BİREYLER VE YÖNTEM 1 Bireyler

Nelder & Wedderburn(1972) propuseram a Famlia de Modelos Lineares Generalizados (MLG), propiciando a unificação em uma classe de vários modelos já existentes de forma

isolada. A idéia central desses modelos consiste em permitir que se tenha várias opções para a distribuição da variável-resposta, permitindo ainda que a mesma pertença a família exponencial de distribuições, e por consequências todas as boas propriedades desta família.

No contexto de séries temporais, a estrutura de correlação das observações não pode ser desprezada. Nesse sentido, uma estrutura mais geral, denominada por Modelos Lin- eares Dinâmicos Generalizados (MLDG), foi proposta por West et al.(1985), gerando a partir de então um significativo interesse nestes modelos devido à sua aplicabilidade em diversas áreas do conhecimento.

Vários trabalhos foram publicados sobre estes modelos, dentre os quais pode-se citar o de Gamerman & West (1987), Grunwald et al. (1993), Fahrmeir (1987), Fruhwirth- Schnatter (1994), Lindsey & Lambert (1995), Gamerman (1991), Gamerman (1998), Chiogna & Gaetan(2002),Hemming & Shaw(2002) eGodolphin & Triantafyllopoulos (2006).

Há na literatura ainda outros trabalhos que tratam de modelos para séries temporais não-gaussianas que não estão sob os MLDG, dentre os quais pode-se citar o de Smith (1979),Smith (1981),Cox(1981),Smith & Miller(1986),Kaufmann(1987),Kitagawa (1987),Harvey & Fernandes(1989), Shephard & Pitt(1997),Jorgensen et al. (1999) e Durbin & Koopman(2000).

O problema com essas classes de modelos é sua tratabilidade analítica que é facil- mente perdida, mesmo para componentes muito simples. Assim, a verossimilhança preditiva, que é fundamental para o processo de inferência, pode apenas ser obtida de forma aproximada. Portanto, a NGSSM proposta por Santos et al. (2010) tem como principal vantagem em relação aos trabalhos citados acima a tratabilidade analítica, onde as equações de evolução e a função de verossimilhança preditiva são exatas.

Modelling Volatility Using State

Space Models with Heavy Tailed

Distributions

Frank M. de Pinhoa_{, Glaura C. Franco}b_{, Ralph S. Silva}c a_{IBMEC, Belo Horizonte, Brasil}

b_{Universidade Federal de Minas Gerais, Belo Horizonte, Brasil} c_{Universidade Federal do Rio de Janeiro, Belo Horizonte, Brasil}

Abstract

This article deals with a non-Gaussian state space model (NGSSM), which is a generalization of the results inSmith & Miller(1986). The NGSSM is at- tractive because the likelihood can be analytically computed, thus avoiding the use of highly demanding computational algorithms such as the particle filter in order to make inference on the parameters. The paper focuses on stochastic volatility models in the NGSSM, where the observation equation is modelled with a heavy tailed distribution such as Log-normal, Log-gamma and Fréchet. Parameter estimation can be accomplished either using classi- cal or Bayesian procedures and a simulation study shows that both methods lead to satisfactory results. In a real data application, the proposed stochas- tic volatility models in the NGSSM are compared with the autoregressive conditionally heteroscedastic and stochastic volatility models using South

and North American stock price indexes.

Keyword: Bayesian and Classical Inference, Heavy Tailed Distributions, Non-Gaussian State Space Model, Stochastic Volatility, Stock price index.

5.1 Introduction

The global financial crisis has generated a significant instability in the prices of financial assets and particularly in the stock market. For this reason, a major concern among economists, fund managers and investment researchers is how long this crisis will impact the variability of asset prices. For this reason, researches focusing on the study and modeling of volatility has been intensified in the last few years.

Relying on the fact that the unconditional distribution of daily returns has fat- ter tails than the normal distribution, the usual time series models that assume nor- mality and homoscedasticity are not appropriate to model volatility. Thus, more adequate procedures, especially the ones presenting conditional variance evolving on time, have been proposed. The most known approaches are the ones concerning con- ditional heteroscedastic models, such as ARCH (Engle, 1982), GARCH (Bollerslev, 1986), EGARCH (Nelson,1991), TGARCH (Zakoian,1994) and multivariate GARCH (Bauwens et al.,2006).

Taylor (1986) proposed the first stochastic volatility model, where the volatility is a stochastic function of the past volatility. Several studies on this approach have been developed, such asMelino & Turnbull(1990),Taylor(1994),Harvey et al.(1994), Jacquier et al.(1994), Eraker et al.(2003) andRaggi & Bordignon (2006).

Recently, a non Gaussian state space model was proposed by Santos et al. (2010). This procedure is a generalization of a result of Smith & Miller(1986), who proposed an exponential observation model with an exact evolution equation for the state. The work ofSantos et al.(2010) allows for analytical computation of the marginal likelihood,

which increases the applicability of the model and enables its use with a wide class of distributions for observational time series. Additionally, this model allows the relaxation of the normality and heteroscedasticity assumptions.

According to Tsay (2005), one of the main characteristics of volatility is that it evolves over time in a continuous way and it always varies within a fixed range. This means that volatility is often stationary. Due to the structure used in the model pro- posed by Santos et al. (2010), the only stochastic component is the level of the series, and it is built in a way similar to the local level model of Harvey (1989). Thus, the model is highly recommended to be applied to stationary series. Any other component, such as seasonality or structural breaks should be inserted as covariates.

There are some recent contributions in the literature that employ the state space approach to handle nonlinear and non Gaussian time series. Some examples are the works of Shephard (1994), extended by Deschamps (2011) for Bayesian estimation, that uses a local scale procedure for modeling volatility. Ferrante & Vidoni(1998) and Vidoni (1999) introduce non-linear and non Gaussian state space models with analytic updating recursions for filtering and prediction.

Thus, the purpose of this work is to present new models in the non-Gaussian state space family that can be used to model volatility. Among them, there is the class of heavy tailed distributions, much employed in the volatility literature, as in the works of Anderson (2001) and Chib et al. (2002). The models introduced here comprise the Log-normal, Log-gamma, Fréchet, Lévy and the Generalized Error Distribution (GED). In addition, the Pareto and Weibull models, already considered in Santos et al.(2010), are also presented.

Monte Carlo results for Bayesian and classical methods of inference in the estima- tion of the non-Gaussian state space model are performed for the distributions cited above. Additionally, the NGSSM addressed here is used to model the most known stock exchange indexes in North and South America and the fits are compared to the clas-

sical generalized autoregressive conditional heteroscedasticity (see GARCH;Bollerslev, 1986) models.

The paper is organized as follows. Section 6.2 defines the NGSSM and presents the inference procedures. Section5.3shows how to write the heavy tailed distributions cited above in the NGSSM form. Section 6.4 shows the results of the Monte Carlo simulation studies and Section 5.5 presents an application of heavy tailed models in the NGSSM to estimate the volatility of several stock exchange indexes. Section 6.5 concludes the work.