Dynamic Conditional Correlation Models with Asymmetric Multivariate Laplace Innovations

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Dynamic Conditional Correlation Models with
Asymmetric Multivariate Laplace Innovations

• Juan P. Cajigas
• Centre for Econometric Analysis (CEA_at_Cass)
• Cass Business School, London

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

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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.

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The AML distribution (...continue)

• Geometric Stable Distributions

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The AML distribution (...continue) NO

• If we have that,
• then

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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

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The AML distribution

• Mixtures of normal distributions representation
• where YN(0,H), Zexp(1) and XAML(m,H).
  Therefore,

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AML distribution
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Two-Step estimation feasible (1)

• Normal case (Engle, 2002 Engle and Sheppard,
  2001)

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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

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AML Two-Step estimation feasible

• FIRST STEP Conditional variances

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AML Two-Step estimation feasible

• SECOND STEP Conditional correlations

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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

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AML Two-Step estimation feasible

• In this case we have

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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

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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

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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