to vec A can have dimensions as high as ${m}^{2}\times {m}^{2}$, whereas a half-vectorization of a symmetric matrix A to vech A can have dimensions as low as $m\left(m-1\right)/2\times m\left(m-1\right)/2$. The matrix A is crucial in the interpretation of symmetric and asymmetric weights attached to the returns shocks.

As the dimension of the unconditional variance of ${\epsilon}_{t}$ in Equation (7) is m, if the variance matrix is not restricted parametrically, the dynamic conditional covariance matrix of (7) would depend on the product of the variance of ${\Phi}_{t}$, with a dimension that lies between $m\left(m-1\right)/2$ and ${m}^{2}$, neither of which would be conformable with the dimension of ${\epsilon}_{t-1}$.

Where A is either a diagonal matrix, or the special case of a scalar matrix,

$A=a{I}_{m}$,

McAleer et al. (

2008) showed that the multivariate extension of GARCH(1,1) from Equation (7), incorporating an infinite geometric lag in terms of the returns shocks, is given as the diagonal (or scalar) BEKK model, namely:

where

A and

B are diagonal (or scalar) matrices.

As in the univariate case, it should be emphasized that the vector random coefficient autoregressive process is a sufficient condition to derive diagonal BEKK, but to date the diagonal BEKK specification has not been derived from any other underlying multivariate stochastic process.

Although the Full BEKK model is always presented in the form of Equation (8), with A and B given as full matrices, as stated above, the specification is not consistent with Equation (8) as the matrices A and B for Full BEKK would have dimensions that lie between $m\left(m-1\right)/2$ and ${m}^{2}$, which would not be conformable for multiplication with the dimension of the vector ${\epsilon}_{t-1}$.

McAleer et al. (

2008) showed that the QMLE of the parameters of the Diagonal BEKK models are consistent and asymptotically normal, so that standard statistical inference on testing hypotheses is valid. The theoretical results can also be obtained from

Nicholls and Quinn (

1980,

1981,

1982). Moreover, as

${Q}_{t}$ in Equation (8) can be estimated consistently,

${\Gamma}_{t}$ in Equation (6) can also be estimated consistently. However, as explained above, asymptotic normality cannot be proved given the definitions in Equations (5) and (6).

However, other special cases of Full BEKK, such as Triangular BEKK and Hadamard BEKK, cannot be obtained from a vector random coefficient autoregressive process, so that any purported asymptotic properties of the QMLE do not exist.

It should be emphasized that the QMLE of the parameters in the conditional means and the conditional variances for univariate GARCH, Diagonal BEKK and Full BEKK will differ as the multivariate models are estimated jointly, whereas the univariate models are estimated individually. The QMLE of the parameters of the conditional means and the conditional variances of Diagonal BEKK and Full BEKK will differ as Diagonal BEKK imposes parametric restrictions on the off-diagonal terms of the conditional covariance matrix of the Full BEKK model.