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Unformatted text preview: covariances as well as the variances are
permitted to be time-varying.
There are 3 main classes of multivariate GARCH formulation that are widely
used: VECH, diagonal VECH and BEKK.
VECH and Diagonal VECH
e.g. suppose that there are two variables used in the model. The conditional
covariance matrix is denoted Ht, and would be 2 × 2. Ht and VECH(Ht) are 57 VECH and Diagonal VECH
• • In the case of the VECH, the conditional variances and covariances would
each depend upon lagged values of all of the variances and covariances
and on lags of the squares of both error terms and their cross products.
In matrix form, it would be written • Writing out all of the elements gives the 3 equations as • Such a model would be hard to estimate. The diagonal VECH is much
simpler and is specified, in the 2 variable case, as follows: • The BEKK Model uses a Quadratic form for the parameter matrices to
ensure a positive definite variance / covariance matrix Ht.
58 BEKK and Model Estimation for M-GARCH •
• • Neither the VECH nor the diagonal VECH ensure a positive definite variancecovariance matrix.
An alternative approach is the BEKK model (Engle & Kroner, 1995).
In matrix form, the BEKK model is Model estimation for all classes of multivariate GARCH model is again
performed using maximum likelihood with the following LLF: where N is the number of variables in the system (assumed 2 above), θ is a
vector containing all of the parameters to be estimated, and T is the number of
59 An Example: Estimating a Time-Varying Hedge Ratio for FTSE
Stock Index Returns
(Brooks, Henry and Persand, 2002).
• 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 β.
– No hedge (β=0)
– Naïve hedge (β=1)
– Multivariate GARCH hedges:
• Symmetric BEKK
• Asymmetric BEKK
In both cases, estimating the OHR involves forming a 1-step ahead
forecast and computing 60 OHR Results 61 Plot of the OHR from Multivariate GARCH Conclusions
- OHR is time-varying and less
- M-GARCH OHR provides a
better hedge, both in-sample
- No role in calculating OHR for
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
- Summer '13
- Financial Markets