Multivariate context

Here - once again - we proceed via the construction of the covariance matrix $\Sigma_t$ but in this case we introduce a different methodology. Instead of making the straightforward bivariate generalization of the GARCH(1,1), we proceed by fitting the 45 sum series,

x_t^(jk) = x_t^(j) + x_t^(k),

(note new notation to reference sum series) to the GARCH(1,1). The fit is done in the same in-sample period and using the same mix of the genetics algorithm and BHHH approach used in the univariate context. The fit parameters for the 45 series are presented in Table 2.

The $\sigma_t^{_{(jk)}}$ are then determined for the sum series using the same recursion expression and build up criteria used in the univariate context. The off-diagonal elements of $\Sigma_t$ are then computed as an implied covariance:  

$$s_t^{_{(jk)}} = s_t^{_{(kj)}} = {\sigma_t^{_{(jk)}}-\sigma_t^{_{(j)}}-\sigma_t^{_{(k)}}\over2},\quad i\ne j.$$ (19)

The following additional specification of diagonal elements:

$$s_t^{_{(kk)}} = {\sigma_t^{_{(k)}}}^2,$$ (20)

where $\sigma_t^{_{(k)}}$ is determined in the univariate context, concludes the full prescription on how $\Sigma_t$ is generated in this model.

Finally, using standard multivariate probability theory the forecast distribution p_t^(P) for x_t+1^(P) is constructed from $\Sigma_t$ in the same manner as expressed at the end of section  (expression ). The 1000 prediction-realization pairs, (p_t^(P),x_t+1^(P)), over the period $t\in[1,1001]$ is the starting point for the evaluation of this model in the multivariate context.

1 4|c|Garch(1,1) 4c|Tail emphasized Garch(1,1)

$$\!\alpha_0\times10^5\! \alpha_1 \beta_1 \!\alpha_0\times10^5\! \alpha_1 \beta_1$$ Series LL LL

$$\! \! \!0.1887\! \!0.04185\! \!0.9279\! \!3.4410\! \!0.6404\! \!0.01669\! \!0.9361\! \!3.1113\!$$

$$\! \! \!0.1806\! \!0.06171\! \!0.9053\! \!3.5187\! \!0.9698\! \!0.02959\! \!0.8880\! \!3.1732\!$$

$$\! \! \!0.1488\! \!0.06580\! \!0.9050\! \!3.5659\! \!0.9808\! \!0.05182\! \!0.8646\! \!3.2155\!$$

$$\! \! \!0.1912\! \!0.07187\! \!0.8925\! \!3.5427\! \!0.9695\! \!0.02959\! \!0.8864\! \!3.1809\!$$

$$\! \! \!0.1601\! \!0.08730\! \!0.8836\! \!3.5648\! \!0.9808\! \!0.05182\! \!0.8658\! \!3.1927\!$$

$$\! \! \!0.1074\! \!0.05991\! \!0.9132\! \!3.6761\! \!0.9748\! \!0.05153\! \!0.8431\! \!3.3196\!$$

$$\! \! \!0.1803\! \!0.06020\! \!0.9061\! \!3.5275\! \!0.9715\! \!0.05133\! \!0.8728\! \!3.1813\!$$

$$\! \! \!0.0069\! \!0.04783\! \!0.9558\! \!3.6714\! \!0.9626\! \!0.02959\! \!0.8670\! \!3.2376\!$$

$$\! \! \!1.2128\! \!0.06719\! \!0.8441\! \!3.0388\! \!0.8269\! \!0.00749\! \!0.9653\! \!2.6915\!$$

$$\! \! \!0.0212\! \!0.02793\! \!0.9680\! \!3.4656\! \!0.4518\! \!0.00840\! \!0.9582\! \!3.0685\!$$

Table 1: Fit parameters ($\alpha_0$, $\alpha_1$, $\beta_1$)for variance forecasts and the mean log-likelihood (LL) over the in-sample for GARCH(1,1) (using 1000 points) and tail emphasized GARCH(1,1) processes (using 500 points). See sections  and for additional details. The series are as defined in section 2.1.

1 4|c|Garch(1,1) 4c|Tail emphasized Garch(1,1)

$$\!\alpha_0\times10^5\! \alpha_1 \beta_1 \!\alpha_0\times10^5\! \alpha_1 \beta_1$$ Series LL LL

$$\! \! \!0.6877\! \!0.05098\! \!0.9184\! \!2.8053\! \!0.9193\! \!0.01906\! \!0.9675\! \!2.4652\!$$

$$\! \! \!0.6354\! \!0.05278\! \!0.9178\! \!2.8283\! \!0.9927\! \!0.01411\! \!0.9675\! \!2.4890\!$$

$$\! \! \!0.6543\! \!0.06359\! \!0.9053\! \!2.8509\! \!0.9926\! \!0.01434\! \!0.9665\! \!2.4990\!$$

$$\! \! \!0.5459\! \!0.05642\! \!0.9178\! \!2.8513\! \!0.9925\! \!0.01406\! \!0.9667\! \!2.5069\!$$

$$\! \! \!0.5400\! \!0.06301\! \!0.9102\! \!2.8814\! \!0.9913\! \!0.01360\! \!0.9657\! \!2.5257\!$$

$$\! \! \!0.4976\! \!0.06509\! \!0.9095\! \!2.9031\! \!0.9884\! \!0.01360\! \!0.9647\! \!2.5450\!$$

$$\! \! \!0.5374\! \!0.07466\! \!0.8984\! \!2.9057\! \!0.9901\! \!0.01360\! \!0.9647\! \!2.5344\!$$

$$\! \! \!0.5536\! \!0.05651\! \!0.9056\! \!3.0168\! \!0.8425\! \!0.00992\! \!0.9644\! \!2.6733\!$$

$$\! \! \!0.5330\! \!0.06163\! \!0.9116\! \!2.8857\! \!0.9914\! \!0.01360\! \!0.9654\! \!2.5303\!$$

$$\! \! \!0.2070\! \!0.05930\! \!0.9320\! \!2.9429\! \!0.9432\! \!0.01360\! \!0.9649\! \!2.5612\!$$

$$\! \! \!0.6234\! \!0.05523\! \!0.9157\! \!2.8355\! \!0.9914\! \!0.01383\! \!0.9673\! \!2.4921\!$$

$$\! \! \!0.6732\! \!0.06993\! \!0.8979\! \!2.8600\! \!0.9912\! \!0.01383\! \!0.9665\! \!2.5014\!$$

$$\! \! \!0.6211\! \!0.07343\! \!0.8960\! \!2.8824\! \!0.9926\! \!0.01406\! \!0.9652\! \!2.5211\!$$

$$\! \! \!0.6008\! \!0.05949\! \!0.8991\! \!3.0180\! \!0.8425\! \!0.00992\! \!0.9642\! \!2.6728\!$$

$$\! \! \!0.5489\! \!0.03985\! \!0.9261\! \!2.9578\! \!0.8897\! \!0.01096\! \!0.9648\! \!2.6301\!$$

$$\! \! \!0.6172\! \!0.04760\! \!0.9114\! \!2.9923\! \!0.8582\! \!0.00980\! \!0.9649\! \!2.6573\!$$

$$\! \! \!0.5591\! \!0.05111\! \!0.9099\! \!3.0193\! \!0.8426\! \!0.00992\! \!0.9639\! \!2.6824\!$$

$$\! \! \!0.6784\! \!0.04999\! \!0.9196\! \!2.8093\! \!0.9194\! \!0.01907\! \!0.9673\! \!2.4691\!$$

$$\! \! \!0.7250\! \!0.06102\! \!0.9055\! \!2.8306\! \!0.9915\! \!0.01383\! \!0.9678\! \!2.4809\!$$

$$\! \! \!0.6535\! \!0.06272\! \!0.9058\! \!2.8553\! \!0.9927\! \!0.01406\! \!0.9665\! \!2.5039\!$$

$$\! \! \!0.6812\! \!0.06939\! \!0.8977\! \!2.8639\! \!0.9915\! \!0.01359\! \!0.9664\! \!2.5058\!$$

$$\! \! \!0.5970\! \!0.04709\! \!0.9128\! \!2.9973\! \!0.8583\! \!0.00992\! \!0.9646\! \!2.6617\!$$

$$\! \! \!0.2163\! \!0.04709\! \!0.9438\! \!2.8810\! \!0.9912\! \!0.01360\! \!0.9660\! \!2.5219\!$$

$$\! \! \!0.1637\! \!0.05014\! \!0.9437\! \!2.9063\! \!0.9802\! \!0.01360\! \!0.9654\! \!2.5314\!$$

$$\! \! \!0.1435\! \!0.05186\! \!0.9429\! \!2.9313\! \!0.9649\! \!0.01360\! \!0.9646\! \!2.5528\!$$

$$\! \! \!0.1439\! \!0.05694\! \!0.9385\! \!2.9369\! \!0.9432\! \!0.01360\! \!0.9652\! \!2.5476\!$$

$$\! \! \!0.3178\! \!0.04254\! \!0.9349\! \!3.0461\! \!0.8165\! \!0.00583\! \!0.9658\! \!2.7015\!$$

$$\! \! \!0.1574\! \!0.04853\! \!0.9454\! \!2.9106\! \!0.9866\! \!0.01360\! \!0.9650\! \!2.5359\!$$

$$\! \! \!1.3111\! \!0.03395\! \!0.9057\! \!2.7977\! \!0.9193\! \!0.01906\! \!0.9680\! \!2.4649\!$$

$$\! \! \!1.1672\! \!0.04607\! \!0.8969\! \!2.8316\! \!0.9928\! \!0.01410\! \!0.9671\! \!2.4979\!$$

$$\! \! \!1.1351\! \!0.04487\! \!0.8978\! \!2.8472\! \!0.9911\! \!0.01371\! \!0.9666\! \!2.5136\!$$

$$\! \! \!0.6276\! \!0.03225\! \!0.9375\! \!2.8249\! \!0.9914\! \!0.01383\! \!0.9678\! \!2.4904\!$$

$$\! \! \!1.2375\! \!0.04736\! \!0.8904\! \!2.8464\! \!0.9913\! \!0.01371\! \!0.9667\! \!2.5127\!$$

$$\! \! \!0.3264\! \!0.02016\! \!0.9616\! \!2.8848\! \!0.9900\! \!0.01360\! \!0.9648\! \!2.5500\!$$

$$\! \! \!1.0297\! \!0.04256\! \!0.9068\! \!2.8339\! \!0.9914\! \!0.01383\! \!0.9672\! \!2.5001\!$$

$$\! \! \!1.4667\! \!0.05619\! \!0.8673\! \!2.8667\! \!0.9912\! \!0.01359\! \!0.9660\! \!2.5266\!$$

$$\! \! \!0.4364\! \!0.02503\! \!0.9462\! \!2.9736\! \!0.8897\! \!0.00922\! \!0.9650\! \!2.6454\!$$

$$\! \! \!0.5821\! \!0.03243\! \!0.9261\! \!3.0152\! \!0.8269\! \!0.00749\! \!0.9655\! \!2.6902\!$$

$$\! \! \!0.5088\! \!0.03200\! \!0.9298\! \!3.0406\! \!0.8198\! \!0.00672\! \!0.9647\! \!2.7147\!$$

$$\! \! \!0.3598\! \!0.03306\! \!0.9406\! \!3.0339\! \!0.8267\! \!0.00749\! \!0.9649\! \!2.6998\!$$

$$\! \! \!0.6591\! \!0.04087\! \!0.9104\! \!3.0367\! \!0.8199\! \!0.00672\! \!0.9651\! \!2.7082\!$$

$$\! \! \!0.0177\! \!0.01519\! \!0.9824\! \!3.1589\! \!0.6367\! \!0.00840\! \!0.9655\! \!2.8000\!$$

$$\! \! \!0.4568\! \!0.02875\! \!0.9383\! \!3.0203\! \!0.8267\! \!0.00749\! \!0.9652\! \!2.6951\!$$

$$\! \! \!11.2385\! \!0.09932\! \!0.0000\! \!3.0703\! \!0.8132\! \!0.00495\! \!0.9642\! \!2.7376\!$$

$$\! \! \!0.1431\! \!0.01737\! \!0.9776\! \!2.6099\! \!0.9908\! \!0.01565\! \!0.9756\! \!2.2537\!$$

Table 2: Fit parameters ($\alpha_0$, $\alpha_1$, $\beta_1$)for variance forecasts and the mean log-likelihood (LL) over the in-sample for GARCH(1,1) (using 1000 points) and tail emphasized GARCH(1,1) processes (using 500 points). See sections  and for additional details. The series are the sum (+) of the series defined in section 2.1. This optimization is needed in the multivariate context to compute the implied covariances used in these methods.