Building Econometric Models



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Ch9 slides

DCC Model Estimation

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • The model may be estimated in a single stage using ML although this will be difficult. So Engle advocates a two-stage procedure where each variable in the system is first modelled separately as a univariate GARCH
  • A joint log-likelihood function for this stage could be constructed, which would simply be the sum (over N) of all of the log-likelihoods for the individual GARCH models
  • In the second stage, the conditional likelihood is maximised with respect to any unknown parameters in the correlation matrix
  • The log-likelihood function for the second stage estimation will be of the form
  • where θ1 and θ2 denote the parameters to be estimated in the 1st and 2nd stages respectively.

Asymmetric Multivariate GARCH

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • Asymmetric models have become very popular in empirical applications, where the conditional variances and / or covariances are permitted to react differently to positive and negative innovations of the same magnitude
  • In the multivariate context, this is usually achieved in the Glosten et al. (1993) framework
  • Kroner and Ng (1998), for example, suggest the following extension to the BEKK formulation (with obvious related modifications for the VECH or diagonal VECH models)
  • where zt−1 is an N-dimensional column vector with elements taking the value −ϵt−1 if the corresponding element of ϵt−1 is negative and zero otherwise.

An Example: Estimating a Time-Varying Hedge Ratio for FTSE Stock Index Returns (Brooks, Henry and Persand, 2002).

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • Data comprises 3580 daily observations on the FTSE 100 stock index and stock index futures contract spanning the period 1 January 1985 - 9 April 1999.
  • Several competing models for determining the optimal hedge ratio are constructed. Define the hedge ratio as .
    • No hedge (=0)
    • Naïve hedge (=1)
    • Multivariate GARCH hedges:
      • Symmetric BEKK
      • Asymmetric BEKK
      • In both cases, estimating the OHR involves forming a 1-step ahead
      • forecast and computing

OHR Results

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

Plot of the OHR from Multivariate GARCH

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • Conclusions
  • - OHR is time-varying and less than 1
  • - M-GARCH OHR provides a
  • better hedge, both in-sample and out-of-sample.
  • - No role in calculating OHR for asymmetries

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