Global Optimization Methods for Estimating GARCH Models

Max E. Jerrell - Northern Arizona University

Nonlinear parameter estimation problems pose significant computational difficulties. Traditional optimization techniques can fail to converge. When they do converge there is no guarantee that the result will be the global rather than a local optimum. Research in global optimization methods has resulted in techniques that show promise in locating the desired optimum. We examine three such methods in this work. Two of these methods, simulated annealing and genetic algorithms can be described as probabilistic methods. In these methods searches are made in neighborhoods of the best current optimum. The best current optimum may be a local rather than global optimum. These methods attempt to avoid being trapped near a local optimum by searching outside the current neighborhood with a defined probability. Given sufficient time, these techniques have a high probability of locating the global optimum.

The third of the methods examined, interval arithmetic, may be described as a covering method. Interval methods allow us to examine the range of the function over a region of the parameter space rather than just evaluating the function at a point in the space. We discuss test that allow us to reject regions of the space where we can show that a global optimum cannot exist. This technique insures that every point in the parameter space is considered but avoids evaluating the function at every point. These test often eliminate large portions of the parameter space in a single step. Regions which can not be eliminated are divided and these new regions are examined. We combine interval methods with point estimation techniques to improve efficiency. We also insure that no point in the sample space is ignored by insuring that no point in a region is ever lost due to roundoff error.

While all of these methods show promise, they also tend to be computationally intensive. We examine these three methods by conducting timing tests on some standard optimization test functions. We also apply them to estimating the parameters of GARCH models.

Scheduled for Session 3.1 Computation And Econometrics - II