Tail emphasized GARCH(1,1)

 The implementation of this model differs from the GARCH(1,1) implementation only in the manner of optimization of the process over the 10 series x_t^(k) (needed in the univariate context) and the additional 45 sum series x_t^(jk) (needed in the multivariate context). Instead of maximizing the average of all 1000 ${\cal L}_t^{_{(k)}}$ or ${\cal L}_t^{_{(jk)}}$ (for the sum series) in the in-sample - we maximize only the average of the most adverse 500. In describing this method as tail emphasized GARCH(1,1), we note that the tail referred to in this context is not the tail of the distributions of x_t^(k) and x_t^(jk), but the tail of the distributions of ${\cal L}_t^{_{(k)}}$ and ${\cal L}_t^{_{(jk)}}$. We however do expect considerable overlap of events $(t\in[-1000,0])$ in the tails of the 2 distributions.

Since the conditional distributions p_t^(k,jk) (notation (k,jk) indicating reference to the series k and the sum series jk) is Gaussian with variance ${\sigma_t^{_{(k,jk)}}}^2$ we can expand ${\cal L}_t^{_{(k,jk)}}$, $t\in[-1000,-1]$, more concretely as:  

$${\cal L}_t^{_{(k,jk)}} = -{1\over2}\ln(2\pi)-\ln(\sigma_t^{_... ...)}}) - {{x_{t+1}^{_{(k,jk)}}}^2\over2{\sigma_t^{_{(k,jk)}}}^2}.$$ (21)

It is clear from the above expression that the dominant contribution to adverse (low) values of ${\cal L}_t^{_{(k,jk)}}$ come from the third term when the model underestimates the risk and predicts a small variance compared to the market move on the next day. It is expected that by optimizing with the objective of maximizing the average of the 500 most adverse ${\cal L}_t^{_{(k,jk)}}$,the model will provide better day to day consistency in forecasting performance than the traditional method which includes all events with equal emphasis.