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To write a BEKK(1,1) model, I would write something like this,

$$H_t=C^*C^{*'}+A_{11}\varepsilon_{t-1}\varepsilon_{t-1}'A_{11}'+ B_{11}H_{t-1}B_{11}' $$

How could I extend this to write the BEKK(1,1,1) i,e with an asymmetric term.

The code for this from Kevin Sheppard is:

The dynamics of a BEKK are given by
%
%   H(:,:,t) = C*C' +
%       A(:,:,1)'*OP(:,:,t-1)*A(:,:,1) + ... + A(:,:,p)'*OP(:,:,t-1)*A(:,:,p) +
%       G(:,:,1)'*OPA(:,:,t-1)*G(:,:,1) + ... + G(:,:,o)'*OPA(:,:,t-1)*G(:,:,o) +
%       B(:,:,1)'*G(:,:,t-1)*B(:,:,1) + ... + B(:,:,q)'*OP(:,:,t-1)*B(:,:,q)
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I believe that Kroner and Ng (1998) - Modelling asymmetric comovements of asset returns is the relevant reference.

You should be able to write an assymmetric BEKK model by writting

$$ H_t = C^* {C^*}^\prime + A_{11}\varepsilon_{t-1}\varepsilon_{t-1}^\prime A_{11}^\prime + B_{11}H_{t-1}B_{11}^\prime + D_{11}\varepsilon_{t-1}^*{\varepsilon_{t-1}^*}^\prime D_{11}^\prime $$

where $\varepsilon^* = \min(\varepsilon, 0)$ according to Asai and McAleer - Dynamic Conditional Correlations for Asymmetric Processes

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