What They Did Not Tell You About Algebraic (Non-)Existence, Mathematical (IR-)Regularity and (Non-)Asymptotic Properties of the Dynamic Conditional Correlation (DCC) Model

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. If there is more than one hedging instrument, multivariate covariances and correlations will have to be calculated. 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. These values can either be determined analytically or numerically on the basis of highly advanced computer simulations. Analytical developments are occasionally promulgated for multivariate conditional volatility models. The primary purpose of the paper is to analyse 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. 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. The paper presents a critical analysis, discussion, evaluation and presentation of caveats relating to the DCC model, and an emphasis on the numerous dos and don’ts in implementing the DCC and related model in practice. Abstract 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. If there is more than one hedging instrument, multivariate covariances and correlations will have to be calculated. 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. These values can either be determined analytically or numerically on the basis of highly advanced computer simulations. Analytical developments are occasionally promulgated for multivariate conditional volatility models. The primary purpose of the paper is to analyse 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. 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. The paper presents a critical analysis, discussion, evaluation and presentation of caveats relating to the DCC model, and an emphasis on the numerous dos and don’ts in implementing the DCC and related model in practice.


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.
Modelling, forecasting and evaluating dynamic covariances between hedging instrument 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 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 the paper is to analyse 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, they will be 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 modelling dynamic conditional covariances on the basis of the Full BEKK model (see McAleer (2019)).

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The remainder of the paper 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 in practice.

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 section, are not available. Much of the basic material relating to the univariate and multivariate specifications in Sections 2.1 -2.3 overlap with the presentation in McAleer (2019).
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, will be presented briefly, followed by DCC. Consider the conditional mean of financial returns, as follows: where the returns, = Δ log , represents the log-difference in financial asset prices ( ), −1 is the information set at time t-1, and 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, .

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-5 used univariate model, GARCH, is presented below as an illustration because the focus of the paper is on estimating and testing DCC.
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 is conditional volatility, and −1 is the information set available at time t-1. The mathematical regularity condition of invertibility is used to relate the conditional variance, ℎ , in equation (3) to the returns shocks, , which has the same measurement as 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 and should be positive as they are the unconditional variances of two independent stochastic processes. The GARCH model is given as: where is the short run ARCH effect, and , which lies in the range (-1,1), is the GARCH contribution to the long run persistence of returns shocks.

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, = /�ℎ . The multivariate extension of equation (1), namely: can remain unchanged by assuming that the three components in the above equation are now × 1 vectors, where is the number of financial assets.
The multivariate definition of the relationship between and is given as: is a diagonal matrix comprising the univariate conditional volatilities. Define the conditional covariance matrix of as . As the × 1 vector, , is assumed to be iid for all elements, the conditional correlation matrix of , which is equivalent to the conditional correlation matrix of , is given by .
Therefore, the conditional expectation of the process in equation (4) is defined as: Equivalently, the conditional correlation matrix, , can be defined as: Equation (5) is useful if a model of is available for purposes of estimating the conditional covariance matrix, , whereas equation (6) is useful if a model of is available for purposes of estimating the conditional correlation matrix, .
Both equations (5) and (6) are instructive for a discussion of asymptotic properties. As the elements of are consistent and asymptotically normal, the consistency of in equation (5) depends on consistent estimation of , whereas the consistency of in equation (6) depends on consistent estimation of . As both and 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 nor 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 and are × 1 vectors, is an × matrix of random coefficients, Technically, a vectorization of a full (that is, non-diagonal) matrix A to vec A can have dimension as high as 2 × 2 , whereas a half-vectorization of a symmetric matrix A to vech A can have dimension as low as ( − 1)/2 × ( − 1)/2. The matrix A is crucial in the interpretation of symmetric and asymmetric weights attached to the returns shocks. 8

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 Q is assumed to be positive definite with unit elements along the main diagonal, the scalar parameters are assumed to satisfy the stability condition, 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) (8), which does not relate the conditional covariance matrix, , in equation (8) to the returns shocks, , which has the same measurement as in equation (1), invertibility does not hold. As there is no connection between and , 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.

Vector Random Coefficient Moving Average Process
The random coefficient moving average process will be 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.

Univariate process
Marek (2005) proposed a linear moving average model with random coefficients (RCMA)..
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 in 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:  (2018), an important special case of the model arises when { } t θ is iid, that is, not measurable with respect to t I , in which case the conditional and unconditional expectations of t ε are zero, and the conditional variance of t ε 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 univariate 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, McAleer (2018) shows that, as t η ~ iid ) , 0 ( ω , the unconditional variance of t ε is given as: The use of an infinite lag length for the random coefficient moving average process in equation (10), with appropriate restrictions on t θ , 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. .

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A sufficient condition for stationarity is that the vector sequence )' , ( The stationarity of 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.

Multivariate process
Extending the analysis to a vector random coefficient moving average (RCMA) model of λ is a scalar random variable, the dynamic conditional covariance matrix is given as: The DCC model in equation (8) is obtained by letting ∞ → p in equations (13) and (14), , and standardizing 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, rather than the standard ARCH model in equation (3), which uses the returns shocks. Moreover, the DCC model adds only one parameter, 1 α , when p = 1, so it is parsimonious in terms of its parametric representation.
The derivation in McAleer (2018) of DCC in equation (14) from a vector random coefficient moving average process is novel and 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 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.

Discussion and Caveats of Dos and Don'ts Regarding DCC
The results in the previous section allow a clear discussion of the caveats associated with (1) Engle (1982) developed an autoregressive model of conditional correlations, ARCH, based on the conditional returns shocks.
(3) The GARCH(1,1) parameters must satisfy the regularity conditions of positivity as they are the unconditional variances from a univariate random coefficient autoregressive process (see Tsay, 1987;McAleer, 2014).
(4) However, the coefficient of the arbitrary lagged conditional variance is a positive or negative fraction (see Bollerslev (1986)).
(5) The results in Tsay (1987)  (7) There is no known underlying stochastic process that leads to the DCC model as a dynamic correlation matrix, so that there are no regularity conditions relating to its specification.
(8) The regularity conditions include invertibility, which is essential in relating the iid standardized residuals to the returns data.
(9) As invertibility does not hold, it follows that there is no likelihood function, and hence no derivatives that would enable the derivation of asymptotic properties for the

Quasi-Maximum Likelihood Estimates (QMLE) of the estimated parameters.
(10) Therefore, any statements regarding the purported "statistical significance" are meaningless and lack statistical validity.
(11) It follows that any empirical results based on the DCC estimates are fatally flawed and lack statistical validity (see McAleer (2018) for a critical analysis).
14 (12) Marek (2005)  (16) As stated in the published versions of both papers, the paper by Tse and Tsui (2002) was submitted to the journal in December 1998, 25 months earlier than the submission of the paper by Engle (2002), and was accepted in May 2001, 7 months earlier than the submission of Engle (2002).
(17) However, as in the case of DCC, the VCC model has no underlying stochastic process that leads to its specification as a dynamic correlation matrix.
(18) Therefore, there are no mathematical regularity conditions relating to VCC, including invertibility, which is essential in relating the iid standardized residuals to the returns data.
Consequently, the QMLE of the estimated parameters of VCC do not possess any asymptotic properties as the likelihood function does not exist.

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(20) It is somewhat surprising that the DCC model has been widely estimated using real data, without any apparent computational difficulties, despite the fact that the model does not actually exist! (21) Such computational outcomes would almost certainly arise from the parsimonious addition of only one parameter when p = 1, even when the value of m is high for the large financial portfolios that are observed in practice.
(22) In short, VCC shares all the existence, specification, mathematical and statistical deficiencies of DCC.
(23) If these two models are to be considered at all, except in connection with the algebraic non-existence, absence of an underlying stochastic process, mathematical irregularity, and unknown asymptotic statistical properties, or alternatively, in the presence of problems that should be avoided at all costs, it is advisable that DCC and VCC be used with extreme and utter caution in empirical practice.