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I estimated an MGARCH-BEKK model (using the R package BEKK, i.e. Baba, Engle, Kraft and Kroner; see Engle and Kroner (1995)) on time series of spot and futures prices. The estimated parameters are:

  =====================================================
               Estimate Std. Error t value Pr(> | t| )
  -----------------------------------------------------
  mu1.DLog_Base  -0.002    0.001    -1.498     0.134   
  mu2.DLog_B3    0.0003    0.001     0.282     0.778   
  A011           0.004     0.003     1.047     0.295   
  A021           0.0004                                
  A022           0.013     0.001    14.475       0     
  A11            0.008     0.027     0.314     0.754   
  A21            -0.096    0.089    -1.077     0.282   
  A12            -0.052    0.088    -0.588     0.557   
  A22            0.661     0.122     5.395    0.00000  
  B11            0.967     0.010    96.058       0     
  B21            0.124                                 
  B12            0.073     0.123     0.596     0.551   
  B22            0.011     0.185     0.058     0.953   
  -----------------------------------------------------

I don't now to calculate the conditional variance and covariance matrix.

$$ \left[ {\begin{array}{cc} \sigma_{ss} & \sigma_{sf} \\ \sigma_{fs} & \sigma_{ff} \\ \end{array} } \right] = \left[ {\begin{array}{cc} c_{11} & 0 \\ c_{21} & c_{22} \\ \end{array} } \right] \left[ {\begin{array}{cc} c_{11} & 0 \\ c_{21} & c_{22} \\ \end{array} } \right] + \left[ {\begin{array}{cc} a_{11} & 0 \\ 0 & a_{22} \\ \end{array} } \right] \left[ {\begin{array}{cc} \epsilon_{s,t-1}^2 & \epsilon_{s,t-1}\epsilon_{f,t-1} \\ \epsilon_{fs,t-1}\epsilon_{s,t-1} & \epsilon_{f,t-1}^2 \\ \end{array} } \right] \left[ {\begin{array}{cc} a_{11} & 0 \\ 0 & a_{22} \\ \end{array} } \right]$$

$$ + \left[ {\begin{array}{cc} b_{11} & 0 \\ 0 & b_{22} \\ \end{array} } \right] \left[ {\begin{array}{cc} \sigma_{ss,t-1} & \sigma_{sf,t-1} \\ \sigma_{fs,t-1} & \sigma_{ff,t-1} \\ \end{array} } \right] \left[ {\begin{array}{cc} b_{11} & 0 \\ 0 & b_{22} \\ \end{array} } \right] $$

My conditional variance and covariance matrix:

$$ \left[ {\begin{array}{cc} \sigma_{ss} & \sigma_{sf} \\ \sigma_{fs} & \sigma_{ff} \\ \end{array} } \right] = $$ $$ \left[ {\begin{array}{cc} 0.004 & 0 \\ 0.0004 & 0.013 \\ \end{array} } \right] \left[ {\begin{array}{cc} 0.004 & 0 \\ 0.0004 & 0.013 \\ \end{array} } \right] + \left[ {\begin{array}{cc} 0.008 & 0 \\ 0 & 0.661 \\ \end{array} } \right] \left[ {\begin{array}{cc} \epsilon_{s,t-1}^2 & \epsilon_{s,t-1}\epsilon_{f,t-1} \\ \epsilon_{fs,t-1}\epsilon_{s,t-1} & \epsilon_{f,t-1}^2 \\ \end{array} } \right] \left[ {\begin{array}{cc} 0.008 & 0 \\ 0 & 0.661 \\ \end{array} } \right]$$

$$ + \left[ {\begin{array}{cc} 0.967 & 0 \\ 0 & 0.011 \\ \end{array} } \right] \left[ {\begin{array}{cc} \sigma_{ss,t-1} & \sigma_{sf,t-1} \\ \sigma_{fs,t-1} & \sigma_{ff,t-1} \\ \end{array} } \right] \left[ {\begin{array}{cc} 0.967 & 0 \\ 0 & 0.011 \\ \end{array} } \right] $$

To calculate the optimal hedge ratio BEKK:

$$h_t = \frac{cov \left( \Delta S_t, \Delta f_t \mid \Omega_{t-1} \right) }{var \left( \Delta f_t \mid \Omega_{t-1} \right)}$$

$\Delta S_t$, $\Delta f_t$ is the return price spot and future, and $\Omega_{t-1}$ is conditional variance and covariance matrix.

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  • $\begingroup$ Unfortunately, comments were not merged :( $\endgroup$ – Bob Jansen Aug 26 at 9:36