A parallel fuzzy GMM-algorithm for approximate VGARCH-modeling with a multi-modal discontinuous merit function

Abstract

General modeling of multivariate stochastic volatility with exact methods is coupled with computational difficulties. Over-fitting and outliers are two central problems in time series modeling, where fuzzy mathematical approaches can be successfully applied. The risk for over-fitting is present also in General Method of Moments-estimation (GMM), but the mathematical flexibility of the approach – in essence based on formulating and solving a consistent system of equations – allows counteracting measures that are difficult or impossible to implement successfully in traditional maximum likelihood based methods. We show that fuzzy mathematical programming can be applied to cope with the problem of over-fitting in GMM-estimation. We introduce a fuzzy component in the equation system derived from the moment conditions. The fuzzy augmented equation system allows an approximate solution that counteracts over-fitting. The penalized objective function is similar to the penalty functions normally used in constrained nonlinear optimization. In order to reduce the risk for over-estimation, the quadratic objective function of the ordinary GMM-formulation is replaced by an augmented exponential fuzzy penalty function. The barrier coefficient and fuzzy tolerance level are estimated via a multi-modal, discontinuous merit function that models hypothetical real world conditions. We show in a large set of simulations, that the fuzzy GMM-algorithm (FGMM) yields estimates mostly closer to the true values than those generated by the crisp GMM. The fuzzy GMM-algorithm is generalized to a multi-computer setting and solved by parallel processors. The multi-computer formulation is fully scalable and independent of mesh size. The study combines high performance computing, fuzzy mathematical programming and GMM in the estimation of volatility.