Dynamic Conditional Correlation Models with Asymmetric Multivariate Laplace Innovations Juan P. Cajigas Centre for Econometric Analysis (CEA@Cass) Cass Business School, London

Dynamic Conditional Correlation (S,L) [Engle (2002), Engle and Sheppard (2001)]

Implications of the assumption of normality Normality-MLE/QMLE = feasible + consistent but inefficient DCC coefficients (Bollerslev and Wooldridge, 1992) Normality is not a satisfactory property for financial time series. Non normal distribution to achieve efficiency with implication for the first stage Importance of efficiency for Portfolio allocation VaR Analysis (Risk)

The main contribution of this paper: (AML)-ADDCC (1,1) We propose an AGDCC (1,1) model and its nested versions using the Asymmetric Multivariate Laplace (AML) distribution for the vector of standardized residuals. This is a special case of the Geometric Stable law (Kotz, Kozubowski and Podgorski, 2003) It preserves convolution properties It has finite variance, It has a closed-form, It allows for leptokurtosis and asymmetries.

The AML distribution (...continue) Geometric Stable Distributions:

The AML distribution (...continue) NO If we have that, then

The AML distribution (...continue) Main properties: Tail behavior governed by the index of stability. For the AML distribution Density function: where v = (2 - n)/2 and Kv(u) is the modified Bessel function of the “third kind”

The AML distribution Mixtures of normal distributions representation: where Y~N(0,H), Z~exp(1) and X~AML(m,H). Therefore,

AML distribution

Two-Step estimation: feasible (1…) Normal case (Engle, 2002; Engle and Sheppard, 2001)

Two-Step estimation: feasible (…2) Normal case Engle (2002): uses Newey-McFadden (1994, HoE) results on GMM to justify the use of MLE for consistency

AML Two-Step estimation: feasible FIRST STEP: Conditional variances

AML Two-Step estimation: feasible SECOND STEP: Conditional correlations

AML Two-Step estimation: feasible For n = 2s + 3, s = 0,1,… the Bessel function has a closed form that transforms the density function to:

AML Two-Step estimation: feasible In this case we have:

Empirical applications (1): Modelling using DCC models Data as in Cappiello, Engle and Sheppard (2004) a) FTSE All-World weekly indices converted to US denominated returns for 21 countries and b) Bond indices of 12 constructed by Datastream. Sample Period: 08/01/1987 - 07/02/2001

Main findings from the empirical application using stocks/bonds Significant differences with Cappiello et al (2004) in the asymmetric models (AGDCC and ADCC): Asymmetric terms much smaller when the AML distribution is used instead of the normal Log-likelihood does not increase with the inclusion of asymmetries when the AML distribution is used

Main findings from the empirical application Distribution of conditional correlations: Much higher kurtosis when the AML distribution is used. The impact of this feature “could” be relevant for VaR applications