A generalized Dynamic Conditional Correlation model for portfolio risk evaluation

TLDR: A generalization of the Dynamic Conditional Correlation multivariate GARCH model, with conditions for positive definiteness of the conditional correlations, and an empirical application to the Italian stock market comparing alternative correlation models for portfolio risk evaluation.

About: This article is published in Mathematics and Computers in Simulation.The article was published on 2009-04-01 and is currently open access. It has received 49 citations till now. The article focuses on the topics: Conditional variance & Autoregressive conditional heteroskedasticity.

Figures

Table 8: The table reports the theoretical and empirical number of exceptions for several confidence levels for the fitted models

Table 1: GARCH parameter estimates and quasi maximum likelihood standard errors (in italics) for each MIBTEL sub-sector – ω is the constant of the variance equation, α represents the ARCH term, β is the GARCH coefficient and γ is the asymmetric effect (the ARCH component is α+γ for negative returns and α for positive returns) – we imposed the restrictions 0<α+½γ+β<1.

Figure 3: rolling correlations over a 250 observation window for a selected Financial sub-sectors

Table 4: Number of exception over last 250 observations for a set of VaR confidence levels

Table 2: DCC estimates over the full sample with Quasi Maximum Likelihood standard errors

Table 3: Diagonal QFDCC estimates over the full sample with Quasi Maximum Likelihood standard errors

Frequently asked questions

Q1. What are the contributions in this paper?

The authors propose a generalization of the Dynamic Conditional Correlation multivariate GARCH model of Engle ( 2002 ) and of the Asymmetric Dynamic Conditional Correlation model of Cappiello et al. ( 2006 ). The model the authors propose introduces a block structure in parameter matrices that allows for interdependence with a reduced number of parameters. In the paper, the authors provide conditions for positive definiteness of the conditional correlations. The authors also present an empirical application to the Italian stock market comparing alternative correlation models for portfolio risk evaluation. 

Q2. What is the common approach to the dynamic correlation literature?

Within the dynamic correlation literature, the most common approach considers univariate specification of the variances, possibly including asymmetric terms following the GJR model of Glosten et al. (1993). 

Q3. What is the way to estimate the correlation matrix?

2The representation with constant conditional correlations allows for a twostep estimation procedure: at first, the authors estimate the conditional variances, which can then be filtered out by premultiplying εt by D −1 t ; then, the correlation matrix can be estimated. 

Q4. What is the starting point for the analysis of dynamic correlation models?

The starting point for the analysis of dynamic correlation models is the Constant Condition Correlation model of Bollerslev (1991). 

Q5. What are the main limitations of the BEKK and Vech models?

BEKK and Vech models are useless in systems with more that 4 or 5 variables since they have3serious optimization problems leading to unstable and inconsistent parameter estimates. 

Q6. What is the main reason why the empirical and theoretical analysis concerning multivariate GARCH models?

In the last few years, the empirical and theoretical analysis concerning multivariate GARCH models attracted a growing interest for two main reasons: the availability of more and more powerful computers that enabled the estimation of complex models with an elevate number of parameters and the introduction of a new class of models: the Dynamic Conditional Correlation multivariate GARCH (DCC) by Engle (2002). 

Q7. What is the GARCH coefficient for each sub-sector?

All sub-sectors conditional variances show a relevant asymmetric effect and only three reports a GARCH coefficient lower than 0.7. 

Q8. What are the main points of the paper?

These papers focused both on the developments of new parameterizations and on their use in empirical applications, demonstrating an high capability to adapt to practical problems. 

Q9. How do the authors compare the performance of the different portfolios?

In order to get directly comparable portfolios in term of returns and avoid any discussion on9the estimation of mean expected returns, the authors consider equally weighted portfolios (i.e. the 5% of the global portfolio is invested in each of the 20 sub-sectors indices of the Mibtel). 

Q10. What is the difference between the two tests?

Both tests are based on the assumption that exceptions follow a binomial distribution and are asymptotically distributed as a chi-square variable with one degree of freedom. 

Q11. What is the way to estimate the conditional variance?

Each conditional variance could be modelled with a standard GARCH model or with more advanced parameterisations such as GARCH models with asymmetry effects as in Glosten et al. (1993) and in Caporin and McAleer (2006), or EGARCH representations as in Nelson (1991). 

Q12. What is the common way to estimate the maximum likelihood?

According to the results of Comte and Lieberman (2003), Ling and McAleer (2003) and McAleer et al. (2006), the maximum likelihood estimators are consistent and asymptotically normally distributed.