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:
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Estimate Std. Error t value Pr(> | t| )
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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
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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.
