Page 145 - Textos de Matemática Vol. 47

PARAMETER ESTIMATION OF BILINEAR PROCESSES USING APPROXIMATE BAYESIAN COMPUTATION
P. DE ZEA BERMUDEZ, M. A. AMARAL TURKMAN, AND K. F. TURKMAN
Dedicated to Nazar´e Lopes
Abstract. Bilinear processes are highly flexible for modeling nonlinear and heavy-tailed features that are often exhibited by financial and envi- ronmental time series. However, intractable likelihood functions, together with the lack of verifiable conditions of invertibility and stationarity, ex- cept for very simple bilinear processes, constraint their use as models. The traditional methods of least squares and conditional maximum like- lihood (CML) do not give satisfactory results, particularly for heavy- tailed series. This paper aims to show the advantages and drawbacks of using simulation-based estimation techniques for bilinear models. Ap- proximate Bayesian Computation (ABC), as well as sequential Markov chain Monte Carlo (MCMC) methods are often applied with success as inferential tools for time series. Due to ease with which one can simulate bilinear processes, ABC is particularly a promising inferential alterna- tive. We assess the viability of ABC as an inferential tool for the bilinear processes. The performance of the ABC algorithm for parameter esti- mation is assessed using several simulated samples from simple bilinear models. The results are compared with the parameter estimates obtained using the CML method.
1. Introduction
Today, we are more aware that many observed time series, particularly those coming from financial markets and environmental sciences, do not conform with the traditional assumptions of linearity and Gaussian error structures, often exhibiting heavy-tailed behaviour. Therefore, models usually applied are increasingly more complex, capturing such features as nonlinear variations both
Accepted: 11 February 2015.
2010 Mathematics Subject Classification. 62M10, 91B84.
Key words and phrases. Bilinear processes, ABC, conditional maximum likelihood, se-
quential MCMC.
The work was financially supported by Funda¸ca˜o para a Ciˆencia e a Tecnologia (FCT),
Portugal, through the projects PEst-OE/MAT/UI0006/2014 and PTDC/MAT/118335/2010.
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