## 1. Introduction

Hedging financial investments is tantamount to insuring against possible losses arising from risky portfolio allocation. In order to hedge efficiently, persistently high negative covariances or equivalently, correlations, between risky assets and the hedging instruments are intended to mitigate against financial risk and subsequent losses.

It is possible to hedge against risky assets using one or more hedging instruments as the benchmark, which requires the calculation of multivariate covariances and correlations. As optimal hedge ratios are unlikely to remain constant using high frequency data, it is essential to specify dynamic time-varying models of covariances and correlations.

Modeling, forecasting, and evaluating dynamic covariances between hedging instruments and risky financial assets requires the specification and estimation of multivariate models of covariances and correlations. These values can either be determined analytically or numerically on the basis of highly advanced computer simulations. High frequency time periods, such as daily data, can lead to either conditional or stochastic volatility, where analytical developments are occasionally promulgated for the former, but always numerically for the latter.

The purpose of this paper is to analyze purported analytical developments for the only multivariate dynamic conditional correlation model to have been developed to date, namely

Engle’s (

2002) widely used Dynamic Conditional Correlation (DCC) model. As dynamic models are not straightforward (or even possible) to translate in terms of the algebraic existence, underlying stochastic processes, specification, mathematical regularity conditions, and asymptotic properties of consistency and asymptotic normality, or the lack thereof, these are evaluated separately.

For the variety of detailed possible outcomes mentioned above, where problematic issues arise constantly and sometimes unexpectedly, a companion paper by the author evaluates the recent developments in modeling dynamic conditional covariances on the basis of the Full BEKK (named for Baba, Engle, Kraft and Kroner) model (see

McAleer 2019). Both papers are intended as Topical Collections to bring the known and unknown results pertaining to DCC into a single collection.

The remainder of the paper presented is as follows. The DCC model is presented in

Section 2, which will enable a subsequent critical analysis and emphasis on a discussion, evaluation, and presentation of caveats in

Section 3 of the numerous dos and don’ts in implementing the DCC model, as well as a related model, in practice.

## 2. Model Specification

Some, though not all, of the results in this section 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 sections, are not available. Much of the basic material relating to the univariate and multivariate specifications in

Section 2.1,

Section 2.2 and

Section 2.3 overlap with the presentation in

McAleer (

2019).

Two earlier papers that questioned the need, usefulness, and practical differences between DCC and the associated multivariate Full BEKK conditional covariance model (see

Baba et al. (

1985) and

Engle and Kroner (

1995) for further details) are given in

Caporin and McAleer (

2012,

2013). Among other queries, the latter authors suggested referring to DCC as a “representation” for purposes of filtering the data, rather than as a model. However, the technical issue and interpretation as to what data are actually being filtered, namely the returns shocks or standardized residuals, and conditional correlations or covariances, was not settled.

The first step in estimating DCC is to estimate the standardized shocks from the univariate conditional mean returns shocks. The most widely used univariate conditional volatility model, namely GARCH (acronym for **G**eneralized **A**uto**R**egressive **C**onditional **H**eteroscedasticity), is presented briefly, followed by DCC.

Consider the conditional mean of financial returns, as follows:

where the returns,

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

$\left({P}_{t}\right),\text{}{I}_{t-1}$, constitute 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,

${\epsilon}_{t}$.

#### 2.1. Univariate Conditional Volatility Models

Univariate conditional volatilities can be used to standardize the conditional covariances in alternative multivariate conditional volatility models to estimate conditional correlations, which are particularly useful in developing dynamic hedging strategies. The most widely used univariate model, GARCH, is presented below as an illustration because the focus of the paper is on estimating and testing DCC.

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

Consider the random coefficient autoregressive process of order one:

where

and

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

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

Tsay (

1987) derived the ARCH(1) (acronym for

**A**uto

**R**egressive

**C**onditional

**H**eteroscedasticity) 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 known 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 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,

$\text{}{\eta}_{t}$, 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,${\Gamma}_{t}$.

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 ${Q}_{t}$. As both ${Q}_{t}$ and ${\Gamma}_{t}$ are products of matrices, and the inverse of the matrix ${D}_{t}$ is not asymptotically normal, even when ${D}_{t}$ is asymptotically normal, neither the Quasi-Maximum Likelihood Estimates (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).

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,

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

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

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.

#### 2.4. DCC Model

This section presents the DCC model, as given in

Engle (

2002), which does not have an underlying stochastic specification that leads to its derivation. Without distinguishing between dynamic conditional covariances and dynamic conditional correlations,

Engle (

2002) presented the DCC specification as:

where

$\overline{Q}$ is assumed to be a positive definite with unit elements along the main diagonal; the scalar parameters are assumed to satisfy the stability condition,

$\alpha +\beta $ < 1, where the two parameters do not have the same interpretation as in the univariate GARCH model; the standardized shocks,

${\eta}_{t}=({\eta}_{1t,\text{\hspace{0.05em}}\dots ,}{\eta}_{mt}){}^{\prime}$, which are not necessarily

iid, are given as

${\eta}_{it}={\epsilon}_{it}/\sqrt{{h}_{it}}$; and

${D}_{t}$ is a diagonal matrix with typical element

$\sqrt{{h}_{it}}$,

i = 1, …,

m. As

m is the number of financial assets, the multivariate definition of the relationship between

${\epsilon}_{t}$ and

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

${\epsilon}_{t}={D}_{t}{\eta}_{t}$.

In view of Equations (5) and (6), as the matrix in Equation (8) does not satisfy the definition of a correlation matrix,

Engle (

2002) uses the following standardization:

There is no clear explanation given for the standardization in Equation (9) or, more recently, in

Aielli (

2013), especially as it does not satisfy the definition of a correlation matrix, as given in Equation (6). The standardization in Equation (9) might make sense if the matrix

${Q}_{t}$ was the conditional covariance matrix of

${\epsilon}_{t}$ or

${\eta}_{t}$, although this is not made clear. Indeed, in the literature relating to DCC, it is not clear whether Equation (8) refers to a conditional covariance or a conditional correlation matrix, although the latter is simply assumed without any clear explanation.

Despite the title of the paper,

Aielli (

2013) also does not provide any stationarity conditions for the DCC model, and does not mention the mathematical regularity condition of invertibility at the univariate of multivariate level. On the basis of Equation (8)—which does not relate the conditional covariance matrix,

${Q}_{t},$ in Equation (8) to the returns shocks,

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

${y}_{t}$ in Equation (1)—invertibility does not hold. As there is no connection between

${Q}_{t}$ and

${y}_{t}$, there is not a valid likelihood function of the parameters given the data.

It follows that there can be no first or second derivatives of the non-existent likelihood function, so the Jacobian and Hessian matrices do not exist. Therefore, there cannot be an analytical derivation of any asymptotic effects relating to consistency and asymptotic normality.

## 3. Vector Random Coefficient Moving Average Process

The random coefficient moving average process is presented in its original univariate form in

Section 3.1, as in

Marek (

2005), with an extension to its multivariate counterpart in

Section 3.2, in order to derive the univariate and multivariate conditional volatility models, respectively, as well as the associated invertibility conditions at the univariate and multivariate levels. Some practical applications of the DCC model are also discussed.

#### 3.1. Univariate Process

Marek (

2005) proposed a linear moving average model with random coefficients (RCMA). It should be emphasized that the random coefficient moving average process is a sufficient condition to derive a novel univariate conditional volatility model.

In this section, we extend the univariate results of

Marek (

2005) using an

m-dimensional vector random coefficient moving average process of order

p, which is used as an underlying stochastic process to derive the DCC model. A novel conditional volatility model is derived, which is given as a function of the standardized shocks rather than of the returns shocks, as in the univariate ARCH model in Equation (3).

Consider a univariate random coefficient moving average process given by:

where

${\eta}_{t}$$~$ iid $(0,\omega )$. The sequence

$\left\{{\theta}_{t}\right\}$ is assumed to be independent of

${\eta}_{t-1},{\eta}_{t},{\eta}_{t+1},\dots $, which is called the Future Independence Condition, with mean zero and variance

$\alpha $. It is also assumed to be measurable with respect to

${I}_{t}$, where

${I}_{t}$ is the information set generated by the random variable, {

${\epsilon}_{t,}{\epsilon}_{t-1,\dots}$}. Furthermore, it is assumed that the process {

${\epsilon}_{t}$} is stationary and invertible, such that

${\eta}_{t}\in {I}_{t}$.

Without the measurability assumption on

$\left\{{\theta}_{t}\right\}$ it would be difficult to obtain results on invertibility. As demonstrated in

McAleer (

2018), an important special case of the model arises when

$\left\{{\theta}_{t}\right\}$ is

iid, that is, not measurable with respect to

${I}_{t}$, in which case the conditional and unconditional expectations of

${\epsilon}_{t}$ are zero, and the conditional variance of

${\epsilon}_{t}$ is given by:

which differs from the ARCH model in Equation (3) in that the returns shock is replaced by the standardized shock. This is a new conditional volatility model, especially as conditional volatility is expressed in terms of a riskless random variable rather than the returns shock, which has the same measurement, and hence risk, as the returns,

${y}_{t}$.

McAleer (

2018) showed that, as

${\eta}_{t}$ ~

iid $(0,\omega )$, the unconditional variance of

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

The use of an infinite lag length for the random coefficient moving average process in Equation (10), with appropriate restrictions on

${\theta}_{t}$, would lead to a generalized ARCH model that differs from the GARCH model of

Bollerslev (

1986) as the returns shock would be replaced with a standardized shock.

A sufficient condition for stationarity is that the vector sequence

${\upsilon}_{t}=({\eta}_{t},{\theta}_{t}{\eta}_{t-1}){}^{\prime}$ is stationary. Moreover, according to Lemma 2.1 of

Marek (

2005), a new sufficient condition for invertibility is that:

The stationarity of ${\upsilon}_{t}=({\eta}_{t},{\theta}_{t}{\eta}_{t-1}){}^{\prime}$ and the invertibility condition in Equation (12) are new results for the univariate ARCH(1) model given in Equation (11), as well as its direct extension to GARCH models.

#### 3.2. Multivariate Process

Extending the analysis to a vector random coefficient moving average (RCMA) model of order

p,

McAleer (

2018) derives a special case of DCC(

p,q), namely DCC(

p,0), as follows:

where

ε_{t} and

η_{t} are both

m × 1 vectors, and

${\theta}_{jt}$,

j = 1,

…, p are random

$m\times m$ matrices, independent of

${\eta}_{t-1},{\eta}_{t},{\eta}_{t+1},\dots $.

As the dimension of the unconditional variance of ${\theta}_{jt}$ is m, if the conditional variance matrix is not restricted parametrically, the dynamic conditional covariance matrix of (13) would depend on the product of the variance of ${\theta}_{jt}$, with dimensions between $m\left(m-1\right)/2$ and ${m}^{2}$, neither of which would be conformable with the dimension of ${\eta}_{t-j}$. Under specific assumptions, it is possible to derive the conditional covariance matrix of ${\epsilon}_{t}$ in Equation (11).

If

${\theta}_{jt}$ in Equation (13) is given as:

where

${\lambda}_{jt}$ is a scalar random variable, the dynamic conditional covariance matrix is given as:

The DCC model in Equation (8) is obtained by letting $p\to \infty $ in Equations (13) and (14), setting ${\alpha}_{j}=\alpha \text{\hspace{0.17em}}{\beta}^{j-1}$, and standardizing ${Q}_{t}$ in Equation (14) to obtain a conditional correlation matrix. For the case p = 1 in Equation (14), the appropriate univariate conditional volatility model is given in the new model in Equation (11), which uses the standardized shocks, in contrast to the standard ARCH in Equation (3), which uses the returns shocks. Moreover, the DCC model adds only one parameter, ${\alpha}_{1}$, when p = 1, so it is parsimonious in terms of its parametric representation.

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

The derivation in

McAleer (

2018) of DCC in Equation (14) from a vector random coefficient moving average process is important as it: (i) demonstrates that DCC is, in fact, a dynamic conditional covariance model of the returns shocks rather than a purported dynamic conditional correlation model; (ii) provides the motivation, which is presently missing, for the standardization of the conditional covariance model in Equation (9) to obtain the conditional correlation model; and (iii) shows that the appropriate ARCH or GARCH model as a first step in calculating DCC is based on the standardized shocks rather than on the returns shocks.