The Pitfalls in Fitting Garch(1,1) Processes

Abstract

Optimization of GARCH(1,1) processes by maximum likelihood has been documented to be numerically difficult and prone to error. In this paper, we show that this difficulty is related to a one dimensional manifold in the parameter space where the likelihood function is large and almost constant. This manifold is revealed by introducing other natural parameterizations for the process, namely other coordinates in the parameter space. In order to avoid spurious fits, we suggest to fix the volatility of the process by a moment estimate and to estimate the remaining parameters with a maximum likelihood procedure. Moreover, the convergence properties of the maximization algorithm are vastly improved by working with one of the new coordinates system. We also investigate the finite sample distribution of the estimated parameters computed with a maximum likelihood. The results indicate that the intercept parameter α0 is strongly biased, whereas the volatility and several of the new coordinates have small bias. The aggregation properties of GARCH(1,1), from 10 minutes to 3.5 days are investigated. The results indicate that the theoretical aggregation relations for this process are not consistent with the FX data. Finally, the process is optimized to deliver the best volatility forecast for the next month.

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The author would like to thank Tim Bollerslev and Ramazan Gencay for their many suggestions, the members of the research group at Olsen & Associates for their comments, and Ralph Baenziger for the implementation of the optimization programs