Optimization Results

The ARMA model optimization results are the points $b$ and $a$ (see Figure 1.1) . These results were defined using a sequence of two global methods referred to as $BAYES1$ and $EXKOR$ (see [14]). $BAYES1$ denotes a search in accordance with a multi-dimensional Bayesian model []. The best result obtained by $BAYES1$ after 50 iterations is a starting point for an one-dimensional coordinate search using 60 iterations by $EXKOR$ [].

The maximal number of auto-regression (AR) parameters $p$ was $10*M$. Here $M=1$ if no external factors are involved. The optimal number $p$ is defined by structural stabilization (see chapter 1.10) . Plotting surfaces and contours the number of moving-average (MA) parameters was fixed $q=2$. The results in Table 1.1 were obtained by optimization of both structural variables $p$ and $q$.

The objective of this work is mainly to show a multi-modality of the problem. Therefore to save the computing time the global optimization was carried out approximately using not many of iterations. The results of global optimization were used as a starting point for local optimization. The reason is that the squared deviation as a function of parameters $b$ is multi-modal considering the wide range of these parameters and becomes uni-modal regarding the narrow range (see Figures 1.16). Thus we guarantee the final results at least as good as that of local optimization.

The high-accuracy global optimization is very expensive. As usual, the computing time is an exponential function of accuracy in the global optimization. Therefore, what happens after the high-accuracy global optimization of the objective function, is not yet clear. However, it seems clear that the investigation of multi-modality of squared deviation and variability of the parameters should be the first step in estimating parameters of non-linear regression models, including the ARFIMA ones. Balancing computing expenses and accuracy of estimation is the important problem of future investigation in both the fields of exchange rate prediction and global optimization.