Session 9A-Overview to Dynamic Time Series Analysis

# Not contained in the garch egarch specifications out

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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 &amp; 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 observations. 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 than 1 - M-GARCH OHR provides a better hedge, both in-sample and out-of-sample. - No role in calculating OHR for asymmetries 62...
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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.

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