Abstract
We provide the analytical gradient of the full model likelihood for the Dynamic Conditional Correlation (DCC) specification by Engle (2002), the generalised version by Cappiello et al. (2006), and of the cDCC model by Aielli(2013). We discuss how the gradient might be further extended by introducing elements related to the conditional variance parameters, and discuss the issue arising from the estimation of constrained and/or reparametrised versions of the model. A computational simulation compares analytical versus numerical gradients, with a view to parameter estimation; we find that analytical differentiation yields more efficiency and improved accuracy.
Highlights
One of most challenging issues related to the class of Multivariate GARCH (MGARCH) models is the so-called curse of dimensionality: as a rule, the number of parameters of a MGARCH model diverges when the number of modelled variables increases
We provide the analytical expression of the gradient for three well-known conditional correlation models: the Dynamic Conditional Correlation (DCC) model by Engle (2002), the generalisation proposed by Cappiello et al (2006), and the more recent Consistent DCC model by Aielli (2013)
In the following we focus on three dynamic conditional correlation models
Summary
One of most challenging issues related to the class of Multivariate GARCH (MGARCH) models is the so-called curse of dimensionality: as a rule, the number of parameters of a MGARCH model diverges when the number of modelled variables increases. We provide the analytical expression of the gradient for three well-known conditional correlation models: the Dynamic Conditional Correlation (DCC) model by Engle (2002), the generalisation proposed by Cappiello et al (2006), and the more recent Consistent DCC (cDCC) model by Aielli (2013) These models are generally estimated by two-step approaches: first, we recover conditional variance parameters by mean of univariate estimations; we maximise the conditional correlation likelihood with respect to correlation parameters, conditionally on the estimated variance parameters.
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