BEKK Conditional Covariance Updates for Spot-Futures Hedge Ratios
Summary
The document asks how to obtain a conditional variance-covariance matrix from an estimated MGARCH-BEKK model fitted to spot and futures price series. It presents a two-asset BEKK-style recursion built from a constant matrix, lagged return residuals, and the previous conditional covariance matrix. The example substitutes estimated coefficients into diagonal shock and persistence matrices, alongside a lower-triangular constant matrix, to illustrate the update.
The stated application is a time-varying minimum-variance hedge ratio: divide the conditional spot-futures covariance by the conditional futures variance. The coefficient table and equations provide a starting point, but the text does not supply residual observations, an initial covariance matrix, or a worked time-series calculation, so they are not enough to produce a numeric hedge ratio. The displayed recursion also uses a simplified diagonal form; a full BEKK specification can include more general coefficient matrices. Users must align the estimated parameterization with the software output and correctly construct the conditional matrix before calculating the ratio.
Key ideas
- A BEKK model updates conditional covariance using a constant term, lagged residual shocks, and the previous covariance matrix.
- The spot-futures minimum-variance hedge ratio is conditional covariance divided by conditional futures variance.
- A numeric ratio requires residual data and a prior covariance matrix in addition to estimated parameters.
- The displayed diagonal shock and persistence matrices are a simplified parameterization that must match the fitted model.
Tags
Full text
# Calculate the conditional variance-covariance matrix to optimal hedge ratio bekk
# Calculate the conditional variance-covariance matrix to optimal hedge ratio bekk
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.