Page 146 - Textos de Matemática Vol. 47

136 P. DE ZEA BERMUDEZ, M. A. AMARAL TURKMAN, AND K. F. TURKMAN
in the mean and variance. Although some of these nonlinear classes of models, such as the GARCH are analytically tractable, likelihood functions for the class of bilinear processes are analytically intractable. For heavy-tailed bilinear processes, least square methods do not give satisfactory results and therefore, inference for such processes can be a challenging task.
The use of MCMC methods in a Bayesian context has proven to be a very powerful solution. The seminal paper by Gelfand and Smith [21] established the use of simulation-based approaches in a Bayesian framework. The use of the Gibbs Sampler, introduced by Geman and Geman [22], become a common way to overcome the di culties of handling complicated posterior distributions. After those ground-breaking papers, many other algorithms were developed to deal with specific problems. The Metropolis Within Gibbs, proposed by Gilks et al. [23] for addressing the simulation from non-logconcave full conditional distributions, the Slice Sampler (Neal [28]) or the Reversible Jump, proposed by Green [16] in order to handle estimation problems associated to models with varying-dimension parameter spaces, are such examples. The above mentioned MCMC methods depend heavily on the concept of likelihood and the existence of analytical expression for the likelihood. The Sequential Monte Carlo (SMC) in general, and Particle Markov chain Monte Carlo in particular, are extensions that enable dealing with complex likelihoods sequentially, bringing some nu- merical ease into the inference. In contrast, the ABC algorithms are suggested for handling situations when it is not possible to give any tractable analytical expression for the likelihood.
The purpose of this paper is to assess the viability of using an ABC algorithm for estimating the parameters of simple bilinear processes.
The structure of the paper is as follows. The bilinear models are reviewed in Section 2. The ABC algorithms are introduced in Section 3. Some simulation results are provided in Section 4 and finally comments and conclusions are given in Section 5.
2. The Bilinear models
2.1. Preliminaries. Bilinear models (Subba Rao and Gabr [36]) are possibly the most natural way to extend the ARMA models for explaining features such as nonlinearity and heavy-tailed behavior. Yt is called a bilinear process, BL(p, q, m, k), if it satisfies the di↵erence equation:
Xp Xq XmXk
Yt ajYt j = cj✏t j + b`1`2 Yt `1 ✏t `2 . (2.1)
j=1 j=0 `1=1 `2=1