Weak Diffusion Limits of Dynamic Conditional Correlation Models

Abstract

The properties of dynamic conditional correlation (DCC) models, introduced more than a decade ago, are still not entirely known. This paper fills one of the gaps by deriving weak diffusion limits of a modified version of the classical DCC model. The limiting system of stochastic differential equations is characterized by a diffusion matrix of reduced rank. The degeneracy is due to perfect collinearity between the innovations of the volatility and correlation dynamics. For the special case of constant conditional correlations, a nondegenerate diffusion limit can be obtained. Alternative sets of conditions are considered for the rate of convergence of the parameters, obtaining time-varying but deterministic variances and/or correlations. A Monte Carlo experiment confirms that the often used quasi-approximate maximum likelihood (QAML) method to estimate the diffusion parameters is inconsistent for any fixed frequency, but that it may provide reasonable approximations for sufficiently large frequencies and sample sizes.

The continuous-time limit of score-driven volatility models

“…The QAML estimates of the parameter θ confirm the result of Wang (2002) that QAML is not necessarily consistent; see also Hafner et al (2017) for further simulation evidences. All the three models provide significantly upward biased estimates of θ for each sampling frequency.…”

“…The problem of temporal aggregation of GARCH models is related to their continuous-time limit, and has been studied by Drost and Nijman (1993) and Drost and Werker (1996) among others. Finally, Hafner et al (2017) derive the weak diffusion limits of a modified version of the dynamic conditional correlation (DCC) model of Engle (2002). The work of Hafner et al (2017) is, to the best of our knowledge, the only one in a multivariate framework.…”

“…Finally, Hafner et al (2017) derive the weak diffusion limits of a modified version of the dynamic conditional correlation (DCC) model of Engle (2002). The work of Hafner et al (2017) is, to the best of our knowledge, the only one in a multivariate framework.…”

Weak Diffusion Limits of Two Real-Time GARCH-type Models

“…Nelson (1992) and Nelson and Foster (1995) prove misspecified GARCH-type models can still consistently filter and forecast volatility when the data generating process is a diffusion or near-diffusion process under mild conditions. The weak convergence results have been extended to other GARCHtype models (Duan (1997), Fornari and Mele (1997), Hafner et al (2017), among others). On the other hand, Corradi (2000) shows there exist alternative conditions that lead to degenerate diffusion limits.…”

Leverage and feedback effects on multifactor Wishart stochastic volatility for option pricing

“…Nelson and Foster (1994), Drost and Werker (1996), Duan (1997) and Fornari and Mele (1997) examined various GARCH-type diffusions. While Nelson (1996) developed the continuous record asymptotic theory for several multivariate GARCH models, Hafner, Laurent and Violante (2010) derived the diffusion limit of the consistent DCC model of Aielli (2006) that modified the DCC model of Engle (2002) in such a way that the unstandardized conditional correlation matrix has martingale difference innovations.…”