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PARAMETER ESTIMATION OF BILINEAR PROCESSES USING ABC 149
5. Comments and conclusions
Bilinear processes form a very flexible class models for capturing heavy- tailed, nonlinear features in time series. However, for time series exhibiting heavy tails, the choice of the model for the innovations and the model for the time series, from the general class given in (2.1), is not straightforward. For example, a choice of a Gaussian structure for the innovations would imply a choice of high value of m in the model. Alternatively, the choice of lower order m may simplify the nonlinear structure. However, this comes at a price of choosing a heavy-tailed model for the innovations (Turkman et al. [38]). In either case, least squares methods or conditional likelihood methods for parameter estimation are no longer satisfactory. In this situation, simulation- based methods may work better and we looked at the viability of using an ABC approach with an appropriate set of summary statistics. The method seems to work well for simple first order bilinear processes and is quite promising. However, the bottleneck for extensions to higher order models is the di culty of verifying the invertibility and the stationary conditions, while searching for admissible solutions within the parameter space.
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