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The purpose of the paper is to discuss ten things potential users should know about the limits of the Dynamic Conditional Correlation (DCC) representation for estimating and forecasting time-varying conditional correlations. The reasons given for caution about the use of DCC include the following: DCC represents the dynamic conditional covariances of the standardized residuals, and hence does not yield dynamic conditional correlations; DCC is stated rather than derived; DCC has no moments; DCC does not have testable regularity conditions; DCC yields inconsistent two step estimators; DCC has no asymptotic properties; DCC is not a special case of Generalized Autoregressive Conditional Correlation (GARCC), which has testable regularity conditions and standard asymptotic properties; DCC is not dynamic empirically as the effect of news is typically extremely small; DCC cannot be distinguished empirically from diagonal Baba, Engle, Kraft and Kroner (BEKK) in small systems; and DCC may be a useful filter or a diagnostic check, but it is not a model.

The 21st century has seen substantial and growing interest in the analysis of dynamic covariances and correlations across investment instruments. In particular, there has been great emphasis paid to the analysis of financial assets (see [

In this research stream, the most widely-used representation is a variation of Multivariate Generalized AutoRegressive Conditional Heteroskedasticity (GARCH), namely Dynamic Conditional Correlation (DCC), as introduced by [

Despite the growing interest in DCC and its central role in the estimation of dynamic correlations, several important issues relating to this representation seem to have been ignored in the financial econometrics literature. These important issues include the absence of any derivation of DCC and its mathematical properties, and a lack of any demonstration of the asymptotic properties of the estimated parameters (for a summary of these issues, see [

Another critical element of DCC is associated with the construction of the dynamic conditional correlations. In fact, the representation seems to provide estimated dynamic correlations as a bi-product of standardization, and not as a direct result of the equation governing the multivariate dynamics. This will be clarified below. An alternative representation which avoids this last criticism, but nevertheless has no discussion of the mathematical properties or demonstration of the asymptotic properties of the estimators, has been proposed by [

It should be mentioned that many empirical applications involving DCC and related representations show that the impact of news can be rather limited, thereby making the estimated conditional correlations similar to those implied by simple BEKK models (see [

This paper highlights some critical issues associated with the use of the DCC and related representations to make potential users aware of the inherent problems they might encounter. The main message is not against the use of DCC, which is the most popular representation of dynamic conditional correlations, but is intended to be cautionary, so that users can understand and appreciate the limits of DCC. In fact, we suggest that DCC be regarded as a filter or as a diagnostic check, as in the Exponentially Weighted Moving Average approach adopted in the first versions of the [

The plan of the paper is to discuss ten things you should know about the DCC representation. These caveats are discussed in

The DCC representation was introduced by [_{t}_{t}_{−1} denotes the information set to time _{t}

Following [_{t}_{1}, _{2},…,_{k}_{t}_{t}

The DCC representation focuses on the dynamic evolution of _{t}

By construction, the standardized residuals have second-order unconditional moment equal to

In practice, the standardized residuals can be used to verify empirically the existence of dynamics in the conditional correlations, for instance, by means of a rolling regression approach. Moreover, if the data generating process of the returns is given in Equations (1) and (2), the dynamic conditional covariance of the standardized residuals is given as:

Without distinguishing between the dynamic conditional covariance and dynamic conditional correlation matrices, [_{t}

However, as the matrix _{t}

It is clear that Equation (7) is a simple standardization, suggesting that the primary statistic of interest, namely the dynamic conditional correlation matrix, can be computed from (6). However, to state the obvious, a dynamic conditional correlation matrix is a standardization of a dynamic conditional covariance matrix, but not every standardization, such as that in Equation (7), is consistent with a dynamic conditional correlation matrix. This lack of equivalence is even more obvious if it cannot be demonstrated (as distinct from being stated) that Equation (6) is a dynamic conditional correlation matrix. (A simple illustration would be to divide 10 elephants by 20 elephants, which is not a correlation despite being a fraction.) It should be clear that, as the second term on the right-hand side of Equation (6) is not a dynamic update of a conditional correlation matrix, the representation in Equation (6) cannot be a dynamic conditional correlation matrix.

Bearing these points in mind, the following caveats should be seriously considered before using the DCC representation.

The simple observation of Equation (6) recognizes the structure of the scalar BEKK model of dynamic conditional correlations (see [

Moreover, using Equation (7), this is equivalent to

However, if we consider the _{t}_{t}_{t}

In addition, a dynamic conditional correlation matrix may be obtained only through the standardization in Equation (7). However, we can also note an inconsistency between the dynamic conditional expectation reported in Equation (5) and the way in which the dynamic conditional correlation matrix is obtained in Equation (7). Such inconsistency causes further problems as _{t}_{t}_{t}

The last remark can easily be verified by visual inspection of the estimates of _{t}_{t}_{t}

From the previous comments, it clearly emerges that DCC is a stated representation, but it is not a derived model that is based on the relationship between the innovations to returns and the standardized residuals. Moreover, the DCC representation does not satisfy the definition that relates dynamic conditional correlations to dynamic conditional covariances, as given in Equation (2). As such, the interpretation of DCC as a representation that may yield dynamic conditional correlations is inherently flawed. This quandary also begs the question as to whether DCC is actually a model, namely a set of assumptions, or alternatively as a representation with explicit and testable mathematical properties and derivable statistical properties.

A further motivation for the previous claim is inherently related to the construction of the conditional correlations within the DCC representation. Generally speaking, conditional correlations can be derived from a conditional covariance model by standardization of the covariances, namely _{t}

The above discussion also affects the many representations which are obtained as generalizations of the DCC representation including, among others, [

This follows from the stated rather than derived properties of the representation (see [

This follows from point (3) above. In particular, [

The absence of explicit regularity conditions and of explicit moment affects also the derivation of asymptotic properties of the parameter estimates. The author of [_{t}

Following the decomposition in (2), we have:
_{V}_{V}_{C}

In [_{t}

The previously outlined approach entails a number of assumptions which are generally not satisfied by empirical data, as follows:

Marginal variances are assumed to be independent, which rules out any form of spillovers or feedback across variances and shocks of the various assets. This is related to the general idea of having dependence across assets governed only by the correlations. However, this is not always the case, and shocks of different assets can affect the variance of a single asset.

The sample correlation matrix is assumed to be an appropriate estimator for the matrix

The approach is called “two step”, when in reality it is a “three step” procedure when sample correlations are used for _{C}_{V}

However, the possible incompatibility between the assumptions leading to the estimation approach described above do not prevent its use, which can be motivated and supported by its computational simplicity, an important issue of which users should be aware. Nevertheless, the asymptotic properties of the “two step” estimator are not discussed in [

We have the additional following caveat:

The author of [_{t}

The primary merit of [

In [

It is clear that the availability of asymptotic properties is still an open question. As a consequence, the reliability of standard inferential procedures, such as statistical significance, or likelihood ratio testing across nested DCC representations, remains unknown, and should be considered only on the basis of appropriate simulation experiments.

In [

Are the purported dynamic conditional correlations real or apparent, and do they arise solely from the standardization of the dynamic conditional covariances? What is the impact of variance misspecification?

With respect to the first question, we refer to the parameter estimates which are generally observed in empirical studies, whereby _{t}_{t}_{t}_{t}

Moving to the second question, we give the underlying intuition starting from a classical example. In the context of the Box-Jenkins procedure, if an ARMA(1,1) is estimated when an ARMA(2,1) representation is correct, then the residuals might still show some AR dynamics. For instance, in the limiting case of real roots for the ARMA(2,1) model, if the roots of the estimated model correspond to those of the true model (such that the AR component captures precisely one of the two roots of the ARMA(2,1) model), then the residuals would still be an AR(1) process. Therefore, estimating the residuals with an AR filter could possibly capture the remaining dynamics.

Transposing the same argument into the GARCH framework, the conditional variance might be estimated as GARCH(1,1), but the correct model might have asymmetry, leverage, jumps, thresholds and/or higher time-varying moments. As a result, the parameter estimates might be biased. The standardized residuals, which are typically not checked for further conditional heteroskedasticity (as the common wisdom is that GARCH(1,1) should be sufficient), may have remaining heteroskedasticity, however mild. Fitting standardized residuals using a GARCH(1,1) model, which is the diagonal term of the DCC representation, will capture some dynamics. Even if the conditional correlations happen to be constant, the conditional covariances across the standardized residuals may appear to be dynamic because of the misspecification. Therefore, standardization does not filter out the dynamics in the covariances due to the biases in the initial GARCH(1,1) estimates. As a result, the conditional correlations may appear to be dynamic (with significant parameter estimates) due to misspecification in the first step. However, no research seems to have followed this line of research, and so it is not clear what the potential impact of the conditional variance misspecification might be on the conditional correlation dynamics.

In [

A significant problem in empirical practice is that many users seem to be under the misapprehension that DCC is a model when it is not. DCC has no obvious or desirable mathematical or statistical properties. Nevertheless, DCC may be a useful filter or a diagnostic check that can capture the dynamics in what are purported to be conditional ‘correlations’, even if they arise through possible model misspecification. In this context, the DCC filter may perform well empirically. In fact, the popularity of the DCC representation is motivated by two main elements, namely the ease in estimation, and the ability of the filter to capture the possible presence of dynamic correlations of conditional variance misspecification.

Consequently, the DCC filter may play a useful role in forecasting out-of-sample dynamic conditional covariances and correlations.

The paper discussed ten things potential users should know about the Dynamic Conditional Correlation (DCC) representation for estimating and forecasting time-varying conditional correlations. The reasons given for being cautious about the use of DCC included the following: DCC represents the dynamic conditional covariances of the standardized residuals, and hence does not yield dynamic conditional correlations; DCC is stated rather than derived; DCC has no moments; DCC does not have testable regularity conditions; DCC yields inconsistent two step estimators; DCC has no asymptotic properties; DCC is not a special case of GARCC, which has testable regularity conditions and standard asymptotic properties; DCC is not dynamic empirically as the effect of news is typically extremely small; DCC cannot be distinguished empirically from diagonal BEKK in small systems; and DCC may be a useful filter or a diagnostic check, but it is not a model.

The computational advantages of the DCC representation might become relevant when focusing on large systems. However, there is no empirical evidence on the comparison of conditional correlations obtained directly from the DCC representation and indirectly from the BEKK model in a large cross section of assets. As a result, we cannot verify if the use of the DCC filter provides conditional paths that are similar to those obtained from a viable alternative model. On the other hand, the BEKK model is more general as it allows for direct spillovers and feedback effects across conditional variance and covariances, as well as indirect spillovers and feedback effects across conditional correlations. The GARCC model is also a viable alternative as it satisfies the definition of a dynamic conditional correlation matrix, and also has demonstrable, as distinct from assumed, regularity conditions and asymptotic properties.

As DCC is presently the most popular representation of dynamic conditional correlations, potential users are strongly encouraged to understand and appreciate the limits of DCC in order to be able to use it as a sensible filter or as a diagnostic check for estimating and forecasting dynamic conditional correlations.

The authors most are grateful to two referees for very helpful comments and suggestions. For financial support, the second author wishes to acknowledge the Australian Research Council, National Science Council, Taiwan, and the Japan Society for the Promotion of Science. An earlier version of this paper was distributed as “Ten Things You Should Know about DCC”.

The authors declare no conflict of interest.

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