The convergence of estimators based on heuristics : theory and application to a GARCH model

Econometric theory describes estimators and their properties, e.g., the convergence of maximum likelihood estimators. However, it is ignored that often the estimators cannot be computed using standard tools, e.g., due to multiple local optima. Then, optimization heuristics might be helpful. The additional random component of heuristics might be analyzed together with the econometric model. A formal framework is proposed for the analysis of the joint convergence of estimator and stochastic optimization algorithm. In an application to a GARCH model, actual rates of convergence are estimated by simulation. The overall quality of the estimates improves compared to conventional approaches.