On the stationarity of Dynamic Conditional Correlation models

Abstract: We provide conditions for the existence and the unicity of strictly stationary solutions of the usual Dynamic Conditional Correlation GARCH models (DCC-GARCH). The proof is based on Tweedie's (1988) criteria, after having rewritten DCC-GARCH models as nonlinear Markov chains. Moreover, we study the existence of their finite moments.

• The paper establishes sufficient conditions for the existence and uniqueness of strictly stationary solutions in DCC-GARCH models using Tweedie’s Markov chain criteria.
• It identifies explicit parameter restrictions—such as the autoregressive coefficients summing to less than one—that ensure finite moments and ergodicity.
• Counterexamples and simulation studies confirm that violating these conditions leads to instability in volatility and correlation dynamics.

Stationarity Conditions for Dynamic Conditional Correlation GARCH Models

Introduction

This paper rigorously investigates the existence, uniqueness, and moment properties of stationary solutions in Dynamic Conditional Correlation (DCC) GARCH models. While DCC-GARCH processes are standard tools for multivariate volatility modeling—especially in financial econometrics—fundamental theoretical aspects, particularly those related to strict stationarity and moment existence, have remained unresolved. Addressing these gaps, the authors establish sufficient conditions for the existence and uniqueness of strictly stationary DCC-GARCH solutions, deploying Tweedie’s (1988) Markov chain criteria after a novel reformulation of the model as a non-linear Markov process.

DCC-GARCH Model Formulation

The DCC-GARCH model is specified as follows. Observed returns $y_t$ are decomposed into conditional means and zero-mean innovations $z_t$, focusing attention on conditional variances and correlations. Volatilities $h_{i,t}$ are assumed to follow vector GARCH dynamics with possible cross effects, while conditional correlations $R_t$ are constructed by normalizing a time-varying covariance matrix $Q_t$, which itself evolves via AR-type recursions involving both lagged $Q_t$ and outer products of standardized residuals.

Unlike BEKK or CCC models, where stationarity results are relatively tractable due to linear structure and explicit Markovian representations, the DCC specification introduces severe nonlinearity as the Markovian structure operates on a state vector whose evolution depends on nonlinear transformations of its own past (via $Q_t \mapsto R_t$). As a result, explicit construction of stationary solutions is generally impossible, necessitating more abstract existence proofs for invariant measures.

Main Theoretical Contributions

Existence of Strictly Stationary Solutions

The authors prove existence of strictly stationary and ergodic solutions to general DCC-GARCH processes. This is achieved by casting the model in the form of a nonlinear Markov chain and applying Tweedie’s criterion for invariant measures without irreducibility requirements. Crucially, this involves bounding the spectral radius of an expected “worst-case” transition matrix $T^*$. The key sufficient condition can, in the diagonal or scalar formulation, be reduced to requiring that the sum of autoregressive coefficients in the volatility and correlation equations respectively are less than one in absolute value (e.g., for scalar-DCC: $\sum a^{(i)} + \sum b^{(j)} < 1$ for volatilities and $\sum |m^{(k)}|^2 < 1$ for correlations).

The proof provides not only existence, but finiteness of moments up to specific orders, given regularity restrictions on the tails of the innovations $\eta_t$. Counterexamples and simulation studies confirm the sharpness of these bounds: when conditions are marginally violated, $Q_t$ trajectories lose stability and ergodicity.

Uniqueness and Ergodicity

Beyond mere existence, uniqueness (and hence ergodicity) of the stationary solution is shown to require more restrictive, but verifiable, moment and contraction conditions. These involve negative Lyapunov exponents for certain random matrix recursions associated with the nonlinear Markov chain structure. The technical results are also developed for various simplified, but practically relevant, DCC parameterizations (diagonal/partially scalar models), further clarifying under what explicit model restrictions both theoretical and empirical econometric procedures are justified.

Practical Implications and Theoretical Impact

The results settle a fundamental theoretical concern long overlooked in the empirical use of DCC-GARCH models: under which parameterizations is statistical inference (consistency and asymptotic normality of QML estimators, for instance) supported by the existence of a unique, ergodic, strictly stationary law for the process. The proved conditions provide direct, checkable constraints on parameter estimates and model specification, particularly clarifying that the persistence parameters for correlation dynamics play an analogous role for stationarity as in univariate GARCH, but must be squared due to the quadratic nature of the recursion.

Importantly, the authors demonstrate that parameters associated with the “outer product” terms in the DCC correlation process (i.e., the $N_l$ coefficients) do not limit stationarity; rather, instability arises solely from the (squared) AR terms. This is counterintuitive but robust, verified both analytically and via simulation.

The analysis also details that finite second-order moments for the innovations are both necessary and sufficient—heavy-tailed innovations with infinite variance will preclude weak stationarity of the process, which is observable in simulated trajectories even when autoregressive coefficients are otherwise admissible.

Future Research Directions

The methodology opens paths for several future developments:

• Derivation of necessary, not merely sufficient, stationarity conditions for more general parameterizations or non-Gaussian innovations.
• Investigation of inferential properties for DCC extensions such as those including asymmetry, regime-switching, or exogenous covariates.
• Establishing analogous stationarity and ergodicity results for multivariate GARCH processes with nonlinear, non-Gaussian innovations or complex transform structures in $Q_t$.
• Application of these theoretical insights to robustify empirical diagnostics and model specification in high-dimensional asset return modeling.

Conclusion

This paper rigorously establishes sufficient (and, in special cases, essentially necessary) parameter constraints for the existence, uniqueness, and moment boundedness of strictly stationary solutions in DCC-GARCH models, resolving a keystone technical issue that underlies their validity for statistical inference and simulation. The theoretical apparatus, based on reformulating DCC-GARCH as a nonlinear Markov process and applying probabilistic invariance measure arguments, yields clear prescriptions for modelers and places empirical work on DCC models on firmer footing.