Univariate context:

In this model the variance of the Gaussian distribution p_t is given by the recursive formula:  

$${\sigma_t^{_{(k)}}}^2 = \alpha_0^{_{(k)}} + \alpha_1^{_{(k)}... ...^2,\quad \sigma_{-249}^{_{(k)}} = \vert x_{-249}^{_{(k)}}\vert.$$ (16)

The recursion is initialized using the same seed value as for the J. P. Morgan RiskMetrics case. As discussed in section , we emphasize that the built up values $\sigma_t^{_{(k)}}$ for $t\in[1,1000]$ are not sensitive to this arbitrary but reasonable choice.

We have already stated that the J. P. Morgan RiskMetrics model is a GARCH(1,1) model. What is different here, however, is that we have separately optimized the GARCH(1,1) for each of the k risk-factors in the in-sample period. Unlike the J. P. Morgan prescription of using $\alpha_0^{_{(k)}} = 0$, $\alpha_1^{_{(k)}} = 0.06$ and $\beta_1^{_{(k)}} = 0.94$ for all k risk factors, in our case we used the values reported in Table 1.

The optimization - reported in Table 1 - was done with the objective of maximizing the average log-likelihood of the 1000 prediction-realization pairs in the in-sample period - corresponding to the events $t\in[-1000,0]$.Mathematically, the average log-likelihood may be expressed as:

$$\bar{\cal L}^{_{(k)}} = {1\over1000}\sum_{t=-1000}^{-1}{\cal... ...over1000}\sum_{t=-1000}^{-1}\ln(p_t^{_{(k)}}(x_{t+1}^{_{(k)}}))$$ (17)

where, $p_t^{_{(k)}}(x_{t+1}^{_{(k)}}) = p_t(\sigma_t^{_{(k)}},x_{t+1}^{_{(k)}})$ is the probability density of the realized event in terms of the forecast distribution and $\sigma_t^{_{(k)}} = \sigma_t^{_{(k)}}(\alpha_0^{_{(k)}},\alpha_1^{_{(k)}},\beta_1^{_{(k)}})$ is the variance of the forecast distribution in accordance with the GARCH(1,1) expression . For stationarity of the GARCH(1,1) process, the optimization is done with constraint $\alpha_0^{_{(k)}} + \alpha_1^{_{(k)}} + \beta_1^{_{(k)}} < 1$.(We note here that the J. P. Morgan RiskMetrics process equation violates this stationarity condition). The solution space was searched using the genetic algorithms method - but convergence was aided by strategic use of the BHHH algorithm [28] at the end of every generation[29].

With Table 1 and the GARCH(1,1) recursion formula for the variance $\sigma_t^{_{(k)}}$ of the Gaussian distribution p_t^(k), we construct the 1000 prediction-realization pairs over the period $t\in[1,1001]$that is the starting point for the evaluation of this model in the univariate context.